[STATEMENT]
lemma \<beta>:
  assumes "iszlfm p"
    and "d_\<beta> p 1"
    and "d_\<delta> p d"
    and dp: "d > 0"
    and "\<not> (\<exists>j::int \<in> {1 .. d}. \<exists>b \<in> Inum (a # bs) ` set (\<beta> p). x = b + j)"
    and p: "Ifm bbs (x # bs) p" (is "?P x")
  shows "?P (x - d)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) p
[PROOF STEP]
using assms
[PROOF STATE]
proof (prove)
using this:
iszlfm p
d_\<beta> p 1
d_\<delta> p d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j)
Ifm bbs (x # bs) p

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) p
[PROOF STEP]
proof (induct p rule: iszlfm.induct)
[PROOF STATE]
proof (state)
goal (75 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Lt (CN 0 c e)); d_\<beta> (Lt (CN 0 c e)) 1; d_\<delta> (Lt (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Lt (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN 0 c e))
 6. \<And>c e. \<lbrakk>iszlfm (Le (CN 0 c e)); d_\<beta> (Le (CN 0 c e)) 1; d_\<delta> (Le (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Le (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Le (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Le (CN 0 c e))
 7. \<And>c e. \<lbrakk>iszlfm (Gt (CN 0 c e)); d_\<beta> (Gt (CN 0 c e)) 1; d_\<delta> (Gt (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Gt (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Gt (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
 8. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 9. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 10. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
A total of 75 subgoals...
[PROOF STEP]
case (5 c e)
[PROOF STATE]
proof (state)
this:
iszlfm (Lt (CN 0 c e))
d_\<beta> (Lt (CN 0 c e)) 1
d_\<delta> (Lt (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Lt (CN 0 c e))

goal (75 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Lt (CN 0 c e)); d_\<beta> (Lt (CN 0 c e)) 1; d_\<delta> (Lt (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Lt (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN 0 c e))
 6. \<And>c e. \<lbrakk>iszlfm (Le (CN 0 c e)); d_\<beta> (Le (CN 0 c e)) 1; d_\<delta> (Le (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Le (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Le (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Le (CN 0 c e))
 7. \<And>c e. \<lbrakk>iszlfm (Gt (CN 0 c e)); d_\<beta> (Gt (CN 0 c e)) 1; d_\<delta> (Gt (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Gt (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Gt (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
 8. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 9. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 10. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
A total of 75 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
iszlfm (Lt (CN 0 c e))
d_\<beta> (Lt (CN 0 c e)) 1
d_\<delta> (Lt (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Lt (CN 0 c e))
[PROOF STEP]
have c1: "c = 1" and  bn: "numbound0 e"
[PROOF STATE]
proof (prove)
using this:
iszlfm (Lt (CN 0 c e))
d_\<beta> (Lt (CN 0 c e)) 1
d_\<delta> (Lt (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Lt (CN 0 c e))

goal (1 subgoal):
 1. c = 1 &&& numbound0 e
[PROOF STEP]
by simp_all
[PROOF STATE]
proof (state)
this:
c = 1
numbound0 e

goal (75 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Lt (CN 0 c e)); d_\<beta> (Lt (CN 0 c e)) 1; d_\<delta> (Lt (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Lt (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN 0 c e))
 6. \<And>c e. \<lbrakk>iszlfm (Le (CN 0 c e)); d_\<beta> (Le (CN 0 c e)) 1; d_\<delta> (Le (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Le (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Le (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Le (CN 0 c e))
 7. \<And>c e. \<lbrakk>iszlfm (Gt (CN 0 c e)); d_\<beta> (Gt (CN 0 c e)) 1; d_\<delta> (Gt (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Gt (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Gt (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
 8. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 9. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 10. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
A total of 75 subgoals...
[PROOF STEP]
with dp p c1 numbound0_I[OF bn,where b = "(x - d)" and b' = "x" and bs = "bs"] 5
[PROOF STATE]
proof (chain)
picking this:
0 < d
Ifm bbs (x # bs) p
c = 1
Inum ((x - d) # bs) e = Inum (x # bs) e
iszlfm (Lt (CN 0 c e))
d_\<beta> (Lt (CN 0 c e)) 1
d_\<delta> (Lt (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Lt (CN 0 c e))
c = 1
numbound0 e
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
0 < d
Ifm bbs (x # bs) p
c = 1
Inum ((x - d) # bs) e = Inum (x # bs) e
iszlfm (Lt (CN 0 c e))
d_\<beta> (Lt (CN 0 c e)) 1
d_\<delta> (Lt (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Lt (CN 0 c e))
c = 1
numbound0 e

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (Lt (CN 0 c e))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
Ifm bbs ((x - d) # bs) (Lt (CN 0 c e))

goal (74 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Le (CN 0 c e)); d_\<beta> (Le (CN 0 c e)) 1; d_\<delta> (Le (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Le (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Le (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Le (CN 0 c e))
 6. \<And>c e. \<lbrakk>iszlfm (Gt (CN 0 c e)); d_\<beta> (Gt (CN 0 c e)) 1; d_\<delta> (Gt (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Gt (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Gt (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
 7. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 8. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 9. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 10. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
A total of 74 subgoals...
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (74 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Le (CN 0 c e)); d_\<beta> (Le (CN 0 c e)) 1; d_\<delta> (Le (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Le (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Le (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Le (CN 0 c e))
 6. \<And>c e. \<lbrakk>iszlfm (Gt (CN 0 c e)); d_\<beta> (Gt (CN 0 c e)) 1; d_\<delta> (Gt (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Gt (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Gt (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
 7. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 8. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 9. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 10. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
A total of 74 subgoals...
[PROOF STEP]
case (6 c e)
[PROOF STATE]
proof (state)
this:
iszlfm (Le (CN 0 c e))
d_\<beta> (Le (CN 0 c e)) 1
d_\<delta> (Le (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Le (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Le (CN 0 c e))

goal (74 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Le (CN 0 c e)); d_\<beta> (Le (CN 0 c e)) 1; d_\<delta> (Le (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Le (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Le (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Le (CN 0 c e))
 6. \<And>c e. \<lbrakk>iszlfm (Gt (CN 0 c e)); d_\<beta> (Gt (CN 0 c e)) 1; d_\<delta> (Gt (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Gt (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Gt (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
 7. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 8. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 9. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 10. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
A total of 74 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
iszlfm (Le (CN 0 c e))
d_\<beta> (Le (CN 0 c e)) 1
d_\<delta> (Le (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Le (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Le (CN 0 c e))
[PROOF STEP]
have c1: "c = 1" and  bn: "numbound0 e"
[PROOF STATE]
proof (prove)
using this:
iszlfm (Le (CN 0 c e))
d_\<beta> (Le (CN 0 c e)) 1
d_\<delta> (Le (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Le (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Le (CN 0 c e))

