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Proof step 1 from proof frag-repla-sametrace (conclusion) (back to tree)

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

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pα), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: pcσ

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;;

Proof step 3 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), pcσ = ⌜pcsl⌝ , Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pα), 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)))(pcσ, pα), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ .l = x; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ finp(σ) ↔ (∃ x. ⌜x⌝ = σ); (as) ⊦ frag(ioa)(⌜sl⌝ , α) ↔ frag*(ioa)(sl, α.l) ∧ finp(α); (s)

Proof step 4 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsl⌝ , ⌜pal⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsl⌝ , ⌜pal⌝ , ⌜as '⌝ + aσ, aα)) ≠ ⌜Π(ext, cext, ren, cren) !(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)))(pcsl, pal), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

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

Additional information

Used terms: aren, ren, cstep, cext, aext, aα, ext, astart, cstart, start, pal, cren, astep, pcsl, ⌜as '⌝ + aσ, step

Proof step 5 from proof frag-repla-sametrace (previous step) (back to tree)

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

Lemma finfrag-repla-sametrace-1

Proof step 6 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsl⌝ , ⌜pal⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsl⌝ , ⌜pal⌝ , ⌜as '⌝ + aσ, aα)) ≠ ⌜Π(ext, cext, ren, cren) !(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)))(pcsl, pal), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ 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) → Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(pal) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 13

Additional information

Proof step 7 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsl⌝ , ⌜pal⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsl⌝ , ⌜pal⌝ , ⌜as '⌝ + aσ, aα)) ≠ ⌜Π(ext, cext, ren, cren) !(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)))(pcsl, pal), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦ nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ 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)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 8 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsl⌝ , ⌜pal⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsl⌝ , ⌜pal⌝ , ⌜as '⌝ + aσ, aα)) ≠ ⌜Π(ext, cext, ren, cren) !(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)))(pcsl, pal), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, 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) ⊦

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, pcsl, pal))⌝ , aext !(aα)]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ , aext !(aα)

Proof step 9 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 10 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsl⌝ , ⌜pal⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsl⌝ , ⌜pal⌝ , ⌜as '⌝ + aσ, aα)) ≠ ⌜Π(ext, cext, ren, cren) !(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)))(pcsl, pal), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ σ1. ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ + σ1 ≠ aext !(aα) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: project-right, name: projaction-fin-def, seq:⊦ # psl = # pal + 1 → πra(lstep, rstep, lren, rren, ⌜psl⌝ , ⌜pal⌝ ) = ⌜πra(lstep, rstep, lren, rren, psl, pal)⌝ , subst:[pal, lstep, lren, rstep, rren, psl] → [pal, step, ren, cstep, cren, pcsl]; ['ls, 'ra, 'la, 'pa, 'rs] → ['s, 'ca, 'a, 'pa, 'cs], <all paths>

Additional information

Used simplifier rules: ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ σ + σ0 = σ ↔ σ0 = [] ∨ # σ = ∞; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜n⌝ ≠ ∞; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ frag*(IOAut(start, step, ext))(sl, al) → (# sl = # al + 1 ↔ true); (ss) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: pcsl, cstep, pal, ren, cren, step

Proof step 11 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ # psl = # pal + 1 → πra(lstep, rstep, lren, rren, ⌜psl⌝ , ⌜pal⌝ ) = ⌜πra(lstep, rstep, lren, rren, psl, pal)⌝

A spec-lemma

Proof step 12 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsl⌝ , ⌜pal⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(pal), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsl⌝ , ⌜pal⌝ , ⌜as '⌝ + aσ, aα)) ≠ ⌜Π(ext, cext, ren, cren) !(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)))(pcsl, pal), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: replaction-fin-def, seq:⊦ # pcsl = # pal + 1 → repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsl⌝ , ⌜pal⌝ , aσ, aα) = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)⌝ , subst:[pal, cext, aα, aσ, cren, ren, cstep, aren, pcsl, aext, step] → [pal, cext, aα, ⌜as '⌝ + aσ, cren, ren, cstep, aren, pcsl, aext, step]; ['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⌝ = ⌜x0⌝ ↔ x = x0; (s) ⊦ frag*(IOAut(start, step, ext))(sl, al) → (# sl = # al + 1 ↔ true); (ss) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: aren, ren, cstep, cext, aext, aα, pal, cren, pcsl, ⌜as '⌝ + aσ, step

Proof step 13 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ # pcsl = # pal + 1 → repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsl⌝ , ⌜pal⌝ , aσ, aα) = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)⌝

Lemma replaction-fin-def

Proof step 14 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), pcσ = ⌜pcsf⌝ , Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pα), 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)))(pcσ, pα), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ .f = nf; (s) ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss) ⊦ frag(ioa)(⌜sf⌝ , α) ↔ fragω(ioa)(sf, α.f) ∧ ¬ finp(α); (s)

Proof step 15 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:repl-frag, seq:nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), frag (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (pcσ, pα), frag(IOAut(astart, astep, aext))(aσ, aα), cext !(πra(step, cstep, ren, cren, pcσ, pα)) = 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, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), subst:[astep, cext, aα, start, pcσ, aσ, cren, ren, astart, cstart, cstep, aren, aext, step, ext, pα] → [astep, cext, aα, start, ⌜pcsf⌝ , ⌜as '⌝ + aσ, cren, ren, astart, cstart, cstep, aren, aext, step, ext, ⌜paf⌝ ]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used simplifier rules: ⊦ frag(ioa)(⌜sf⌝ , ⌜af⌝ ) ↔ fragω(ioa)(sf, af); (s) Used terms: aren, ren, cstep, cext, aext, aα, ext, ⌜pcsf⌝ , astart, cstart, start, cren, astep, ⌜paf⌝ , ⌜as '⌝ + aσ, step

Proof step 16 from proof frag-repla-sametrace (previous step) (back to tree)

nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(astart, astep, aext))(aσ, aα), cext !(πra(step, cstep, ren, cren, pcσ, pα)) = 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, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα))

