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

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

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

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

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: pcσ

Additional information

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

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

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 ⌜ ⌝, ⌜ ⌝;; ⊦ frag(ioa)(σ, ⌜af⌝ ) ↔ fragω(ioa)(σ.f, af) ∧ ¬ finp(σ); (s) ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss) ⊦ σ0 + ⌜x⌝ = ⌜nf0⌝ ↔ σ0 = ⌜nf0⌝ ; (s) ⊦ ⌜x⌝ ≠ ⌜nf⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: append-finfin, seq: ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ , subst:[x, y] → [(s, cs) ', pcsl]; ['a] → [('s, 'cs)], <rotated> <all paths>

Additional information

Used terms: (s, cs) ', pcsl

Proof step 7 from proof frag-repls-head (previous step) (back to tree)

⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝

A spec-lemma

Proof step 8 from proof frag-repls-head (previous step) (back to tree)

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

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, ⌜as '⌝ + aσ, cren, ren, cstep, aren, (s, cs) ' + 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: ⊦ #(a ' + x) = (# x)+1; (s) ⊦ m +1 = m + 1; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m + n = m + n0 ↔ n = n0; (s) ⊦ frag*(IOAut(start, step, ext))(s ' + sl, al) → (# sl = # al ↔ 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, (s, cs) ' + pcsl, ⌜as '⌝ + aσ, step

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

#((s, cs) ' + pcsl) = # pal + 1, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + pcsl, pal), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) ' + pcsl⌝ , ⌜pal⌝ )) ⊑ aext !(aα) ⊦ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα)⌝ ≠ [] ∧ hd(⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα)⌝ ) = (s, as)

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, (s, cs) ' + pcsl]; ['ls, 'ra, 'la, 'pa, 'rs] → ['s, 'ca, 'a, 'pa, 'cs], <all paths>

Additional information

Used terms: (s, cs) ' + pcsl, cstep, pal, ren, cren, step

Proof step 11 from proof frag-repls-head (previous step) (back to tree)

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

A spec-lemma

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

#((s, cs) ' + pcsl) = # pal + 1, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + pcsl, pal), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(⌜πra(step, cstep, ren, cren, (s, cs) ' + pcsl, pal)⌝ ) ⊑ aext !(aα) ⊦ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα)⌝ ≠ [] ∧ hd(⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα)⌝ ) = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

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

User applied rule apply lemma

And obtained the following new goals

Rule argument

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

Additional information

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

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

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

Lemma finfrag-repls-head

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

Heuristic elimination applied rule apply elim lemma

And closed the goal!

Rule argument

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

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ hd(⌜a '⌝ + ⌜x⌝ ) = a; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα), pas, x0

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

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 ⌜ ⌝, ⌜ ⌝;; ⊦ frag(ioa)(σ, ⌜al⌝ ) ↔ frag*(ioa)(σ.l, al) ∧ finp(σ); (s) ⊦ finp(σ) ↔ (∃ x. ⌜x⌝ = σ); (as) ⊦ ⌜x⌝ ≠ σ + ⌜nf0⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream

Proof step 19 from proof frag-repls-head (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (⌜x⌝ + ⌜nf⌝ ).f = x + nf; (s) ⊦ frag(ioa)(σ, ⌜af⌝ ) ↔ fragω(ioa)(σ.f, af) ∧ ¬ finp(σ); (s)

Proof step 20 from proof frag-repls-head (previous step) (back to tree)

⌜pcsf0⌝ = ⌜(s, cs) '⌝ + ⌜pcsf⌝ , frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf0⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf0⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf0⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: pcsf0 = (s, cs) ' + pcsf

Additional information

Used formulas: pcsf0 = (s, cs) ' + pcsf

Proof step 21 from proof frag-repls-head (previous step) (back to tree)

pcsf0 = (s, cs) ' + pcsf, ⌜pcsf0⌝ = ⌜(s, cs) '⌝ + ⌜pcsf⌝ , frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf0⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf0⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf0⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x + nf⌝ = ⌜x⌝ + ⌜nf⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # x = # y → (⌜x⌝ + σ = ⌜y⌝ + σ0 ↔ x = y ∧ σ = σ0); (s)

Proof step 22 from proof frag-repls-head (previous step) (back to tree)

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:inffrag-rec1, seq:⊦ fragω(IOAut(start, step, ext))(s ' + sf, af) ↔ step(s, af(0), sf(0)) ∧ fragω(IOAut(start, step, ext))(sf, λ n. af(n + 1)), subst:[step, s, start, sf, ext, af] → [Π(step, cstep, ren, cren, ext, cext), (s, cs), Π(start, cstart), pcsf, Π(ext, cext, ren, cren), paf]; ['s, 'a, 'e] → [('s, 'cs), 'pa, 'e], <all paths>

