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Proof step 1 from proof finfrag-repl-frag (conclusion) (back to tree)

⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ oinj(cext) ∧ oinj(aext) ∧ oinj(ext) ∧ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(⌜πra(step, cstep, ren, cren, pcsl, pal)⌝ ) ⊑ aext !(aα) ∧ sameshared(cren, aren, ren) → frag*(Π(IOAut(start, step, ext), IOAut(astart, astep, aext), ren, aren)) (repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof finfrag-repl-frag (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), cext !(⌜πra(step, cstep, ren, cren, pcsl, pal)⌝ ) ⊑ aext !(aα), ¬ frag*(Π(IOAut(start, step, ext), IOAut(astart, astep, aext), ren, aren)) (repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren) = IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)); (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 3 from proof finfrag-repl-frag (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα)) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: pcsl

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s)

Proof step 4 from proof finfrag-repl-frag (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcs + pcsl0, pal), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, pcs + pcsl0, pal))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcs + pcsl0, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcs + pcsl0, pal, ⌜as '⌝ + aσ, aα)), ∀ astart, astep, aext, as, aσ, aα, start, cstart, step, cstep, ren, cren, ext, cext, pal, aren. ¬ ( frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl0, pal) ∧ nounsharedextR(cren, cext, ren) ∧ oinj(ext) ∧ oinj(aext) ∧ oinj(cext) ∧ sameshared(cren, aren, ren) ∧ Πok(ext, aext, ren, aren) ∧ Πok(ext, cext, ren, cren) ∧ ⌜cext !(πra(step, cstep, ren, cren, pcsl0, pal))⌝ ⊑ aext !(aα) ∧ ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcsl0, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsl0, pal, ⌜as '⌝ + aσ, aα))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s)

Proof step 5 from proof finfrag-repl-frag (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcs ' + pcsl0, pal), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, pcs ' + pcsl0, pal))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))(pcsl0, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: # pcsl0 = # pal

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ frag*(IOAut(start, step, ext))(s ' + sl, al) → (# sl = # al ↔ true); (ss) Used formulas: # pcsl0 = # pal

Proof step 6 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl0 = # pal, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcs ' + pcsl0, pal), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, pcs ' + pcsl0, pal))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))(pcsl0, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: pcs

Additional information

Proof step 7 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl0 = # pal, pcs = (s, cs), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcs ' + pcsl0, pal), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, pcs ' + pcsl0, pal))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))(pcsl0, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 8 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl0 = # pal, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + pcsl0, pal), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + pcsl0, pal))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl0, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl0, pal, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))(pcsl0, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: pcsl0

Additional information

Used simplifier rules: repls-base: ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ', [], ⌜as '⌝ + aσ, aα) = (s, as) '; repla-base: ⊦ repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ', [], ⌜as '⌝ + aσ, aα) = []; List('a) freely generated by [], +;; ⊦ # [] = 0; (s) ⊦ # x = 0 ↔ x = []; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s) ⊦ πra(lstep, rstep, lren, rren, (ls, rs) ', []) = []; (s) ⊦ ext !([]) = []; (s) ⊦ ⌜[]⌝ = []; (s) ⊦ [] ⊑ σ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ frag*(IOAut(start, step, ext))(s ', []); (s)

Proof step 9 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl0 = # pal, pcsl0 = pcs + pcsl, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + pcsl0, pal), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + pcsl0, pal))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl0, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl0, pal, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))(pcsl0, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 10 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl + 1 = # pal, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + pcs ' + pcsl, pal), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + pcs ' + pcsl, pal))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcs ' + pcsl, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcs ' + pcsl, pal, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))(pcs ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, pcs ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, pcs ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, pcs ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: pcs

Additional information

Proof step 11 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl + 1 = # pal, pcs = (s0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + pcs ' + pcsl, pal), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + pcs ' + pcsl, pal))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcs ' + pcsl, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcs ' + pcsl, pal, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))(pcs ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, pcs ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, pcs ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, pcs ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 12 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl + 1 = # pal, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (s0, cs0) ' + pcsl, pal), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + (s0, cs0) ' + pcsl, pal))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pal, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: pal

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ # [] = 0; (s) ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws) ⊦ # x = 0 ↔ x = []; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 13 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl + 1 = # pal, pal = pa + pal1, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (s0, cs0) ' + pcsl, pal), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + (s0, cs0) ' + pcsl, pal))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pal, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + n = m + n0 ↔ n = n0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 14 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:finfrag-rec, seq:⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al), subst:[a, al, step, s, start, ext, sl] → [pa, pal1, Π(step, cstep, ren, cren, ext, cext), (s, cs), Π(start, cstart), Π(ext, cext, ren, cren), (s0, cs0) ' + pcsl]; ['s, 'a, 'e] → [('s, 'cs), 'pa, 'e], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: pal1, (s, cs), Π(ext, cext, ren, cren), (s0, cs0) ' + pcsl, pa, Π(start, cstart), Π(step, cstep, ren, cren, ext, cext)

Proof step 15 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al)

A spec-lemma

Proof step 16 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), pa, ((s0, cs0) ' + pcsl).head), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (a ' + x).head = a; (s)

Proof step 17 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), pa, (s0, cs0)), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: product, name: lrstep-def, seq:⊦ Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)) ↔ rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra)), subst:[pa, ls0, rext, lstep, rs0, ls, rs, rren, lren, lext, rstep] → [pa, s0, cext, step, cs0, s, cs, cren, ren, ext, cstep]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['s, 'ca, 'a, 'pa, 'cs, 'e], <all paths>

Additional information

Used terms: cext, pa, cs, cstep, ren, ext, cren, cs0, step, s, s0

Proof step 18 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)) ↔ rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra))

A spec-lemma

Proof step 19 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)), cs = cs0 ∧ (∃ la. step(s, la, s0) ∧ pa = ren(la) ∧ ¬ cod(cren)(pa)) ∨ s = s0 ∧ (∃ ra. cstep(cs, ra, cs0) ∧ pa = cren(ra) ∧ ¬ cod(ren)(pa)) ∨ (∃ la, ra. step(s, la, s0) ∧ ext(la) ≠ None ∧ pa = ren(la) ∧ cstep(cs, ra, cs0) ∧ cext(ra) ≠ None ∧ pa = cren(ra)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 20 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)), cs = cs0 ∧ step(s, a, s0) ∧ pa = ren(a) ∧ ¬ cod(cren)(ren(a)) ∨ s = s0 ∧ cstep(cs, ca, cs0) ∧ pa = cren(ca) ∧ ¬ cod(ren)(cren(ca)) ∨ step(s, a0, s0) ∧ ext(a0) ≠ None ∧ cren(ca0) = ren(a0) ∧ cstep(cs, ca0, cs0) ∧ cext(ca0) ≠ None ∧ pa = cren(ca0) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 14

Additional information

Proof step 21 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, cs = cs0, pa = ren(a), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1))⌝ ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: repls-rec-addS: ⊦ step(s, a, s0) ∧ ¬ cod(cren)(ren(a)) → repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs) ' + pcsl, ren(a) ' + pal, ⌜as '⌝ + aσ, aα) = (s, as) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα); repla-rec-addS: ⊦ step(s, a, s0) ∧ ¬ cod(cren)(ren(a)) → repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs) ' + pcsl, ren(a) ' + pal, ⌜as '⌝ + aσ, aα) = ren(a) ' + repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα); a: ⊦ (x + y) + z = x + y + z; from specification list append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # psl = # pal ∧ lstep(ls, la, ls0) ∧ ¬ cod(rren)(lren(la)) → πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psl, lren(la) ' + pal) = πra(lstep, rstep, lren, rren, (ls0, rs) ' + psl, pal); (s)

Proof step 22 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) ((s, as) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα), ren(a) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:finfrag-rec, seq:⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al), subst:[a, al, step, s, start, ext, sl] → [ren(a), repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα), Π(step, astep, ren, aren, ext, aext), (s, as), Π(start, astart), Π(ext, aext, ren, aren), repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα)]; ['s, 'a, 'e] → [('s, 'as), 'pa, 'e], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα), (s, as), Π(ext, aext, ren, aren), repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα), ren(a), Π(start, astart), Π(step, astep, ren, aren, ext, aext)

Proof step 23 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al)

A spec-lemma

Proof step 24 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)), ¬ Π(step, astep, ren, aren, ext, aext) ((s, as), ren(a), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα).head) ∨ ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα)) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:finfrag-repls-head, seq:nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), frag (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (⌜(s, cs) '⌝ + ⌜pcsl⌝ , ⌜pal⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), cext !(⌜πra(step, cstep, ren, cren, (s, cs) ' + pcsl, pal)⌝ ) ⊑ aext !(aα) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα) .head = (s, as), subst:[astep, cext, as, start, aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsl, aext, step, pal, cs, aα, s] → [astep, cext, as, start, aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsl, aext, step, pal1, cs0, aα, s0]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used simplifier rules: ⊦ frag(ioa)(σ, ⌜al⌝ ) ↔ frag*(ioa)(σ.l, al) ∧ finp(σ); (s) ⊦ (⌜x⌝ + ⌜y⌝ ).l = x + y; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss) ⊦ σ0 + ⌜x⌝ = ⌜nf0⌝ ↔ σ0 = ⌜nf0⌝ ; (s) ⊦ ⌜x⌝ ≠ ⌜nf⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: aren, ren, cstep, cext, s0, as, aext, aα, ext, astart, cstart, start, pal1, cren, astep, pcsl, cs0, aσ, step

Proof step 25 from proof finfrag-repl-frag (previous step) (back to tree)

nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜(s, cs) '⌝ + ⌜pcsl⌝ , ⌜pal⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), cext !(⌜πra(step, cstep, ren, cren, (s, cs) ' + pcsl, pal)⌝ ) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα).head = (s, as)

Lemma finfrag-repls-head

Proof step 26 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα).head = (s0, as), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα) ≠ [], frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)), ¬ Π(step, astep, ren, aren, ext, aext) ((s, as), ren(a), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα).head) ∨ ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα)) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-add, seq: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail), subst:[x, head, tail] → [repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα), pas, x0]; ['a] → [('s, 'as)]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα), pas, x0

Proof step 27 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, pas = (s0, as), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα) = pas + x0, repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα) ≠ [], frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)), ¬ Π(step, astep, ren, aren, ext, aext)((s, as), ren(a), pas) ∨ ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ a + x = a ' + x; (s)

Proof step 28 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα) = (s0, as) ' + x0, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)), ¬ Π(step, astep, ren, aren, ext, aext)((s, as), ren(a), (s0, as)) ∨ ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) ((s0, as) ' + x0, repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: product, name: lrstep-def, seq:⊦ Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)) ↔ rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra)), subst:[pa, ls0, rext, lstep, rs0, ls, rs, rren, lren, lext, rstep] → [ren(a), s0, aext, step, as, s, as, aren, ren, ext, astep]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['s, 'aa, 'a, 'pa, 'as, 'e], <all paths>

