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Proof step 1 from proof repls-infrec-skip (conclusion) (back to tree)

⊦ 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)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof repls-infrec-skip (previous step) (back to tree)

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), 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)))((s, cs0) ' + pcsf, paf), 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), ¬ cod(ren)(cren(ca)) ⊦

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, (s, cs0), ⌜as '⌝ + aσ, ren, cstep, aren, aext, step, aα, pcsf, cren, paf]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as], <all paths>

Additional information

Used terms: aren, ren, cstep, cext, aext, aα, cren, (s, cs0), paf, ⌜as '⌝ + aσ, step, pcsf

Proof step 3 from proof repls-infrec-skip (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 4 from proof repls-infrec-skip (previous step) (back to tree)

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), 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)))((s, cs0) ' + pcsf, paf), 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), ¬ cod(ren)(cren(ca)), ¬ cext !(πra(step, cstep, ren, cren, ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦

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] → [pcsf, start, cstart, cext, step, cs0, ca, s, cren, ren, ext, cstep, cs, paf]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['s, 'ca, 'a, 'pa, 'cs, 'e]

Additional information

Used terms: cext, cs, cstart, cstep, pcsf, ca, ren, ext, cren, cs0, step, start, paf, s

Proof step 5 from proof repls-infrec-skip (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 6 from proof repls-infrec-skip (previous step) (back to tree)

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), 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)))((s, cs0) ' + pcsf, paf), 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), ¬ cod(ren)(cren(ca)), ¬ cext !(πra(step, cstep, ren, cren, ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), Πok(ext, cext, ren, cren) ∧ 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)) → πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , cren(ca) + ⌜paf⌝ ) = ⌜ca '⌝ + πra(step, cstep, ren, cren, (s, cs0) + ⌜pcsf⌝ , ⌜paf⌝ ) ⊦

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 7 from proof repls-infrec-skip (previous step) (back to tree)

πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ ) = ⌜ca '⌝ + πra(step, cstep, ren, cren, ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ ), 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), 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)))((s, cs0) ' + pcsf, paf), 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), ¬ cod(ren)(cren(ca)), ¬ cext !(πra(step, cstep, ren, cren, ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦

User applied rule subst equation

And closed the goal!

Rule argument

pos: left 1, flags: discard

Additional information

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

Proof step 8 from proof repls-infrec-skip (previous step) (back to tree)

repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs0) + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), 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)))((s, cs0) ' + pcsf, paf), 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), cext !(πra(step, cstep, ren, cren, ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)) ⊦

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 9 from proof repls-infrec-skip (previous step) (back to tree)

repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs0) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), 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)))((s, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

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

Additional information

Used terms: aren, ren, cstep, cext, aext, aα, cren, (⌜cren(ca) '⌝ + ⌜paf⌝ ).f, ⌜as '⌝ + aσ, step, (⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ ).f

Proof step 10 from proof repls-infrec-skip (previous step) (back to tree)

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

Lemma replstate-inf-def

Proof step 11 from proof repls-infrec-skip (previous step) (back to tree)

repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs0) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), 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)))((s, cs0) ' + pcsf, paf), 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), cext !(πra(step, cstep, ren, cren, ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), cext !(πra(step, cstep, ren, cren, ⌜(⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ ).f⌝ , ⌜(⌜cren(ca) '⌝ + ⌜paf⌝ ).f⌝ )) ⊑ aext !(aα) → repls(step, cstep, cren, ren, aren, cext, aext, ⌜(⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ ).f⌝ , ⌜(⌜cren(ca) '⌝ + ⌜paf⌝ ).f⌝ , ⌜as '⌝ + aσ, aα) = lim (⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ ).f.take(n + 1), (⌜cren(ca) '⌝ + ⌜paf⌝ ).f.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (⌜x⌝ + ⌜nf⌝ ).f = x + nf; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (⌜a '⌝ + ⌜x⌝ + ⌜nf⌝ ).f.take(n + 1) = a ' + (⌜x⌝ + ⌜nf⌝ ).f.take(n); (s) ⊦ ⌜x + nf⌝ = ⌜x⌝ + ⌜nf⌝ ; (s) ⊦ ⌜(⌜x⌝ + ⌜y⌝ + ⌜nf⌝ ).f⌝ = ⌜x⌝ + ⌜y⌝ + ⌜nf⌝ ; (s)

