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

nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), oinj(cext), oinj(aext), oinj(ext), 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))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

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

User applied rule structural induction

And obtained the following new goals

Rule argument

term: pcσ

Additional information

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

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

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

User applied rule structural induction

And obtained the following new goals

Rule argument

term: pα

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;; ⊦ ¬ frag(ioa)(⌜sl⌝ , ⌜af⌝ ); (s)

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: replstate-fin-def, seq:⊦ # pcsl = # pal + 1 → repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcsl⌝ , ⌜pal⌝ , aσ, aα) = ⌜repls(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, 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: ⊦ frag(ioa)(⌜sl⌝ , ⌜al⌝ ) ↔ frag*(ioa)(sl, al); (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, aσ, step

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

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

Lemma replstate-fin-def

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

cext !(πra(step, cstep, ren, cren, ⌜pcsl⌝ , ⌜pal⌝ )) = aext !(aα), # pcsl = # pal + 1, 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), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)⌝ , repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsl⌝ , ⌜pal⌝ , aσ, aα)) ⊦

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, 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 terms: aren, ren, cstep, cext, aext, aα, pal, cren, pcsl, aσ, step

Proof step 7 from proof repl-frag (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 8 from proof repl-frag (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsl⌝ , ⌜pal⌝ )) = aext !(aα), # pcsl = # pal + 1, frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜pcsl⌝ , ⌜pal⌝ ), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)⌝ , ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, 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 terms: pcsl, cstep, pal, ren, cren, step

Proof step 9 from proof repl-frag (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 10 from proof repl-frag (previous step) (back to tree)

cext !(⌜πra(step, cstep, ren, cren, pcsl, pal)⌝ ) = aext !(aα), # pcsl = # pal + 1, 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), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)⌝ , ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)⌝ ) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:finfrag-repl-frag, seq:⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ oinj(cext) ∧ oinj(aext) ∧ oinj(ext) ∧ frag* (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(⌜πra(step, cstep, ren, cren, pcsl, pal)⌝ ) ⊑ aext !(aα) ∧ sameshared(cren, aren, ren) → frag*(Π(IOAut(start, step, ext), IOAut(astart, astep, aext), ren, aren)) (repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα)), subst:[astep, pal, cext, aα, as, start, aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsl, aext, step] → [astep, pal, cext, aα, hd(aσ), start, tl(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, hd(aσ), aext, aα, ext, astart, cstart, start, pal, cren, astep, pcsl, tl(aσ), step

Proof step 11 from proof repl-frag (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) ∧ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(⌜πra(step, cstep, ren, cren, pcsl, pal)⌝ ) ⊑ aext !(aα) ∧ sameshared(cren, aren, ren) → frag*(Π(IOAut(start, step, ext), IOAut(astart, astep, aext), ren, aren)) (repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα))

Lemma finfrag-repl-frag

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

cext !(⌜πra(step, cstep, ren, cren, pcsl, pal)⌝ ) = aext !(aα), # pcsl = # pal + 1, frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜pcsl⌝ , ⌜pal⌝ ), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)⌝ , ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)⌝ ), nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ oinj(cext) ∧ oinj(aext) ∧ oinj(ext) ∧ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ cext !(⌜πra(step, cstep, ren, cren, pcsl, pal)⌝ ) ⊑ aext !(aα) ∧ sameshared(cren, aren, ren) → frag*(Π(IOAut(start, step, ext), IOAut(astart, astep, aext), ren, aren)) (repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜hd(aσ) '⌝ + tl(aσ), aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜hd(aσ) '⌝ + tl(aσ), aα)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ σ ⊑ σ; (s) ⊦ Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren) = IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)); (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ frag(ioa)(⌜sl⌝ , ⌜al⌝ ) ↔ frag*(ioa)(sl, al); (s)

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

# pcsl = # pal + 1, ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ = aext !(aα), frag(IOAut(astart, astep, aext))(aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)), frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(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, pcsl, pal, ⌜hd(aσ) '⌝ + tl(aσ), aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜hd(aσ) '⌝ + tl(aσ), aα)) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 13

Additional information

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

# pcsl = # pal + 1, ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ = aext !(aα), frag(IOAut(astart, astep, aext))(aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(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, pcsl, pal, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) ⊦

User applied rule apply elim lemma

And closed the goal!

Rule argument

spec: stream, name:elim-hd-tl, seq: ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0), subst:[σ, a, σ0] → [aσ, as, σ0]; ['a] → ['as]

Additional information

Used simplifier rules: ⊦ ⌜a '⌝ + σ ≠ []; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ a + σ = ⌜a '⌝ + σ; (s) ⊦ ¬ frag(IOAut(start, step, ext))([], α); (s) ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0); (elim) Used terms: aσ, as, σ0

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

# pcsl = # pal + 1, ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ = aext !(aα), frag(IOAut(astart, astep, aext))(aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal), frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜hd(aσ) '⌝ + tl(aσ), aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜hd(aσ) '⌝ + tl(aσ), aα)), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag*(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) ⊦

User applied rule apply elim lemma

And closed the goal!

Rule argument

spec: stream, name:elim-hd-tl, seq: ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0), subst:[σ, a, σ0] → [aσ, as, σ0]; ['a] → ['as]

Additional information

Used simplifier rules: ⊦ ⌜a '⌝ + σ ≠ []; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ a + σ = ⌜a '⌝ + σ; (s) ⊦ ¬ frag(IOAut(start, step, ext))([], α); (s) ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0); (elim) Used terms: aσ, as, σ0

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

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

User applied rule structural induction

And obtained the following new goals

Rule argument

term: pα

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;; ⊦ ¬ frag(ioa)(⌜sf⌝ , ⌜al⌝ ); (s)

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ frag(ioa)(⌜sf⌝ , ⌜af⌝ ) ↔ fragω(ioa)(sf, af); (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf⌝ , ⌜paf⌝ , aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf⌝ , ⌜paf⌝ , aσ, aα)) ⊦

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, 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, aσ, step, pcsf