Additional information

Used terms: (s, cs), Π(ext, cext, ren, cren), Π(start, cstart), pcsf, paf, Π(step, cstep, ren, cren, ext, cext)

Proof step 23 from proof frag-repls-head (previous step) (back to tree)

⊦ fragω(IOAut(start, step, ext))(s ' + sf, af) ↔ step(s, af(0), sf(0)) ∧ fragω(IOAut(start, step, ext))(sf, λ n. af(n + 1))

A spec-lemma

Proof step 24 from proof frag-repls-head (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), paf(0), pcsf(0)), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), paf(0), pcsf(0)), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

User applied rule structural induction

And obtained the following new goals

Rule argument

term: pcsf(0)

Additional information

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

(s0, cs0) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), paf(0), pcsf(0)), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 27 from proof frag-repls-head (previous step) (back to tree)

(s0, cs0) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), paf(0), (s0, cs0)), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: pcsf = (s0, cs0) ' + (λ n. pcsf(n + 1))

Additional information

Used formulas: pcsf = (s0, cs0) ' + (λ n. pcsf(n + 1))

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

(s0, cs0) = pcsf(0), pcsf = (s0, cs0) ' + (λ n. pcsf(n + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), paf(0), (s0, cs0)), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 2, flags: discard

Additional information

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

(s0, cs0) = (((s0, cs0) ' + (λ n. pcsf(n + 1))))(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), paf(0), (s0, cs0)), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) ' + (λ n. pcsf(n + 1))⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) ' + (λ n. pcsf(n + 1))⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) ' + (λ n. pcsf(n + 1))⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

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 + nf⌝ = ⌜x⌝ + ⌜nf⌝ ; (s) ⊦ ((a ' + nf))(0) = a; (s)

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

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), paf(0), (s0, cs0)), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

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] → [paf]; ['a] → ['pa], <all paths>

Additional information

Used terms: paf

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

⊦ ⌜nf⌝ = nf(0) + ⌜λ n. nf(n + 1)⌝

A spec-lemma

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

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), paf(0), (s0, cs0)), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , paf(0) + ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , paf(0) + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , paf(0) + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

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 33 from proof frag-repls-head (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), paf(0), (s0, cs0)), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf(0) '⌝ + ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf(0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf(0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: product, name: lrstep-def, seq:⊦ Π(lstep, rstep, lren, rren, lext, rext)((ls, rs), pa, (ls0, rs0)) ↔ rs = rs0 ∧ (∃ la. lstep(ls, la, ls0) ∧ pa = lren(la) ∧ ¬ cod(rren)(pa)) ∨ ls = ls0 ∧ (∃ ra. rstep(rs, ra, rs0) ∧ pa = rren(ra) ∧ ¬ cod(lren)(pa)) ∨ (∃ la, ra. lstep(ls, la, ls0) ∧ lext(la) ≠ None ∧ pa = lren(la) ∧ rstep(rs, ra, rs0) ∧ rext(ra) ≠ None ∧ pa = rren(ra)), subst:[pa, ls0, rext, lstep, rs0, ls, rs, rren, lren, lext, rstep] → [paf(0), s0, cext, step, cs0, s, cs, cren, ren, ext, cstep]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['s, 'ca, 'a, 'pa, 'cs, 'e], <all paths>

Additional information

Used terms: cext, paf(0), cs, cstep, ren, ext, cren, cs0, step, s, s0

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

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

A spec-lemma

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

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf(0) '⌝ + ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), cs = cs0 ∧ (∃ la. step(s, la, s0) ∧ paf(0) = ren(la) ∧ ¬ cod(cren)(paf(0))) ∨ s = s0 ∧ (∃ ra. cstep(cs, ra, cs0) ∧ paf(0) = cren(ra) ∧ ¬ cod(ren)(paf(0))) ∨ (∃ la, ra. step(s, la, s0) ∧ ext(la) ≠ None ∧ paf(0) = ren(la) ∧ cstep(cs, ra, cs0) ∧ cext(ra) ≠ None ∧ paf(0) = cren(ra)) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf(0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf(0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf(0) '⌝ + ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), cs = cs0 ∧ step(s, a, s0) ∧ paf(0) = ren(a) ∧ ¬ cod(cren)(paf(0)) ∨ s = s0 ∧ cstep(cs, ca, cs0) ∧ paf(0) = cren(ca) ∧ ¬ cod(ren)(paf(0)) ∨ step(s, a0, s0) ∧ ext(a0) ≠ None ∧ paf(0) = ren(a0) ∧ cstep(cs, ca0, cs0) ∧ cext(ca0) ≠ None ∧ paf(0) = cren(ca0) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf(0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf(0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 6