goal (1 subgoal):
 1. c = 1 &&& numbound0 e
[PROOF STEP]
by simp_all
[PROOF STATE]
proof (state)
this:
c = 1
numbound0 e

goal (74 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Le (CN 0 c e)); d_\<beta> (Le (CN 0 c e)) 1; d_\<delta> (Le (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Le (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Le (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Le (CN 0 c e))
 6. \<And>c e. \<lbrakk>iszlfm (Gt (CN 0 c e)); d_\<beta> (Gt (CN 0 c e)) 1; d_\<delta> (Gt (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Gt (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Gt (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
 7. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 8. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 9. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 10. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
A total of 74 subgoals...
[PROOF STEP]
with dp p c1 numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] 6
[PROOF STATE]
proof (chain)
picking this:
0 < d
Ifm bbs (x # bs) p
c = 1
Inum ((x - d) # bs) e = Inum (x # bs) e
iszlfm (Le (CN 0 c e))
d_\<beta> (Le (CN 0 c e)) 1
d_\<delta> (Le (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Le (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Le (CN 0 c e))
c = 1
numbound0 e
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
0 < d
Ifm bbs (x # bs) p
c = 1
Inum ((x - d) # bs) e = Inum (x # bs) e
iszlfm (Le (CN 0 c e))
d_\<beta> (Le (CN 0 c e)) 1
d_\<delta> (Le (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Le (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Le (CN 0 c e))
c = 1
numbound0 e

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (Le (CN 0 c e))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
Ifm bbs ((x - d) # bs) (Le (CN 0 c e))

goal (73 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Gt (CN 0 c e)); d_\<beta> (Gt (CN 0 c e)) 1; d_\<delta> (Gt (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Gt (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Gt (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
 6. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 7. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 8. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 9. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 10. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
A total of 73 subgoals...
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (73 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Gt (CN 0 c e)); d_\<beta> (Gt (CN 0 c e)) 1; d_\<delta> (Gt (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Gt (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Gt (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
 6. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 7. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 8. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 9. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 10. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
A total of 73 subgoals...
[PROOF STEP]
case (7 c e)
[PROOF STATE]
proof (state)
this:
iszlfm (Gt (CN 0 c e))
d_\<beta> (Gt (CN 0 c e)) 1
d_\<delta> (Gt (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Gt (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Gt (CN 0 c e))

goal (73 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Gt (CN 0 c e)); d_\<beta> (Gt (CN 0 c e)) 1; d_\<delta> (Gt (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Gt (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Gt (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
 6. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 7. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 8. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 9. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 10. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
A total of 73 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
iszlfm (Gt (CN 0 c e))
d_\<beta> (Gt (CN 0 c e)) 1
d_\<delta> (Gt (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Gt (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Gt (CN 0 c e))
[PROOF STEP]
have p: "Ifm bbs (x # bs) (Gt (CN 0 c e))" and c1: "c=1" and bn: "numbound0 e"
[PROOF STATE]
proof (prove)
using this:
iszlfm (Gt (CN 0 c e))
d_\<beta> (Gt (CN 0 c e)) 1
d_\<delta> (Gt (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Gt (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Gt (CN 0 c e))

goal (1 subgoal):
 1. Ifm bbs (x # bs) (Gt (CN 0 c e)) &&& c = 1 &&& numbound0 e
[PROOF STEP]
by simp_all
[PROOF STATE]
proof (state)
this:
Ifm bbs (x # bs) (Gt (CN 0 c e))
c = 1
numbound0 e

goal (73 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Gt (CN 0 c e)); d_\<beta> (Gt (CN 0 c e)) 1; d_\<delta> (Gt (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Gt (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Gt (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
 6. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 7. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 8. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 9. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 10. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
A total of 73 subgoals...
[PROOF STEP]
let ?e = "Inum (x # bs) e"
[PROOF STATE]
proof (state)
goal (73 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Gt (CN 0 c e)); d_\<beta> (Gt (CN 0 c e)) 1; d_\<delta> (Gt (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Gt (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Gt (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
 6. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 7. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 8. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 9. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 10. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
A total of 73 subgoals...
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
proof (cases "(x - d) + ?e > 0")
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. 0 < x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
 2. \<not> 0 < x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
case True
[PROOF STATE]
proof (state)
this:
0 < x - d + Inum (x # bs) e

goal (2 subgoals):
 1. 0 < x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
 2. \<not> 0 < x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
0 < x - d + Inum (x # bs) e
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
0 < x - d + Inum (x # bs) e

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
using c1 numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"]
[PROOF STATE]
proof (prove)
using this:
0 < x - d + Inum (x # bs) e
c = 1
Inum ((x - d) # bs) e = Inum (x # bs) e

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))

goal (1 subgoal):
 1. \<not> 0 < x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<not> 0 < x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
case False
[PROOF STATE]
proof (state)
this:
\<not> 0 < x - d + Inum (x # bs) e

goal (1 subgoal):
 1. \<not> 0 < x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
let ?v = "Neg e"
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<not> 0 < x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
have vb: "?v \<in> set (\<beta> (Gt (CN 0 c e)))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. Neg e \<in> set (\<beta> (Gt (CN 0 c e)))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
Neg e \<in> set (\<beta> (Gt (CN 0 c e)))

goal (1 subgoal):
 1. \<not> 0 < x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
from 7(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="x" and bs="bs"]]
[PROOF STATE]
proof (chain)
picking this:
\<forall>j\<in>{1..d}. x \<noteq> - Inum (x # bs) e + j
[PROOF STEP]
have nob: "\<not> (\<exists>j\<in> {1 ..d}. x = - ?e + j)"
[PROOF STATE]
proof (prove)
using this:
\<forall>j\<in>{1..d}. x \<noteq> - Inum (x # bs) e + j

goal (1 subgoal):
 1. \<not> (\<exists>j\<in>{1..d}. x = - Inum (x # bs) e + j)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<not> (\<exists>j\<in>{1..d}. x = - Inum (x # bs) e + j)

goal (1 subgoal):
 1. \<not> 0 < x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
from False p
[PROOF STATE]
proof (chain)
picking this:
\<not> 0 < x - d + Inum (x # bs) e
Ifm bbs (x # bs) (Gt (CN 0 c e))
[PROOF STEP]
have "x + ?e > 0 \<and> x + ?e \<le> d"
[PROOF STATE]
proof (prove)
using this:
\<not> 0 < x - d + Inum (x # bs) e
Ifm bbs (x # bs) (Gt (CN 0 c e))