Proof step 19 from proof repl-frag (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 20 from proof repl-frag (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), frag(IOAut(astart, astep, aext))(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), 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), ¬ 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), aσ, aα)⌝ ⌝ ), repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf⌝ , ⌜paf⌝ , aσ, aα)) ⊦

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, 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 terms: aren, ren, cstep, cext, aext, aα, cren, paf, aσ, step, pcsf

Proof step 21 from proof repl-frag (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 22 from proof repl-frag (previous step) (back to tree)

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

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. ⌜repls (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα) ⌝ ⌝ ]; ['a] → [('s, 'as)]

Additional information

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

Proof step 23 from proof repl-frag (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 24 from proof repl-frag (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

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

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 12

Additional information

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

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

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:mon-repls, 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))) ((s, cs) ' + pcsf, paf) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, (s, cs) + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ), subst:[astep, cs, cext, as, start, aσ, ext, ren, astart, cstart, cstep, aren, aext, step, aα, s, pcsf, cren, paf] → [astep, pcsf(0)._2, cext, hd(aσ), start, tl(aσ), ext, ren, astart, cstart, cstep, aren, aext, step, aα, pcsf(0)._1, λ n. pcsf(n + 1), cren, paf]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used terms: aren, ren, cstep, cext, pcsf(0)._1, hd(aσ), aext, aα, ext, astart, cstart, start, cren, astep, paf, pcsf(0)._2, tl(aσ), step, λ n. pcsf(n + 1)

Proof step 27 from proof repl-frag (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)))((s, cs) ' + pcsf, paf) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, (s, cs) + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )

Lemma mon-repls

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), frag(IOAut(astart, astep, aext))(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), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), 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(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ cext !(πra(step, cstep, ren, cren, (pcsf(0)._1, pcsf(0)._2) + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜hd(aσ) '⌝ + tl(aσ), aα) ⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcsf(0)._1, pcsf(0)._2) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜hd(aσ) '⌝ + tl(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] → [aσ, as, σ0]; ['a] → ['as]

Additional information

Used simplifier rules: ⊦ ¬ frag(IOAut(start, step, ext))([], α); (s) ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0); (elim) Used terms: aσ, as, σ0

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), aσ = as + σ0, aσ ≠ [], frag(IOAut(astart, astep, aext))(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), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcsf(0)._1, pcsf(0)._2) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + σ = ⌜a '⌝ + σ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜a '⌝ + σ ≠ []; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcsf(0)._1, pcsf(0)._2) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: intrace-expand, seq: ⊦ ⌜nf⌝ = nf(0) + ⌜λ n. nf(n + 1)⌝ , subst:[nf] → [pcsf]; ['a] → [('s, 'cs)], <all paths>

Additional information

Used terms: pcsf

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

⊦ ⌜nf⌝ = nf(0) + ⌜λ n. nf(n + 1)⌝

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, pcsf(0) + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcsf(0)._1, pcsf(0)._2) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf(0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcsf(0)._1, pcsf(0)._2) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: pcsf(0)

Additional information

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf(0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcsf(0)._1, pcsf(0)._2) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ σ ⊑ σ; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (λ n. pcsf(n + 1)), paf) → mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: streamseq-mon-lim, name: mon-def, seq: ⊦ mon(σσ) ↔ (∀ n. ⌜n⌝ + ⌜1⌝ < # σσ → σσ[n] ⊑ σσ[n + 1]), subst:[σσ] → [⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα) ⌝ ⌝ ]; ['a] → [('s, 'as)], <all paths>

Additional information

Used terms: ⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα) ⌝ ⌝

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

⊦ mon(σσ) ↔ (∀ n. ⌜n⌝ + ⌜1⌝ < # σσ → σσ[n] ⊑ σσ[n + 1])

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (λ n. pcsf(n + 1)), paf) → mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∃ n. ¬ ( ⌜n⌝ + ⌜1⌝ < # ⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ → ⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ [n] ⊑ ⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ [n + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ⌜nf⌝ [n] = nf(n); (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ n∞ < ∞ ↔ n∞ ≠ ∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ , fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (λ n. pcsf(n + 1)), paf) → mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: streamseq-mon-lim, name: mon-def, seq: ⊦ mon(σσ) ↔ (∀ n. ⌜n⌝ + ⌜1⌝ < # σσ → σσ[n] ⊑ σσ[n + 1]), subst:[σσ] → [⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα) ⌝ ⌝ ]; ['a] → [('s, 'as)], <all paths>

Additional information

Used terms: ⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα) ⌝ ⌝

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

⊦ mon(σσ) ↔ (∀ n. ⌜n⌝ + ⌜1⌝ < # σσ → σσ[n] ⊑ σσ[n + 1])

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ , fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (λ n. pcsf(n + 1)), paf) → (∀ n. ⌜n⌝ + ⌜1⌝ < # ⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ → ⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ [n] ⊑ ⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ [n + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ [n] = nf(n); (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ n∞ < ∞ ↔ n∞ ≠ ∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ , fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (λ n. pcsf(n + 1)), paf) → (∀ n0. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0), paf.take(n0), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0 + 1), paf.take(n0 + 1), ⌜as '⌝ + σ0, aα)⌝ ) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: pcsf = (s, cs) ' + (λ n. pcsf(n + 1))

Additional information

Used formulas: pcsf = (s, cs) ' + (λ n. pcsf(n + 1))

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

cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), pcsf = (s, cs) ' + (λ n. pcsf(n + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ , fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (λ n. pcsf(n + 1)), paf) → (∀ n0. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0), paf.take(n0), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0 + 1), paf.take(n0 + 1), ⌜as '⌝ + σ0, aα)⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), pcsf = (s, cs) ' + (λ n. pcsf(n + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ , ∀ n0. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0), paf.take(n0), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0 + 1), paf.take(n0 + 1), ⌜as '⌝ + σ0, aα)⌝ ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 15 terms: n abort: F discard: T computed: F

Additional information

Used terms: n

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

cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), pcsf = (s, cs) ' + (λ n. pcsf(n + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + σ0, 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 '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: to-succ-nf, seq: ⊦ nf.take(n + 1) = nf(0) + (λ n. nf(n + 1)).take(n), subst:[nf, n] → [pcsf, n]; ['a] → [('s, 'cs)], <all paths>