Additional information

Used simplifier rules: repls-infrec-addS: ⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ step(s, a, s0) ∧ ¬ cod(cren)(ren(a)) ∧ cext ! (πra (step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ )) ⊑ aext !(aα) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ fragω (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((s0, cs) ' + pcsf, paf) → repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ , ⌜as '⌝ + aσ, aα) = ⌜(s, as) '⌝ + repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα); ⊦ ⌜a '⌝ + σ ≠ []; (s) ⊦ hd(⌜a '⌝ + σ) = a; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream

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

paf(0) = cren(ca), s = s0, cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf(0) '⌝ + ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), ¬ cod(ren)(paf(0)) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf(0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf(0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: repls-infrec-skip: ⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ cext ! (πra (step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ aext !(aα) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ fragω (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((s, cs0) ' + pcsf, paf) ∧ cstep(cs, ca, cs0) ∧ ¬ cod(ren)(cren(ca)) → repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , ⌜as '⌝ + aσ, aα) = repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα); append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream

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

paf(0) = cren(ca), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca) '⌝ + ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s0, as)

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: project-right, name:projaction-infrec-addR, seq:⊦ Πok(lext, rext, lren, rren) ∧ fragω (IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))) ((ls, rs0) ' + psf, paf) ∧ rstep(rs, ra, rs0) ∧ ¬ cod(lren)(rren(ra)) → πra (lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls, rs0) '⌝ + ⌜psf⌝ , rren(ra) + ⌜paf⌝ ) = ⌜ra '⌝ + πra(lstep, rstep, lren, rren, (ls, rs0) + ⌜psf⌝ , ⌜paf⌝ ), subst:[psf, lstart, rstart, rext, lstep, rs0, ra, ls, rren, lren, lext, rstep, rs, paf] → [λ n. pcsf(n + 1), start, cstart, cext, step, cs0, ca, s0, cren, ren, ext, cstep, cs, λ n. paf(n + 1)]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['s, 'ca, 'a, 'pa, 'cs, 'e]

Additional information

Used terms: cext, cs, cstart, cstep, λ n. pcsf(n + 1), ca, ren, ext, cren, cs0, step, start, λ n. paf(n + 1), s0

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

⊦ Πok(lext, rext, lren, rren) ∧ fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls, rs0) ' + psf, paf) ∧ rstep(rs, ra, rs0) ∧ ¬ cod(lren)(rren(ra)) → πra(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls, rs0) '⌝ + ⌜psf⌝ , rren(ra) + ⌜paf⌝ ) = ⌜ra '⌝ + πra(lstep, rstep, lren, rren, (ls, rs0) + ⌜psf⌝ , ⌜paf⌝ )

A spec-lemma

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

paf(0) = cren(ca), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca) '⌝ + ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), Πok(ext, cext, ren, cren) ∧ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)) ∧ cstep(cs, ca, cs0) ∧ ¬ cod(ren)(cren(ca)) → πra(step, cstep, ren, cren, ⌜(s0, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , cren(ca) + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca '⌝ + πra(step, cstep, ren, cren, (s0, cs0) + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s0, as)

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 41 from proof frag-repls-head (previous step) (back to tree)

paf(0) = cren(ca), πra(step, cstep, ren, cren, ⌜(s0, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca) '⌝ + ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s0, as)

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 2, flags: discard

Additional information

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

paf(0) = cren(ca), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(⌜ca '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s0, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: nounsharedextR-notcodren-internal: ⊦ ¬ cod(ren)(cren(ca)) ∧ nounsharedextR(cren, cext, ren) → (cext(ca) = None ↔ true); append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ [] + σ = σ; (s) ⊦ oe + β = (oe = None ⊃ [];⌜oe.get '⌝ ) + β; (s) ⊦ ext !(⌜a '⌝ + α) = ext(a) + ext !(α); (s)