Lemma repl-frag

Proof step 17 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, 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, ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: replaction-inf-def, seq:⊦ cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf⌝ , ⌜paf⌝ , aσ, aα) = lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα) ⌝ ⌝ ), subst:[cext, aσ, ren, cstep, aren, aext, step, aα, pcsf, cren, paf] → [cext, ⌜as '⌝ + aσ, ren, cstep, aren, aext, step, aα, pcsf, cren, paf]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as], <all paths>

Additional information

Used simplifier rules: ⊦ σ ⊑ σ; (s) Used terms: aren, ren, cstep, cext, aext, aα, cren, paf, ⌜as '⌝ + aσ, step, pcsf

Proof step 18 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf⌝ , ⌜paf⌝ , aσ, aα) = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )

Lemma replaction-inf-def

Proof step 19 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ Π(ext, cext, ren, cren) !(⌜paf⌝ ), 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, ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), 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, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ σ ⊑ σ; (s)

Proof step 20 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ Π(ext, cext, ren, cren) !(⌜paf⌝ ), 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, ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: replstate-inf-def, seq:⊦ cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf⌝ , ⌜paf⌝ , aσ, aα) = lim (⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα) ⌝ ⌝ ), subst:[cext, aσ, ren, cstep, aren, aext, step, aα, pcsf, cren, paf] → [cext, ⌜as '⌝ + aσ, ren, cstep, aren, aext, step, aα, pcsf, cren, paf]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as], <all paths>

Additional information

Used simplifier rules: ⊦ σ ⊑ σ; (s) Used terms: aren, ren, cstep, cext, aext, aα, cren, paf, ⌜as '⌝ + aσ, step, pcsf

Proof step 21 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf⌝ , ⌜paf⌝ , aσ, aα) = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )

Lemma replstate-inf-def

Proof step 22 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, 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, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ σ ⊑ σ; (s)

Proof step 23 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: streamseq-mon-lim, name:lim-def, seq:⊦ mon(σσ) ∧ σσ ≠ [] → (∀ n. ⌜n⌝ < # σσ → σσ[n] ⊑ lim(σσ)) ∧ (∀ m. ¬ # lim(σσ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # σσ ∧ ⌜lim(σσ).take(m)⌝ ⊑ σσ[n])), subst:[σσ] → [⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ]; ['a] → ['pa]

Additional information

Used terms: ⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝

Proof step 24 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ mon(σσ) ∧ σσ ≠ [] → (∀ n. ⌜n⌝ < # σσ → σσ[n] ⊑ lim(σσ)) ∧ (∀ m. ¬ # lim(σσ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # σσ ∧ ⌜lim(σσ).take(m)⌝ ⊑ σσ[n]))

A spec-lemma

Proof step 25 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ∧ ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ≠ [] → (∀ n. ⌜n⌝ < # ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ → ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ [n] ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ∧ (∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ∧ ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ [n])) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ [n] = nf(n); (s) ⊦ ⌜m⌝ < ∞; (s) ⊦ # ⌜nf⌝ = ∞; (s) ⊦ [] ≠ ⌜nf⌝ ; (s)

Proof step 26 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) → (∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ∧ (∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ )) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 14

Additional information

Proof step 27 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: mon-repla-1, seq:⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ fragω (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (pcsf, paf) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ), subst:[astep, cext, as, start, aσ, ext, ren, astart, cstart, cstep, aren, aext, step, aα, pcsf, cren, paf] → [astep, cext, as, start, aσ, ext, ren, astart, cstart, cstep, aren, aext, step, aα, pcsf, cren, paf]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as], <all paths>

Additional information

Used simplifier rules: ⊦ σ ⊑ σ; (s) Used terms: aren, ren, cstep, cext, as, aext, aα, ext, astart, cstart, start, cren, astep, paf, aσ, step, pcsf

Proof step 28 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )

Lemma mon-repla-1

Proof step 29 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: extensionality-to, seq:⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n)), subst:[σ0, σ1] → [Π(ext, aext, ren, aren) ! (lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )), Π(ext, cext, ren, cren) !(⌜paf⌝ )]; ['a] → ['e], <all paths>

Additional information

Used terms: Π(ext, aext, ren, aren) ! (lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )), Π(ext, cext, ren, cren) !(⌜paf⌝ )

Proof step 30 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n))

A spec-lemma

Proof step 31 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) ∨ (∃ n. ¬ ( ¬ # Π(ext, aext, ren, aren) ! (lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) < ⌜n⌝ → Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) .take (n ) = Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 32 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) ∨ ¬ ( ¬ # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) < ⌜n⌝ → Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) .take (n ) = Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n)) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: # Π(ext, aext, ren, aren) ! (lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )) ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ )

Additional information

Used formulas: # Π(ext, aext, ren, aren) ! (lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )) ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ )

Proof step 33 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) ∨ ¬ ( ¬ # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) < ⌜n⌝ → Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) .take (n ) = Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 34 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: # Π(ext, aext, ren, aren) ! (lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )) < # Π(ext, cext, ren, cren) !(⌜paf⌝ )

Additional information

Used formulas: # Π(ext, aext, ren, aren) ! (lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )) < # Π(ext, cext, ren, cren) !(⌜paf⌝ )

Proof step 35 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) < # Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: # Π(ext, aext, ren, aren) ! (lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ))

Additional information

Used simplifier rules: Natinf freely generated by ∞, ⌜ ⌝;; ⊦ ¬ ∞ < n∞; (s)

Proof step 36 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), ⌜n⌝ = # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) < # Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 37 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), ⌜n⌝ = # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), ⌜n⌝ ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜n⌝ < # Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: remtau, name:remtau_of_to-to_of_rmtau, seq:⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m))), subst:[ext, α, n] → [Π(ext, cext, ren, cren), ⌜paf⌝ , n + 1]; ['a, 'e] → ['pa, 'e]

Additional information

Used terms: Π(ext, cext, ren, cren), ⌜paf⌝ , n + 1

Proof step 38 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m)))

A spec-lemma

Proof step 39 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), ⌜n⌝ = # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), ⌜n⌝ ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜n⌝ < # Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n + 1⌝ → (∃ m. ¬ # ⌜paf⌝ < ⌜m⌝ ∧ Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n + 1) = Π(ext, cext, ren, cren) !(⌜paf⌝ .take(m))), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ¬ ∞ < n∞; (s) ⊦ # ⌜nf⌝ = ∞; (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 ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (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)