Additional information

Used terms: pcsf, n

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

⊦ nf.take(n + 1) = nf(0) + (λ n. nf(n + 1)).take(n)

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), pcsf = (s, cs) ' + (λ n. pcsf(n + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + σ0, 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 '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf(0) + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), pcsf = (s, cs) ' + (λ n. pcsf(n + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + σ0, 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 '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: to-succ2-nf, seq: ⊦ nf.take(n + 2) = nf(0) + (λ n. nf(n + 1)).take(n + 1), subst:[nf, n] → [pcsf, n]; ['a] → [('s, 'cs)], <all paths>

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) c: ⊦ m + n = n + m; from specification nat-basic Used terms: pcsf, n

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

⊦ nf.take(n + 2) = nf(0) + (λ n. nf(n + 1)).take(n + 1)

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), pcsf ≠ (s, cs) ' + (λ n. pcsf(n + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ , fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (λ n. pcsf(n + 1)), paf) → (∀ n0. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0), paf.take(n0), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0 + 1), paf.take(n0 + 1), ⌜as '⌝ + σ0, aα)⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:add-list-fun-def, seq: ⊦ x + nf = (λ n. (n < # x ⊃ x[n];nf(n - # x))), subst:[x, nf] → [(s, cs) ', λ n. pcsf(n + 1)]; ['a] → [('s, 'cs)], <all paths>

Additional information

Used terms: (s, cs) ', λ n. pcsf(n + 1)

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

⊦ x + nf = (λ n. (n < # x ⊃ x[n];nf(n - # x)))

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), pcsf ≠ (λ n. (n < # (s, cs) ' ⊃ (s, cs) '[n];pcsf((n - # (s, cs) ') + 1))), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ , fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (λ n. (n < # (s, cs) ' ⊃ (s, cs) '[n];pcsf((n - # (s, cs) ') + 1)), paf) → (∀ n0. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0), paf.take(n0), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0 + 1), paf.take(n0 + 1), ⌜as '⌝ + σ0, aα)⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # a ' = 0 +1; (s) ⊦ a '[0] = a; (s) ⊦ m < 1 ↔ m = 0; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), pcsf(n0) ≠ (n0 = 0 ⊃ (s, cs);pcsf((n0 - 1) + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ , fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(λ n1. (n1 = 0 ⊃ (s, cs);pcsf((n1 - 1) + 1)), paf) → (∀ n0. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0), paf.take(n0), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0 + 1), paf.take(n0 + 1), ⌜as '⌝ + σ0, aα)⌝ ) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 3

Additional information

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

cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), n0 ≠ 0, pcsf(n0) ≠ pcsf((n0 - 1) + 1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ , fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(λ n1. (n1 = 0 ⊃ (s, cs);pcsf((n1 - 1) + 1)), paf) → (∀ n0. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0), paf.take(n0), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0 + 1), paf.take(n0 + 1), ⌜as '⌝ + σ0, aα)⌝ ) ⊦

Heuristic elimination applied rule apply elim lemma

And closed the goal!

Rule argument

spec: nat, name:elim-pred-c, seq:n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1, subst:[n, m] → [n0, n2]

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: n0, n2

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

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

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), 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), aσ, aα) ⌝ ⌝

Proof step 56 from proof repl-frag (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 57 from proof repl-frag (previous step) (back to tree)

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

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

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

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 12

Additional information

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

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

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:mon-repla, 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))) ((s, cs) ' + pcsf, paf) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, (s, cs) + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ), subst:[astep, cs, cext, as, start, aσ, ext, ren, astart, cstart, cstep, aren, aext, step, aα, s, pcsf, cren, paf] → [astep, pcsf(0)._2, cext, hd(aσ), start, tl(aσ), ext, ren, astart, cstart, cstep, aren, aext, step, aα, pcsf(0)._1, λ n. pcsf(n + 1), cren, paf]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used terms: aren, ren, cstep, cext, pcsf(0)._1, hd(aσ), aext, aα, ext, astart, cstart, start, cren, astep, paf, pcsf(0)._2, tl(aσ), step, λ n. pcsf(n + 1)

Proof step 60 from proof repl-frag (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)))((s, cs) ' + pcsf, paf) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, (s, cs) + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )

Lemma mon-repla

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), 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(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ cext !(πra(step, cstep, ren, cren, (pcsf(0)._1, pcsf(0)._2) + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜hd(aσ) '⌝ + tl(aσ), aα) ⌝ ⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), frag(IOAut(astart, astep, aext))(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), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcsf(0)._1, pcsf(0)._2) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜hd(aσ) '⌝ + tl(aσ), aα) ⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), 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] → [aσ, as, σ0]; ['a] → ['as]

Additional information

Used simplifier rules: ⊦ ¬ frag(IOAut(start, step, ext))([], α); (s) ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0); (elim) Used terms: aσ, as, σ0

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), aσ = as + σ0, aσ ≠ [], frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcsf(0)._1, pcsf(0)._2) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + σ = ⌜a '⌝ + σ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜a '⌝ + σ ≠ []; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcsf(0)._1, pcsf(0)._2) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: pcsf(0)

Additional information

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcsf(0)._1, pcsf(0)._2) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (λ n. pcsf(n + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 14

Additional information

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) ⊦ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (λ n. pcsf(n + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα)

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: pcsf = (s, cs) ' + (λ n. pcsf(n + 1))

Additional information

Used formulas: pcsf = (s, cs) ' + (λ n. pcsf(n + 1))

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), pcsf = (s, cs) ' + (λ n. pcsf(n + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ) ⊦ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (λ n. pcsf(n + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), pcsf = (s, cs) ' + (λ n. pcsf(n + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ¬ cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: intrace-expand, seq: ⊦ ⌜nf⌝ = nf(0) + ⌜λ n. nf(n + 1)⌝ , subst:[nf] → [pcsf]; ['a] → [('s, 'cs)], <all paths>