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

paf(0) = cren(ca), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s0, as)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: replstate-inf, seq:⊦ cext !(πra(step, cstep, ren, cren, ⌜pcs '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → repls (step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , aσ, aα) = lim (⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, pcs + pcsf.take(n), paf.take(n), aσ, aα) ⌝ ⌝ ), subst:[cext, pcs, aσ, ren, cstep, aren, aext, step, aα, pcsf, cren, paf] → [cext, (s0, cs0), ⌜as '⌝ + aσ, ren, cstep, aren, aext, step, aα, λ n. pcsf(n + 1), cren, λ n. paf(n + 1)]; ['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, (s0, cs0), λ n. paf(n + 1), ⌜as '⌝ + aσ, step, λ n. pcsf(n + 1)

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

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

Lemma replstate-inf

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

paf(0) = cren(ca), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)) ⊦ lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ [] ∧ hd (lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = (s0, as)

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 46 from proof frag-repls-head (previous step) (back to tree)

paf(0) = cren(ca), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)) ⊦ lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ [] ∧ hd (lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = (s0, as)

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, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ]; ['a] → [('s, 'as)]

Additional information

Used terms: ⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝

Proof step 47 from proof frag-repls-head (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 48 from proof frag-repls-head (previous step) (back to tree)

paf(0) = cren(ca), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), mon (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ∧ ⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ≠ [] → (∀ n. ⌜n⌝ < # ⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ → ⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ [n ] ⊑ lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )) ∧ (∀ m. ¬ # lim (⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # ⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ∧ ⌜lim (⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ) .take (m ) ⌝ ⊑ ⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ [n ])) ⊦ lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ [] ∧ hd (lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = (s0, as)

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 49 from proof frag-repls-head (previous step) (back to tree)

paf(0) = cren(ca), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), mon (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) → (∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) ∧ (∀ m. ¬ # lim (⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim (⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ) .take (m ) ⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα) ⌝ )) ⊦ lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ [] ∧ hd (lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = (s0, as)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 9

Additional information

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

paf(0) = cren(ca), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), ¬ mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦ lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ [] ∧ hd (lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = (s0, as)

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, cs0, cext, as, start, aσ, ext, ren, astart, cstart, cstep, aren, aext, step, aα, s0, λ n. pcsf(n + 1), cren, λ n. paf(n + 1)]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used simplifier rules: ⊦ a + σ = ⌜a '⌝ + σ; (s) Used terms: aren, ren, cstep, cext, s0, as, aext, aα, ext, astart, cstart, start, cren, astep, λ n. paf(n + 1), cs0, aσ, step, λ n. pcsf(n + 1)

Proof step 51 from proof frag-repls-head (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 52 from proof frag-repls-head (previous step) (back to tree)

paf(0) = cren(ca), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ m. ¬ # lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ) .take (m ) ⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ) ⊦ lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ [] ∧ hd (lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = (s0, as)

User applied rule weakening

And obtained the following new goals

Rule argument

poslist: left 10

Additional information

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

paf(0) = cren(ca), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), ∀ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦ lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ [] ∧ hd (lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = (s0, as)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 9 terms: 0 abort: F discard: T computed: F

Additional information

Used terms: 0

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

paf(0) = cren(ca), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(0), (λ n. paf(n + 1)).take(0), ⌜as '⌝ + aσ, aα)⌝ ⊑ lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ¬ cod(ren)(cren(ca)) ⊦ lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ [] ∧ hd (lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = (s0, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: repls-base: ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ', [], ⌜as '⌝ + aσ, aα) = (s, as) '; append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ nf.take(0) = []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + [] = x; (s)

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

paf(0) = cren(ca), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), ⌜(s0, as) '⌝ ⊑ lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ¬ cod(ren)(cren(ca)) ⊦ lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ [] ∧ hd (lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )) = (s0, as)

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] → [⌜(s0, as) '⌝ , lim (⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )]; ['a] → [('s, 'as)], <all paths>

Additional information

Used simplifier rules: ⊦ ⌜a '⌝ + σ ≠ []; (s) ⊦ hd(⌜a '⌝ + σ) = a; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: ⌜(s0, as) '⌝ , lim (⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + (λ n. pcsf(n + 1)).take(n), (λ n. paf(n + 1)).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ )

Proof step 56 from proof frag-repls-head (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 57 from proof frag-repls-head (previous step) (back to tree)

paf(0) = ren(a0), paf(0) = cren(ca0), cext(ca0) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf(0) '⌝ + ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf(0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf(0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 58 from proof frag-repls-head (previous step) (back to tree)