Proof step 40 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n + 1) = Π(ext, cext, ren, cren) !(paf.take(n0)), ⌜n⌝ = # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), ⌜n⌝ ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜n⌝ < # Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:finfrag-repla-sametrace-1, seq:⊦ nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ frag* (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(aσ, aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ ⊑ aext !(aα) → Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) = Π(ext, cext, ren, cren) !(pal), subst:[astep, pal, cext, aα, start, aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsl, aext, step] → [astep, paf.take(n0), cext, aα, start, ⌜as '⌝ + aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsf.take(n0 + 1), aext, step]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used terms: aren, ren, cstep, cext, aext, aα, ext, astart, cstart, start, paf.take(n0), cren, astep, pcsf.take(n0 + 1), ⌜as '⌝ + aσ, step

Proof step 41 from proof frag-repla-sametrace (previous step) (back to tree)

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

Lemma finfrag-repla-sametrace-1

Proof step 42 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n + 1) = Π(ext, cext, ren, cren) !(paf.take(n0)), ⌜n⌝ = # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), ⌜n⌝ ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜n⌝ < # Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf.take(n0 + 1), paf.take(n0)) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, pcsf.take(n0 + 1), paf.take(n0)))⌝ ⊑ aext !(aα) → Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 17

Additional information

Used simplifier rules: ⊦ fragω(IOAut(start, step, ext))(sf, af) → (frag*(IOAut(start, step, ext))(sf.take(n + 1), af.take(n)) ↔ true); (ss) ⊦ rext !(πra(lstep, rstep, lren, rren, ⌜psf⌝ , ⌜paf⌝ )) = β ∧ fragω (IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))) (psf, paf) → ⌜rext !(πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n)))⌝ ⊑ β; (ss) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 43 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n + 1) = Π(ext, cext, ren, cren) !(paf.take(n0)), ⌜n⌝ = # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), ⌜n⌝ ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜n⌝ < # Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 44 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n + 1) = Π(ext, cext, ren, cren) !(paf.take(n0)), ⌜n⌝ = # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), ⌜n⌝ ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜n⌝ < # Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-getput, name:extension, seq: ⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n]), subst:[y, x] → [Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n + 1), Π(ext, cext, ren, cren) !(paf.take(n0))]; ['a] → ['e], <all paths>

Additional information

Used terms: Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n + 1), Π(ext, cext, ren, cren) !(paf.take(n0))

Proof step 45 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n])

A spec-lemma

Proof step 46 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), # Π(ext, cext, ren, cren) !(paf.take(n0)) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n + 1), ⌜n⌝ = # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), ⌜n⌝ ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜n⌝ < # Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ n1. n1 < # Π(ext, cext, ren, cren) !(paf.take(n0)) → Π(ext, cext, ren, cren) !(paf.take(n0))[n1] = Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n + 1)[n1] ⊦

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 + 1 ↔ ¬ n < m; (s) m < n, ¬ # σ < ⌜n⌝ ⊦ σ.take(n)[m] = σ[m]; (s) ⊦ m < n → (n ≤ m ↔ false); (s) ⊦ m < n → n ≠ 0; (s) ⊦ # x = 0 ↔ x = []; (s) trans-Nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat 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 ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s)

Proof step 47 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), # Π(ext, cext, ren, cren) !(paf.take(n0)) = n + 1, ⌜n⌝ = # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), ⌜n⌝ ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜n⌝ < # Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ n1. ¬ n < n1 → Π(ext, cext, ren, cren) !(paf.take(n0))[n1] = Π(ext, cext, ren, cren) !(⌜paf⌝ )[n1] ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 19 terms: n0 abort: F discard: T computed: F

Additional information

Used terms: n0

Proof step 48 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, cext, ren, cren) !(paf.take(n0)) = n + 1, ⌜n⌝ = # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), ⌜n⌝ ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜n⌝ < # Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ n1. ¬ n < n1 → Π(ext, cext, ren, cren) !(paf.take(n0))[n1] = Π(ext, cext, ren, cren) !(⌜paf⌝ )[n1], ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

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] → [⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα) ⌝ , lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )]; ['a] → ['pa], <all paths>

Additional information

Used simplifier rules: ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ⌜n⌝ + n∞ = ⌜n⌝ ↔ n∞ = ⌜0⌝ ; (s) ⊦ ⌜n⌝ + n∞ = ⌜0⌝ ↔ n∞ = ⌜0⌝ ∧ n = 0; (ws) ⊦ # σ = ⌜0⌝ ↔ σ = []; (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 Used terms: ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα) ⌝ , lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )

Proof step 49 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 50 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) < # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: nat-inf, name: lele-refl, seq: ⊦ ¬ n∞1 < n∞ → (n∞ < n∞1 ↔ n∞ ≠ n∞1), subst:[n∞, n∞1] → [# Π(ext, aext, ren, aren) ! (lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )), # Π(ext, cext, ren, cren) !(⌜paf⌝ )], <all paths>

Additional information

Used terms: # Π(ext, aext, ren, aren) ! (lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )), # Π(ext, cext, ren, cren) !(⌜paf⌝ )

Proof step 51 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ ¬ n∞1 < n∞ → (n∞ < n∞1 ↔ n∞ ≠ n∞1)

A spec-lemma

Proof step 52 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) < # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ n∞ < n∞0 → ¬ n∞0 < n∞; (s)

Proof step 53 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ), # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule weakening

And obtained the following new goals

Rule argument

poslist: left 2

Additional information

Proof step 54 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: Π(ext, cext, ren, cren) !(⌜paf⌝ )

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;; ⊦ # ⌜nf⌝ = ∞; (s) ⊦ ¬ ∞ < n∞; (s)

Proof step 55 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), ⌜el⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 56 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), ⌜el⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜# el⌝ < # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: remtau, name:remtau_of_to-to_of_rmtau, seq:⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m))), subst:[ext, α, n] → [Π(ext, cext, ren, cren), ⌜paf⌝ , # el]; ['a, 'e] → ['pa, 'e]

Additional information

Used terms: Π(ext, cext, ren, cren), ⌜paf⌝ , # el

Proof step 57 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m)))