Additional information

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

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

⊦ ⌜nf⌝ = nf(0) + ⌜λ n. nf(n + 1)⌝

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), pcsf ≠ (s, cs) ' + (λ n. pcsf(n + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ) ⊦ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (λ n. pcsf(n + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:add-list-fun-def, seq: ⊦ x + nf = (λ n. (n < # x ⊃ x[n];nf(n - # x))), subst:[x, nf] → [(s, cs) ', λ n. pcsf(n + 1)]; ['a] → [('s, 'cs)], <all paths>

Additional information

Used terms: (s, cs) ', λ n. pcsf(n + 1)

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

⊦ x + nf = (λ n. (n < # x ⊃ x[n];nf(n - # x)))

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), pcsf ≠ (λ n. (n < # (s, cs) ' ⊃ (s, cs) '[n];pcsf((n - # (s, cs) ') + 1))), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) ⊦ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (λ n. (n < # (s, cs) ' ⊃ (s, cs) '[n];pcsf((n - # (s, cs) ') + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # a ' = 0 +1; (s) ⊦ a '[0] = a; (s) ⊦ m < 1 ↔ m = 0; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), pcsf(n) ≠ (n = 0 ⊃ (s, cs);pcsf((n - 1) + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ) ⊦ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(λ n0. (n0 = 0 ⊃ (s, cs);pcsf((n0 - 1) + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 3

Additional information

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), n ≠ 0, pcsf(n) ≠ pcsf((n - 1) + 1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) ⊦ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(λ n0. (n0 = 0 ⊃ (s, cs);pcsf((n0 - 1) + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα)

Heuristic elimination applied rule apply elim lemma

And closed the goal!

Rule argument

spec: nat, name:elim-pred-c, seq:n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1, subst:[n, m] → [n, n1]

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: n, n1

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: streamseq-mon-lim, name: mon-def, seq: ⊦ mon(σσ) ↔ (∀ n. ⌜n⌝ + ⌜1⌝ < # σσ → σσ[n] ⊑ σσ[n + 1]), subst:[σσ] → [⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα) ⌝ ⌝ ]; ['a] → ['pa], <all paths>

Additional information

Used terms: ⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα) ⌝ ⌝

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

⊦ mon(σσ) ↔ (∀ n. ⌜n⌝ + ⌜1⌝ < # σσ → σσ[n] ⊑ σσ[n + 1])

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ), ∃ n. ¬ ( ⌜n⌝ + ⌜1⌝ < # ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ → ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ [n] ⊑ ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ [n + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ⌜nf⌝ [n] = nf(n); (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ n∞ < ∞ ↔ n∞ ≠ ∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ , ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: streamseq-mon-lim, name: mon-def, seq: ⊦ mon(σσ) ↔ (∀ n. ⌜n⌝ + ⌜1⌝ < # σσ → σσ[n] ⊑ σσ[n + 1]), subst:[σσ] → [⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα) ⌝ ⌝ ]; ['a] → ['pa], <all paths>

Additional information

Used terms: ⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα) ⌝ ⌝

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

⊦ mon(σσ) ↔ (∀ n. ⌜n⌝ + ⌜1⌝ < # σσ → σσ[n] ⊑ σσ[n + 1])

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ , ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ), ∀ n. ⌜n⌝ + ⌜1⌝ < # ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ → ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ [n] ⊑ ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ [n + 1] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ [n] = nf(n); (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ n∞ < ∞ ↔ n∞ ≠ ∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ , ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ n0. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0), paf.take(n0), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n0 + 1), paf.take(n0 + 1), ⌜as '⌝ + σ0, aα)⌝ ⊦

User applied rule all left

And obtained the following new goals

Rule argument

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

Additional information

Used terms: n

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, 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, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + σ0, 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 '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ , ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: to-succ-nf, seq: ⊦ nf.take(n + 1) = nf(0) + (λ n. nf(n + 1)).take(n), subst:[nf, n] → [pcsf, n]; ['a] → [('s, 'cs)], <all paths>

Additional information

Used terms: pcsf, n

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

⊦ nf.take(n + 1) = nf(0) + (λ n. nf(n + 1)).take(n)

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), 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, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + σ0, 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 '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf(0) + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ , ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), (s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, 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, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + σ0, 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 '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ )), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + σ0, aα)⌝ , ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + σ0, aα)⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: to-succ2-nf, seq: ⊦ nf.take(n + 2) = nf(0) + (λ n. nf(n + 1)).take(n + 1), subst:[nf, n] → [pcsf, n]; ['a] → [('s, 'cs)], <all paths>

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) c: ⊦ m + n = n + m; from specification nat-basic Used terms: pcsf, n

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

⊦ nf.take(n + 2) = nf(0) + (λ n. nf(n + 1)).take(n + 1)

A spec-lemma

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

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

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:repl-size, seq:⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ frag (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (pcσ, pα) ∧ frag(IOAut(astart, astep, aext))(aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, pcσ, pα)) ⊑ aext !(aα) → # repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα) = # repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα) + ⌜1⌝ , subst:[astep, cext, aα, start, pcσ, aσ, cren, ren, astart, cstart, cstep, aren, aext, step, ext, pα] → [astep, cext, aα, start, ⌜pcsf⌝ , 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 terms: aren, ren, cstep, cext, aext, aα, ext, ⌜pcsf⌝ , astart, cstart, start, cren, astep, ⌜paf⌝ , aσ, step

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

⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα) ∧ frag(IOAut(astart, astep, aext))(aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, pcσ, pα)) ⊑ aext !(aα) → # repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα) = # repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα) + ⌜1⌝

Lemma repl-size

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

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

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 12

Additional information

Used simplifier rules: ⊦ frag(ioa)(⌜sf⌝ , ⌜af⌝ ) ↔ fragω(ioa)(sf, af); (s) ⊦ σ ⊑ σ; (s)

Proof step 91 from proof repl-frag (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) = # repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf⌝ , ⌜paf⌝ , aσ, aα) + ⌜1⌝ , frag(IOAut(astart, astep, aext))(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), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ) ⊦

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, 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, aσ, step, pcsf