goal (1 subgoal):
 1. 0 < x + Inum (x # bs) e \<and> x + Inum (x # bs) e \<le> d
[PROOF STEP]
by (simp add: c1)
[PROOF STATE]
proof (state)
this:
0 < x + Inum (x # bs) e \<and> x + Inum (x # bs) e \<le> d

goal (1 subgoal):
 1. \<not> 0 < x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
0 < x + Inum (x # bs) e \<and> x + Inum (x # bs) e \<le> d
[PROOF STEP]
have "x + ?e \<ge> 1 \<and> x + ?e \<le> d"
[PROOF STATE]
proof (prove)
using this:
0 < x + Inum (x # bs) e \<and> x + Inum (x # bs) e \<le> d

goal (1 subgoal):
 1. 1 \<le> x + Inum (x # bs) e \<and> x + Inum (x # bs) e \<le> d
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
1 \<le> x + Inum (x # bs) e \<and> x + Inum (x # bs) e \<le> d

goal (1 subgoal):
 1. \<not> 0 < x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
1 \<le> x + Inum (x # bs) e \<and> x + Inum (x # bs) e \<le> d
[PROOF STEP]
have "\<exists>j::int \<in> {1 .. d}. j = x + ?e"
[PROOF STATE]
proof (prove)
using this:
1 \<le> x + Inum (x # bs) e \<and> x + Inum (x # bs) e \<le> d

goal (1 subgoal):
 1. \<exists>j\<in>{1..d}. j = x + Inum (x # bs) e
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>j\<in>{1..d}. j = x + Inum (x # bs) e

goal (1 subgoal):
 1. \<not> 0 < x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<exists>j\<in>{1..d}. j = x + Inum (x # bs) e
[PROOF STEP]
have "\<exists>j::int \<in> {1 .. d}. x = (- ?e + j)"
[PROOF STATE]
proof (prove)
using this:
\<exists>j\<in>{1..d}. j = x + Inum (x # bs) e

goal (1 subgoal):
 1. \<exists>j\<in>{1..d}. x = - Inum (x # bs) e + j
[PROOF STEP]
by (simp add: algebra_simps)
[PROOF STATE]
proof (state)
this:
\<exists>j\<in>{1..d}. x = - Inum (x # bs) e + j

goal (1 subgoal):
 1. \<not> 0 < x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
with nob
[PROOF STATE]
proof (chain)
picking this:
\<not> (\<exists>j\<in>{1..d}. x = - Inum (x # bs) e + j)
\<exists>j\<in>{1..d}. x = - Inum (x # bs) e + j
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<not> (\<exists>j\<in>{1..d}. x = - Inum (x # bs) e + j)
\<exists>j\<in>{1..d}. x = - Inum (x # bs) e + j

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
Ifm bbs ((x - d) # bs) (Gt (CN 0 c e))

goal (72 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 6. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 7. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 8. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 9. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 10. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
A total of 72 subgoals...
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (72 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 6. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 7. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 8. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 9. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 10. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
A total of 72 subgoals...
[PROOF STEP]
case (8 c e)
[PROOF STATE]
proof (state)
this:
iszlfm (Ge (CN 0 c e))
d_\<beta> (Ge (CN 0 c e)) 1
d_\<delta> (Ge (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Ge (CN 0 c e))

goal (72 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 6. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 7. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 8. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 9. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 10. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
A total of 72 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
iszlfm (Ge (CN 0 c e))
d_\<beta> (Ge (CN 0 c e)) 1
d_\<delta> (Ge (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Ge (CN 0 c e))
[PROOF STEP]
have p: "Ifm bbs (x # bs) (Ge (CN 0 c e))" and c1: "c = 1" and bn: "numbound0 e"
[PROOF STATE]
proof (prove)
using this:
iszlfm (Ge (CN 0 c e))
d_\<beta> (Ge (CN 0 c e)) 1
d_\<delta> (Ge (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Ge (CN 0 c e))

goal (1 subgoal):
 1. Ifm bbs (x # bs) (Ge (CN 0 c e)) &&& c = 1 &&& numbound0 e
[PROOF STEP]
by simp_all
[PROOF STATE]
proof (state)
this:
Ifm bbs (x # bs) (Ge (CN 0 c e))
c = 1
numbound0 e

goal (72 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 6. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 7. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 8. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 9. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 10. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
A total of 72 subgoals...
[PROOF STEP]
let ?e = "Inum (x # bs) e"
[PROOF STATE]
proof (state)
goal (72 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>c e. \<lbrakk>iszlfm (Ge (CN 0 c e)); d_\<beta> (Ge (CN 0 c e)) 1; d_\<delta> (Ge (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Ge (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Ge (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 6. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 7. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 8. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 9. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 10. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
A total of 72 subgoals...
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
proof (cases "(x - d) + ?e \<ge> 0")
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. 0 \<le> x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 2. \<not> 0 \<le> x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
case True
[PROOF STATE]
proof (state)
this:
0 \<le> x - d + Inum (x # bs) e

goal (2 subgoals):
 1. 0 \<le> x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
 2. \<not> 0 \<le> x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
0 \<le> x - d + Inum (x # bs) e
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
0 \<le> x - d + Inum (x # bs) e

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
using c1 numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"]
[PROOF STATE]
proof (prove)
using this:
0 \<le> x - d + Inum (x # bs) e
c = 1
Inum ((x - d) # bs) e = Inum (x # bs) e