paf(0) = cren(ca0), ren(a0) = cren(ca0), cext(ca0) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: project-right, name:projaction-infrec-addboth, seq:pa = rren(ra) ⊦ Πok(lext, rext, lren, rren) ∧ fragω (IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))) ((ls0, rs0) ' + psf, paf) ∧ lstep(ls, la, ls0) ∧ rstep(rs, ra, rs0) ∧ lren(la) = rren(ra) → πra (lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls0, rs0) '⌝ + ⌜psf⌝ , pa + ⌜paf⌝ ) = ⌜ra '⌝ + πra(lstep, rstep, lren, rren, ⌜(ls0, rs0) '⌝ + ⌜psf⌝ , ⌜paf⌝ ), subst:[psf, lstart, pa, ls0, rstart, rext, lstep, ra, ls, rren, rs0, la, lren, lext, rstep, rs, paf] → [λ n. pcsf(n + 1), start, ren(a0), s0, cstart, cext, step, ca0, s, cren, cs0, a0, ren, ext, cstep, cs, λ n. paf(n + 1)]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['s, 'ca, 'a, 'pa, 'cs, 'e]

Additional information

Used terms: cext, ren(a0), cs, cstart, cstep, λ n. pcsf(n + 1), ca0, ren, ext, cren, a0, cs0, step, start, λ n. paf(n + 1), s, s0

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

pa = rren(ra) ⊦ Πok(lext, rext, lren, rren) ∧ fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs0) ' + psf, paf) ∧ lstep(ls, la, ls0) ∧ rstep(rs, ra, rs0) ∧ lren(la) = rren(ra) → πra(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls0, rs0) '⌝ + ⌜psf⌝ , pa + ⌜paf⌝ ) = ⌜ra '⌝ + πra(lstep, rstep, lren, rren, ⌜(ls0, rs0) '⌝ + ⌜psf⌝ , ⌜paf⌝ )

A spec-lemma

Proof step 60 from proof frag-repls-head (previous step) (back to tree)

paf(0) = cren(ca0), ren(a0) = cren(ca0), cext(ca0) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα), Πok(ext, cext, ren, cren) ∧ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)) ∧ step(s, a0, s0) ∧ cstep(cs, ca0, cs0) ∧ ren(a0) = cren(ca0) → πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ren(a0) + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

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 61 from proof frag-repls-head (previous step) (back to tree)

paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), cext(ca0) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 3, flags:

Additional information

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

paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), cext(ca0) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), cext !(⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(aα) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

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 ! ( ⌜ca0 '⌝ + πra (step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )), aext !(aα)]; ['a] → ['e], <all paths>

Additional information

Used terms: cext ! ( ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )), aext !(aα)

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

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

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

paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), cext(ca0) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), ∃ σ1. cext !(⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + σ1 = aext !(aα) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ext(a) ≠ None → ext !(⌜a '⌝ + α) = ⌜ext(a).get '⌝ + ext !(α); (s)

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

(⌜cext(ca0).get '⌝ + cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ))) + β = aext !(aα), paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), cext(ca0) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

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

Additional information

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

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

(⌜e '⌝ + cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ))) + β = aext !(aα), cext(ca0) = Some(e), paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), cext(ca0) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

(⌜e '⌝ + cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ))) + β = aext !(aα), cext(ca0) = Some(e), paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name: remtau-split-one, seq:⊦ ext !(α) = ⌜e '⌝ + β ↔ (∃ al, a, α0. α = ⌜al⌝ + ⌜a '⌝ + α0 ∧ ext(a) = Some(e) ∧ ext !(⌜al⌝ ) = [] ∧ ext !(α0) = β), subst:[e, β, ext, α] → [e, cext ! (πra (step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + β, aext, aα]; ['e, 'a] → ['e, 'aa], <all paths>

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: e, cext ! (πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + β, aext, aα

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

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

A spec-lemma

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

cext(ca0) = Some(e), paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), ∃ al, aa, α0. aα = ⌜al⌝ + ⌜aa '⌝ + α0 ∧ aext(aa) = Some(e) ∧ aext !(⌜al⌝ ) = [] ∧ aext !(α0) = cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + β ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

aext !(aα0) = cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + β, aext !(aal) = [], aext(aa) = Some(e), cext(ca0) = Some(e), paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)) = (s, as)