Proof step 92 from proof repl-frag (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 93 from proof repl-frag (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = # repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf⌝ , ⌜paf⌝ , aσ, aα) + ⌜1⌝ , frag(IOAut(astart, astep, aext))(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), 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), ¬ 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), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ) ⊦

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, 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 terms: aren, ren, cstep, cext, aext, aα, cren, paf, aσ, step, pcsf

Proof step 94 from proof repl-frag (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 95 from proof repl-frag (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) + ⌜1⌝ , frag(IOAut(astart, astep, aext))(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), 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), ¬ 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), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ σ ⊑ σ; (s)

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name: frag, seq:⊦ frag(IOAut(start, step, ext))(σ, α) ↔ # σ = # α + ⌜1⌝ ∧ (∀ n. ⌜n⌝ < # α → step(σ[n], α[n], σ[n +1])), subst:[σ, step, α, start, ext] → [lim (⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα) ⌝ ⌝ ), Π(step, astep, ren, aren, ext, aext), lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα) ⌝ ⌝ ), Π(start, astart), Π(ext, aext, ren, aren)]; ['s, 'a, 'e] → [('s, 'as), 'pa, '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), aσ, aα) ⌝ ⌝ ), Π(start, astart), lim (⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα) ⌝ ⌝ ), Π(step, astep, ren, aren, ext, aext)

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

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

A spec-lemma

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)

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

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

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 17 terms: n + 1 abort: F discard: T computed: F

Additional information

Used terms: n + 1

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) + ⌜1⌝ , ⌜n⌝ < # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1]), ¬ # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜n + 1⌝ → (∃ n0. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(n + 1)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), 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 < ⌜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 101 from proof repl-frag (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) + ⌜1⌝ , ⌜n⌝ < # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), frag(IOAut(astart, astep, aext))(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), 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), aσ, aα)⌝ ⌝ ).take(n + 1)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ , ¬ Π(step, astep, ren, aren, ext, aext) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1]), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), 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), aσ, aα) ⌝ ⌝ ) .take (n + 1 ) ⌝ , ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), 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), aσ, aα) ⌝ ⌝ ) .take (n + 1 ) ⌝ , ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) ⌝

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

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) + ⌜1⌝ , ⌜n⌝ < # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1]), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∃ σ1. ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(n + 1)⌝ + σ1 = ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: extensionality, seq: ⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0[n] = σ1[n]), subst:[σ0, σ1] → [ ⌜lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα) ⌝ ⌝ ) .take (n + 1 ) ⌝ + pα, ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), 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), aσ, aα) ⌝ ⌝ ) .take (n + 1 ) ⌝ + pα, ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) ⌝

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

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

A spec-lemma

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ [n] = x[n]; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (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 ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) ⊦ # ⌜σ.take(n)⌝ = ⌜# σ.take(n)⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

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

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

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 18 terms: n abort: F discard: T computed: F

Additional information

Used terms: n

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # pα = ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ , ⌜n⌝ < # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1]), ¬ ⌜n⌝ + # pα < ⌜n⌝ → (⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(n + 1)⌝ + pα)[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), 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∞ + n∞1 < n∞1; (s)

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

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

Additional information

Used simplifier rules: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]; (s) ⊦ ⌜n⌝ ≠ ∞; (s) n∞ ≠ ∞ ⊦ (⌜m⌝ + n∞).n = m + n∞.n; (s) ⊦ ⌜n⌝ .n = n; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # ⌜σ.take(n)⌝ = ⌜# σ.take(n)⌝ ; (s) ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ ⌜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⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ ¬ n < n; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf Used terms: ⌜lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα) ⌝ ⌝ ) .take (n + 1 ) ⌝ , pα, n

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

⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # pα = ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ , ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(n + 1)⌝ [n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], ⌜n⌝ < # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ⌜n⌝ < # ⌜lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(n + 1)⌝ , frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1]), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), 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 < n; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (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) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) ⊦ # ⌜σ.take(n)⌝ = ⌜# σ.take(n)⌝ ; (s) m < n, ¬ # σ < ⌜n⌝ ⊦ σ.take(n)[m] = σ[m]; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ⌜x⌝ [n] = x[n]; (s)

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

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

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:repl-fin-size, seq:⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ frag* (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, ⌜pcsl⌝ , ⌜pal⌝ )) ⊑ aext !(aα) → # repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα) = # repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα) + 1, subst:[astep, pal, cext, aα, start, aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsl, aext, step] → [astep, paf.take(n0), cext, aα, start, 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), aσ, step

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

⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, ⌜pcsl⌝ , ⌜pal⌝ )) ⊑ aext !(aα) → # repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα) = # repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα) + 1

Lemma repl-fin-size

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

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

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 16

Additional information

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

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

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 ⊦ fragω(IOAut(start, step, ext))(sf, af) → (frag*(IOAut(start, step, ext))(sf.take(n + 1), af.take(n)) ↔ true); (ss)

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: project-right, name: projaction-inf-def, seq:⊦ πra(lstep, rstep, lren, rren, ⌜psf⌝ , ⌜paf⌝ ) = lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), subst:[psf, lstep, paf, rren, lren, rstep] → [pcsf, step, paf, cren, ren, cstep]; ['ls, 'ra, 'la, 'pa, 'rs] → ['s, 'ca, 'a, 'pa, 'cs], <all paths>

Additional information

Used terms: cstep, pcsf, ren, cren, step, paf

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

⊦ πra(lstep, rstep, lren, rren, ⌜psf⌝ , ⌜paf⌝ ) = lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )

A spec-lemma

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

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

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. ⌜πra(step, cstep, ren, cren, pcsf.take(n + 1), paf.take(n))⌝ ⌝ ]; ['a] → ['ca]

Additional information

Used terms: ⌜λ n. ⌜πra(step, cstep, ren, cren, pcsf.take(n + 1), paf.take(n))⌝ ⌝

Proof step 119 from proof repl-frag (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 120 from proof repl-frag (previous step) (back to tree)

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

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

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

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 17

Additional information

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: project-right, name: mon-pia, seq:⊦ fragω(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren)) (psf, paf) → mon(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), subst:[psf, lstart, rstart, rext, lstep, paf, rren, lren, lext, rstep] → [pcsf, start, cstart, cext, step, paf, cren, ren, ext, cstep]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['s, 'ca, 'a, 'pa, 'cs, 'e], <all paths>

