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

⊦ 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)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

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), 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), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(cren)(ren(a)) ⊦

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, cs), ⌜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, (s0, cs), paf, ⌜as '⌝ + aσ, step, pcsf

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

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), 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), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ¬ cext !(πra(step, cstep, ren, cren, ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

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

Additional information

Used simplifier rules: ⊦ a + σ = ⌜a '⌝ + σ; (s) Used terms: cext, cs, cstart, cstep, pcsf, ren, ext, cren, a, step, start, paf, s, s0

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

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

A spec-lemma

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

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

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

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

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) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ ).f, cren, (⌜ren(a) '⌝ + ⌜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, (⌜ren(a) '⌝ + ⌜paf⌝ ).f, ⌜as '⌝ + aσ, step, (⌜(s, cs) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ ).f

Proof step 8 from proof repls-infrec-addS (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 9 from proof repls-infrec-addS (previous step) (back to tree)

repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ , ⌜as '⌝ + aσ, aα) ≠ ⌜(s, as) '⌝ + lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(cren)(ren(a)), cext !(πra(step, cstep, ren, cren, ⌜(⌜(s, cs) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ ).f⌝ , ⌜(⌜ren(a) '⌝ + ⌜paf⌝ ).f⌝ )) ⊑ aext !(aα) → repls(step, cstep, cren, ren, aren, cext, aext, ⌜(⌜(s, cs) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ ).f⌝ , ⌜(⌜ren(a) '⌝ + ⌜paf⌝ ).f⌝ , ⌜as '⌝ + aσ, aα) = lim (⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (⌜(s, cs) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ ).f.take(n + 1), (⌜ren(a) '⌝ + ⌜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 10 from proof repls-infrec-addS (previous step) (back to tree)

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

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags: discard

Additional information

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

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

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) ' + ((s0, cs) ' + pcsf).take(n), (ren(a) ' + paf).take(n), ⌜as '⌝ + aσ, aα) ⌝ , 1]; ['a] → [('s, 'as)], <all paths>

Additional information

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

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

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

A spec-lemma

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: streamseq-mon-lim, name: mon-def, seq: ⊦ mon(σσ) ↔ (∀ n. ⌜n⌝ + ⌜1⌝ < # σσ → σσ[n] ⊑ σσ[n + 1]), subst:[σσ] → [⌜λ n. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs) ' + pcsf).take(n), (ren(a) ' + paf).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ]; ['a] → [('s, 'as)], <all paths>

Additional information

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

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

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

A spec-lemma

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: repls-rec-addS: ⊦ step(s, a, s0) ∧ ¬ cod(cren)(ren(a)) → repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs) ' + pcsl, ren(a) ' + pal, ⌜as '⌝ + aσ, aα) = (s, as) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + nf).take(n + 1) = a ' + nf.take(n); (s) ⊦ ⌜nf⌝ [n] = nf(n); (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ n∞ < ∞ ↔ n∞ ≠ ∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:infapp-to-ge, seq: ⊦ # x ≤ n → (x + nf).take(n) = x + nf.take(n - # x), subst:[x, nf, n] → [(s0, cs) ', pcsf, n]; ['a] → [('s, 'cs)], <all paths>

Additional information

Used simplifier rules: repls-base: ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ', [], ⌜as '⌝ + aσ, aα) = (s, as) '; ⊦ # a ' = 0 +1; (s) ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ m < 1 ↔ m = 0; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ nf.take(0) = []; (s) ⊦ σ1 ⊑ σ1 + σ2; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: (s0, cs) ', pcsf, n

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

⊦ # x ≤ n → (x + nf).take(n) = x + nf.take(n - # x)

A spec-lemma

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

lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs) ' + pcsf).take(n), (ren(a) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ ⌜(s, as) '⌝ + lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), # (s0, cs) ' ≤ n, cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ¬ ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs) ' + pcsf.take(n - # (s0, cs) '), (ren(a) ' + paf).take(n), ⌜as '⌝ + aσ, aα) ⌝ ⊑ ⌜(s, as) '⌝ + ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # a ' = 0 +1; (s)