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))

goal (1 subgoal):
 1. \<not> 0 \<le> x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<not> 0 \<le> x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
case False
[PROOF STATE]
proof (state)
this:
\<not> 0 \<le> x - d + Inum (x # bs) e

goal (1 subgoal):
 1. \<not> 0 \<le> x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
let ?v = "Sub (C (- 1)) e"
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<not> 0 \<le> x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
have vb: "?v \<in> set (\<beta> (Ge (CN 0 c e)))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. Sub (C (- 1)) e \<in> set (\<beta> (Ge (CN 0 c e)))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
Sub (C (- 1)) e \<in> set (\<beta> (Ge (CN 0 c e)))

goal (1 subgoal):
 1. \<not> 0 \<le> x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
from 8(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="x" and bs="bs"]]
[PROOF STATE]
proof (chain)
picking this:
\<forall>j\<in>{1..d}. x \<noteq> - 1 - Inum (x # bs) e + j
[PROOF STEP]
have nob: "\<not> (\<exists>j\<in> {1 ..d}. x =  - ?e - 1 + j)"
[PROOF STATE]
proof (prove)
using this:
\<forall>j\<in>{1..d}. x \<noteq> - 1 - Inum (x # bs) e + j

goal (1 subgoal):
 1. \<not> (\<exists>j\<in>{1..d}. x = - Inum (x # bs) e - 1 + j)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<not> (\<exists>j\<in>{1..d}. x = - Inum (x # bs) e - 1 + j)

goal (1 subgoal):
 1. \<not> 0 \<le> x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
from False p
[PROOF STATE]
proof (chain)
picking this:
\<not> 0 \<le> x - d + Inum (x # bs) e
Ifm bbs (x # bs) (Ge (CN 0 c e))
[PROOF STEP]
have "x + ?e \<ge> 0 \<and> x + ?e < d"
[PROOF STATE]
proof (prove)
using this:
\<not> 0 \<le> x - d + Inum (x # bs) e
Ifm bbs (x # bs) (Ge (CN 0 c e))

goal (1 subgoal):
 1. 0 \<le> x + Inum (x # bs) e \<and> x + Inum (x # bs) e < d
[PROOF STEP]
by (simp add: c1)
[PROOF STATE]
proof (state)
this:
0 \<le> x + Inum (x # bs) e \<and> x + Inum (x # bs) e < d

goal (1 subgoal):
 1. \<not> 0 \<le> x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
0 \<le> x + Inum (x # bs) e \<and> x + Inum (x # bs) e < d
[PROOF STEP]
have "x + ?e +1 \<ge> 1 \<and> x + ?e + 1 \<le> d"
[PROOF STATE]
proof (prove)
using this:
0 \<le> x + Inum (x # bs) e \<and> x + Inum (x # bs) e < d

goal (1 subgoal):
 1. 1 \<le> x + Inum (x # bs) e + 1 \<and> x + Inum (x # bs) e + 1 \<le> d
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
1 \<le> x + Inum (x # bs) e + 1 \<and> x + Inum (x # bs) e + 1 \<le> d

goal (1 subgoal):
 1. \<not> 0 \<le> x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
1 \<le> x + Inum (x # bs) e + 1 \<and> x + Inum (x # bs) e + 1 \<le> d
[PROOF STEP]
have "\<exists>j::int \<in> {1 .. d}. j = x + ?e + 1"
[PROOF STATE]
proof (prove)
using this:
1 \<le> x + Inum (x # bs) e + 1 \<and> x + Inum (x # bs) e + 1 \<le> d

goal (1 subgoal):
 1. \<exists>j\<in>{1..d}. j = x + Inum (x # bs) e + 1
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>j\<in>{1..d}. j = x + Inum (x # bs) e + 1

goal (1 subgoal):
 1. \<not> 0 \<le> x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<exists>j\<in>{1..d}. j = x + Inum (x # bs) e + 1
[PROOF STEP]
have "\<exists>j::int \<in> {1 .. d}. x= - ?e - 1 + j"
[PROOF STATE]
proof (prove)
using this:
\<exists>j\<in>{1..d}. j = x + Inum (x # bs) e + 1

goal (1 subgoal):
 1. \<exists>j\<in>{1..d}. x = - Inum (x # bs) e - 1 + j
[PROOF STEP]
by (simp add: algebra_simps)
[PROOF STATE]
proof (state)
this:
\<exists>j\<in>{1..d}. x = - Inum (x # bs) e - 1 + j

goal (1 subgoal):
 1. \<not> 0 \<le> x - d + Inum (x # bs) e \<Longrightarrow> Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
with nob
[PROOF STATE]
proof (chain)
picking this:
\<not> (\<exists>j\<in>{1..d}. x = - Inum (x # bs) e - 1 + j)
\<exists>j\<in>{1..d}. x = - Inum (x # bs) e - 1 + j
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<not> (\<exists>j\<in>{1..d}. x = - Inum (x # bs) e - 1 + j)
\<exists>j\<in>{1..d}. x = - Inum (x # bs) e - 1 + j

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
Ifm bbs ((x - d) # bs) (Ge (CN 0 c e))

goal (71 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 6. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 7. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 8. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 9. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
A total of 71 subgoals...
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (71 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 6. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 7. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 8. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 9. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
A total of 71 subgoals...
[PROOF STEP]
case (3 c e)
[PROOF STATE]
proof (state)
this:
iszlfm (Eq (CN 0 c e))
d_\<beta> (Eq (CN 0 c e)) 1
d_\<delta> (Eq (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Eq (CN 0 c e))

goal (71 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 6. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 7. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 8. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 9. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
A total of 71 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
iszlfm (Eq (CN 0 c e))
d_\<beta> (Eq (CN 0 c e)) 1
d_\<delta> (Eq (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Eq (CN 0 c e))
[PROOF STEP]
have p: "Ifm bbs (x #bs) (Eq (CN 0 c e))" (is "?p x")
    and c1: "c = 1"
    and bn: "numbound0 e"
[PROOF STATE]
proof (prove)
using this:
iszlfm (Eq (CN 0 c e))
d_\<beta> (Eq (CN 0 c e)) 1
d_\<delta> (Eq (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (Eq (CN 0 c e))

goal (1 subgoal):
 1. Ifm bbs (x # bs) (Eq (CN 0 c e)) &&& c = 1 &&& numbound0 e
[PROOF STEP]
by simp_all
[PROOF STATE]
proof (state)
this:
Ifm bbs (x # bs) (Eq (CN 0 c e))
c = 1
numbound0 e

goal (71 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 6. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 7. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 8. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 9. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
A total of 71 subgoals...
[PROOF STEP]
let ?e = "Inum (x # bs) e"
[PROOF STATE]
proof (state)
goal (71 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 6. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 7. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 8. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 9. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
A total of 71 subgoals...
[PROOF STEP]
let ?v="(Sub (C (- 1)) e)"
[PROOF STATE]
proof (state)
goal (71 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 6. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 7. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 8. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 9. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
A total of 71 subgoals...
[PROOF STEP]
have vb: "?v \<in> set (\<beta> (Eq (CN 0 c e)))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. Sub (C (- 1)) e \<in> set (\<beta> (Eq (CN 0 c e)))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
Sub (C (- 1)) e \<in> set (\<beta> (Eq (CN 0 c e)))

goal (71 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 6. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 7. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 8. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 9. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
A total of 71 subgoals...
[PROOF STEP]
from p
[PROOF STATE]
proof (chain)
picking this:
Ifm bbs (x # bs) (Eq (CN 0 c e))
[PROOF STEP]
have "x= - ?e"
[PROOF STATE]
proof (prove)
using this:
Ifm bbs (x # bs) (Eq (CN 0 c e))

goal (1 subgoal):
 1. x = - Inum (x # bs) e
[PROOF STEP]
by (simp add: c1)
[PROOF STATE]
proof (state)
this:
x = - Inum (x # bs) e

goal (71 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (Eq (CN 0 c e)); d_\<beta> (Eq (CN 0 c e)) 1; d_\<delta> (Eq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Eq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
 4. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 5. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 6. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 7. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 8. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 9. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
A total of 71 subgoals...
[PROOF STEP]
with 3(5)
[PROOF STATE]
proof (chain)
picking this:
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j)
x = - Inum (x # bs) e
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j)
x = - Inum (x # bs) e