A spec-lemma

Proof step 58 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), ⌜el⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜# el⌝ < # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜# el⌝ → (∃ m. ¬ # ⌜paf⌝ < ⌜m⌝ ∧ Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(# el) = Π(ext, cext, ren, cren) !(⌜paf⌝ .take(m))), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = n → x.take(n) = x; (s) ⊦ ⌜x⌝ .take(n) = x.take(n); (s) ⊦ ¬ n < n; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ¬ ∞ < n∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 59 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜# Π(ext, cext, ren, cren) !(paf.take(n))⌝ < # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:finfrag-repla-sametrace-1, seq:⊦ nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ frag* (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(aσ, aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ ⊑ aext !(aα) → Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) = Π(ext, cext, ren, cren) !(pal), subst:[astep, pal, cext, aα, start, aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsl, aext, step] → [astep, paf.take(n), cext, aα, start, ⌜as '⌝ + aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsf.take(n + 1), aext, step]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used terms: aren, ren, cstep, cext, aext, aα, ext, astart, cstart, start, paf.take(n), cren, astep, pcsf.take(n + 1), ⌜as '⌝ + aσ, step

Proof step 60 from proof frag-repla-sametrace (previous step) (back to tree)

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

Lemma finfrag-repla-sametrace-1

Proof step 61 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜# Π(ext, cext, ren, cren) !(paf.take(n))⌝ < # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf.take(n + 1), paf.take(n)) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, pcsf.take(n + 1), paf.take(n)))⌝ ⊑ aext !(aα) → Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 15

Additional information

Used simplifier rules: ⊦ fragω(IOAut(start, step, ext))(sf, af) → (frag*(IOAut(start, step, ext))(sf.take(n + 1), af.take(n)) ↔ true); (ss) ⊦ rext !(πra(lstep, rstep, lren, rren, ⌜psf⌝ , ⌜paf⌝ )) = β ∧ fragω (IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))) (psf, paf) → ⌜rext !(πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n)))⌝ ⊑ β; (ss) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 62 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜# Π(ext, cext, ren, cren) !(paf.take(n))⌝ < # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 63 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜# Π(ext, cext, ren, cren) !(paf.take(n))⌝ < # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 16 terms: n abort: F discard: F computed: F

Additional information

Used terms: n

Proof step 64 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜# Π(ext, cext, ren, cren) !(paf.take(n))⌝ < # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

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] → [⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ , lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )]; ['a] → ['pa], <all paths>

Additional information

Used terms: ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ , lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )

Proof step 65 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 66 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜# Π(ext, cext, ren, cren) !(paf.take(n))⌝ < # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∃ σ1. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + σ1 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

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 ⊦ # σ = ⌜0⌝ ↔ σ = []; (s) ⊦ ⌜n⌝ < n∞ + ⌜n⌝ ↔ n∞ ≠ ⌜0⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s)

Proof step 67 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + pα = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), Π(ext, aext, ren, aren) !(pα) ≠ [], frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + pα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + # pα < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: Π(ext, aext, ren, aren) ! ( ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ + pα) = Π(ext, aext, ren, aren) ! (lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ))

Additional information

Used simplifier rules: ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used formulas: Π(ext, aext, ren, aren) ! ( ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ + pα) = Π(ext, aext, ren, aren) ! (lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ))

Proof step 68 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + pα) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + pα = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), Π(ext, aext, ren, aren) !(pα) ≠ [], frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + pα), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + # pα < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name: remtau-append, seq: ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2), subst:[ext, al, α2] → [Π(ext, aext, ren, aren), repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα), pα]; ['a, 'e] → ['pa, 'e], <all paths>

Additional information

Used terms: Π(ext, aext, ren, aren), repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα), pα

Proof step 69 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2)

A spec-lemma

Proof step 70 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + pα = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα))⌝ + Π(ext, aext, ren, aren) !(pα) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), Π(ext, aext, ren, aren) !(pα) ≠ [], frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + pα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + # pα < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: Π(ext, aext, ren, aren) !(pα) = []

Additional information

Used formulas: Π(ext, aext, ren, aren) !(pα) = []

Proof step 71 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), ⌜Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα))⌝ + Π(ext, aext, ren, aren) !(pα) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + pα = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), Π(ext, aext, ren, aren) !(pα) ≠ [], frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + pα), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + # pα < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

User applied rule apply 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] → [Π(ext, aext, ren, aren), pα, hd(Π(ext, aext, ren, aren) !(pα)), tl(Π(ext, aext, ren, aren) !(pα))]; ['a, 'e] → ['pa, 'e]

Additional information

Used terms: Π(ext, aext, ren, aren), pα, hd(Π(ext, aext, ren, aren) !(pα)), tl(Π(ext, aext, ren, aren) !(pα))

Proof step 72 from proof frag-repla-sametrace (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 73 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + pα = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα))⌝ + Π(ext, aext, ren, aren) !(pα) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), Π(ext, aext, ren, aren) !(pα) ≠ [], frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + pα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), Π(ext, aext, ren, aren) !(pα) = ⌜hd(Π(ext, aext, ren, aren) !(pα)) '⌝ + tl(Π(ext, aext, ren, aren) !(pα)) ↔ (∃ al, pa, α0. pα = ⌜al⌝ + ⌜pa '⌝ + α0 ∧ Π(ext, aext, ren, aren)(pa) = Some(hd(Π(ext, aext, ren, aren) !(pα))) ∧ Π(ext, aext, ren, aren) !(⌜al⌝ ) = [] ∧ Π(ext, aext, ren, aren) !(α0) = tl(Π(ext, aext, ren, aren) !(pα))), ∀ m. ¬ ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + # pα < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-hd-tl, seq: ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0), subst:[σ, a, σ0] → [Π(ext, aext, ren, aren) !(pα), e, σ0]; ['a] → ['e]

Additional information

Used simplifier rules: ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0); (elim) Used terms: Π(ext, aext, ren, aren) !(pα), e, σ0