Additional information

Used terms: aext, ren(a), as, astep, ren, ext, aren, as, step, s, s0

Proof step 29 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)) ↔ rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra))

A spec-lemma

Proof step 30 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα) = (s0, as) ' + x0, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)), ¬ ( as = as ∧ (∃ la. step(s, la, s0) ∧ ren(a) = ren(la) ∧ ¬ cod(aren)(ren(a))) ∨ s = s0 ∧ (∃ ra. astep(as, ra, as) ∧ ren(a) = aren(ra) ∧ ¬ cod(ren)(ren(a))) ∨ (∃ la, ra. step(s, la, s0) ∧ ext(la) ≠ None ∧ ren(a) = ren(la) ∧ astep(as, ra, as) ∧ aext(ra) ≠ None ∧ ren(a) = aren(ra))) ∨ ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) ((s0, as) ' + x0, repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ Πok(lext, rext, lren, rren) ∧ lren(la) = rren(ra) → (lext(la) = None ↔ false); (as) ⊦ Πok(lext, rext, lren, rren) ∧ lren(la) = rren(ra) → (rext(ra) = None ↔ false); (as) ⊦ Πok(lext, rext, lren, rren) → (lren(la1) = lren(la2) ↔ la1 = la2); (s) ⊦ cod(f)(f(a)); (s)

Proof step 31 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα) = (s0, as) ' + x0, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)), cod(aren)(ren(a)) ∧ ¬(∃ ra. astep(as, ra, as) ∧ ren(a) = aren(ra)) ∨ ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) ((s0, as) ' + x0, repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα)) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 16

Additional information

Used simplifier rules: sameshared-notcodc-notcoda: ⊦ sameshared(cren, aren, ren) ∧ ¬ cod(cren)(ren(a)) → aren(aa) ≠ ren(a); ⊦ cod(f)(b) ↔ (∃ a. f(a) = b); (as)

Proof step 32 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα) = (s0, as) ' + x0, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) ((s0, as) ' + x0, repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 16 terms: astart, astep, aext, as, aσ, aα, start, cstart, step, cstep, ren, cren, ext, cext, pal1, aren abort: F discard: T computed: T

Additional information

Used terms: astart, astep, aext, as, aσ, aα, start, cstart, step, cstep, ren, cren, ext, cext, pal1, aren

Proof step 33 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, pa = cren(ca), s = s0, cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1))⌝ ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: repls-rec-skip: ⊦ cstep(cs, ca, cs0) ∧ ¬ cod(ren)(cren(ca)) → repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s, cs0) ' + pcsl, cren(ca) ' + pal, ⌜as '⌝ + aσ, aα) = repls (step, cstep, cren, ren, aren, cext, aext, (s, cs0) ' + pcsl, pal, ⌜as '⌝ + aσ, aα); repla-rec-skip: ⊦ cstep(cs, ca, cs0) ∧ ¬ cod(ren)(cren(ca)) → repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s, cs0) ' + pcsl, cren(ca) ' + pal, ⌜as '⌝ + aσ, aα) = repla (step, cstep, cren, ren, aren, cext, aext, (s, cs0) ' + pcsl, pal, ⌜as '⌝ + aσ, aα); a: ⊦ (x + y) + z = x + y + z; from specification list append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s) ⊦ a + x = a ' + x; (s) ⊦ # psl = # pal ∧ rstep(rs, ra, rs0) ∧ ¬ cod(lren)(rren(ra)) → πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls, rs0) ' + psl, rren(ra) ' + pal) = ra + πra(lstep, rstep, lren, rren, (ls, rs0) + psl, pal); (s)

Proof step 34 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(ca ')⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 15 terms: astart, astep, aext, as, aσ, aα, start, cstart, step, cstep, ren, cren, ext, cext, pal1, aren abort: F discard: T computed: T

Additional information

Used terms: astart, astep, aext, as, aσ, aα, start, cstart, step, cstep, ren, cren, ext, cext, pal1, aren

Proof step 35 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(ca ')⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα)), ¬ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∨ ¬ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1) ∨ ¬ nounsharedextR(cren, cext, ren) ∨ ¬ oinj(ext) ∨ ¬ oinj(aext) ∨ ¬ oinj(cext) ∨ ¬ sameshared(cren, aren, ren) ∨ ¬ Πok(ext, aext, ren, aren) ∨ ¬ Πok(ext, cext, ren, cren) ∨ ¬ ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα) ∨ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 36 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(ca ')⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜as '⌝ + aσ, aα)), ¬ ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name:finremtau-one, seq: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '), subst:[ext, a] → [cext, ca]; ['a, 'e] → ['ca, 'e], <all paths>

Additional information

Used simplifier rules: nounsharedextR-notcodren-internal: ⊦ ¬ cod(ren)(cren(ca)) ∧ nounsharedextR(cren, cext, ren) → (cext(ca) = None ↔ true); ⊦ ⌜[]⌝ = []; (s) ⊦ [] + σ = σ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: cext, ca

Proof step 37 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get ')

A spec-lemma

Proof step 38 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, cren(ca0) = ren(a0), pa = cren(ca0), cext(ca0) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, pa ' + pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) ⊦ ext(a) ≠ None → ext !(a ' + al) = ext(a).get ' + ext !(al); (s) ⊦ a + x = a ' + x; (s) pa = lren(la) ⊦ # psl = # pal ∧ rstep(rs, ra, rs0) ∧ rren(ra) = lren(la) ∧ lstep(ls, la, ls0) → πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ' + psl, pa ' + pal) = ra + πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl, pal); (s)

Proof step 39 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, cren(ca0) = ren(a0), cext(ca0) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext(ca0).get '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: option, name:elim-Some, seq: ⊦ opt ≠ None → (get = opt.get ↔ opt = Some(get)), subst:[opt, get] → [cext(ca0), e]; ['a] → ['e]

Additional information

Used simplifier rules: ⊦ opt ≠ None → (get = opt.get ↔ opt = Some(get)); (elim) Used terms: cext(ca0), e

Proof step 40 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, cext(ca0) = Some(e), cren(ca0) = ren(a0), cext(ca0) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜e '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 41 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜e '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: prefix-def, seq: ⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0), subst:[σ, σ0] → [⌜e '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ , aext !(aα)]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜e '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ , aext !(aα)

Proof step 42 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 43 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)), ∃ σ1. (⌜e '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ) + σ1 = aext !(aα) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 44 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, (⌜e '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ) + β = aext !(aα), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name:remtau-split-one, seq:⊦ ext !(α) = ⌜e '⌝ + β ↔ (∃ al, a, α0. α = ⌜al⌝ + ⌜a '⌝ + α0 ∧ ext(a) = Some(e) ∧ ext !(⌜al⌝ ) = [] ∧ ext !(α0) = β), subst:[ext, α, e, β] → [aext, aα, e, ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β]; ['e, 'a] → ['e, 'aa], <all paths>

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: aext, aα, e, ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β

Proof step 45 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ ext !(α) = ⌜e '⌝ + β ↔ (∃ al, a, α0. α = ⌜al⌝ + ⌜a '⌝ + α0 ∧ ext(a) = Some(e) ∧ ext !(⌜al⌝ ) = [] ∧ ext !(α0) = β)

A spec-lemma

Proof step 46 from proof finfrag-repl-frag (previous step) (back to tree)

# pcsl = # pal1, cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)), ∃ al, aa, α0. aα = ⌜al⌝ + ⌜aa '⌝ + α0 ∧ aext(aa) = Some(e) ∧ aext !(⌜al⌝ ) = [] ∧ aext !(α0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ = [] ↔ x = []; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 47 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:frag-from-1, seq:⊦ frag(IOAut(start, step, ext))(σ, α) ∧ ¬ # α < ⌜n⌝ → (frag(IOAut(start, step, ext))(σ.drop(n), α.drop(n)) ↔ true), subst:[step, α, start, ext, n, σ] → [astep, ⌜aal⌝ + ⌜aa '⌝ + aα0, astart, aext, # aal + 1, ⌜as '⌝ + aσ]; ['s, 'a, 'e] → ['as, 'aa, 'e]

Additional information

Used terms: aext, ⌜aal⌝ + ⌜aa '⌝ + aα0, astart, ⌜as '⌝ + aσ, # aal + 1, astep

Proof step 48 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ frag(IOAut(start, step, ext))(σ, α) ∧ ¬ # α < ⌜n⌝ → (frag(IOAut(start, step, ext))(σ.drop(n), α.drop(n)) ↔ true)

A spec-lemma

Proof step 49 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0) ∧ ¬ #(⌜aal⌝ + ⌜aa '⌝ + aα0) < ⌜# aal + 1⌝ → (frag(IOAut(astart, astep, aext))((⌜as '⌝ + aσ).drop(# aal + 1), (⌜aal⌝ + ⌜aa '⌝ + aα0).drop(# aal + 1)) ↔ true), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (⌜a '⌝ + σ).drop(1) = σ; (s) ⊦ (⌜x⌝ + σ).drop(m + # x) = σ.drop(m); (s) ⊦ n + 0 = n; (s) ⊦ n ≠ 0 → (⌜a '⌝ + σ).drop((* n) + m) = σ.drop((n - 1) + m); (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) ⊦ n∞ + ⌜0⌝ = n∞; (s) ⊦ m ≤ n → (⌜(* m)⌝ + n∞2 < ⌜(* n)⌝ + n∞1 ↔ n∞2 < ⌜n - m⌝ + n∞1); (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 50 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(aσ.drop(# aal), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)), ∀ astart0, astep0, aext0, as0, aσ0, aα0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, pal0, aren0. ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl, pal0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ oinj(ext0) ∨ ¬ oinj(aext0) ∨ ¬ oinj(cext0) ∨ ¬ sameshared(cren0, aren0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl, pal0))⌝ ⊑ aext0 !(aα0) ∨ frag*(IOAut(Π(start0, astart0), Π(step0, astep0, ren0, aren0, ext0, aext0), Π(ext0, aext0, ren0, aren0))) (repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0), repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl, pal0, ⌜as0 '⌝ + aσ0, aα0)) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 22 terms: astart, astep, aext, aσ[# aal], aσ.drop(# aal + 1), aα0, start, cstart, step, cstep, ren, cren, ext, cext, pal1, aren abort: F discard: T computed: F

Additional information

Used terms: astart, astep, aext, aσ[# aal], aσ.drop(# aal + 1), aα0, start, cstart, step, cstep, ren, cren, ext, cext, pal1, aren