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:repls-infrec-addboth, seq:⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ cext ! (πra (step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ )) ⊑ aext !(⌜aa '⌝ + aα) ∧ frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ∧ aext !(⌜aal⌝ ) = [] ∧ n = # aal ∧ fragω (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((s0, cs0) ' + pcsf, paf) ∧ step(s, a, s0) ∧ cstep(cs, ca, cs0) ∧ ren(a) = cren(ca) ∧ cext(ca) = aext(aa) → repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) = ⌜lift(aσ.take(n), s)⌝ + ⌜(s, aσ[n]) '⌝ + repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , aσ.drop(n + 1), aα), subst:[ca, a, astep, cs, aa, aσ, ext, ren, astart, cstart, n, cstep, aren, aext, step, cext, s0, cs0, aα, s, pcsf, start, aal, cren, paf] → [ca0, a0, astep, cs, aa, ⌜as '⌝ + aσ, ext, ren, astart, cstart, # aal, cstep, aren, aext, step, cext, s0, cs0, aα0, s, λ n. pcsf(n + 1), start, aal, cren, λ n. paf(n + 1)]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used terms: aren, ren, cstep, cext, aa, s, aext, aα0, ext, ca0, astart, a0, cstart, start, cren, aal, astep, cs0, s0, λ n. paf(n + 1), cs, # aal, ⌜as '⌝ + aσ, step, λ n. pcsf(n + 1)

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

⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ )) ⊑ aext !(⌜aa '⌝ + aα) ∧ frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ∧ aext !(⌜aal⌝ ) = [] ∧ n = # aal ∧ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf) ∧ step(s, a, s0) ∧ cstep(cs, ca, cs0) ∧ ren(a) = cren(ca) ∧ cext(ca) = aext(aa) → repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) = ⌜lift(aσ.take(n), s)⌝ + ⌜(s, aσ[n]) '⌝ + repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , aσ.drop(n + 1), aα)

Lemma repls-infrec-addboth

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

aext !(aal) = [], aext !(aα0) = cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + β, aext(aa) = Some(e), cext(ca0) = Some(e), paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜ren(a0) '⌝ + ⌜λ n. paf(n + 1)⌝ )) ⊑ aext !(⌜aa '⌝ + aα0) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0) ∧ aext !(⌜aal⌝ ) = [] ∧ # aal = # aal ∧ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)) ∧ step(s, a0, s0) ∧ cstep(cs, ca0, cs0) ∧ ren(a0) = cren(ca0) ∧ cext(ca0) = aext(aa) → repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜ren(a0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0) = ⌜lift((⌜as '⌝ + aσ).take(# aal), s)⌝ + ⌜(s, (⌜as '⌝ + aσ)[# aal]) '⌝ + repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , (⌜as '⌝ + aσ).drop(# aal + 1), aα0) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)) = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ n + 0 = n; (s) ⊦ n ≠ 0 → (⌜a '⌝ + σ).drop((* n) + m) = σ.drop((n - 1) + m); (s) ⊦ ⌜x⌝ = [] ↔ x = []; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ oe + β = (oe = None ⊃ [];⌜oe.get '⌝ ) + β; (s) ⊦ ext !(⌜a '⌝ + α) = ext(a) + ext !(α); (s)

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

aext !(aal) = [], aext !(aα0) = cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + β, aext(aa) = Some(e), cext(ca0) = Some(e), paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ )) ⊑ ⌜Some(e).get '⌝ + aext !(aα0) → repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0) = ⌜lift((⌜as '⌝ + aσ).take(# aal), s)⌝ + ⌜(s, (⌜as '⌝ + aσ)[# aal]) '⌝ + repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , aσ.drop(# aal), aα0) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)) = (s, as)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 16

Additional information

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

aext !(aα0) = cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + β, aext !(aal) = [], aext(aa) = Some(e), cext(ca0) = Some(e), paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), ¬ cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ )) ⊑ ⌜Some(e).get '⌝ + aext !(aα0) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)) = (s, as)

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, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ )), ⌜Some(e).get '⌝ + aext !(aα0)]; ['a] → ['e], <all paths>

Additional information

Used terms: cext ! (πra (step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ )), ⌜Some(e).get '⌝ + aext !(aα0)

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

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

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

aext !(aal) = [], aext !(aα0) = cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + β, aext(aa) = Some(e), cext(ca0) = Some(e), paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), ∀ σ1. cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ )) + σ1 ≠ ⌜Some(e).get '⌝ + aext !(aα0) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)) = (s, as)

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 7, flags:

Additional information

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

aext !(aα0) = cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + β, aext !(aal) = [], aext(aa) = Some(e), cext(ca0) = Some(e), paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), ∀ σ1. cext !(⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + σ1 ≠ ⌜Some(e).get '⌝ + aext !(aα0) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)) = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # x = # y → (⌜x⌝ + σ = ⌜y⌝ + σ0 ↔ x = y ∧ σ = σ0); (s) ⊦ oe + β = (oe = None ⊃ [];⌜oe.get '⌝ ) + β; (s) ⊦ ext !(⌜a '⌝ + α) = ext(a) + ext !(α); (s)