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
[PROOF STEP]
using dp
[PROOF STATE]
proof (prove)
using this:
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Eq (CN 0 c e))). x = b + j)
x = - Inum (x # bs) e
0 < d

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))
[PROOF STEP]
apply simp
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>\<forall>j\<in>{1..d}. x \<noteq> - 1 - Inum (a # bs) e + j; x = - Inum (x # bs) e; 0 < d\<rbrakk> \<Longrightarrow> c * (x - d) + Inum ((x - d) # bs) e = 0
[PROOF STEP]
apply (erule ballE[where x="1"])
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<lbrakk>x = - Inum (x # bs) e; 0 < d; x \<noteq> - 1 - Inum (a # bs) e + 1\<rbrakk> \<Longrightarrow> c * (x - d) + Inum ((x - d) # bs) e = 0
 2. \<lbrakk>x = - Inum (x # bs) e; 0 < d; 1 \<notin> {1..d}\<rbrakk> \<Longrightarrow> c * (x - d) + Inum ((x - d) # bs) e = 0
[PROOF STEP]
apply (simp_all add:algebra_simps numbound0_I[OF bn,where b="x"and b'="a"and bs="bs"])
[PROOF STATE]
proof (prove)
goal:
No subgoals!
[PROOF STEP]
done
[PROOF STATE]
proof (state)
this:
Ifm bbs ((x - d) # bs) (Eq (CN 0 c e))

goal (70 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 5. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 6. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 7. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 8. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 10. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
A total of 70 subgoals...
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (70 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 5. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 6. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 7. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 8. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 10. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
A total of 70 subgoals...
[PROOF STEP]
case (4 c e)
[PROOF STATE]
proof (state)
this:
iszlfm (NEq (CN 0 c e))
d_\<beta> (NEq (CN 0 c e)) 1
d_\<delta> (NEq (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (NEq (CN 0 c e))

goal (70 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 5. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 6. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 7. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 8. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 10. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
A total of 70 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
iszlfm (NEq (CN 0 c e))
d_\<beta> (NEq (CN 0 c e)) 1
d_\<delta> (NEq (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (NEq (CN 0 c e))
[PROOF STEP]
have p: "Ifm bbs (x # bs) (NEq (CN 0 c e))" (is "?p x")
    and c1: "c = 1"
    and bn: "numbound0 e"
[PROOF STATE]
proof (prove)
using this:
iszlfm (NEq (CN 0 c e))
d_\<beta> (NEq (CN 0 c e)) 1
d_\<delta> (NEq (CN 0 c e)) d
0 < d
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j)
Ifm bbs (x # bs) (NEq (CN 0 c e))

goal (1 subgoal):
 1. Ifm bbs (x # bs) (NEq (CN 0 c e)) &&& c = 1 &&& numbound0 e
[PROOF STEP]
by simp_all
[PROOF STATE]
proof (state)
this:
Ifm bbs (x # bs) (NEq (CN 0 c e))
c = 1
numbound0 e

goal (70 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 5. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 6. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 7. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 8. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 10. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
A total of 70 subgoals...
[PROOF STEP]
let ?e = "Inum (x # bs) e"
[PROOF STATE]
proof (state)
goal (70 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 5. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 6. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 7. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 8. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 10. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
A total of 70 subgoals...
[PROOF STEP]
let ?v="Neg e"
[PROOF STATE]
proof (state)
goal (70 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 5. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 6. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 7. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 8. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 10. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
A total of 70 subgoals...
[PROOF STEP]
have vb: "?v \<in> set (\<beta> (NEq (CN 0 c e)))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. Neg e \<in> set (\<beta> (NEq (CN 0 c e)))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
Neg e \<in> set (\<beta> (NEq (CN 0 c e)))

goal (70 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>c e. \<lbrakk>iszlfm (NEq (CN 0 c e)); d_\<beta> (NEq (CN 0 c e)) 1; d_\<delta> (NEq (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NEq (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 5. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 6. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 7. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 8. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 10. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
A total of 70 subgoals...
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
[PROOF STEP]
proof (cases "x - d + Inum ((x - d) # bs) e = 0")
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. x - d + Inum ((x - d) # bs) e = 0 \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 2. x - d + Inum ((x - d) # bs) e \<noteq> 0 \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
[PROOF STEP]
case False
[PROOF STATE]
proof (state)
this:
x - d + Inum ((x - d) # bs) e \<noteq> 0

goal (2 subgoals):
 1. x - d + Inum ((x - d) # bs) e = 0 \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
 2. x - d + Inum ((x - d) # bs) e \<noteq> 0 \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
x - d + Inum ((x - d) # bs) e \<noteq> 0
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
x - d + Inum ((x - d) # bs) e \<noteq> 0

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
[PROOF STEP]
by (simp add: c1)
[PROOF STATE]
proof (state)
this:
Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))

goal (1 subgoal):
 1. x - d + Inum ((x - d) # bs) e = 0 \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. x - d + Inum ((x - d) # bs) e = 0 \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
[PROOF STEP]
case True
[PROOF STATE]
proof (state)
this:
x - d + Inum ((x - d) # bs) e = 0

goal (1 subgoal):
 1. x - d + Inum ((x - d) # bs) e = 0 \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
x - d + Inum ((x - d) # bs) e = 0
[PROOF STEP]
have "x = - Inum ((x - d) # bs) e + d"
[PROOF STATE]
proof (prove)
using this:
x - d + Inum ((x - d) # bs) e = 0

goal (1 subgoal):
 1. x = - Inum ((x - d) # bs) e + d
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
x = - Inum ((x - d) # bs) e + d

goal (1 subgoal):
 1. x - d + Inum ((x - d) # bs) e = 0 \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
x = - Inum ((x - d) # bs) e + d
[PROOF STEP]
have "x = - Inum (a # bs) e + d"
[PROOF STATE]
proof (prove)
using this:
x = - Inum ((x - d) # bs) e + d

goal (1 subgoal):
 1. x = - Inum (a # bs) e + d
[PROOF STEP]
by (simp add: numbound0_I[OF bn,where b="x - d"and b'="a"and bs="bs"])
[PROOF STATE]
proof (state)
this:
x = - Inum (a # bs) e + d

goal (1 subgoal):
 1. x - d + Inum ((x - d) # bs) e = 0 \<Longrightarrow> Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
[PROOF STEP]
with 4(5)
[PROOF STATE]
proof (chain)
picking this:
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j)
x = - Inum (a # bs) e + d
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j)
x = - Inum (a # bs) e + d