Proof step 51 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(aσ.drop(# aal), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)), ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ∨ ¬ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1) ∨ ¬ nounsharedextR(cren, cext, ren) ∨ ¬ oinj(ext) ∨ ¬ oinj(aext) ∨ ¬ oinj(cext) ∨ ¬ sameshared(cren, aren, ren) ∨ ¬ Πok(ext, aext, ren, aren) ∨ ¬ Πok(ext, cext, ren, cren) ∨ ¬ ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα0) ∨ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 52 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(aσ.drop(# aal), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)), ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ∨ ¬ ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα0) ∨ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: from-succ, seq: ⊦ ⌜n⌝ < # σ → σ.drop(n) = σ[n] + σ.drop(n + 1), subst:[σ, n] → [aσ, # aal]; ['a] → ['as], <all paths>

Additional information

Used terms: aσ, # aal

Proof step 53 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ ⌜n⌝ < # σ → σ.drop(n) = σ[n] + σ.drop(n + 1)

A spec-lemma

Proof step 54 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(aσ.drop(# aal), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ ⌜# aal⌝ < # aσ, ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)), ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ∨ ¬ ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα0) ∨ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name: frag, seq:⊦ frag(IOAut(start, step, ext))(σ, α) ↔ # σ = # α + ⌜1⌝ ∧ (∀ n. ⌜n⌝ < # α → step(σ[n], α[n], σ[n +1])), subst:[σ, step, α, start, ext] → [⌜as '⌝ + aσ, astep, ⌜aal⌝ + ⌜aa '⌝ + aα0, astart, aext]; ['s, 'a, 'e] → ['as, 'aa, 'e], <all paths>

Additional information

Used terms: aext, ⌜aal⌝ + ⌜aa '⌝ + aα0, astart, ⌜as '⌝ + aσ, astep

Proof step 55 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ frag(IOAut(start, step, ext))(σ, α) ↔ # σ = # α + ⌜1⌝ ∧ (∀ n. ⌜n⌝ < # α → step(σ[n], α[n], σ[n +1]))

A spec-lemma

Proof step 56 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], #(⌜as '⌝ + aσ) = #(⌜aal⌝ + ⌜aa '⌝ + aα0) + ⌜1⌝ , # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(aσ.drop(# aal), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ ⌜# aal⌝ < # aσ, ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)), ∀ n. ⌜n⌝ < #(⌜aal⌝ + ⌜aa '⌝ + aα0) → astep((⌜as '⌝ + aσ)[n], (⌜aal⌝ + ⌜aa '⌝ + aα0)[n], (⌜as '⌝ + aσ)[n +1]), ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ∨ ¬ ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα0) ∨ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (⌜a '⌝ + σ)[n + 1] = σ[n]; (s) ⊦ m +1 = m + 1; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ n < m → (⌜(* n)⌝ + n∞1 = ⌜(* m)⌝ + n∞2 ↔ n∞1 = ⌜m - n⌝ + n∞2); (s) ⊦ ⌜(* n)⌝ + ⌜(* m)⌝ = ⌜n + m⌝ ; (s)

Proof step 57 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # pcsl = # pal1, # aσ = ⌜1⌝ + ⌜# aal⌝ + # aα0, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(aσ.drop(# aal), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ ⌜# aal⌝ < # aσ, ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)), ∀ n. ¬ ⌜# aal⌝ + # aα0 < ⌜n⌝ → astep((⌜as '⌝ + aσ)[n], (⌜aal⌝ + ⌜aa '⌝ + aα0)[n], aσ[n]), ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ∨ ¬ ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα0) ∨ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) ⊦

User applied rule subst equation

And closed the goal!

Rule argument

pos: left 4, flags: discard

Additional information

Used simplifier rules: ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) ⊦ ¬ n∞ + n∞1 < n∞1; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf

Proof step 58 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(aσ[# aal] + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)), ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ∨ ¬ ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα0) ∨ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + σ = ⌜a '⌝ + σ; (s)

Proof step 59 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)), ¬ ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα0) ∨ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 23

Additional information

Proof step 60 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)), ¬ ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ ⊑ aext !(aα0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: prefix-def, seq: ⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0), subst:[σ, σ0] → [⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ , aext !(aα0)]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ , aext !(aα0)

Proof step 61 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 62 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)), ∀ σ1. ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + σ1 ≠ aext !(aα0) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 23 terms: β

Additional information

Used terms: β

Proof step 63 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: repls-rec-addboth, seq:⊦ step(s, a, s0) ∧ cstep(cs, ca, cs0) ∧ aext !(⌜aal⌝ ) = [] ∧ cext(ca) = aext(aa) ∧ cren(ca) = ren(a) ∧ n = # aal → repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a) ' + pal, aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) = lift(aσ.take(n), s) + (s, aσ[n]) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal, aσ.drop(n + 1), aα), subst:[ca, a, aal, aa, aσ, cren, ren, n, cstep, aren, pcsl, aext, step, pal, cs, cext, s0, cs0, aα, s] → [ca0, a0, aal, aa, ⌜as '⌝ + aσ, cren, ren, # aal, cstep, aren, pcsl, aext, step, pal1, cs, cext, s0, cs0, aα0, s]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as], <all paths>

Additional information

Used simplifier rules: ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ ⌜x⌝ = [] ↔ x = []; (s) Used terms: aren, ren, cstep, cext, aa, s, aext, aα0, ca0, a0, pal1, cren, aal, pcsl, cs0, s0, cs, # aal, ⌜as '⌝ + aσ, step

Proof step 64 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ step(s, a, s0) ∧ cstep(cs, ca, cs0) ∧ aext !(⌜aal⌝ ) = [] ∧ cext(ca) = aext(aa) ∧ cren(ca) = ren(a) ∧ n = # aal → repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a) ' + pal, aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) = lift(aσ.take(n), s) + (s, aσ[n]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal, aσ.drop(n + 1), aα)

Lemma repls-rec-addboth

Proof step 65 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], aext !(⌜aal⌝ ) = [], # pcsl = # pal1, # aal = # aal, aext(aa) = Some(e), cext(ca0) = Some(e), cext(ca0) = aext(aa), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, (⌜as '⌝ + aσ).drop(# aal + 1), aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ n + 0 = n; (s) ⊦ n ≠ 0 → (⌜a '⌝ + σ).drop((* n) + m) = σ.drop((n - 1) + m); (s) ⊦ ⌜x⌝ = [] ↔ x = []; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 66 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: repla-rec-addboth, seq:⊦ step(s, a, s0) ∧ cstep(cs, ca, cs0) ∧ cren(ca) = ren(a) ∧ aext !(⌜aal⌝ ) = [] ∧ cext(ca) = aext(aa) ∧ n = # aal → repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a) ' + pal, aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) = aal.map(aren) + aren(aa) ' + repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal, aσ.drop(n + 1), aα), subst:[ca, a, aal, aa, aσ, cren, ren, n, cstep, aren, pcsl, aext, step, pal, cs, cext, s0, cs0, aα, s] → [ca0, a0, aal, aa, ⌜as '⌝ + aσ, cren, ren, # aal, cstep, aren, pcsl, aext, step, pal1, cs, cext, s0, cs0, aα0, s]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as], <all paths>

Additional information

Used simplifier rules: ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ ⌜x⌝ = [] ↔ x = []; (s) Used terms: aren, ren, cstep, cext, aa, s, aext, aα0, ca0, a0, pal1, cren, aal, pcsl, cs0, s0, cs, # aal, ⌜as '⌝ + aσ, step

Proof step 67 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ step(s, a, s0) ∧ cstep(cs, ca, cs0) ∧ cren(ca) = ren(a) ∧ aext !(⌜aal⌝ ) = [] ∧ cext(ca) = aext(aa) ∧ n = # aal → repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a) ' + pal, aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) = aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal, aσ.drop(n + 1), aα)

Lemma repla-rec-addboth

Proof step 68 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(⌜aal⌝ ) = [], # pcsl = # pal1, # aal = # aal, aext(aa) = Some(e), cext(ca0) = Some(e), cext(ca0) = aext(aa), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, (⌜as '⌝ + aσ).drop(# aal + 1), aα0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ n + 0 = n; (s) ⊦ n ≠ 0 → (⌜a '⌝ + σ).drop((* n) + m) = σ.drop((n - 1) + m); (s) ⊦ ⌜x⌝ = [] ↔ x = []; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 69 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:fragstar, seq:⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1])), subst:[ioa, sl, al] → [IOAut (Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)), lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aal.map(aren) + aren(aa) ' + repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)]; ['s, 'a, 'e] → [('s, 'as), 'pa, 'e], <all paths>

Additional information

Used terms: IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)), lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aal.map(aren) + aren(aa) ' + repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)

Proof step 70 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1]))

A spec-lemma

Proof step 71 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), # ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)) ≠ #(aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)) + 1 ∨ (∃ n. ¬ ( n < #(aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)) → IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)).step (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n ], (aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n], ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n +1 ]))) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:fragstar, seq:⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1])), subst:[ioa, sl, al] → [IOAut (Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)), repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)]; ['s, 'a, 'e] → [('s, 'as), 'pa, 'e], <all paths>

Additional information

Used terms: IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)), repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)

Proof step 72 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1]))

A spec-lemma

Proof step 73 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ n. n < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)).step (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n +1]), # ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)) ≠ #(aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)) + 1 ∨ (∃ n. ¬ ( n < #(aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)) → IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)).step (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n ], (aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n], ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n +1 ]))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-size: ⊦ # lift(asl, s) = # asl; ⊦ IOAut(start, step, ext).step = step; (s) ⊦ m +1 = m + 1; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ #(a ' + x) = (# x)+1; (s) ⊦ # x.map(f) = # x; (s) ⊦ #(x + y) = # x + # y; (s) n < n0 ⊦ m + (* n) = m0 + (* n0) ↔ m = m0 + (n0 - n); (s) ⊦ m + n = m + n0 ↔ n = n0; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 74 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]), # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) ≠ # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) + 1 ∨ ¬ ( ¬ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) < n → Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n ], (aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n], ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n + 1 ])) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: 1 + # repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) ≠ # repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)

Additional information

Used formulas: 1 + # repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) ≠ # repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)

Proof step 75 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) ≠ # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]), # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) ≠ # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) + 1 ∨ ¬ ( ¬ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) < n → Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n ], (aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n], ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n + 1 ])) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic

Proof step 76 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) ≠ # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: from-succ, seq: ⊦ ⌜n⌝ < # σ → σ.drop(n) = σ[n] + σ.drop(n + 1), subst:[σ, n] → [aσ, # aal]; ['a] → ['as], <all paths>

Additional information

Used simplifier rules: ⊦ a + σ = ⌜a '⌝ + σ; (s) c: ⊦ m + n = n + m; from specification nat-basic Used terms: aσ, # aal

Proof step 77 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ ⌜n⌝ < # σ → σ.drop(n) = σ[n] + σ.drop(n + 1)