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

lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs) ' + pcsf).take(n), (ren(a) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ ⌜(s, as) '⌝ + lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), n ≠ 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, cs) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), 1 ≤ n, cext !(πra(step, cstep, ren, cren, ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ¬ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs) ' + pcsf.take(n - 1), (ren(a) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜(s, as) '⌝ + ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

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

n = n0 + 1, lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs) ' + pcsf).take(n), (ren(a) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ ⌜(s, as) '⌝ + lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), n ≠ 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, cs) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), 1 ≤ n, cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ¬ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs) ' + pcsf.take(n0), (ren(a) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜(s, as) '⌝ + ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: repls-rec-addS: ⊦ step(s, a, s0) ∧ ¬ cod(cren)(ren(a)) → repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs) ' + pcsl, ren(a) ' + pal, ⌜as '⌝ + aσ, aα) = (s, as) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα); append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜x⌝ + σ1 ⊑ ⌜x⌝ + σ2 ↔ σ1 ⊑ σ2; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + nf).take(n + 1) = a ' + nf.take(n); (s) ⊦ m ≤ m + n; (s) ⊦ m + 1 ≠ 0; (s)

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

lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs) ' + pcsf).take(n), (ren(a) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ ⌜(s, as) '⌝ + lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ¬ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsf.take(n0), paf.take(n0), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsf.take(n0 + 1), paf.take(n0 + 1), ⌜as '⌝ + aσ, aα)⌝ ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:repls-mon, 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) → ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⊑ ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) + pcsf.take(n + 1), paf.take(n + 1), ⌜as '⌝ + aσ, aα) ⌝ , subst:[astep, cs, cext, as, start, aσ, ext, ren, astart, cstart, n, cstep, aren, aext, step, aα, s, pcsf, cren, paf] → [astep, cs, cext, as, start, aσ, ext, ren, astart, cstart, n0, cstep, aren, aext, step, aα, s0, pcsf, cren, paf]; ['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) ⊦ a + x = a ' + x; (s) Used terms: aren, ren, cstep, cext, s0, as, aext, aα, ext, astart, cstart, start, cren, astep, paf, cs, n0, aσ, step, pcsf

Proof step 22 from proof repls-infrec-addS (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) → ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) + pcsf.take(n + 1), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝

Lemma repls-mon

Proof step 23 from proof repls-infrec-addS (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: repls-rec-addS: ⊦ step(s, a, s0) ∧ ¬ cod(cren)(ren(a)) → repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs) ' + pcsl, ren(a) ' + pal, ⌜as '⌝ + aσ, aα) = (s, as) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + nf).take(n + 1) = a ' + nf.take(n); (s)

Proof step 24 from proof repls-infrec-addS (previous step) (back to tree)

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: streamseq-mon-lim, name: lim-extractprefix, seq: ⊦ mon(⌜σf⌝ ) → lim(⌜λ n. ⌜x⌝ + σf(n)⌝ ) = ⌜x⌝ + lim(⌜σf⌝ ), subst:[x, σf] → [(s, as) ', λ n0. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsf.take(n0), paf.take(n0), ⌜as '⌝ + aσ, aα) ⌝ ]; ['a] → [('s, 'as)], <all paths>

Additional information

Used terms: (s, as) ', λ n0. ⌜repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsf.take(n0), paf.take(n0), ⌜as '⌝ + aσ, aα) ⌝

Proof step 25 from proof repls-infrec-addS (previous step) (back to tree)

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

A spec-lemma

Proof step 26 from proof repls-infrec-addS (previous step) (back to tree)

lim(⌜λ n. ⌜(s, as) '⌝ + ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ≠ ⌜(s, as) '⌝ + lim(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs) ' + pcsf, paf), mon(⌜λ n. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs) ' + pcsf).take(n), (ren(a) ' + paf).take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ )) ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ¬ mon(⌜λ n0. ⌜repls(step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsf.take(n0), paf.take(n0), ⌜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, s0, pcsf, cren, paf]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as], <all paths>

Additional information

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

Proof step 27 from proof repls-infrec-addS (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 Tree for repls-infrec-addS (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