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
[PROOF STEP]
using dp
[PROOF STATE]
proof (prove)
using this:
\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NEq (CN 0 c e))). x = b + j)
x = - Inum (a # bs) e + d
0 < d

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
Ifm bbs ((x - d) # bs) (NEq (CN 0 c e))

goal (69 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 5. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 6. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 7. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 8. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 9. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
A total of 69 subgoals...
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (69 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 5. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 6. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 7. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 8. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 9. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
A total of 69 subgoals...
[PROOF STEP]
case (9 j c e)
[PROOF STATE]
proof (state)
this:
iszlfm (Dvd j (CN 0 c e))
d_\<beta> (Dvd j (CN 0 c e)) 1
d_\<delta> (Dvd j (CN 0 c e)) d
0 < d
\<not> (\<exists>ja\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd j (CN 0 c e))). x = b + ja)
Ifm bbs (x # bs) (Dvd j (CN 0 c e))

goal (69 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 5. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 6. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 7. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 8. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 9. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
A total of 69 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
iszlfm (Dvd j (CN 0 c e))
d_\<beta> (Dvd j (CN 0 c e)) 1
d_\<delta> (Dvd j (CN 0 c e)) d
0 < d
\<not> (\<exists>ja\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd j (CN 0 c e))). x = b + ja)
Ifm bbs (x # bs) (Dvd j (CN 0 c e))
[PROOF STEP]
have p: "Ifm bbs (x # bs) (Dvd j (CN 0 c e))" (is "?p x")
    and c1: "c = 1"
    and bn: "numbound0 e"
[PROOF STATE]
proof (prove)
using this:
iszlfm (Dvd j (CN 0 c e))
d_\<beta> (Dvd j (CN 0 c e)) 1
d_\<delta> (Dvd j (CN 0 c e)) d
0 < d
\<not> (\<exists>ja\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd j (CN 0 c e))). x = b + ja)
Ifm bbs (x # bs) (Dvd j (CN 0 c e))

goal (1 subgoal):
 1. Ifm bbs (x # bs) (Dvd j (CN 0 c e)) &&& c = 1 &&& numbound0 e
[PROOF STEP]
by simp_all
[PROOF STATE]
proof (state)
this:
Ifm bbs (x # bs) (Dvd j (CN 0 c e))
c = 1
numbound0 e

goal (69 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 5. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 6. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 7. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 8. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 9. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
A total of 69 subgoals...
[PROOF STEP]
let ?e = "Inum (x # bs) e"
[PROOF STATE]
proof (state)
goal (69 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 5. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 6. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 7. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 8. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 9. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
A total of 69 subgoals...
[PROOF STEP]
from 9
[PROOF STATE]
proof (chain)
picking this:
iszlfm (Dvd j (CN 0 c e))
d_\<beta> (Dvd j (CN 0 c e)) 1
d_\<delta> (Dvd j (CN 0 c e)) d
0 < d
\<not> (\<exists>ja\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd j (CN 0 c e))). x = b + ja)
Ifm bbs (x # bs) (Dvd j (CN 0 c e))
[PROOF STEP]
have id: "j dvd d"
[PROOF STATE]
proof (prove)
using this:
iszlfm (Dvd j (CN 0 c e))
d_\<beta> (Dvd j (CN 0 c e)) 1
d_\<delta> (Dvd j (CN 0 c e)) d
0 < d
\<not> (\<exists>ja\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd j (CN 0 c e))). x = b + ja)
Ifm bbs (x # bs) (Dvd j (CN 0 c e))

goal (1 subgoal):
 1. j dvd d
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
j dvd d

goal (69 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 5. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 6. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 7. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 8. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 9. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
A total of 69 subgoals...
[PROOF STEP]
from c1
[PROOF STATE]
proof (chain)
picking this:
c = 1
[PROOF STEP]
have "?p x \<longleftrightarrow> j dvd (x + ?e)"
[PROOF STATE]
proof (prove)
using this:
c = 1

goal (1 subgoal):
 1. Ifm bbs (x # bs) (Dvd j (CN 0 c e)) = (j dvd x + Inum (x # bs) e)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
Ifm bbs (x # bs) (Dvd j (CN 0 c e)) = (j dvd x + Inum (x # bs) e)

goal (69 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 5. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 6. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 7. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 8. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 9. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
A total of 69 subgoals...
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
Ifm bbs (x # bs) (Dvd j (CN 0 c e)) = (j dvd x + Inum (x # bs) e)

goal (69 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 5. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 6. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 7. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 8. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 9. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
A total of 69 subgoals...
[PROOF STEP]
have "\<dots> \<longleftrightarrow> j dvd x - d + ?e"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (j dvd x + Inum (x # bs) e) = (j dvd x - d + Inum (x # bs) e)
[PROOF STEP]
using zdvd_period[OF id, where x="x" and c="-1" and t="?e"]
[PROOF STATE]
proof (prove)
using this:
(j dvd x + Inum (x # bs) e) = (j dvd x + - 1 * d + Inum (x # bs) e)

goal (1 subgoal):
 1. (j dvd x + Inum (x # bs) e) = (j dvd x - d + Inum (x # bs) e)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(j dvd x + Inum (x # bs) e) = (j dvd x - d + Inum (x # bs) e)

goal (69 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (Dvd i (CN 0 c e)); d_\<beta> (Dvd i (CN 0 c e)) 1; d_\<delta> (Dvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Dvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (Dvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Dvd i (CN 0 c e))
 4. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 5. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 6. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 7. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 8. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 9. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 10. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
A total of 69 subgoals...
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
Ifm bbs (x # bs) (Dvd j (CN 0 c e)) = (j dvd x - d + Inum (x # bs) e)
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
Ifm bbs (x # bs) (Dvd j (CN 0 c e)) = (j dvd x - d + Inum (x # bs) e)