A spec-lemma

Proof step 78 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]), # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) ≠ # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) + 1 ∨ ¬ ( ¬ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) < n → Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n ], (aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n], ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n + 1 ])) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss) c: ⊦ m + n = n + m; from specification nat-basic

Proof step 79 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n ], (aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n], ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n + 1 ]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: aren(aa) = ren(a0)

Additional information

Used simplifier rules: product-ok-refine-cext-imp-aext: ⊦ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ oinj(aext) ∧ oinj(ext) ∧ aext(aa) = Some(e) ∧ cext(ca0) = Some(e) ∧ cren(ca0) = ren(a0) → (aren(aa) = ren(a0) ↔ true); Used formulas: aren(aa) = ren(a0)

Proof step 80 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n ], (aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n], ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n + 1 ]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 81 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n ], (aal.map(aren) + ren(a0) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n], ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n + 1 ]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name: get-append-less, seq: ⊦ n < # x → (x + y)[n] = x[n], subst:[x, y, n] → [aal.map(aren), ren(a0) ' + repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), n]; ['a] → ['pa], <all paths>

Additional information

Used terms: aal.map(aren), ren(a0) ' + repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), n

Proof step 82 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ n < # x → (x + y)[n] = x[n]

A spec-lemma

Proof step 83 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ n < # aal.map(aren), ¬ Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n ], (aal.map(aren) + ren(a0) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n], ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n + 1 ]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m ≤ m + n; (s) let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ # x.map(f) = # x; (s)

Proof step 84 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), # aal ≤ n, n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n ], (aal.map(aren) + ren(a0) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n], ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n + 1 ]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name: get-append-grt, seq: ⊦ ¬ n < # x → (x + y)[n] = y[n - # x], subst:[x, y, n] → [lift((⌜as '⌝ + aσ).take(# aal), s), (s, (⌜as '⌝ + aσ)[# aal]) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), n + 1]; ['a] → [('s, 'as)], <all paths>

Additional information

Used simplifier rules: lift-size: ⊦ # lift(asl, s) = # asl; ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ # a ' = 0 +1; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat ⊦ # x = 0 ↔ x = []; (s) ⊦ ¬ m + n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: lift((⌜as '⌝ + aσ).take(# aal), s), (s, (⌜as '⌝ + aσ)[# aal]) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), n + 1

Proof step 85 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ ¬ n < # x → (x + y)[n] = y[n - # x]

A spec-lemma

Proof step 86 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), # aal ≤ n, ¬ n + 1 < # lift((⌜as '⌝ + aσ).take(# aal), s), ¬ Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n ], (aal.map(aren) + ren(a0) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n], ((s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n + 1 - # lift((⌜as '⌝ + aσ).take(# aal), s) ]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-size: ⊦ # lift(asl, s) = # asl; a: ⊦ (x + y) + z = x + y + z; from specification list c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + x)[n + 1] = x[n]; (s) ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 87 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), # aal ≤ n, n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), # aal ≤ n + 1, ¬ Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n ], (aal.map(aren) + ren(a0) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[n - # aal]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-sub, seq:n ≤ m ⊦ n0 = m - n ↔ m = n0 + n, subst:[m, n, n0] → [n, # aal, n1]

Additional information

Used simplifier rules: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: n, # aal, n1

Proof step 88 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), n = n1 + # aal, Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), # aal ≤ n + 1, n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), # aal ≤ n, ¬ Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n ], (aal.map(aren) + ren(a0) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[n1]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m ≤ m + n; (s) ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)

Proof step 89 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), n1 ≤ # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n1 + # aal ], (aal.map(aren) + ren(a0) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n1 + # aal], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[n1]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name: get-append-grt, seq: ⊦ ¬ n < # x → (x + y)[n] = y[n - # x], subst:[x, y, n] → [aal.map(aren), ren(a0) ' + repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), n1 + # aal]; ['a] → ['pa], <all paths>

Additional information

Used simplifier rules: ⊦ # x.map(f) = # x; (s) ⊦ ¬ m + n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: aal.map(aren), ren(a0) ' + repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), n1 + # aal

Proof step 90 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ ¬ n < # x → (x + y)[n] = y[n - # x]

A spec-lemma

Proof step 91 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n1 ≤ # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ n1 + # aal < # aal.map(aren), ¬ Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n1 + # aal ], (ren(a0) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n1 + # aal - # aal.map(aren) ], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[n1]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + n - m = n; (s) ⊦ # x.map(f) = # x; (s) ⊦ ¬ m + n < m; (s)

Proof step 92 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), n1 ≤ # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n1 + # aal ], (ren(a0) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n1], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[n1]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name: get-append-grt, seq: ⊦ ¬ n < # x → (x + y)[n] = y[n - # x], subst:[x, y, n] → [lift((⌜as '⌝ + aσ).take(# aal), s), (s, (⌜as '⌝ + aσ)[# aal]) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), n1 + # aal]; ['a] → [('s, 'as)], <all paths>

Additional information

Used simplifier rules: lift-size: ⊦ # lift(asl, s) = # asl; ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ # a ' = 0 +1; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ ¬ m + n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: lift((⌜as '⌝ + aσ).take(# aal), s), (s, (⌜as '⌝ + aσ)[# aal]) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), n1 + # aal

Proof step 93 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ ¬ n < # x → (x + y)[n] = y[n - # x]

A spec-lemma

Proof step 94 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n1 ≤ # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ n1 + # aal < # lift((⌜as '⌝ + aσ).take(# aal), s), ¬ Π(step, astep, ren, aren, ext, aext) (((s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n1 + # aal - # lift((⌜as '⌝ + aσ).take(# aal), s) ], (ren(a0) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n1], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[n1]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-size: ⊦ # lift(asl, s) = # asl; c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + n - m = n; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ ¬ m + n < m; (s)

Proof step 95 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), n1 ≤ # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext) (((s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n1], (ren(a0) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n1], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[n1]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: n1

Additional information

Used simplifier rules: Nat freely generated by 0, +1;;

Proof step 96 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), n1 = 0, Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n1 ≤ # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext) (((s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n1], (ren(a0) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n1], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[n1]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (a ' + x)[0] = a; (s) ⊦ 0 ≤ m; (s)

Proof step 97 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext) ((s, (⌜as '⌝ + aσ)[# aal]), ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[0]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:finfrag-repls-head, seq:nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), frag (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (⌜(s, cs) '⌝ + ⌜pcsl⌝ , ⌜pal⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), cext !(⌜πra(step, cstep, ren, cren, (s, cs) ' + pcsl, pal)⌝ ) ⊑ aext !(aα) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα) .head = (s, as), subst:[astep, cext, as, start, aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsl, aext, step, pal, cs, aα, s] → [astep, cext, aσ[# aal], start, aσ.drop(# aal + 1), ext, cren, ren, astart, cstart, cstep, aren, pcsl, aext, step, pal1, cs0, aα0, s0]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used terms: aren, ren, cstep, cext, s0, aσ[# aal], aext, aα0, ext, astart, cstart, start, pal1, cren, astep, pcsl, cs0, aσ.drop(# aal + 1), step

Proof step 98 from proof finfrag-repl-frag (previous step) (back to tree)

nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜(s, cs) '⌝ + ⌜pcsl⌝ , ⌜pal⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), cext !(⌜πra(step, cstep, ren, cren, (s, cs) ' + pcsl, pal)⌝ ) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα).head = (s, as)

Lemma finfrag-repls-head

Proof step 99 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) ((s, (⌜as '⌝ + aσ)[# aal]), ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[0]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦ frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜(s0, cs0) '⌝ + ⌜pcsl⌝ , ⌜pal1⌝ ) ∧ cext !(⌜πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1)⌝ ) ⊑ aext !(aα0)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: prefix-def, seq: ⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0), subst:[σ, σ0] → [cext !(⌜πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1)⌝ ), aext !(aα0)]; ['a] → ['e], <all paths>

Additional information

Used terms: cext !(⌜πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1)⌝ ), aext !(aα0)

Proof step 100 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 101 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext) ((s, (⌜as '⌝ + aσ)[# aal]), ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[0]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦ frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜(s0, cs0) '⌝ + ⌜pcsl⌝ , ⌜pal1⌝ ) ∧ (∃ σ1. cext !(⌜πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1)⌝ ) + σ1 = aext !(aα0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜x⌝ + ⌜y⌝ = ⌜z⌝ ↔ x + y = z; (s) ⊦ finp(σ) ↔ (∃ x. ⌜x⌝ = σ); (as) ⊦ ⌜x⌝ ≠ ⌜nf⌝ ; (s) ⊦ σ0 + ⌜x⌝ = ⌜nf0⌝ ↔ σ0 = ⌜nf0⌝ ; (s) ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ (⌜x⌝ + ⌜y⌝ ).l = x + y; (s) ⊦ frag(ioa)(σ, ⌜al⌝ ) ↔ frag*(ioa)(σ.l, al) ∧ finp(σ); (s)

Proof step 102 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) ((s, (⌜as '⌝ + aσ)[# aal]), ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[0]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦ ∃ σ1. ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + σ1 = aext !(aα0)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: β

Additional information

Used terms: β

Proof step 103 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0).head = (s0, aσ[# aal]), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ≠ [], ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) ((s, (⌜as '⌝ + aσ)[# aal]), ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[0]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-add, seq: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail), subst:[x, head, tail] → [repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), pas, x0]; ['a] → [('s, 'as)]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), pas, x0

Proof step 104 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), pas = (s0, aσ[# aal]), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = pas + x0, Some(e) ≠ None, ext(a0) ≠ None, repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ≠ [], ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext) ((s, (⌜as '⌝ + aσ)[# aal]), ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[0]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + n = m + n0 ↔ n = n0; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ x = a ' + x0 → x[m + 1] = x0[m]; (s) ⊦ a + x = a ' + x; (s)

Proof step 105 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # x0 = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = (s0, aσ[# aal]) ' + x0, Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) ((s, (⌜as '⌝ + aσ)[# aal]), ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[0]), ∀ n0. n0 < # x0 → Π(step, astep, ren, aren, ext, aext) (((s0, aσ[# aal]) ' + x0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], x0[n0]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: from-succ, seq: ⊦ ⌜n⌝ < # σ → σ.drop(n) = σ[n] + σ.drop(n + 1), subst:[σ, n] → [aσ, # aal]; ['a] → ['as], <all paths>

Additional information

Used terms: aσ, # aal

Proof step 106 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ ⌜n⌝ < # σ → σ.drop(n) = σ[n] + σ.drop(n + 1)

A spec-lemma

Proof step 107 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # pcsl = # pal1, # x0 = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ[# aal] + aσ.drop(# aal + 1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ[# aal] + aσ.drop(# aal + 1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = (s0, aσ[# aal]) ' + x0, Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext) ((s, (⌜as '⌝ + aσ)[# aal]), ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ[# aal] + aσ.drop(# aal + 1), aα0)[0]), ∀ n0. n0 < # x0 → Π(step, astep, ren, aren, ext, aext) (((s0, aσ[# aal]) ' + x0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], x0[n0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (a ' + x)[0] = a; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + n = m + n0 ↔ n = n0; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s) ⊦ a + σ = ⌜a '⌝ + σ; (s)