Proof step 74 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(pα) = e + σ0, ⌜Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα))⌝ + Π(ext, aext, ren, aren) !(pα) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + pα = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), Π(ext, aext, ren, aren) !(pα) ≠ [], frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + pα), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), Π(ext, aext, ren, aren) !(pα) = ⌜e '⌝ + σ0 ↔ (∃ al, pa, α0. pα = ⌜al⌝ + ⌜pa '⌝ + α0 ∧ Π(ext, aext, ren, aren)(pa) = Some(e) ∧ Π(ext, aext, ren, aren) !(⌜al⌝ ) = [] ∧ Π(ext, aext, ren, aren) !(α0) = σ0), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + # pα < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 19

Additional information

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

Proof step 75 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(pα) = e + σ0, cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), Π(ext, aext, ren, aren) !(pα) = ⌜e '⌝ + σ0, ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + pα = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα))⌝ + Π(ext, aext, ren, aren) !(pα) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), Π(ext, aext, ren, aren) !(pα) ≠ [], frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + pα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + # pα < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∃ al, pa, α0. pα = ⌜al⌝ + ⌜pa '⌝ + α0 ∧ Π(ext, aext, ren, aren)(pa) = Some(e) ∧ Π(ext, aext, ren, aren) !(⌜al⌝ ) = [] ∧ Π(ext, aext, ren, aren) !(α0) = σ0 ⊦

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 ⊦ # x = # y → (⌜x⌝ + σ = ⌜y⌝ + σ0 ↔ x = y ∧ σ = σ0); (s) ⊦ [] + σ = σ; (s) ⊦ ⌜[]⌝ = []; (s) ⊦ ⌜a '⌝ + σ ≠ []; (s) ⊦ oe + β = (oe = None ⊃ [];⌜oe.get '⌝ ) + β; (s) ⊦ ext !(⌜a '⌝ + α) = ext(a) + ext !(α); (s) ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ a + σ = ⌜a '⌝ + σ; (s) ⊦ ⌜x⌝ = [] ↔ x = []; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 76 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(pal) = [], ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜# pal⌝ + ⌜1⌝ + # pα0 < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 20 terms: # repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) + # pal + 1 abort: F discard: T computed: F

Additional information

Used terms: # repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) + # pal + 1

Proof step 77 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(pal) = [], cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜# pal⌝ + ⌜1⌝ + # pα0 < ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) + # pal + 1⌝ → (∃ n0. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) .take (# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) + # pal + 1 ) ⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜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 ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) 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).take(1) = a '; (s) ⊦ (⌜x⌝ + σ).take(m + # x) = x + σ.take(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⌝ + n∞1 < ⌜n⌝ + n∞2 ↔ n∞1 < n∞2; (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)

Proof step 78 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(pal) = [], cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

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] → [ ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ + ⌜pal⌝ + ⌜pa '⌝ , ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα) ⌝ ]; ['a] → ['pa], <all paths>

Additional information

Used terms: ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ + ⌜pal⌝ + ⌜pa '⌝ , ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα) ⌝

Proof step 79 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 80 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(pal) = [], ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∃ σ1. (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + σ1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 81 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(pal) = [], cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 20 terms: n0 abort: F discard: T computed: F

Additional information

Used terms: n0

Proof step 82 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(pal) = [], ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 83 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(pal) = [], cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 ⊦

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] → [⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα) ⌝ , ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0]; ['a] → ['pa], <all paths>

Additional information

Used terms: ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα) ⌝ , ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0

Proof step 84 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 85 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(pal) = [], ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∃ σ1. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + σ1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 86 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(pal) = [], cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:finfrag-repla-sametrace-1, seq:⊦ nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ frag* (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(aσ, aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ ⊑ aext !(aα) → Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) = Π(ext, cext, ren, cren) !(pal), subst:[astep, pal, cext, aα, start, aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsl, aext, step] → [astep, paf.take(n0), cext, aα, start, ⌜as '⌝ + aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsf.take(n0 + 1), aext, step]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used terms: aren, ren, cstep, cext, aext, aα, ext, astart, cstart, start, paf.take(n0), cren, astep, pcsf.take(n0 + 1), ⌜as '⌝ + aσ, step

Proof step 87 from proof frag-repla-sametrace (previous step) (back to tree)

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

Lemma finfrag-repla-sametrace-1

Proof step 88 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(pal) = [], ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf.take(n0 + 1), paf.take(n0)) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, pcsf.take(n0 + 1), paf.take(n0)))⌝ ⊑ aext !(aα) → Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 21

Additional information

Used simplifier rules: ⊦ fragω(IOAut(start, step, ext))(sf, af) → (frag*(IOAut(start, step, ext))(sf.take(n + 1), af.take(n)) ↔ true); (ss) ⊦ rext !(πra(lstep, rstep, lren, rren, ⌜psf⌝ , ⌜paf⌝ )) = β ∧ fragω (IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))) (psf, paf) → ⌜rext !(πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n)))⌝ ⊑ β; (ss) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 89 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(pal) = [], cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: Π(ext, cext, ren, cren) !(⌜paf.take(n0)⌝ ) ⊑ Π(ext, cext, ren, cren) !(⌜paf⌝ )

Additional information

Used formulas: Π(ext, cext, ren, cren) !(⌜paf.take(n0)⌝ ) ⊑ Π(ext, cext, ren, cren) !(⌜paf⌝ )

Proof step 90 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(pal) = [], ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), Π(ext, cext, ren, cren) !(⌜paf.take(n0)⌝ ) ⊑ Π(ext, cext, ren, cren) !(⌜paf⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 91 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(pal) = [], cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜Π(ext, cext, ren, cren) !(paf.take(n0))⌝ ⊑ ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ ⊦

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] → [⌜Π(ext, cext, ren, cren) !(paf.take(n0))⌝ , ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ ]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜Π(ext, cext, ren, cren) !(paf.take(n0))⌝ , ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝

Proof step 92 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 93 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(pal) = [], ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∃ σ1. ⌜Π(ext, cext, ren, cren) !(paf.take(n0))⌝ + σ1 = ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 94 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(pal) = [], cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), ⌜Π(ext, cext, ren, cren) !(paf.take(n0))⌝ + β = ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ , ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: β

