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Proof step 1 from proof repla-infrec-addboth (conclusion) (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) → repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) = ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , aσ.drop(n + 1), aα)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(⌜aal⌝ ) = [], cext(ca) = aext(aa), n = # aal, ren(a) = cren(ca), repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , aσ.drop(n + 1), aα), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜ren(a) '⌝ + ⌜paf⌝ )) ⊑ aext !(⌜aa '⌝ + 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 ⊦ oe + β = (oe = None ⊃ [];⌜oe.get '⌝ ) + β; (s) ⊦ ext !(⌜a '⌝ + α) = ext(a) + ext !(α); (s) ⊦ ⌜x⌝ = [] ↔ x = []; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 3 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , aσ.drop(# aal + 1), aα), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: replaction-inf, seq:⊦ cext !(πra(step, cstep, ren, cren, ⌜pcs '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → repla (step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , aσ, aα) = lim (⌜λ n. ⌜repla (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), aσ.drop(# aal + 1), 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, cs0), paf, aσ.drop(# aal + 1), step, pcsf

Proof step 4 from proof repla-infrec-addboth (previous step) (back to tree)

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

Lemma replaction-inf

Proof step 5 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , aσ.drop(# aal + 1), aα), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ¬ cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦

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] → [pcsf, start, ren(a), s0, cstart, cext, step, ca, s, cren, cs0, a, ren, ext, cstep, cs, paf]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['s, 'ca, 'a, 'pa, 'cs, 'e]

Additional information

Used terms: cext, ren(a), cs, cstart, cstep, pcsf, ca, ren, ext, cren, a, cs0, step, start, paf, s, s0

Proof step 6 from proof repla-infrec-addboth (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 7 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , aσ.drop(# aal + 1), aα), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ¬ cext !(πra(step, cstep, ren, cren, ⌜(s0, 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)))((s0, cs0) ' + pcsf, paf) ∧ step(s, a, s0) ∧ cstep(cs, ca, cs0) ∧ ren(a) = cren(ca) → πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ren(a) + ⌜paf⌝ ) = ⌜ca '⌝ + πra(step, cstep, ren, cren, ⌜(s0, 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 8 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ ) = ⌜ca '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ ), repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , aσ.drop(# aal + 1), aα), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ¬ cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 4, flags: discard

Additional information

Proof step 9 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , aσ.drop(# aal + 1), aα), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(⌜ca '⌝ + πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ¬ cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜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 ⊦ Πok(lext, rext, lren, rren) ∧ lren(la) = rren(ra) → (rext(ra) = None ↔ false); (as) ⊦ oe + β = (oe = None ⊃ [];⌜oe.get '⌝ ) + β; (s) ⊦ ext !(⌜a '⌝ + α) = ext(a) + ext !(α); (s)

Proof step 10 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ , aσ.drop(# aal + 1), aα), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), ⌜aext(aa).get '⌝ + cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ¬ cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦

User applied rule case distinction

And closed the goal!

Rule argument

pos: left 11

Additional information

Used simplifier rules: ⊦ ⌜x⌝ + σ1 ⊑ ⌜x⌝ + σ2 ↔ σ1 ⊑ σ2; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ [] + σ = σ; (s) ⊦ Πok(lext, rext, lren, rren) ∧ lren(la) = rren(ra) → (rext(ra) = None ↔ false); (as)

Proof step 11 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜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 ⊦ a + x = a ' + x; (s)

Proof step 12 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:replaction-inf-def, seq:⊦ cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcsf⌝ , ⌜paf⌝ , aσ, aα) = lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), aσ, aα) ⌝ ⌝ ), subst:[cext, aσ, ren, cstep, aren, aext, step, aα, pcsf, cren, paf] → [cext, aσ, ren, cstep, aren, aext, step, ⌜aal⌝ + ⌜aa '⌝ + aα, (⌜(s, cs) '⌝ + ⌜(s0, 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, ⌜aal⌝ + ⌜aa '⌝ + aα, cren, (⌜cren(ca) '⌝ + ⌜paf⌝ ).f, aσ, step, (⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ ).f

Proof step 13 from proof repla-infrec-addboth (previous step) (back to tree)

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

Lemma replaction-inf-def

Proof step 14 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ ).f⌝ , ⌜(⌜cren(ca) '⌝ + ⌜paf⌝ ).f⌝ )) ⊑ aext !(⌜aal⌝ + ⌜aa '⌝ + aα) → repla (step, cstep, cren, ren, aren, cext, aext, ⌜(⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ ).f⌝ , ⌜(⌜cren(ca) '⌝ + ⌜paf⌝ ).f⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) = lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ ).f.take(n + 1), (⌜cren(ca) '⌝ + ⌜paf⌝ ).f.take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + 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 ⊦ [] + σ = σ; (s) ⊦ ⌜[]⌝ = []; (s) ⊦ (⌜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) ⊦ oe + β = (oe = None ⊃ [];⌜oe.get '⌝ ) + β; (s) ⊦ ext !(⌜a '⌝ + α) = ext(a) + ext !(α); (s) ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s)

Proof step 15 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) = lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ , aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 4, flags: discard

Additional information

Proof step 16 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα) ⊦

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. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ , 1]; ['a] → ['pa], <all paths>

Additional information

Used terms: λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ , 1

Proof step 17 from proof repla-infrec-addboth (previous step) (back to tree)