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (Dvd j (CN 0 c e))
[PROOF STEP]
using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p
[PROOF STATE]
proof (prove)
using this:
Ifm bbs (x # bs) (Dvd j (CN 0 c e)) = (j dvd x - d + Inum (x # bs) e)
Inum ((x - d) # bs) e = Inum (x # bs) e
c = 1
Ifm bbs (x # bs) (Dvd j (CN 0 c e))

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (Dvd j (CN 0 c e))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
Ifm bbs ((x - d) # bs) (Dvd j (CN 0 c e))

goal (68 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 4. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 5. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 6. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 7. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 8. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
 10. \<And>va vb. \<lbrakk>iszlfm (Lt (Add va vb)); d_\<beta> (Lt (Add va vb)) 1; d_\<delta> (Lt (Add va vb)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Add va vb))). x = b + j); Ifm bbs (x # bs) (Lt (Add va vb))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Add va vb))
A total of 68 subgoals...
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (68 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 4. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 5. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 6. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 7. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 8. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
 10. \<And>va vb. \<lbrakk>iszlfm (Lt (Add va vb)); d_\<beta> (Lt (Add va vb)) 1; d_\<delta> (Lt (Add va vb)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Add va vb))). x = b + j); Ifm bbs (x # bs) (Lt (Add va vb))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Add va vb))
A total of 68 subgoals...
[PROOF STEP]
case (10 j c e)
[PROOF STATE]
proof (state)
this:
iszlfm (NDvd j (CN 0 c e))
d_\<beta> (NDvd j (CN 0 c e)) 1
d_\<delta> (NDvd j (CN 0 c e)) d
0 < d
\<not> (\<exists>ja\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd j (CN 0 c e))). x = b + ja)
Ifm bbs (x # bs) (NDvd j (CN 0 c e))

goal (68 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 4. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 5. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 6. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 7. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 8. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
 10. \<And>va vb. \<lbrakk>iszlfm (Lt (Add va vb)); d_\<beta> (Lt (Add va vb)) 1; d_\<delta> (Lt (Add va vb)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Add va vb))). x = b + j); Ifm bbs (x # bs) (Lt (Add va vb))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Add va vb))
A total of 68 subgoals...
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
iszlfm (NDvd j (CN 0 c e))
d_\<beta> (NDvd j (CN 0 c e)) 1
d_\<delta> (NDvd j (CN 0 c e)) d
0 < d
\<not> (\<exists>ja\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd j (CN 0 c e))). x = b + ja)
Ifm bbs (x # bs) (NDvd j (CN 0 c e))
[PROOF STEP]
have p: "Ifm bbs (x # bs) (NDvd j (CN 0 c e))" (is "?p x")
    and c1: "c = 1"
    and bn: "numbound0 e"
[PROOF STATE]
proof (prove)
using this:
iszlfm (NDvd j (CN 0 c e))
d_\<beta> (NDvd j (CN 0 c e)) 1
d_\<delta> (NDvd j (CN 0 c e)) d
0 < d
\<not> (\<exists>ja\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd j (CN 0 c e))). x = b + ja)
Ifm bbs (x # bs) (NDvd j (CN 0 c e))

goal (1 subgoal):
 1. Ifm bbs (x # bs) (NDvd j (CN 0 c e)) &&& c = 1 &&& numbound0 e
[PROOF STEP]
by simp_all
[PROOF STATE]
proof (state)
this:
Ifm bbs (x # bs) (NDvd j (CN 0 c e))
c = 1
numbound0 e

goal (68 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 4. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 5. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 6. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 7. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 8. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
 10. \<And>va vb. \<lbrakk>iszlfm (Lt (Add va vb)); d_\<beta> (Lt (Add va vb)) 1; d_\<delta> (Lt (Add va vb)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Add va vb))). x = b + j); Ifm bbs (x # bs) (Lt (Add va vb))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Add va vb))
A total of 68 subgoals...
[PROOF STEP]
let ?e = "Inum (x # bs) e"
[PROOF STATE]
proof (state)
goal (68 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 4. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 5. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 6. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 7. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 8. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
 10. \<And>va vb. \<lbrakk>iszlfm (Lt (Add va vb)); d_\<beta> (Lt (Add va vb)) 1; d_\<delta> (Lt (Add va vb)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Add va vb))). x = b + j); Ifm bbs (x # bs) (Lt (Add va vb))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Add va vb))
A total of 68 subgoals...
[PROOF STEP]
from 10
[PROOF STATE]
proof (chain)
picking this:
iszlfm (NDvd j (CN 0 c e))
d_\<beta> (NDvd j (CN 0 c e)) 1
d_\<delta> (NDvd j (CN 0 c e)) d
0 < d
\<not> (\<exists>ja\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd j (CN 0 c e))). x = b + ja)
Ifm bbs (x # bs) (NDvd j (CN 0 c e))
[PROOF STEP]
have id: "j dvd d"
[PROOF STATE]
proof (prove)
using this:
iszlfm (NDvd j (CN 0 c e))
d_\<beta> (NDvd j (CN 0 c e)) 1
d_\<delta> (NDvd j (CN 0 c e)) d
0 < d
\<not> (\<exists>ja\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd j (CN 0 c e))). x = b + ja)
Ifm bbs (x # bs) (NDvd j (CN 0 c e))

goal (1 subgoal):
 1. j dvd d
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
j dvd d

goal (68 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 4. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 5. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 6. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 7. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 8. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
 10. \<And>va vb. \<lbrakk>iszlfm (Lt (Add va vb)); d_\<beta> (Lt (Add va vb)) 1; d_\<delta> (Lt (Add va vb)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Add va vb))). x = b + j); Ifm bbs (x # bs) (Lt (Add va vb))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Add va vb))
A total of 68 subgoals...
[PROOF STEP]
from c1
[PROOF STATE]
proof (chain)
picking this:
c = 1
[PROOF STEP]
have "?p x \<longleftrightarrow> \<not> j dvd (x + ?e)"
[PROOF STATE]
proof (prove)
using this:
c = 1

goal (1 subgoal):
 1. Ifm bbs (x # bs) (NDvd j (CN 0 c e)) = (\<not> j dvd x + Inum (x # bs) e)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
Ifm bbs (x # bs) (NDvd j (CN 0 c e)) = (\<not> j dvd x + Inum (x # bs) e)