Proof step 108 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # pcsl = # pal1, # x0 = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = (s0, aσ[# aal]) ' + x0, Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext)((s, (⌜as '⌝ + aσ)[# aal]), ren(a0), (s0, aσ[# aal])), ∀ n0. n0 < # x0 → Π(step, astep, ren, aren, ext, aext) (((s0, aσ[# aal]) ' + x0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], x0[n0]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: product, name: lrstep-def, seq:⊦ Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)) ↔ rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra)), subst:[pa, ls0, rext, lstep, rs0, ls, rs, rren, lren, lext, rstep] → [ren(a0), s0, aext, step, aσ[# aal], s, (⌜as '⌝ + aσ)[# aal], aren, ren, ext, astep]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['s, 'aa, 'a, 'pa, 'as, 'e], <all paths>

Additional information

Used terms: aext, ren(a0), (⌜as '⌝ + aσ)[# aal], astep, ren, ext, aren, aσ[# aal], step, s, s0

Proof step 109 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)) ↔ rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra))

A spec-lemma

Proof step 110 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # x0 = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), # pcsl = # pal1, aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = (s0, aσ[# aal]) ' + x0, Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ n0. n0 < # x0 → Π(step, astep, ren, aren, ext, aext) (((s0, aσ[# aal]) ' + x0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], x0[n0]), ∀ la, ra. ¬(step(s, la, s0) ∧ ext(la) ≠ None ∧ ren(a0) = ren(la) ∧ astep((⌜as '⌝ + aσ)[# aal], ra, aσ[# aal]) ∧ aext(ra) ≠ None ∧ ren(a0) = aren(ra)), (⌜as '⌝ + aσ)[# aal] ≠ aσ[# aal] ∨ (∀ la. ¬(step(s, la, s0) ∧ ren(a0) = ren(la) ∧ ¬ cod(aren)(ren(a0)))), s ≠ s0 ∨ (∀ ra. ¬(astep((⌜as '⌝ + aσ)[# aal], ra, aσ[# aal]) ∧ ren(a0) = aren(ra) ∧ ¬ cod(ren)(ren(a0)))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ cod(f)(b) ↔ (∃ a. f(a) = b); (as) ⊦ ¬ cod(f)(b) → b ≠ f(a); (as) ⊦ Πok(lext, rext, lren, rren) ∧ lren(la) = rren(ra) → (rext(ra) = None ↔ false); (as) ⊦ Πok(lext, rext, lren, rren) → (lren(la1) = lren(la2) ↔ la1 = la2); (s) ⊦ cod(f)(f(a)); (s)

Proof step 111 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # x0 = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), # pcsl = # pal1, aext(aa) = Some(e), aren(aa) = ren(a0), aren(aa0) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = (s0, aσ[# aal]) ' + x0, Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ n0. n0 < # x0 → Π(step, astep, ren, aren, ext, aext) (((s0, aσ[# aal]) ' + x0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], x0[n0]), ∀ ra. ¬ astep((⌜as '⌝ + aσ)[# aal], ra, aσ[# aal]) ∨ ren(a0) ≠ aren(ra) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 27 terms: aa abort: F discard: F computed: T

Additional information

Used terms: aa

Proof step 112 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # pcsl = # pal1, # x0 = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), aext(aa) = Some(e), aren(aa0) = ren(a0), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = (s0, aσ[# aal]) ' + x0, Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ ra. ¬ astep((⌜as '⌝ + aσ)[# aal], ra, aσ[# aal]) ∨ ren(a0) ≠ aren(ra), ∀ n0. n0 < # x0 → Π(step, astep, ren, aren, ext, aext) (((s0, aσ[# aal]) ' + x0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], x0[n0]), ¬ astep((⌜as '⌝ + aσ)[# aal], aa, aσ[# aal]) ∨ ren(a0) ≠ aren(aa) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 113 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # x0 = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), # pcsl = # pal1, aext(aa) = Some(e), aren(aa) = ren(a0), aren(aa0) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = (s0, aσ[# aal]) ' + x0, Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ astep((⌜as '⌝ + aσ)[# aal], aa, aσ[# aal]), ∀ n0. n0 < # x0 → Π(step, astep, ren, aren, ext, aext) (((s0, aσ[# aal]) ' + x0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], x0[n0]), ∀ ra. ¬ astep((⌜as '⌝ + aσ)[# aal], ra, aσ[# aal]) ∨ ren(a0) ≠ aren(ra) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name: frag-step, seq:⌜n⌝ < # α ⊦ ¬ step(σ[n], α[n], σ[n + 1]) → (frag(IOAut(start, step, ext))(σ, α) ↔ false), subst:[step, α, start, ext, n, σ] → [astep, ⌜aal⌝ + ⌜aa '⌝ + aα0, astart, aext, # aal, ⌜as '⌝ + aσ]; ['s, 'a, 'e] → ['as, 'aa, 'e], <all paths>

Additional information

Used terms: aext, ⌜aal⌝ + ⌜aa '⌝ + aα0, astart, ⌜as '⌝ + aσ, # aal, astep

Proof step 114 from proof finfrag-repl-frag (previous step) (back to tree)

⌜n⌝ < # α ⊦ ¬ step(σ[n], α[n], σ[n + 1]) → (frag(IOAut(start, step, ext))(σ, α) ↔ false)

A spec-lemma

Proof step 115 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # pcsl = # pal1, # x0 = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), aext(aa) = Some(e), aren(aa0) = ren(a0), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = (s0, aσ[# aal]) ' + x0, Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ astep((⌜as '⌝ + aσ)[# aal], aa, aσ[# aal]), ∀ ra. ¬ astep((⌜as '⌝ + aσ)[# aal], ra, aσ[# aal]) ∨ ren(a0) ≠ aren(ra), ∀ n0. n0 < # x0 → Π(step, astep, ren, aren, ext, aext) (((s0, aσ[# aal]) ' + x0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], x0[n0]) ⊦ ⌜# aal⌝ < #(⌜aal⌝ + ⌜aa '⌝ + aα0) ∧ ¬ astep((⌜as '⌝ + aσ)[# aal], (⌜aal⌝ + ⌜aa '⌝ + aα0)[# aal], (⌜as '⌝ + aσ)[# aal + 1])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (⌜a '⌝ + σ)[n + 1] = σ[n]; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 116 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # x0 = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), # pcsl = # pal1, aext(aa) = Some(e), aren(aa) = ren(a0), aren(aa0) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = (s0, aσ[# aal]) ' + x0, Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, astep((⌜as '⌝ + aσ)[# aal], (⌜aal⌝ + ⌜aa '⌝ + aα0)[# aal], aσ[# aal]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ astep((⌜as '⌝ + aσ)[# aal], aa, aσ[# aal]), ∀ n0. n0 < # x0 → Π(step, astep, ren, aren, ext, aext) (((s0, aσ[# aal]) ' + x0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], x0[n0]), ∀ ra. ¬ astep((⌜as '⌝ + aσ)[# aal], ra, aσ[# aal]) ∨ ren(a0) ≠ aren(ra) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: get-append-notls, seq: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n], subst:[σ, σ1, m] → [⌜aal⌝ , ⌜aa '⌝ + aα0, # aal]; ['a] → ['aa], <all paths>

Additional information

Used simplifier rules: ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜n⌝ .n = n; (s) ⊦ m - m = 0; (s) ⊦ (⌜a '⌝ + σ)[0] = a; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]; (s) ⊦ ⌜x⌝ [n] = x[n]; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ ¬ n < n; (s) Used terms: ⌜aal⌝ , ⌜aa '⌝ + aα0, # aal

Proof step 117 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]

A spec-lemma

Proof step 118 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), n1 = n +1, Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n1 ≤ # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext) (((s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n1], (ren(a0) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0))[n1], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[n1]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m +1 = m + 1; (s) a: ⊦ (x + y) + z = x + y + z; from specification list c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + x)[n + 1] = x[n]; (s) ⊦ m + 1 ≤ n ↔ m < n; (s)

Proof step 119 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) ≠ [], n < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[n], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[n], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[n + 1]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 28 terms: n abort: F discard: T computed: F

Additional information

Used terms: n

Proof step 120 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) ≠ [], ⌜# aal⌝ < # aσ, n < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[n], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[n], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0)[n + 1]), n < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: from-succ, seq: ⊦ ⌜n⌝ < # σ → σ.drop(n) = σ[n] + σ.drop(n + 1), subst:[σ, n] → [aσ, # aal]; ['a] → ['as], <all paths>

Additional information

Used simplifier rules: ⊦ a + σ = ⌜a '⌝ + σ; (s) Used terms: aσ, # aal

Proof step 121 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ ⌜n⌝ < # σ → σ.drop(n) = σ[n] + σ.drop(n + 1)

A spec-lemma

Proof step 122 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, n < # aal.map(aren), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n ], aal.map(aren)[n], ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n + 1 ]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ n < # x → x.map(f)[n] = f(x[n]); (s) ⊦ # x.map(f) = # x; (s)

Proof step 123 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, n < # aal, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext) (( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n ], aren(aal[n]), ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n + 1 ]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name: get-append-less, seq: ⊦ n < # x → (x + y)[n] = x[n], subst:[x, y, n] → [lift((⌜as '⌝ + aσ).take(# aal), s), (s, (⌜as '⌝ + aσ)[# aal]) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), n]; ['a] → [('s, 'as)], <all paths>

Additional information

Used simplifier rules: lift-size: ⊦ # lift(asl, s) = # asl; ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ # a ' = 0 +1; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m < n → (n ≤ m ↔ false); (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: lift((⌜as '⌝ + aσ).take(# aal), s), (s, (⌜as '⌝ + aσ)[# aal]) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), n

Proof step 124 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ n < # x → (x + y)[n] = x[n]

A spec-lemma

Proof step 125 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜# aal⌝ < # aσ, n < # aal, n < # lift((⌜as '⌝ + aσ).take(# aal), s), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext) (lift((⌜as '⌝ + aσ).take(# aal), s)[n], aren(aal[n]), ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n + 1 ]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-size: ⊦ # lift(asl, s) = # asl; ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 126 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜# aal⌝ < # aσ, n < # aal, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext) (lift((⌜as '⌝ + aσ).take(# aal), s)[n], aren(aal[n]), ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n + 1 ]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name: get-append-less, seq: ⊦ n < # x → (x + y)[n] = x[n], subst:[x, y, n] → [lift((⌜as '⌝ + aσ).take(# aal), s), (s, (⌜as '⌝ + aσ)[# aal]) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), n + 1]; ['a] → [('s, 'as)], <all paths>