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

aext !(aα0) = cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + β, aext !(aal) = [], aext(aa) = Some(e), cext(ca0) = Some(e), paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren), ∀ σ1. cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + σ1 ≠ aext !(aα0) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)) = (s, as)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 16 terms: β

Additional information

Used terms: β

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

aext !(aα0) = cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + β, aext !(aal) = [], aext(aa) = Some(e), cext(ca0) = Some(e), paf(0) = cren(ca0), ren(a0) = cren(ca0), repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0) = ⌜lift((⌜as '⌝ + aσ).take(# aal), s)⌝ + ⌜(s, (⌜as '⌝ + aσ)[# aal]) '⌝ + repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , aσ.drop(# aal), aα0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ , ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)) = (s, as)

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 7, flags: discard

Additional information

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

aext !(aal) = [], aext !(aα0) = cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + β, aext(aa) = Some(e), cext(ca0) = Some(e), paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren) ⊦ ⌜lift((⌜as '⌝ + aσ).take(# aal), s)⌝ + ⌜(s, (⌜as '⌝ + aσ)[# aal]) '⌝ + repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , aσ.drop(# aal), aα0) ≠ [] ∧ hd ( ⌜lift((⌜as '⌝ + aσ).take(# aal), s)⌝ + ⌜(s, (⌜as '⌝ + aσ)[# aal]) '⌝ + repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , aσ.drop(# aal), aα0)) = (s, as)

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 ⊦ ⌜a '⌝ + σ ≠ []; (s) ⊦ ⌜x⌝ = [] ↔ x = []; (s) ⊦ σ0 + σ1 = [] ↔ σ0 = [] ∧ σ1 = []; (ws)

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

aext !(aal) = [], aext !(aα0) = cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + β, aext(aa) = Some(e), cext(ca0) = Some(e), paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), ⌜lift((⌜as '⌝ + aσ).take(# aal), s)⌝ + ⌜(s, (⌜as '⌝ + aσ)[# aal]) '⌝ + repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , aσ.drop(# aal), aα0) ≠ [], Some(e) ≠ None, ext(a0) ≠ None, hd ( ⌜lift((⌜as '⌝ + aσ).take(# aal), s)⌝ + ⌜(s, (⌜as '⌝ + aσ)[# aal]) '⌝ + repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , aσ.drop(# aal), aα0)) ≠ (s, as), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren) ⊦

Heuristic elimination 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] → [ ⌜lift((⌜as '⌝ + aσ).take(# aal), s)⌝ + ⌜(s, (⌜as '⌝ + aσ)[# aal]) '⌝ + repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , aσ.drop(# aal), aα0), pas, σ0]; ['a] → [('s, 'as)]

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0); (elim) Used terms: ⌜lift((⌜as '⌝ + aσ).take(# aal), s)⌝ + ⌜(s, (⌜as '⌝ + aσ)[# aal]) '⌝ + repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , aσ.drop(# aal), aα0), pas, σ0

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

aext !(aα0) = cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + β, aext !(aal) = [], ⌜lift((⌜as '⌝ + aσ).take(# aal), s)⌝ + ⌜(s, (⌜as '⌝ + aσ)[# aal]) '⌝ + repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , aσ.drop(# aal), aα0) = pas + σ0, aext(aa) = Some(e), cext(ca0) = Some(e), paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), ⌜lift((⌜as '⌝ + aσ).take(# aal), s)⌝ + ⌜(s, (⌜as '⌝ + aσ)[# aal]) '⌝ + repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , aσ.drop(# aal), aα0) ≠ [], Some(e) ≠ None, ext(a0) ≠ None, pas ≠ (s, as), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

aext !(aα0) = cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ )) + β, aext !(aal) = [], ⌜lift((⌜as '⌝ + aσ).take(# aal), s)⌝ + ⌜(s, (⌜as '⌝ + aσ)[# aal]) '⌝ + repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ , aσ.drop(# aal), aα0) = ⌜pas '⌝ + σ0, aext(aa) = Some(e), cext(ca0) = Some(e), paf(0) = cren(ca0), ren(a0) = cren(ca0), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜cren(ca0) '⌝ + ⌜λ n. paf(n + 1)⌝ ) = ⌜ca0 '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜λ n. paf(n + 1)⌝ ), Some(e) ≠ None, ext(a0) ≠ None, pas ≠ (s, as), cstep(cs, ca0, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + (λ n. pcsf(n + 1)), λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), step(s, a0, s0), Πok(ext, cext, ren, cren) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: cons-to-num, seq: ⊦ n ≠ 0 → (⌜a '⌝ + σ).take((* n)) = a ' + σ.take(n - 1), subst:[a, σ, n] → [as, aσ, # aal]; ['a] → ['as], <all paths>