Proof step 12 from proof repls-infrec-skip (previous step) (back to tree)

repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , ⌜as '⌝ + aσ, aα) = lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs0) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), 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)))((s, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags: discard

Additional information

Proof step 13 from proof repls-infrec-skip (previous step) (back to tree)

lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs0) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), 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)))((s, cs0) ' + pcsf, paf), 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), cext !(πra(step, cstep, ren, cren, ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: streamseq-mon-lim, name:lim-postfix, seq: ⊦ mon(⌜σf⌝ ) → lim(⌜σf⌝ ) = lim(⌜λ n. σf(n + m)⌝ ), subst:[σf, m] → [λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + aσ, aα) ⌝ , 1]; ['a] → [('s, 'as)], <all paths>

Additional information

Used simplifier rules: repls-rec-skip: ⊦ cstep(cs, ca, cs0) ∧ ¬ cod(ren)(cren(ca)) → repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s, cs0) ' + pcsl, cren(ca) ' + pal, ⌜as '⌝ + aσ, aα) = repls (step, cstep, cren, ren, aren, cext, aext, (s, cs0) ' + pcsl, pal, ⌜as '⌝ + aσ, aα); ⊦ (a ' + nf).take(n + 1) = a ' + nf.take(n); (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + aσ, aα) ⌝ , 1

Proof step 14 from proof repls-infrec-skip (previous step) (back to tree)

⊦ mon(⌜σf⌝ ) → lim(⌜σf⌝ ) = lim(⌜λ n. σf(n + m)⌝ )

A spec-lemma

Proof step 15 from proof repls-infrec-skip (previous step) (back to tree)

lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs0) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), 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)))((s, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), ¬ mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply rewrite 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, cs, cext, as, start, aσ, ext, ren, astart, cstart, cstep, aren, aext, step, aα, s, (s, cs0) ' + pcsf, cren, cren(ca) ' + paf]; ['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 + y) + z = x + y + z; from specification list 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, cren, astep, cren(ca) ' + paf, cs, aσ, step, (s, cs0) ' + pcsf

Proof step 16 from proof repls-infrec-skip (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 17 from proof repls-infrec-skip (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 18 from proof repls-infrec-skip (previous step) (back to tree)

lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs0) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), 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)))((s, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), ¬ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (s, cs0) ' + pcsf, cren(ca) ' + paf), ¬ mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:inffrag-rec, seq:⊦ fragω(IOAut(start, step, ext))(s ' + sf, a ' + af) ↔ step(s, a, sf(0)) ∧ fragω(IOAut(start, step, ext))(sf, af), subst:[a, step, s, start, sf, ext, af] → [cren(ca), Π(step, cstep, ren, cren, ext, cext), (s, cs), Π(start, cstart), (s, cs0) ' + 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), cren(ca), Π(start, cstart), (s, cs0) ' + pcsf, paf, Π(step, cstep, ren, cren, ext, cext)

Proof step 19 from proof repls-infrec-skip (previous step) (back to tree)

⊦ fragω(IOAut(start, step, ext))(s ' + sf, a ' + af) ↔ step(s, a, sf(0)) ∧ fragω(IOAut(start, step, ext))(sf, af)

A spec-lemma

Proof step 20 from proof repls-infrec-skip (previous step) (back to tree)

lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs0) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), 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)))((s, cs0) ' + pcsf, paf), 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), cext !(πra(step, cstep, ren, cren, ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), ¬ mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ¬ Π(step, cstep, ren, cren, ext, cext)((s, cs), cren(ca), (((s, cs0) ' + pcsf))(0)) ∨ ¬ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs0) ' + pcsf, paf) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ((a ' + nf))(0) = a; (s)

Proof step 21 from proof repls-infrec-skip (previous step) (back to tree)

lim (⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs0) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), 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)))((s, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), ¬ mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ¬ Π(step, cstep, ren, cren, ext, cext)((s, cs), cren(ca), (s, cs0)) ⊦

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] → [cren(ca), s, 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 simplifier rules: ⊦ Πok(lext, rext, lren, rren) → (rren(ra1) = rren(ra2) ↔ ra1 = ra2); (s) Used terms: cext, cren(ca), cs, cstep, ren, ext, cren, cs0, step, s, s

Proof step 22 from proof repls-infrec-skip (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 Tree for repls-infrec-skip (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