Additional information

Used simplifier rules: ⊦ Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren) = IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)); (s) Used terms: cext, cstart, cstep, pcsf, ren, ext, cren, step, start, paf

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

⊦ fragω(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren))(psf, paf) → mon(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )

A spec-lemma

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

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

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

cext !(lim(⌜λ n. ⌜πra(step, cstep, ren, cren, pcsf.take(n + 1), paf.take(n))⌝ ⌝ )) = aext !(aα), # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # pα = ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ , lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], ⌜n⌝ < # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), frag(IOAut(astart, astep, aext))(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), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜πra(step, cstep, ren, cren, pcsf.take(n0 + 1), paf.take(n0))⌝ ⊑ lim(⌜λ n. ⌜πra(step, cstep, ren, cren, pcsf.take(n + 1), paf.take(n))⌝ ⌝ ), ¬ Π(step, astep, ren, aren, ext, aext) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1]), ¬ cext !(πra(step, cstep, ren, cren, ⌜pcsf.take(n0 + 1)⌝ , ⌜paf.take(n0)⌝ )) ⊑ aext !(aα), ∀ m. ¬ # lim(⌜λ n. ⌜πra(step, cstep, ren, cren, pcsf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n1. ⌜lim(⌜λ n. ⌜πra(step, cstep, ren, cren, pcsf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πra(step, cstep, ren, cren, pcsf.take(n1 + 1), paf.take(n1))⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), 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] → [⌜πra(step, cstep, ren, cren, pcsf.take(n0 + 1), paf.take(n0))⌝ , lim(⌜λ n. ⌜πra(step, cstep, ren, cren, pcsf.take(n + 1), paf.take(n))⌝ ⌝ )]; ['a] → ['ca], <all paths>

Additional information

Used terms: ⌜πra(step, cstep, ren, cren, pcsf.take(n0 + 1), paf.take(n0))⌝ , lim(⌜λ n. ⌜πra(step, cstep, ren, cren, pcsf.take(n + 1), paf.take(n))⌝ ⌝ )

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

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

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

cext !(lim(⌜λ n. ⌜πra(step, cstep, ren, cren, pcsf.take(n + 1), paf.take(n))⌝ ⌝ )) = aext !(aα), # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # pα = ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ , lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], ⌜n⌝ < # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1]), ¬ cext !(πra(step, cstep, ren, cren, ⌜pcsf.take(n0 + 1)⌝ , ⌜paf.take(n0)⌝ )) ⊑ aext !(aα), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πra(step, cstep, ren, cren, pcsf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n1. ⌜lim(⌜λ n. ⌜πra(step, cstep, ren, cren, pcsf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πra(step, cstep, ren, cren, pcsf.take(n1 + 1), paf.take(n1))⌝ ), ∃ σ1. ⌜πra(step, cstep, ren, cren, pcsf.take(n0 + 1), paf.take(n0))⌝ + σ1 = lim(⌜λ n. ⌜πra(step, cstep, ren, cren, pcsf.take(n + 1), 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 ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s)

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

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

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, ⌜pcsf.take(n0 + 1)⌝ , ⌜paf.take(n0)⌝ )), aext !(aα)]; ['a] → ['e], <all paths>

Additional information

Used terms: cext !(πra(step, cstep, ren, cren, ⌜pcsf.take(n0 + 1)⌝ , ⌜paf.take(n0)⌝ )), aext !(aα)

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

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

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

# lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) + ⌜1⌝ , ⌜cext !(πra(step, cstep, ren, cren, pcsf.take(n0 + 1), paf.take(n0)))⌝ + cext !(cα) = aext !(aα), ⌜πra(step, cstep, ren, cren, pcsf.take(n0 + 1), paf.take(n0))⌝ + cα = lim(⌜λ n. ⌜πra(step, cstep, ren, cren, pcsf.take(n + 1), paf.take(n))⌝ ⌝ ), (⌜n⌝ + ⌜1⌝ ) + # pα = ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ , lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], ⌜n⌝ < # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), frag(IOAut(astart, astep, aext))(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), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1]), ∀ m. ¬ ⌜# πra(step, cstep, ren, cren, pcsf.take(n0 + 1), paf.take(n0))⌝ + # cα < ⌜m⌝ → (∃ n1. ⌜(⌜πra(step, cstep, ren, cren, pcsf.take(n0 + 1), paf.take(n0))⌝ + cα).take(m)⌝ ⊑ ⌜πra(step, cstep, ren, cren, pcsf.take(n1 + 1), paf.take(n1))⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ σ1. cext !(πra(step, cstep, ren, cren, ⌜pcsf.take(n0 + 1)⌝ , ⌜paf.take(n0)⌝ )) + σ1 ≠ aext !(aα) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 21 terms: cext !(cα) abort: F discard: T computed: F

Additional information

Used terms: cext !(cα)

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

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

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] → [paf.take(n0), step, ren, cstep, cren, pcsf.take(n0 + 1)]; ['ls, 'ra, 'la, 'pa, 'rs] → ['s, 'ca, 'a, 'pa, 'cs], <all paths>

Additional information

Used simplifier rules: ⊦ # nf.take(n) = n; (s) ⊦ m + n = m + n0 ↔ n = n0; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) Used terms: pcsf.take(n0 + 1), cstep, paf.take(n0), ren, cren, step

Proof step 132 from proof repl-frag (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 133 from proof repl-frag (previous step) (back to tree)

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

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 18 terms: n0 abort: F discard: T computed: F

Additional information

Used terms: n0

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) + 1, # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # pα = ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ , lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], ⌜n⌝ < # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ¬ Π(step, astep, ren, aren, ext, aext) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1]), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), 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] → [⌜repls (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) ⌝ , lim (⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα) ⌝ ⌝ )]; ['a] → [('s, 'as)], <all paths>