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;; ⊦ ⌜x⌝ ≠ σ + ⌜nf0⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream

Proof step 95 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(pal) = [], ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, ⌜Π(ext, cext, ren, cren) !(paf.take(n0))⌝ + β = ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ , Π(ext, aext, ren, aren)(pa) = Some(e), β = ⌜el⌝ , ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦

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 ⊦ ⌜x⌝ + ⌜y⌝ = ⌜z⌝ ↔ x + y = z; (s)

Proof step 96 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(pal) = [], cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), Π(ext, cext, ren, cren) !(paf.take(n0)) + el = Π(ext, cext, ren, cren) !(paf.take(n)), Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: Π(ext, aext, ren, aren) ! ( ( ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα) = Π(ext, aext, ren, aren) ! (⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα) ⌝ )

Additional information

Used formulas: Π(ext, aext, ren, aren) ! ( ( ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα) = Π(ext, aext, ren, aren) ! (⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα) ⌝ )

Proof step 97 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(pal) = [], Π(ext, aext, ren, aren) ! ((⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα) = Π(ext, aext, ren, aren) !(⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ ), Π(ext, cext, ren, cren) !(paf.take(n0)) + el = Π(ext, cext, ren, cren) !(paf.take(n)), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name: remtau-append, seq: ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2), subst:[ext, al, α2] → [Π(ext, aext, ren, aren), repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα), ⌜pal⌝ + ⌜pa '⌝ + pα]; ['a, 'e] → ['pa, 'e], <all paths>

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: Π(ext, aext, ren, aren), repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα), ⌜pal⌝ + ⌜pa '⌝ + pα

Proof step 98 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2)

A spec-lemma

Proof step 99 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(pal) = [], cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), Π(ext, cext, ren, cren) !(paf.take(n0)) + el = Π(ext, cext, ren, cren) !(paf.take(n)), ⌜Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα))⌝ + Π(ext, aext, ren, aren) !(⌜pal⌝ + ⌜pa '⌝ + pα) = Π(ext, aext, ren, aren) !(⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ ), Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren) ⊦

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) ⊦ ⌜x⌝ .l = x; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (⌜x⌝ + ⌜y⌝ ).l = x + y; (s) ⊦ [] + σ = σ; (s) ⊦ ⌜[]⌝ = []; (s) ⊦ (⌜x⌝ + ⌜y⌝ + ⌜y0⌝ ).l = x + y + y0; (s) ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s) ⊦ Some(a).get = a; (s) ⊦ Some(a) ≠ None; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ finp(σ) ↔ (∃ x. ⌜x⌝ = σ); (as) ⊦ finp(⌜x⌝ ); (s) ⊦ σ0 + σ = ⌜z⌝ ↔ finp(σ0) ∧ finp(σ) ∧ σ0.l + σ.l = z; (as) ⊦ # x = # y → (⌜x⌝ + σ = ⌜y⌝ + σ0 ↔ x = y ∧ σ = σ0); (s) ⊦ oe + β = (oe = None ⊃ [];⌜oe.get '⌝ ) + β; (s) ⊦ ext !(⌜a '⌝ + α) = ext(a) + ext !(α); (s) ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s)

Proof step 100 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(pal) = [], Π(ext, cext, ren, cren) !(paf.take(n)) + e ' + Π(ext, aext, ren, aren) !(pal2) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, cext, ren, cren) !(paf.take(n0)) + el = Π(ext, cext, ren, cren) !(paf.take(n)), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) + pal + pa ' + pal2 = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦

User applied rule subst equation

And closed the goal!

Rule argument

pos: left 5, flags: rotate, discard

Additional information

Used simplifier rules: ⊦ x = x + y ↔ y = []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 101 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(pal) = [], ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(ext, cext, ren, cren) !(⌜paf.take(n0)⌝ ) ⊑ Π(ext, cext, ren, cren) !(⌜paf⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 102 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(pal) = [], cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ ⌜Π(ext, cext, ren, cren) !(paf.take(n0))⌝ ⊑ ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ ⊦

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] → [⌜Π(ext, cext, ren, cren) !(paf.take(n0))⌝ , ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ ]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜Π(ext, cext, ren, cren) !(paf.take(n0))⌝ , ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝

Proof step 103 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 104 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(pal) = [], ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ σ1. ⌜Π(ext, cext, ren, cren) !(paf.take(n0))⌝ + σ1 ≠ ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: to-plus-from-inf, seq: ⊦ ⌜nf.take(n)⌝ + ⌜nf.drop(n)⌝ = ⌜nf⌝ , subst:[nf, n] → [paf, n0]; ['a] → ['pa], <rotated> <all paths>

Additional information

Used terms: paf, n0

Proof step 105 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ ⌜nf.take(n)⌝ + ⌜nf.drop(n)⌝ = ⌜nf⌝

A spec-lemma

Proof step 106 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(pal) = [], cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf.take(n0)⌝ + ⌜paf.drop(n0)⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = Π(ext, cext, ren, cren) !(⌜paf.take(n0)⌝ + ⌜paf.drop(n0)⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ σ1. ⌜Π(ext, cext, ren, cren) !(paf.take(n0))⌝ + σ1 ≠ ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ ⊦

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 ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s)

Proof step 107 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf.take(n0)⌝ + ⌜paf.drop(n0)⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(pal) = [], ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), (Some(e) = None ⊃ [];⌜Some(e).get '⌝ ) + Π(ext, aext, ren, aren) !(pα0) = ⌜e '⌝ + Π(ext, aext, ren, aren) !(pα0), (⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ ) + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ , ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0, Π(ext, aext, ren, aren)(pa) = Some(e), ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ = ⌜Π(ext, cext, ren, cren) !(paf.take(n0))⌝ + Π(ext, cext, ren, cren) !(⌜paf.drop(n0)⌝ ), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ + ⌜pal⌝ + ⌜pa '⌝ + pα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ σ1. ⌜Π(ext, cext, ren, cren) !(paf.take(n0))⌝ + σ1 ≠ ⌜Π(ext, cext, ren, cren) !(paf.take(n))⌝ ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 22 terms: Π(ext, cext, ren, cren) !(⌜paf.drop(n0)⌝ )