Additional information

Used terms: lift((⌜as '⌝ + aσ).take(# aal), s), (s, (⌜as '⌝ + aσ)[# aal]) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), n + 1

Proof step 127 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ n < # x → (x + y)[n] = x[n]

A spec-lemma

Proof step 128 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, n < # aal, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ n + 1 < # lift((⌜as '⌝ + aσ).take(# aal), s), ¬ Π(step, astep, ren, aren, ext, aext) (lift((⌜as '⌝ + aσ).take(# aal), s)[n], aren(aal[n]), ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) )[n + 1 ]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-size: ⊦ # lift(asl, s) = # asl; lift-rec: ⊦ lift(as ' + asl, s) = (s, as) ' + lift(asl, s); c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ¬ n < n; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (a ' + x)[n + 1] = x[n]; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (⌜a '⌝ + σ)[n + 1] = σ[n]; (s) ⊦ n + 0 = n; (s) ⊦ n ≠ 0 → (⌜a '⌝ + σ).take((* n) + m) = a ' + σ.take((n - 1) + m); (s) ⊦ m ≤ m + n; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ m < n → (n ≤ m + 1 ↔ n = m + 1); (s) ⊦ m + 1 ≠ 0; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat ⊦ m < n ↔ ¬ n ≤ m; (ss) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 129 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) + 1, # pcsl = # pal1, # aal = n + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜n⌝ + ⌜1⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext) (((s, as) ' + lift(aσ.take(n), s))[n], aren(aal[n]), (lift(aσ.take(n), s) + (s, aσ[n]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0))[n]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name: get-append-grt, seq: ⊦ ¬ n < # x → (x + y)[n] = y[n - # x], subst:[x, y, n] → [lift(aσ.take(n), s), (s, aσ[n]) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0), n]; ['a] → [('s, 'as)], <all paths>

Additional information

Used simplifier rules: lift-size: ⊦ # lift(asl, s) = # asl; ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ⊦ ¬ m + n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ ¬ n < n; (s) Used terms: lift(aσ.take(n), s), (s, aσ[n]) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0), n

Proof step 130 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ ¬ n < # x → (x + y)[n] = y[n - # x]

A spec-lemma

Proof step 131 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # aal = n + 1, # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜n⌝ + ⌜1⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ n < # lift(aσ.take(n), s), ¬ Π(step, astep, ren, aren, ext, aext) (((s, as) ' + lift(aσ.take(n), s))[n], aren(aal[n]), ((s, aσ[n]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0))[n - # lift(aσ.take(n), s)]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-size: ⊦ # lift(asl, s) = # asl; a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (a ' + x)[0] = a; (s) ⊦ m - m = 0; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ¬ m + n < m; (s) ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf ⊦ m ≤ m; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 132 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) + 1, # pcsl = # pal1, # aal = n + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜n⌝ + ⌜1⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext)(((s, as) ' + lift(aσ.take(n), s))[n], aren(aal[n]), (s, aσ[n])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name: frag-step, seq:⌜n⌝ < # α ⊦ ¬ step(σ[n], α[n], σ[n + 1]) → (frag(IOAut(start, step, ext))(σ, α) ↔ false), subst:[step, α, start, ext, n, σ] → [astep, ⌜aal⌝ + ⌜aa '⌝ + aα0, astart, aext, n, ⌜as '⌝ + aσ]; ['s, 'a, 'e] → ['as, 'aa, 'e], <all paths>

Additional information

Used terms: aext, ⌜aal⌝ + ⌜aa '⌝ + aα0, astart, ⌜as '⌝ + aσ, n, astep

Proof step 133 from proof finfrag-repl-frag (previous step) (back to tree)

⌜n⌝ < # α ⊦ ¬ step(σ[n], α[n], σ[n + 1]) → (frag(IOAut(start, step, ext))(σ, α) ↔ false)

A spec-lemma

Proof step 134 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # aal = n + 1, # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜n⌝ + ⌜1⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext)(((s, as) ' + lift(aσ.take(n), s))[n], aren(aal[n]), (s, aσ[n])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0 + 1]) ⊦ ⌜n⌝ < #(⌜aal⌝ + ⌜aa '⌝ + aα0) ∧ ¬ astep((⌜as '⌝ + aσ)[n], (⌜aal⌝ + ⌜aa '⌝ + aα0)[n], (⌜as '⌝ + aσ)[n + 1])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (⌜a '⌝ + σ)[n + 1] = σ[n]; (s) ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 135 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # aal = n + 1, # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜n⌝ + ⌜1⌝ < # aσ, astep((⌜as '⌝ + aσ)[n], (⌜aal⌝ + ⌜aa '⌝ + aα0)[n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext)(((s, as) ' + lift(aσ.take(n), s))[n], aren(aal[n]), (s, aσ[n])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: get-append-ls, seq: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m], subst:[σ, σ1, m] → [⌜aal⌝ , ⌜aa '⌝ + aα0, n]; ['a] → ['aa], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]; (s) ⊦ ⌜n⌝ .n = n; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m = n + 1 → ¬ m ≤ n; (s) Used terms: ⌜aal⌝ , ⌜aa '⌝ + aα0, n

Proof step 136 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]

A spec-lemma

Proof step 137 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) + 1, # pcsl = # pal1, # aal = n + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜n⌝ + ⌜1⌝ < # aσ, ⌜n⌝ < # ⌜aal⌝ , astep((⌜as '⌝ + aσ)[n], ⌜aal⌝ [n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext)(((s, as) ' + lift(aσ.take(n), s))[n], aren(aal[n]), (s, aσ[n])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ¬ n < n; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜x⌝ [n] = x[n]; (s)

Proof step 138 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) + 1, # pcsl = # pal1, # aal = n + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜n⌝ + ⌜1⌝ < # aσ, astep((⌜as '⌝ + aσ)[n], aal[n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext)(((s, as) ' + lift(aσ.take(n), s))[n], aren(aal[n]), (s, aσ[n])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0 + 1]) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: n

Additional information

Used simplifier rules: Nat freely generated by 0, +1;;

Proof step 139 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # aal = n + 1, # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), n = 0, Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜n⌝ + ⌜1⌝ < # aσ, astep((⌜as '⌝ + aσ)[n], aal[n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext)(((s, as) ' + lift(aσ.take(n), s))[n], aren(aal[n]), (s, aσ[n])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-base: ⊦ lift([], s) = []; c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ x ≠ [] → x[0] = x.head; (s) ⊦ a '[0] = a; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + [] = x; (s) ⊦ σ.take(0) = []; (s) ⊦ # σ = ⌜0⌝ ↔ σ = []; (s) ⊦ ⌜1⌝ < n∞ ↔ n∞ ≠ ⌜0⌝ ∧ n∞ ≠ ⌜1⌝ ; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜(* n)⌝ + ⌜(* m)⌝ = ⌜n + m⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (⌜a '⌝ + σ)[0] = a; (s) ⊦ a + x = a ' + x; (s) ⊦ # x = 1 ↔ x = x.head + []; (as)

Proof step 140 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(1), aα0), aal = aal.head ', aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), # aσ ≠ ⌜1⌝ , Some(e) ≠ None, aal ≠ [], aσ ≠ [], ext(a0) ≠ None, astep(as, aal.head, aσ[0]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[1] '⌝ + aσ.drop(2), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext)((s, as), aren(aal.head), (s, aσ[0])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0)[n0 + 1]) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-add, seq: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail), subst:[x, head, tail] → [aal, aa0, aal0]; ['a] → ['aa]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: aal, aa0, aal0

Proof step 141 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(1), aα0), aal = aa0 ', aal = aa0 + aal0, aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), # aσ ≠ ⌜1⌝ , Some(e) ≠ None, aal ≠ [], aσ ≠ [], ext(a0) ≠ None, astep(as, aa0, aσ[0]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[1] '⌝ + aσ.drop(2), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext)((s, as), aren(aa0), (s, aσ[0])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ [] + σ = σ; (s) ⊦ ⌜[]⌝ = []; (s) ⊦ ext !([]) = []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) ⊦ a + x = a ' + x; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ ext !(a ') = [] ↔ ext(a) = None; (s) ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s) ⊦ a ' = a0 ' + x ↔ a = a0 ∧ x = []; (ws)

Proof step 142 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(1), aα0), aext(aa0) = None, aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), # aσ ≠ ⌜1⌝ , Some(e) ≠ None, aσ ≠ [], ext(a0) ≠ None, astep(as, aa0, aσ[0]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[1] '⌝ + aσ.drop(2), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aa0 '⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext)((s, as), aren(aa0), (s, aσ[0])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[1] '⌝ + aσ.drop(2), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: product, name: lrstep-def, seq:⊦ Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)) ↔ rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra)), subst:[pa, ls0, rext, lstep, rs0, ls, rs, rren, lren, lext, rstep] → [aren(aa0), s, aext, step, aσ[0], s, as, aren, ren, ext, astep]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['s, 'aa, 'a, 'pa, 'as, 'e], <all paths>

Additional information

Used simplifier rules: ⊦ Πok(lext, rext, lren, rren) → (rren(ra1) = rren(ra2) ↔ ra1 = ra2); (s) ⊦ cod(f)(b) ↔ (∃ a. f(a) = b); (as) ⊦ Πok(lext, rext, lren, rren) ∧ lren(la) = rren(ra) → (rext(ra) = None ↔ false); (as) Used terms: aext, aren(aa0), as, astep, ren, ext, aren, aσ[0], step, s, s

Proof step 143 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)) ↔ rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra))

A spec-lemma

Proof step 144 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # aal = n + 1, # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(n + 1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), n = n1 +1, Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜n⌝ + ⌜1⌝ < # aσ, astep((⌜as '⌝ + aσ)[n], aal[n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext)(((s, as) ' + lift(aσ.take(n), s))[n], aren(aal[n]), (s, aσ[n])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[n + 1] '⌝ + aσ.drop(2 + n), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m +1 = m + 1; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (a ' + x)[n + 1] = x[n]; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ⌜(* n)⌝ + ⌜(* m)⌝ = ⌜n + m⌝ ; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (⌜a '⌝ + σ)[n + 1] = σ[n]; (s)

Proof step 145 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) + 1, # pcsl = # pal1, # aal = 2 + n1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(2 + n1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(2 + n1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜2⌝ + ⌜n1⌝ < # aσ, astep(aσ[n1], aal[n1 + 1], aσ[n1 + 1]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext)(lift(aσ.take(n1 + 1), s)[n1], aren(aal[n1 + 1]), (s, aσ[n1 + 1])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: lift-get, seq: ⊦ n < # asl → lift(asl, s)[n] = (s, asl[n]), subst:[asl, s, n] → [aσ.take(n1 + 1), s, n1]; ['s, 'as] → ['s, 'as], <all paths>