Additional information

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

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

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) + ⌜1⌝ , # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) + 1, (⌜n⌝ + ⌜1⌝ ) + # pα = ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ , lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], ⌜n⌝ < # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), frag(IOAut(astart, astep, aext))(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), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext) (lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1]), ∀ m. ¬ # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∃ σ1. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + σ1 = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), 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)

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

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

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: pα

Additional information

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

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) + 1, ⌜# repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + # paσ = # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) + ⌜1⌝ , ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), (⌜n⌝ + ⌜1⌝ ) + # pα = ⌜# repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ , lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], pα = ⌜pal⌝ , ⌜n⌝ < # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) ((⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n + 1]), ∀ m. ¬ ⌜# repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + # paσ < ⌜m⌝ → (∃ n. ⌜(⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), 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∞ + n∞1 < n∞1; (s) ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (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) ⊦ n < m → (⌜(* n)⌝ + n∞1 = ⌜(* m)⌝ + n∞2 ↔ n∞1 = ⌜m - n⌝ + n∞2); (s) ⊦ ⌜n1⌝ + ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n1 + n + m = n0; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# pal⌝ + # paσ, # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = 2 + n + # pal, n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα), ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], frag(IOAut(astart, astep, aext))(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), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext) ((⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n + 1]), ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ < ⌜m⌝ → (∃ n. ⌜(⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: extensionality, seq: ⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0[n] = σ1[n]), subst:[σ0, σ1] → [ ⌜repls (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) ⌝ + paσ, lim (⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα) ⌝ ⌝ )]; ['a] → [('s, 'as)], <all paths>

Additional information

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

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

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

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), #(⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ) = # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = 2 + n + # pal, # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# pal⌝ + # paσ, n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) ((⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n + 1]), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ < ⌜m⌝ → (∃ n. ⌜(⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n. ⌜n⌝ < #(⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ) → (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = 2 + n + # pal, # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# pal⌝ + # paσ, n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα), (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) ((⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n + 1]), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ < ⌜m⌝ → (∃ n. ⌜(⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n1. ⌜n1⌝ < (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ → (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n1] ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 19 terms: n + 1 abort: F discard: F computed: F

Additional information

Used terms: n + 1

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# pal⌝ + # paσ, # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = 2 + n + # pal, (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], frag(IOAut(astart, astep, aext))(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), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext) ((⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n + 1]), ⌜n + 1⌝ < (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ → (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n + 1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1], ∀ n1. ⌜n1⌝ < (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ → (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n1], ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ < ⌜m⌝ → (∃ n. ⌜(⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), 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∞ + n∞1 < n∞1; (s) ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) ⊦ m ≤ n → (⌜(* m)⌝ + n∞2 < ⌜(* n)⌝ + n∞1 ↔ n∞2 < ⌜n - m⌝ + n∞1); (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = 2 + n + # pal, # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# pal⌝ + # paσ, n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα), (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n + 1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1], frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) ((⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n + 1]), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ < ⌜m⌝ → (∃ n. ⌜(⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n1. ⌜n1⌝ < (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ → (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n1] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: get-append-ls, seq: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m], subst:[σ, σ1, m] → [⌜repls (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) ⌝ , paσ, n + 1]; ['a] → [('s, 'as)], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]; (s) ⊦ ⌜n⌝ .n = n; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) 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 ⊦ m + n - n + n1 = m - n1; (s) ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s) ⊦ m ≠ 0 → (m + n ≤ 1 ↔ m = 1 ∧ n = 0); (ws) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) ⌝ , paσ, n + 1

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

⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# pal⌝ + # paσ, # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = 2 + n + # pal, (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα), ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ [n + 1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1], lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], ⌜n + 1⌝ < # ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ , frag(IOAut(astart, astep, aext))(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), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext) ((⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ [n + 1]), ∀ n1. ⌜n1⌝ < (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ → (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n1], ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ < ⌜m⌝ → (∃ n. ⌜(⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic n < n0 ⊦ (* n) ≠ m0 + (* n0); (s) ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s) ⊦ m + n < m + n0 ↔ n < n0; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ ⌜x⌝ [n] = x[n]; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = 2 + n + # pal, # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# pal⌝ + # paσ, n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα), (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1], frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) ((⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1]), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ < ⌜m⌝ → (∃ n. ⌜(⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n1. ⌜n1⌝ < (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ → (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n1] ⊦

User applied rule all left

And obtained the following new goals

Rule argument

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

Additional information

Used terms: n

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# pal⌝ + # paσ, # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = 2 + n + # pal, (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα), repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1], lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], frag(IOAut(astart, astep, aext))(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), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext) ((⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1]), ⌜n⌝ < (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ → (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ < ⌜m⌝ → (∃ n. ⌜(⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), 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 → (⌜(* n)⌝ = ⌜(* m)⌝ + n∞2 ↔ false); (s) ⊦ ⌜n⌝ < n∞ + ⌜n⌝ ↔ n∞ ≠ ⌜0⌝ ; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = 2 + n + # pal, # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# pal⌝ + # paσ, n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα), (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1], (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) ((⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1]), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ < ⌜m⌝ → (∃ n. ⌜(⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: get-append-ls, seq: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m], subst:[σ, σ1, m] → [⌜repls (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) ⌝ , paσ, n]; ['a] → [('s, 'as)], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]; (s) ⊦ ⌜n⌝ .n = n; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m + m0 ≤ m ↔ m0 = 0; (s) ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) ⌝ , paσ, n

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

⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# pal⌝ + # paσ, # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = 2 + n + # pal, (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα), ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ [n] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1], lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], ⌜n⌝ < # ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ , frag(IOAut(astart, astep, aext))(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), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext) (⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ [n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1]), ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ < ⌜m⌝ → (∃ n. ⌜(⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic n < n0 ⊦ (* n) ≠ m0 + (* n0); (s) ⊦ m < m + n ↔ n ≠ 0; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ ⌜x⌝ [n] = x[n]; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = 2 + n + # pal, # lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# pal⌝ + # paσ, n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα), (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1]), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⊑ lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ < ⌜m⌝ → (∃ n. ⌜(⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)⌝ + paσ).take(m)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ) ⊦