Additional information

Used terms: Π(ext, cext, ren, cren) !(⌜paf.drop(n0)⌝ )

Proof step 108 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ≠ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) ∨ ¬ ( ¬ # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) < ⌜n⌝ → Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) .take (n ) = Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 109 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: remtau, name:remtau_of_to-to_of_rmtau, seq:⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m))), subst:[ext, α, n] → [Π(ext, cext, ren, cren), ⌜paf⌝ , n]; ['a, 'e] → ['pa, 'e]

Additional information

Used terms: Π(ext, cext, ren, cren), ⌜paf⌝ , n

Proof step 110 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m)))

A spec-lemma

Proof step 111 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ → (∃ m. ¬ # ⌜paf⌝ < ⌜m⌝ ∧ Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(⌜paf⌝ .take(m))), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ¬ ∞ < n∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 112 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: remtau, name:remtau_of_to-to_of_rmtau, seq:⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m))), subst:[ext, α, n] → [Π(ext, aext, ren, aren), lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ), n]; ['a, 'e] → ['pa, 'e]

Additional information

Used terms: Π(ext, aext, ren, aren), lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ), n

Proof step 113 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m)))

A spec-lemma

Proof step 114 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ¬ # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) < ⌜n⌝ → (∃ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ ∧ Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) .take (n ) = Π(ext, aext, ren, aren) ! (lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m))), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 115 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 19 terms: n1 abort: F discard: T computed: T

Additional information

Used terms: n1

Proof step 116 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 117 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

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] → [⌜lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ) .take (n1 ) ⌝ , ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα) ⌝ ]; ['a] → ['pa], <all paths>

Additional information

Used terms: ⌜lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ) .take (n1 ) ⌝ , ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα) ⌝

Proof step 118 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 119 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∃ σ1. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + σ1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 120 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: # Π(ext, cext, ren, cren) ! ( ⌜lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ) .take (n1 ) ⌝ + pα) = # Π(ext, cext, ren, cren) ! (⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα) ⌝ )

Additional information

Used formulas: # Π(ext, cext, ren, cren) ! ( ⌜lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ) .take (n1 ) ⌝ + pα) = # Π(ext, cext, ren, cren) ! (⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα) ⌝ )

Proof step 121 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), # Π(ext, cext, ren, cren) ! (⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα) = # Π(ext, cext, ren, cren) !(⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ ), ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name: remtau-append, seq: ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2), subst:[ext, al, α2] → [Π(ext, cext, ren, cren), lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ) .take (n1 ), pα]; ['a, 'e] → ['pa, 'e], <all paths>

Additional information

Used terms: Π(ext, cext, ren, cren), lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ) .take (n1 ), pα

Proof step 122 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2)

A spec-lemma

Proof step 123 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # ( ⌜Π(ext, cext, ren, cren) ! (lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)) ⌝ + Π(ext, cext, ren, cren) !(pα)) = # Π(ext, cext, ren, cren) !(⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ ), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: size-append, seq: ⊦ #(σ + σ1) = # σ + # σ1, subst:[σ, σ1] → [⌜Π(ext, cext, ren, cren) ! (lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ) .take (n1 )) ⌝ , Π(ext, cext, ren, cren) !(pα)]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜Π(ext, cext, ren, cren) ! (lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ) .take (n1 )) ⌝ , Π(ext, cext, ren, cren) !(pα)

Proof step 124 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ #(σ + σ1) = # σ + # σ1

A spec-lemma

Proof step 125 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , # ⌜Π(ext, cext, ren, cren) ! (lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)) ⌝ + # Π(ext, cext, ren, cren) !(pα) = # Π(ext, cext, ren, cren) !(⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ ), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: Π(ext, cext, ren, cren) !(pα)

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;; ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ # ⌜nf⌝ = ∞; (s) ⊦ ⌜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 ⊦ ⌜n⌝ ≠ ∞; (s)

Proof step 126 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), # ⌜Π(ext, cext, ren, cren) ! (lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)) ⌝ + # Π(ext, cext, ren, cren) !(pα) = # Π(ext, cext, ren, cren) !(⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ ), ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), ⌜el⌝ = Π(ext, cext, ren, cren) !(pα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

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⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n + m = n0; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 127 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , # Π(ext, cext, ren, cren) ! (lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)) + # el = # Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), ⌜el⌝ = Π(ext, cext, ren, cren) !(pα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:finfrag-repla-sametrace-1, seq:⊦ nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ frag* (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(aσ, aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ ⊑ aext !(aα) → Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) = Π(ext, cext, ren, cren) !(pal), subst:[astep, pal, cext, aα, start, aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsl, aext, step] → [astep, paf.take(n2), cext, aα, start, ⌜as '⌝ + aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsf.take(n2 + 1), aext, step]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used terms: aren, ren, cstep, cext, aext, aα, ext, astart, cstart, start, paf.take(n2), cren, astep, pcsf.take(n2 + 1), ⌜as '⌝ + aσ, step

Proof step 128 from proof frag-repla-sametrace (previous step) (back to tree)

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

Lemma finfrag-repla-sametrace-1

Proof step 129 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), # Π(ext, cext, ren, cren) ! (lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)) + # el = # Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)), ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), ⌜el⌝ = Π(ext, cext, ren, cren) !(pα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf.take(n2 + 1), paf.take(n2)) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, pcsf.take(n2 + 1), paf.take(n2)))⌝ ⊑ aext !(aα) → Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n2)), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 22

Additional information

Used simplifier rules: ⊦ fragω(IOAut(start, step, ext))(sf, af) → (frag*(IOAut(start, step, ext))(sf.take(n + 1), af.take(n)) ↔ true); (ss) ⊦ rext !(πra(lstep, rstep, lren, rren, ⌜psf⌝ , ⌜paf⌝ )) = β ∧ fragω (IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))) (psf, paf) → ⌜rext !(πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n)))⌝ ⊑ β; (ss) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 130 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n2)), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , # Π(ext, cext, ren, cren) ! (lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)) + # el = # Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), ⌜el⌝ = Π(ext, cext, ren, cren) !(pα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:finfrag-repla-sametrace-1, seq:⊦ nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ frag* (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(aσ, aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ ⊑ aext !(aα) → Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) = Π(ext, cext, ren, cren) !(pal), subst:[astep, pal, cext, aα, start, aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsl, aext, step] → [astep, paf.take(n0), cext, aα, start, ⌜as '⌝ + aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsf.take(n0 + 1), aext, step]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used terms: aren, ren, cstep, cext, aext, aα, ext, astart, cstart, start, paf.take(n0), cren, astep, pcsf.take(n0 + 1), ⌜as '⌝ + aσ, step