Additional information

Used simplifier rules: ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) lsle-trans: ⊦ n∞ < n∞1 ∧ ¬ n∞2 < n∞1 → n∞ < n∞2; (forward) from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ⊦ ¬ m + n < m; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ¬ n < n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: aσ.take(n1 + 1), s, n1

Proof step 146 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ n < # asl → lift(asl, s)[n] = (s, asl[n])

Lemma lift-get

Proof step 147 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # aal = 2 + n1, # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(2 + n1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(2 + n1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜2⌝ + ⌜n1⌝ < # aσ, n1 < # aσ.take(n1 + 1), astep(aσ[n1], aal[n1 + 1], aσ[n1 + 1]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext)((s, aσ.take(n1 + 1)[n1]), aren(aal[n1 + 1]), (s, aσ[n1 + 1])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: m < n, ¬ # σ < ⌜n⌝ ⊦ σ.take(n)[m] = σ[m]; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ¬ m + n < m; (s) ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) lsle-trans: ⊦ n∞ < n∞1 ∧ ¬ n∞2 < n∞1 → n∞ < n∞2; (forward) from specification nat-inf ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ ¬ n < n; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s)

Proof step 148 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) + 1, # pcsl = # pal1, # aal = 2 + n1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(2 + n1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(2 + n1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜2⌝ + ⌜n1⌝ < # aσ, astep(aσ[n1], aal[n1 + 1], aσ[n1 + 1]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext)((s, aσ[n1]), aren(aal[n1 + 1]), (s, aσ[n1 + 1])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: product, name: lrstep-def, seq:⊦ Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)) ↔ rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra)), subst:[pa, ls0, rext, lstep, rs0, ls, rs, rren, lren, lext, rstep] → [aren(aal[n1 + 1]), s, aext, step, aσ[n1 + 1], s, aσ[n1], aren, ren, ext, astep]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['s, 'aa, 'a, 'pa, 'as, 'e], <all paths>

Additional information

Used terms: aext, aren(aal[n1 + 1]), aσ[n1], astep, ren, ext, aren, aσ[n1 + 1], step, s, s

Proof step 149 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)) ↔ rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra))

A spec-lemma

Proof step 150 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # aal = 2 + n1, # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(2 + n1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(2 + n1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜2⌝ + ⌜n1⌝ < # aσ, astep(aσ[n1], aal[n1 + 1], aσ[n1 + 1]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0 + 1]), ∀ la, ra. ¬(step(s, la, s) ∧ ext(la) ≠ None ∧ aren(aal[n1 + 1]) = ren(la) ∧ astep(aσ[n1], ra, aσ[n1 + 1]) ∧ aext(ra) ≠ None ∧ aren(aal[n1 + 1]) = aren(ra)), aσ[n1] ≠ aσ[n1 + 1] ∨ (∀ la. ¬(step(s, la, s) ∧ aren(aal[n1 + 1]) = ren(la) ∧ ¬ cod(aren)(aren(aal[n1 + 1])))), s ≠ s ∨ (∀ ra. ¬(astep(aσ[n1], ra, aσ[n1 + 1]) ∧ aren(aal[n1 + 1]) = aren(ra) ∧ ¬ cod(ren)(aren(aal[n1 + 1])))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ Πok(lext, rext, lren, rren) → (lren(la1) = lren(la2) ↔ la1 = la2); (s) ⊦ Πok(lext, rext, lren, rren) ∧ lren(la) = rren(ra) → (rext(ra) = None ↔ false); (as) ⊦ Πok(lext, rext, lren, rren) → (rren(ra1) = rren(ra2) ↔ ra1 = ra2); (s) ⊦ Πok(lext, rext, lren, rren) ∧ lren(la) = rren(ra) → (lext(la) = None ↔ false); (as) ⊦ cod(f)(f(a)); (s) ⊦ cod(f)(b) ↔ (∃ a. f(a) = b); (as)

Proof step 151 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # aal = 2 + n1, # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(2 + n1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(2 + n1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), ren(a) = aren(aal[n1 + 1]), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜2⌝ + ⌜n1⌝ < # aσ, astep(aσ[n1], aal[n1 + 1], aσ[n1 + 1]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ step(s, a, s), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name:finremtau-empty-equiv, seq: ⊦ ext !(al) = [] ↔ (∀ n. n < # al → ext(al[n]) = None), subst:[ext, al] → [aext, aal]; ['e, 'a] → ['e, 'aa], <all paths>

Additional information

Used terms: aext, aal

Proof step 152 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ ext !(al) = [] ↔ (∀ n. n < # al → ext(al[n]) = None)

A spec-lemma

Proof step 153 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) + 1, # pcsl = # pal1, # aal = 2 + n1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(2 + n1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(2 + n1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), ren(a) = aren(aal[n1 + 1]), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜2⌝ + ⌜n1⌝ < # aσ, astep(aσ[n1], aal[n1 + 1], aσ[n1 + 1]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ step(s, a, s), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0 + 1]), ∀ n. n < # aal → aext(aal[n]) = None ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 154 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # aal = 2 + n1, # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(2 + n1), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(2 + n1), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), ren(a) = aren(aal[n1 + 1]), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜2⌝ + ⌜n1⌝ < # aσ, astep(aσ[n1], aal[n1 + 1], aσ[n1 + 1]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ step(s, a, s), ∀ n. n < 2 + n1 → aext(aal[n]) = None, ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[2 + n1] '⌝ + aσ.drop(3 + n1), aα0)[n0 + 1]) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 29 terms: n1 + 1 abort: F discard: T computed: F

Additional information

Used simplifier rules: ⊦ m + n < m + n0 ↔ n < n0; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ Πok(lext, rext, lren, rren) ∧ lren(la) = rren(ra) → (rext(ra) = None ↔ false); (as) Used terms: n1 + 1

Proof step 155 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, n < # aal, ⌜# aal⌝ < # aσ, n + 1 < # lift((⌜as '⌝ + aσ).take(# aal), s), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext)(lift((⌜as '⌝ + aσ).take(# aal), s)[n], aren(aal[n]), lift((⌜as '⌝ + aσ).take(# aal), s)[n + 1]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-size: ⊦ # lift(asl, s) = # asl; ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 156 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜# aal⌝ < # aσ, n < # aal, n + 1 < # aal, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext)(lift((⌜as '⌝ + aσ).take(# aal), s)[n], aren(aal[n]), lift((⌜as '⌝ + aσ).take(# aal), s)[n + 1]), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: lift-get, seq: ⊦ n < # asl → lift(asl, s)[n] = (s, asl[n]), subst:[asl, s, n] → [(⌜as '⌝ + aσ).take(# aal), s, n + 1]; ['s, 'as] → ['s, 'as], <all paths>

Additional information

Used simplifier rules: ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ # a ' = 0 +1; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic trans-Nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic ⊦ m + 1 ≠ 0; (s) ⊦ m < n → (n ≤ m + 1 ↔ n = m + 1); (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: (⌜as '⌝ + aσ).take(# aal), s, n + 1

Proof step 157 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ n < # asl → lift(asl, s)[n] = (s, asl[n])

Lemma lift-get

Proof step 158 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, n + 1 < # aal, n < # aal, ⌜# aal⌝ < # aσ, n + 1 < # (⌜as '⌝ + aσ).take(# aal), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext)(lift((⌜as '⌝ + aσ).take(# aal), s)[n], aren(aal[n]), (s, (⌜as '⌝ + aσ).take(# aal)[n + 1])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (⌜a '⌝ + σ)[n + 1] = σ[n]; (s) m < n, ¬ # σ < ⌜n⌝ ⊦ σ.take(n)[m] = σ[m]; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s)

Proof step 159 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜# aal⌝ < # aσ, n < # aal, n + 1 < # aal, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext)(lift((⌜as '⌝ + aσ).take(# aal), s)[n], aren(aal[n]), (s, aσ[n])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: lift-get, seq: ⊦ n < # asl → lift(asl, s)[n] = (s, asl[n]), subst:[asl, s, n] → [(⌜as '⌝ + aσ).take(# aal), s, n]; ['s, 'as] → ['s, 'as], <all paths>

Additional information

Used simplifier rules: ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ # a ' = 0 +1; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: (⌜as '⌝ + aσ).take(# aal), s, n

Proof step 160 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ n < # asl → lift(asl, s)[n] = (s, asl[n])

Lemma lift-get

Proof step 161 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, n + 1 < # aal, n < # aal, ⌜# aal⌝ < # aσ, n < # (⌜as '⌝ + aσ).take(# aal), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext)((s, (⌜as '⌝ + aσ).take(# aal)[n]), aren(aal[n]), (s, aσ[n])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: m < n, ¬ # σ < ⌜n⌝ ⊦ σ.take(n)[m] = σ[m]; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s)

Proof step 162 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜# aal⌝ < # aσ, n + 1 < # aal, n < # aal, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext)((s, (⌜as '⌝ + aσ)[n]), aren(aal[n]), (s, aσ[n])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name: frag-step, seq:⌜n⌝ < # α ⊦ ¬ step(σ[n], α[n], σ[n + 1]) → (frag(IOAut(start, step, ext))(σ, α) ↔ false), subst:[step, α, start, ext, n, σ] → [astep, ⌜aal⌝ + ⌜aa '⌝ + aα0, astart, aext, n, ⌜as '⌝ + aσ]; ['s, 'a, 'e] → ['as, 'aa, 'e], <all paths>

Additional information

Used terms: aext, ⌜aal⌝ + ⌜aa '⌝ + aα0, astart, ⌜as '⌝ + aσ, n, astep

Proof step 163 from proof finfrag-repl-frag (previous step) (back to tree)

⌜n⌝ < # α ⊦ ¬ step(σ[n], α[n], σ[n + 1]) → (frag(IOAut(start, step, ext))(σ, α) ↔ false)

A spec-lemma

Proof step 164 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, n < # aal, n + 1 < # aal, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext)((s, (⌜as '⌝ + aσ)[n]), aren(aal[n]), (s, aσ[n])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦ ⌜n⌝ < #(⌜aal⌝ + ⌜aa '⌝ + aα0) ∧ ¬ astep((⌜as '⌝ + aσ)[n], (⌜aal⌝ + ⌜aa '⌝ + aα0)[n], (⌜as '⌝ + aσ)[n + 1])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (⌜a '⌝ + σ)[n + 1] = σ[n]; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ m < n → ⌜m⌝ < ⌜n⌝ + n∞0; (s) ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 165 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜# aal⌝ < # aσ, n + 1 < # aal, n < # aal, astep((⌜as '⌝ + aσ)[n], (⌜aal⌝ + ⌜aa '⌝ + aα0)[n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext)((s, (⌜as '⌝ + aσ)[n]), aren(aal[n]), (s, aσ[n])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: get-append-ls, seq: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m], subst:[σ, σ1, m] → [⌜aal⌝ , ⌜aa '⌝ + aα0, n]; ['a] → ['aa], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]; (s) ⊦ ⌜n⌝ .n = n; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: ⌜aal⌝ , ⌜aa '⌝ + aα0, n