User applied rule weakening

And obtained the following new goals

Rule argument

poslist: left 3, left 19, left 20

Additional information

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = 2 + n + # pal, (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα), repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1], lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], frag(IOAut(astart, astep, aext))(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), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1]) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:finfrag-repl-frag, seq:⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ oinj(cext) ∧ oinj(aext) ∧ oinj(ext) ∧ frag* (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(⌜πra(step, cstep, ren, cren, pcsl, pal)⌝ ) ⊑ aext !(aα) ∧ sameshared(cren, aren, ren) → frag*(Π(IOAut(start, step, ext), IOAut(astart, astep, aext), ren, aren)) (repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα)), subst:[astep, pal, cext, aα, as, start, aσ, ext, cren, ren, astart, cstart, cstep, aren, pcsl, aext, step] → [astep, paf.take(n0), cext, aα, hd(aσ), start, tl(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, hd(aσ), aext, aα, ext, astart, cstart, start, paf.take(n0), cren, astep, pcsf.take(n0 + 1), tl(aσ), step

Proof step 154 from proof repl-frag (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) ∧ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(⌜πra(step, cstep, ren, cren, pcsl, pal)⌝ ) ⊑ aext !(aα) ∧ sameshared(cren, aren, ren) → frag*(Π(IOAut(start, step, ext), IOAut(astart, astep, aext), ren, aren)) (repls(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, ⌜as '⌝ + aσ, aα))

Lemma finfrag-repl-frag

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = 2 + n + # pal, n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα), (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1]), nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ oinj(cext) ∧ oinj(aext) ∧ oinj(ext) ∧ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf.take(n0 + 1), paf.take(n0)) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ cext !(⌜πra(step, cstep, ren, cren, pcsf.take(n0 + 1), paf.take(n0))⌝ ) ⊑ aext !(aα) ∧ sameshared(cren, aren, ren) → frag*(Π(IOAut(start, step, ext), IOAut(astart, astep, aext), ren, aren)) (repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜hd(aσ) '⌝ + tl(aσ), aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜hd(aσ) '⌝ + tl(aσ), aα)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = 2 + n + # pal, (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα), repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1], lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], frag(IOAut(astart, astep, aext))(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), oinj(ext), oinj(aext), oinj(cext), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1]), frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, pcsf.take(n0 + 1), paf.take(n0)))⌝ ⊑ 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, pcsf.take(n0 + 1), paf.take(n0), ⌜hd(aσ) '⌝ + tl(aσ), aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜hd(aσ) '⌝ + tl(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] → [aσ, as, σ0]; ['a] → ['as]

Additional information

Used simplifier rules: ⊦ ¬ frag(IOAut(start, step, ext))([], α); (s) ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0); (elim) Used terms: aσ, as, σ0

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα) = 2 + n + # pal, n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα), (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ ), lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n + 1], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n] = lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα)⌝ ⌝ )[n], aσ = as + σ0, aσ ≠ [], frag(IOAut(astart, astep, aext))(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), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), aσ, aα)[n + 1]), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, pcsf.take(n0 + 1), paf.take(n0)))⌝ ⊑ 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, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)) ⊦

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 ⊦ 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 + σ = ⌜a '⌝ + σ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜a '⌝ + σ ≠ []; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα) = 2 + n + # pal, (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n1. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n1 + 1), paf.take(n1), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα), repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n] = lim(⌜λ n1. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n1 + 1), paf.take(n1), ⌜as '⌝ + σ0, aα)⌝ ⌝ )[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n + 1] = lim(⌜λ n1. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n1 + 1), paf.take(n1), ⌜as '⌝ + σ0, aα)⌝ ⌝ )[n + 1], lim(⌜λ n1. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n1 + 1), paf.take(n1), ⌜as '⌝ + σ0, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n], frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, 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.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), 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(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:fragstar, seq:⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1])), subst:[ioa, sl, al] → [IOAut (Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)), repls (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα), repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)]; ['s, 'a, 'e] → [('s, 'as), 'pa, 'e], <all paths>

Additional information

Used terms: IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)), repls (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα), repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)

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

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

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα) = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα) + 1, # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα) = 2 + n + # pal, n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα), (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n1. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n1 + 1), paf.take(n1), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n1. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n1 + 1), paf.take(n1), ⌜as '⌝ + σ0, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n + 1] = lim(⌜λ n1. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n1 + 1), paf.take(n1), ⌜as '⌝ + σ0, aα)⌝ ⌝ )[n + 1], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n] = lim(⌜λ n1. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n1 + 1), paf.take(n1), ⌜as '⌝ + σ0, aα)⌝ ⌝ )[n], frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n + 1]), ∀ n. n < # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα) → IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)).step (repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n +1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m + n = m + n0 ↔ n = n0; (s) ⊦ IOAut(start, step, ext).step = step; (s) ⊦ m +1 = m + 1; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) = aext !(aα), # repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα) = 2 + n + # pal, n + 1 + # pal = # repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα), (⌜2⌝ + ⌜n⌝ + ⌜# pal⌝ ) + # paσ = # lim(⌜λ n1. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n1 + 1), paf.take(n1), ⌜as '⌝ + σ0, aα)⌝ ⌝ ), lim(⌜λ n1. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n1 + 1), paf.take(n1), ⌜as '⌝ + σ0, aα)⌝ ⌝ )[n] = repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n + 1] = lim(⌜λ n1. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n1 + 1), paf.take(n1), ⌜as '⌝ + σ0, aα)⌝ ⌝ )[n + 1], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n] = lim(⌜λ n1. ⌜repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n1 + 1), paf.take(n1), ⌜as '⌝ + σ0, aα)⌝ ⌝ )[n], frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n + 1]), ∀ n1. ¬ n + # pal < n1 → Π(step, astep, ren, aren, ext, aext) (repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n1], repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n1], repls(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n0 + 1), paf.take(n0), ⌜as '⌝ + σ0, aα)[n1 + 1]) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 18 terms: n

Additional information

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

Proof Tree for repl-frag (View conclusion)

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

the proof tree