Additional information

Used simplifier rules: lift-base: ⊦ lift([], s) = []; lift-rec: ⊦ lift(as ' + asl, s) = (s, as) ' + lift(asl, s); ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ⌜a '⌝ + σ = ⌜a0 '⌝ + σ0 ↔ a = a0 ∧ σ = σ0; (ws) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # x = 0 ↔ x = []; (s) ⊦ σ.take(0) = []; (s) ⊦ ⌜[]⌝ = []; (s) ⊦ (⌜a '⌝ + σ)[0] = a; (s) ⊦ # [] = 0; (s) ⊦ σ.drop(0) = σ; (s) ⊦ [] + σ = σ; (s) Used terms: as, aσ, # aal

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

⊦ n ≠ 0 → (⌜a '⌝ + σ).take((* n)) = a ' + σ.take(n - 1)

A spec-lemma

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

(s0, cs0) = pcsf(0), pcsf ≠ (s0, cs0) ' + (λ n. pcsf(n + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), paf(0), (s0, cs0)), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

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] → [(s0, cs0) ', λ n. pcsf(n + 1)]; ['a] → [('s, 'cs)], <all paths>

Additional information

Used terms: (s0, cs0) ', λ n. pcsf(n + 1)

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

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

A spec-lemma

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

(s0, cs0) = pcsf(0), pcsf ≠ (λ n. (n < # (s0, cs0) ' ⊃ (s0, cs0) '[n];pcsf((n - # (s0, cs0) ') + 1))), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), paf(0), (s0, cs0)), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

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 88 from proof frag-repls-head (previous step) (back to tree)

(s0, cs0) = pcsf(0), pcsf(n) ≠ (n = 0 ⊃ (s0, cs0);pcsf((n - 1) + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), paf(0), (s0, cs0)), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 2

Additional information

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

(s0, cs0) = pcsf(0), n ≠ 0, pcsf(n) ≠ pcsf((n - 1) + 1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, λ n. paf(n + 1)), nounsharedextR(cren, cext, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), paf(0), (s0, cs0)), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

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, n0]

Additional information

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

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

⌜pcsf0⌝ = ⌜(s, cs) '⌝ + ⌜pcsf⌝ , pcsf0 ≠ (s, cs) ' + pcsf, frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf0⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf0⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf0⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

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) ', pcsf]; ['a] → [('s, 'cs)], <all paths>

Additional information

Used terms: (s, cs) ', pcsf

Proof step 91 from proof frag-repls-head (previous step) (back to tree)

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

A spec-lemma

Proof step 92 from proof frag-repls-head (previous step) (back to tree)

⌜pcsf0⌝ = ⌜(s, cs) '⌝ + ⌜pcsf⌝ , pcsf0 ≠ (λ n. (n < # (s, cs) ' ⊃ (s, cs) '[n];pcsf(n - # (s, cs) '))), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (λ n. (n < # (s, cs) ' ⊃ (s, cs) '[n];pcsf(n - # (s, cs) ')), paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf0⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf0⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf0⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

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 93 from proof frag-repls-head (previous step) (back to tree)

⌜pcsf0⌝ = ⌜(s, cs) '⌝ + ⌜pcsf⌝ , pcsf0(n) ≠ (n = 0 ⊃ (s, cs);pcsf(n - 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(λ n0. (n0 = 0 ⊃ (s, cs);pcsf(n0 - 1)), paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf0⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf0⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf0⌝ , ⌜paf⌝ , ⌜as '⌝ + aσ, aα)) = (s, as)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:append-fininf, seq: ⊦ ⌜x⌝ + ⌜nf⌝ = ⌜λ n. (n < # x ⊃ x[n];nf(n - # x))⌝ , subst:[x, nf] → [(s, cs) ', pcsf]; ['a] → [('s, 'cs)], <all paths>

Additional information

Used simplifier rules: ⊦ m < 1 ↔ m = 0; (s) ⊦ a '[0] = a; (s) ⊦ # a ' = 0 +1; (s) ⊦ ⌜nf⌝ = ⌜nf0⌝ ↔ nf = nf0; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: (s, cs) ', pcsf

Proof step 94 from proof frag-repls-head (previous step) (back to tree)

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

A spec-lemma

Proof Tree for frag-repls-head (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