goal (68 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 4. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 5. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 6. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 7. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 8. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
 10. \<And>va vb. \<lbrakk>iszlfm (Lt (Add va vb)); d_\<beta> (Lt (Add va vb)) 1; d_\<delta> (Lt (Add va vb)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Add va vb))). x = b + j); Ifm bbs (x # bs) (Lt (Add va vb))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Add va vb))
A total of 68 subgoals...
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
Ifm bbs (x # bs) (NDvd j (CN 0 c e)) = (\<not> j dvd x + Inum (x # bs) e)

goal (68 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 4. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 5. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 6. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 7. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 8. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
 10. \<And>va vb. \<lbrakk>iszlfm (Lt (Add va vb)); d_\<beta> (Lt (Add va vb)) 1; d_\<delta> (Lt (Add va vb)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Add va vb))). x = b + j); Ifm bbs (x # bs) (Lt (Add va vb))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Add va vb))
A total of 68 subgoals...
[PROOF STEP]
have "\<dots> \<longleftrightarrow> \<not> j dvd x - d + ?e"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<not> j dvd x + Inum (x # bs) e) = (\<not> j dvd x - d + Inum (x # bs) e)
[PROOF STEP]
using zdvd_period[OF id, where x="x" and c="-1" and t="?e"]
[PROOF STATE]
proof (prove)
using this:
(j dvd x + Inum (x # bs) e) = (j dvd x + - 1 * d + Inum (x # bs) e)

goal (1 subgoal):
 1. (\<not> j dvd x + Inum (x # bs) e) = (\<not> j dvd x - d + Inum (x # bs) e)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<not> j dvd x + Inum (x # bs) e) = (\<not> j dvd x - d + Inum (x # bs) e)

goal (68 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<And>i c e. \<lbrakk>iszlfm (NDvd i (CN 0 c e)); d_\<beta> (NDvd i (CN 0 c e)) 1; d_\<delta> (NDvd i (CN 0 c e)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (NDvd i (CN 0 c e))). x = b + j); Ifm bbs (x # bs) (NDvd i (CN 0 c e))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (NDvd i (CN 0 c e))
 4. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 5. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 6. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 7. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 8. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 9. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
 10. \<And>va vb. \<lbrakk>iszlfm (Lt (Add va vb)); d_\<beta> (Lt (Add va vb)) 1; d_\<delta> (Lt (Add va vb)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Add va vb))). x = b + j); Ifm bbs (x # bs) (Lt (Add va vb))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Add va vb))
A total of 68 subgoals...
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
Ifm bbs (x # bs) (NDvd j (CN 0 c e)) = (\<not> j dvd x - d + Inum (x # bs) e)
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
Ifm bbs (x # bs) (NDvd j (CN 0 c e)) = (\<not> j dvd x - d + Inum (x # bs) e)

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (NDvd j (CN 0 c e))
[PROOF STEP]
using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p
[PROOF STATE]
proof (prove)
using this:
Ifm bbs (x # bs) (NDvd j (CN 0 c e)) = (\<not> j dvd x - d + Inum (x # bs) e)
Inum ((x - d) # bs) e = Inum (x # bs) e
c = 1
Ifm bbs (x # bs) (NDvd j (CN 0 c e))

goal (1 subgoal):
 1. Ifm bbs ((x - d) # bs) (NDvd j (CN 0 c e))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
Ifm bbs ((x - d) # bs) (NDvd j (CN 0 c e))

goal (67 subgoals):
 1. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (And p q); d_\<beta> (And p q) 1; d_\<delta> (And p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (And p q)). x = b + j); Ifm bbs (x # bs) (And p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (And p q)
 2. \<And>p q. \<lbrakk>\<lbrakk>iszlfm p; d_\<beta> p 1; d_\<delta> p d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). x = b + j); Ifm bbs (x # bs) p\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) p; \<lbrakk>iszlfm q; d_\<beta> q 1; d_\<delta> q d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> q). x = b + j); Ifm bbs (x # bs) q\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) q; iszlfm (Or p q); d_\<beta> (Or p q) 1; d_\<delta> (Or p q) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Or p q)). x = b + j); Ifm bbs (x # bs) (Or p q)\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Or p q)
 3. \<lbrakk>iszlfm T; d_\<beta> T 1; d_\<delta> T d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> T). x = b + j); Ifm bbs (x # bs) T\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) T
 4. \<lbrakk>iszlfm F; d_\<beta> F 1; d_\<delta> F d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> F). x = b + j); Ifm bbs (x # bs) F\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) F
 5. \<And>va. \<lbrakk>iszlfm (Lt (C va)); d_\<beta> (Lt (C va)) 1; d_\<delta> (Lt (C va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (C va))). x = b + j); Ifm bbs (x # bs) (Lt (C va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (C va))
 6. \<And>va. \<lbrakk>iszlfm (Lt (Bound va)); d_\<beta> (Lt (Bound va)) 1; d_\<delta> (Lt (Bound va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Bound va))). x = b + j); Ifm bbs (x # bs) (Lt (Bound va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Bound va))
 7. \<And>vd vb vc. \<lbrakk>iszlfm (Lt (CN (Suc vd) vb vc)); d_\<beta> (Lt (CN (Suc vd) vb vc)) 1; d_\<delta> (Lt (CN (Suc vd) vb vc)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (CN (Suc vd) vb vc))). x = b + j); Ifm bbs (x # bs) (Lt (CN (Suc vd) vb vc))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (CN (Suc vd) vb vc))
 8. \<And>va. \<lbrakk>iszlfm (Lt (Neg va)); d_\<beta> (Lt (Neg va)) 1; d_\<delta> (Lt (Neg va)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Neg va))). x = b + j); Ifm bbs (x # bs) (Lt (Neg va))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Neg va))
 9. \<And>va vb. \<lbrakk>iszlfm (Lt (Add va vb)); d_\<beta> (Lt (Add va vb)) 1; d_\<delta> (Lt (Add va vb)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Add va vb))). x = b + j); Ifm bbs (x # bs) (Lt (Add va vb))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Add va vb))
 10. \<And>va vb. \<lbrakk>iszlfm (Lt (Sub va vb)); d_\<beta> (Lt (Sub va vb)) 1; d_\<delta> (Lt (Sub va vb)) d; 0 < d; \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> (Lt (Sub va vb))). x = b + j); Ifm bbs (x # bs) (Lt (Sub va vb))\<rbrakk> \<Longrightarrow> Ifm bbs ((x - d) # bs) (Lt (Sub va vb))
A total of 67 subgoals...
[PROOF STEP]
qed (auto simp add: numbound0_I[where bs="bs" and b="(x - d)" and b'="x"] gr0_conv_Suc)