Proof step 131 from proof frag-repla-sametrace (previous step) (back to tree)

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

Lemma finfrag-repla-sametrace-1

Proof step 132 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n2)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), # Π(ext, cext, ren, cren) ! (lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)) + # el = # Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)), ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), ⌜el⌝ = Π(ext, cext, ren, cren) !(pα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf.take(n0 + 1), paf.take(n0)) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, pcsf.take(n0 + 1), paf.take(n0)))⌝ ⊑ aext !(aα) → Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 23

Additional information

Used simplifier rules: ⊦ fragω(IOAut(start, step, ext))(sf, af) → (frag*(IOAut(start, step, ext))(sf.take(n + 1), af.take(n)) ↔ true); (ss) ⊦ rext !(πra(lstep, rstep, lren, rren, ⌜psf⌝ , ⌜paf⌝ )) = β ∧ fragω (IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))) (psf, paf) → ⌜rext !(πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n)))⌝ ⊑ β; (ss) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 133 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n2)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , # Π(ext, cext, ren, cren) ! (lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)) + # el = # Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), ⌜el⌝ = Π(ext, cext, ren, cren) !(pα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 134 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n2)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), # Π(ext, cext, ren, cren) ! (lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)) + # el = # Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)), ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), ⌜el⌝ = Π(ext, cext, ren, cren) !(pα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 24 terms: n0 abort: F discard: T computed: F

Additional information

Used terms: n0

Proof step 135 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n2)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , # Π(ext, cext, ren, cren) ! (lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)) + # el = # Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), ⌜el⌝ = Π(ext, cext, ren, cren) !(pα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ ⊦

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] → [⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα) ⌝ , lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )]; ['a] → ['pa], <all paths>

Additional information

Used terms: ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα) ⌝ , lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )

Proof step 136 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 137 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n2)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), # Π(ext, cext, ren, cren) ! (lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)) + # el = # Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)), ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), ⌜el⌝ = Π(ext, cext, ren, cren) !(pα), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ∃ σ1. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + σ1 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

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 ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s)

Proof step 138 from proof frag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n0)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n2)), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜# Π(ext, cext, ren, cren) !(paf.take(n0))⌝ + # Π(ext, aext, ren, aren) !(pα0) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), # Π(ext, cext, ren, cren) !((⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα0).take(n1)) + # el = # Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)), ⌜(⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα0).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), (⌜Π(ext, cext, ren, cren) !(paf.take(n0))⌝ + Π(ext, aext, ren, aren) !(pα0)).take(n) = Π(ext, aext, ren, aren) !((⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα0).take(n1)), ⌜el⌝ = Π(ext, cext, ren, cren) !(pα), (⌜Π(ext, cext, ren, cren) !(paf.take(n0))⌝ + Π(ext, aext, ren, aren) !(pα0)).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ¬ ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + # pα0 < ⌜n1⌝ ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 8, flags: rotate, discard

Additional information

Proof step 139 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(paf.take(n2)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n), ⌜(⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα0).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , # Π(ext, cext, ren, cren) !((⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα0).take(n1)) + # el = # Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)), ⌜# Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n)⌝ + # Π(ext, aext, ren, aren) !(pα0) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα0 = lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), (⌜Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n)⌝ + Π(ext, aext, ren, aren) !(pα0)).take(n) = Π(ext, aext, ren, aren) !((⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα0).take(n1)), ⌜el⌝ = Π(ext, cext, ren, cren) !(pα), (⌜Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n)⌝ + Π(ext, aext, ren, aren) !(pα0)).take(n) ≠ Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + pα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ¬ ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ + # pα0 < ⌜n1⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:append-le-to, seq: ⊦ ¬ # σ < ⌜n⌝ → (σ + σ0).take(n) = σ.take(n), subst:[n, σ, σ0] → [n, ⌜Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n)⌝ , Π(ext, aext, ren, aren) !(pα0)]; ['a] → ['e], <all paths>

Additional information

Used simplifier rules: ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ ⌜x⌝ .take(n) = x.take(n); (s) ¬ # σ < ⌜n⌝ ⊦ σ.take(n).take(n) = σ.take(n); (s) ⊦ # σ < ⌜n⌝ → (σ + σ0).take(n) = σ.l + σ0.take(n - # σ.l); (s) ⊦ ⌜x⌝ .l = x; (s) ⊦ m - m = 0; (s) ⊦ σ.take(0) = []; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: n, ⌜Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n)⌝ , Π(ext, aext, ren, aren) !(pα0)

Proof step 140 from proof frag-repla-sametrace (previous step) (back to tree)

⊦ ¬ # σ < ⌜n⌝ → (σ + σ0).take(n) = σ.take(n)

A spec-lemma

Proof step 141 from proof frag-repla-sametrace (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = # Π(ext, cext, ren, cren) !(⌜paf⌝ ), ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ , Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) = Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)), Π(ext, cext, ren, cren) !(⌜paf⌝ ).take(n) = Π(ext, cext, ren, cren) !(paf.take(n0)), # Π(ext, cext, ren, cren) ! (⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ).take(n1)⌝ + pα) ≠ # Π(ext, cext, ren, cren) !(⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n2 + 1), paf.take(n2), ⌜as '⌝ + aσ, aα)⌝ ), Π(ext, aext, ren, aren) !(lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )).take(n) ≠ Π(ext, cext, ren, cren) !(paf.take(n0)), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ # Π(ext, cext, ren, cren) !(⌜paf⌝ ) < ⌜n⌝ , ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜n1⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule subst equation

And closed the goal!

Rule argument

pos: left 3, flags: discard

Additional information

Proof Tree for frag-repla-sametrace (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