Proof step 166 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]

A spec-lemma

Proof step 167 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, n < # aal, n + 1 < # aal, ⌜# aal⌝ < # aσ, ⌜n⌝ < # ⌜aal⌝ , astep((⌜as '⌝ + aσ)[n], ⌜aal⌝ [n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext)((s, (⌜as '⌝ + aσ)[n]), aren(aal[n]), (s, aσ[n])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜x⌝ [n] = x[n]; (s)

Proof step 168 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜# aal⌝ < # aσ, n + 1 < # aal, n < # aal, astep((⌜as '⌝ + aσ)[n], aal[n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ Π(step, astep, ren, aren, ext, aext)((s, (⌜as '⌝ + aσ)[n]), aren(aal[n]), (s, aσ[n])), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: product, name: lrstep-def, seq:⊦ Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)) ↔ rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra)), subst:[pa, ls0, rext, lstep, rs0, ls, rs, rren, lren, lext, rstep] → [aren(aal[n]), s, aext, step, aσ[n], s, (⌜as '⌝ + aσ)[n], aren, ren, ext, astep]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['s, 'aa, 'a, 'pa, 'as, 'e], <all paths>

Additional information

Used terms: aext, aren(aal[n]), (⌜as '⌝ + aσ)[n], astep, ren, ext, aren, aσ[n], step, s, s

Proof step 169 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)) ↔ rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra))

A spec-lemma

Proof step 170 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, n < # aal, n + 1 < # aal, ⌜# aal⌝ < # aσ, astep((⌜as '⌝ + aσ)[n], aal[n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]), ∀ la, ra. ¬(step(s, la, s) ∧ ext(la) ≠ None ∧ aren(aal[n]) = ren(la) ∧ astep((⌜as '⌝ + aσ)[n], ra, aσ[n]) ∧ aext(ra) ≠ None ∧ aren(aal[n]) = aren(ra)), (⌜as '⌝ + aσ)[n] ≠ aσ[n] ∨ (∀ la. ¬(step(s, la, s) ∧ aren(aal[n]) = ren(la) ∧ ¬ cod(aren)(aren(aal[n])))), s ≠ s ∨ (∀ ra. ¬(astep((⌜as '⌝ + aσ)[n], ra, aσ[n]) ∧ aren(aal[n]) = aren(ra) ∧ ¬ cod(ren)(aren(aal[n])))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ Πok(lext, rext, lren, rren) → (lren(la1) = lren(la2) ↔ la1 = la2); (s) ⊦ Πok(lext, rext, lren, rren) ∧ lren(la) = rren(ra) → (rext(ra) = None ↔ false); (as) ⊦ Πok(lext, rext, lren, rren) → (rren(ra1) = rren(ra2) ↔ ra1 = ra2); (s) ⊦ Πok(lext, rext, lren, rren) ∧ lren(la) = rren(ra) → (lext(la) = None ↔ false); (as) ⊦ cod(f)(f(a)); (s) ⊦ cod(f)(b) ↔ (∃ a. f(a) = b); (as)

Proof step 171 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), ren(a) = aren(aal[n]), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, n < # aal, n + 1 < # aal, ⌜# aal⌝ < # aσ, astep((⌜as '⌝ + aσ)[n], aal[n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ step(s, a, s), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: product, name: productok-def, seq:⊦ Πok(lext, rext, lren, rren) ↔ inj(lren) ∧ inj(rren) ∧ surj(lren, rren) ∧ sharingok(lext, rext, lren, rren), subst:[lext, rext, lren, rren] → [ext, aext, ren, aren]; ['la, 'e, 'ra, 'pa] → ['a, 'e, 'aa, 'pa], <all paths>

Additional information

Used terms: ext, aext, ren, aren

Proof step 172 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ Πok(lext, rext, lren, rren) ↔ inj(lren) ∧ inj(rren) ∧ surj(lren, rren) ∧ sharingok(lext, rext, lren, rren)

A spec-lemma

Proof step 173 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), ren(a) = aren(aal[n]), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜# aal⌝ < # aσ, n + 1 < # aal, n < # aal, astep((⌜as '⌝ + aσ)[n], aal[n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), inj(ren), inj(aren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), sharingok(ext, aext, ren, aren), step(s, a0, s0), surj(ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ step(s, a, s), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: product, name: sharingok-def, seq:⊦ sharingok(lext, rext, lren, rren) ↔ (∀ la, ra. ( lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ ( ¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra))), subst:[lext, rext, lren, rren] → [ext, aext, ren, aren]; ['la, 'e, 'ra, 'pa] → ['a, 'e, 'aa, 'pa], <all paths>

Additional information

Used terms: ext, aext, ren, aren

Proof step 174 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ sharingok(lext, rext, lren, rren) ↔ (∀ la, ra. (lren(la) = rren(ra) → lext(la) ≠ None ∧ rext(ra) ≠ None ∧ lext(la) = rext(ra)) ∧ (¬ cod(rren)(lren(la)) ∧ ¬ cod(lren)(rren(ra)) ∧ lext(la) ≠ None ∧ rext(ra) ≠ None → lext(la) ≠ rext(ra)))

A spec-lemma

Proof step 175 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, aext !(aal) = [], # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), ren(a) = aren(aal[n]), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, n < # aal, n + 1 < # aal, ⌜# aal⌝ < # aσ, astep((⌜as '⌝ + aσ)[n], aal[n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), inj(aren), inj(ren), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), surj(ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ step(s, a, s), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]), ∀ la, ra. (ren(la) = aren(ra) → ext(la) ≠ None ∧ aext(ra) ≠ None ∧ ext(la) = aext(ra)) ∧ (¬ cod(aren)(ren(la)) ∧ ¬ cod(ren)(aren(ra)) ∧ ext(la) ≠ None ∧ aext(ra) ≠ None → ext(la) ≠ aext(ra)) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 35 terms: a, aal[n] abort: F discard: F computed: T

Additional information

Used terms: a, aal[n]

Proof step 176 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), ren(a) = aren(aal[n]), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, ⌜# aal⌝ < # aσ, n + 1 < # aal, n < # aal, astep((⌜as '⌝ + aσ)[n], aal[n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), inj(ren), inj(aren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), surj(ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ step(s, a, s), ren(a) = aren(aal[n]) → ext(a) ≠ None ∧ aext(aal[n]) ≠ None ∧ ext(a) = aext(aal[n]), ¬ cod(aren)(ren(a)) ∧ ¬ cod(ren)(aren(aal[n])) ∧ ext(a) ≠ None ∧ aext(aal[n]) ≠ None → ext(a) ≠ aext(aal[n]), ∀ la, ra. (ren(la) = aren(ra) → ext(la) ≠ None ∧ aext(ra) ≠ None ∧ ext(la) = aext(ra)) ∧ (¬ cod(aren)(ren(la)) ∧ ¬ cod(ren)(aren(ra)) ∧ ext(la) ≠ None ∧ aext(ra) ≠ None → ext(la) ≠ aext(ra)), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ cod(f)(f(a)); (s)

Proof step 177 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), ext(a) = aext(aal[n]), ren(a) = aren(aal[n]), Some(e) ≠ None, aal ≠ [], ext(a) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, n + 1 < # aal, n < # aal, astep((⌜as '⌝ + aσ)[n], aal[n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), inj(ren), inj(aren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), surj(ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ step(s, a, s), ∀ la, ra. (ren(la) = aren(ra) → ext(la) ≠ None ∧ aext(ra) ≠ None ∧ ext(la) = aext(ra)) ∧ (¬ cod(aren)(ren(la)) ∧ ¬ cod(ren)(aren(ra)) ∧ ext(la) ≠ None ∧ aext(ra) ≠ None → ext(la) ≠ aext(ra)), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name:finremtau-empty-equiv, seq: ⊦ ext !(al) = [] ↔ (∀ n. n < # al → ext(al[n]) = None), subst:[ext, al] → [aext, aal]; ['e, 'a] → ['e, 'aa], <all paths>

Additional information

Used terms: aext, aal

Proof step 178 from proof finfrag-repl-frag (previous step) (back to tree)

⊦ ext !(al) = [] ↔ (∀ n. n < # al → ext(al[n]) = None)

A spec-lemma

Proof step 179 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # pcsl = # pal1, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), ext(a) = aext(aal[n]), ren(a) = aren(aal[n]), Some(e) ≠ None, aal ≠ [], ext(a) ≠ None, ext(a0) ≠ None, n < # aal, n + 1 < # aal, ⌜# aal⌝ < # aσ, astep((⌜as '⌝ + aσ)[n], aal[n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), inj(aren), inj(ren), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), step(s, a0, s0), surj(ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ step(s, a, s), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]), ∀ la, ra. (ren(la) = aren(ra) → ext(la) ≠ None ∧ aext(ra) ≠ None ∧ ext(la) = aext(ra)) ∧ (¬ cod(aren)(ren(la)) ∧ ¬ cod(ren)(aren(ra)) ∧ ext(la) ≠ None ∧ aext(ra) ≠ None → ext(la) ≠ aext(ra)), ∀ n. n < # aal → aext(aal[n]) = None ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 180 from proof finfrag-repl-frag (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s0, cs0) ' + pcsl, pal1))⌝ + β, # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) = # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) + 1, # pcsl = # pal1, 1 + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0) = # repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), aext(aa) = Some(e), aren(aa) = ren(a0), cext(ca0) = Some(e), cren(ca0) = ren(a0), ext(a) = aext(aal[n]), ren(a) = aren(aal[n]), Some(e) ≠ None, aal ≠ [], ext(a) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, n + 1 < # aal, n < # aal, astep((⌜as '⌝ + aσ)[n], aal[n], aσ[n]), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsl, pal1), inj(ren), inj(aren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), step(s, a0, s0), surj(ren, aren), Πok(ext, cext, ren, cren), n ≤ # aal + # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, aσ.drop(# aal), aα0), ¬ step(s, a, s), ∀ n. n < # aal → aext(aal[n]) = None, ∀ la, ra. (ren(la) = aren(ra) → ext(la) ≠ None ∧ aext(ra) ≠ None ∧ ext(la) = aext(ra)) ∧ (¬ cod(aren)(ren(la)) ∧ ¬ cod(ren)(aren(ra)) ∧ ext(la) ≠ None ∧ aext(ra) ≠ None → ext(la) ≠ aext(ra)), ∀ n0. n0 < # repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0], repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)[n0 + 1]) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 35 terms: n abort: F discard: F computed: T

Additional information

Used terms: n

Proof Tree for finfrag-repl-frag (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.

the proof tree