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

A spec-lemma

Proof step 18 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: streamseq-mon-lim, name:mon-def, seq: ⊦ mon(σσ) ↔ (∀ n. ⌜n⌝ + ⌜1⌝ < # σσ → σσ[n] ⊑ σσ[n + 1]), subst:[σσ] → [⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ]; ['a] → ['pa], <all paths>

Additional information

Used terms: ⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝

Proof step 19 from proof repla-infrec-addboth (previous step) (back to tree)

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

A spec-lemma

Proof step 20 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ∃ n. ¬ ( ⌜n⌝ + ⌜1⌝ < # ⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ → ⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ [n ] ⊑ ⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ [n + 1 ]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + nf).take(n + 1) = a ' + nf.take(n); (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ⌜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 21 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα)⌝ ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:repla-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) → ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⊑ ⌜repla (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, hd(aσ), start, tl(aσ), ext, ren, astart, cstart, n, cstep, aren, aext, step, ⌜aal⌝ + ⌜aa '⌝ + aα, s, (⌜(s0, 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, s, hd(aσ), aext, ⌜aal⌝ + ⌜aa '⌝ + aα, ext, astart, cstart, start, cren, astep, (⌜cren(ca) '⌝ + ⌜paf⌝ ).f, cs, n, tl(aσ), step, (⌜(s0, cs0) '⌝ + ⌜pcsf⌝ ).f

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

Lemma repla-mon

Proof step 23 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + 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) ' + (⌜(s0, cs0) '⌝ + ⌜pcsf⌝ ).f, (⌜cren(ca) '⌝ + ⌜paf⌝ ).f) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), ⌜aal⌝ + ⌜aa '⌝ + aα) ∧ cext !(πra(step, cstep, ren, cren, (s, cs) + ⌜(⌜(s0, cs0) '⌝ + ⌜pcsf⌝ ).f⌝ , ⌜(⌜cren(ca) '⌝ + ⌜paf⌝ ).f⌝ )) ⊑ aext !(⌜aal⌝ + ⌜aa '⌝ + aα) → ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) + (⌜(s0, cs0) '⌝ + ⌜pcsf⌝ ).f.take(n), (⌜cren(ca) '⌝ + ⌜paf⌝ ).f.take(n), ⌜hd(aσ) '⌝ + tl(aσ), ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) + (⌜(s0, cs0) '⌝ + ⌜pcsf⌝ ).f.take(n + 1), (⌜cren(ca) '⌝ + ⌜paf⌝ ).f.take(n + 1), ⌜hd(aσ) '⌝ + tl(aσ), ⌜aal⌝ + ⌜aa '⌝ + 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 ⊦ [] + σ = σ; (s) ⊦ ⌜[]⌝ = []; (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) ⊦ (⌜x⌝ + ⌜nf⌝ ).f = x + nf; (s) ⊦ a + x = a ' + x; (s) ⊦ oe + β = (oe = None ⊃ [];⌜oe.get '⌝ ) + β; (s) ⊦ ext !(⌜a '⌝ + α) = ext(a) + ext !(α); (s) ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s) ⊦ ⌜x + nf⌝ = ⌜x⌝ + ⌜nf⌝ ; (s) ⊦ a + σ = ⌜a '⌝ + σ; (s)

Proof step 24 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα)⌝ , fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (s0, cs0) ' + pcsf, cren(ca) ' + paf) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), ⌜aal⌝ + ⌜aa '⌝ + aα) → ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜hd(aσ) '⌝ + tl(aσ), ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), ⌜hd(aσ) '⌝ + tl(aσ), ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

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

Additional information

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

Proof step 25 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], aσ = as + σ0, cext(ca) = aext(aa), ren(a) = cren(ca), aσ ≠ [], lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα)⌝ , fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (s0, cs0) ' + pcsf, cren(ca) ' + paf) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) → ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a + σ = ⌜a '⌝ + σ; (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 ⊦ n + 0 = n; (s) ⊦ n ≠ 0 → (⌜a '⌝ + σ).drop((* n) + m) = σ.drop((n - 1) + m); (s) ⊦ ⌜a '⌝ + σ ≠ []; (s)

Proof step 26 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n0. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n0), (cren(ca) ' + paf).take(n0), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n0. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n0), paf.take(n0), σ0.drop(# aal), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ¬ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (s0, cs0) ' + pcsf, cren(ca) ' + paf), ¬ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + 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), (s0, 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), (s0, cs0) ' + pcsf, paf, Π(step, cstep, ren, cren, ext, cext)

Proof step 27 from proof repla-infrec-addboth (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 28 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n0. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n0), (cren(ca) ' + paf).take(n0), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n0. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n0), paf.take(n0), σ0.drop(# aal), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ , ¬ Π(step, cstep, ren, cren, ext, cext)((s, cs), cren(ca), (((s0, cs0) ' + pcsf))(0)) ∨ ¬ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, 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 29 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n0. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n0), (cren(ca) ' + paf).take(n0), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n0. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n0), paf.take(n0), σ0.drop(# aal), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ¬ Π(step, cstep, ren, cren, ext, cext)((s, cs), cren(ca), (s0, cs0)), ¬ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⊦

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), 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, cren(ca), cs, cstep, ren, ext, cren, cs0, step, s, s0

Proof step 30 from proof repla-infrec-addboth (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 31 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n0. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n0), (cren(ca) ' + paf).take(n0), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n0. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n0), paf.take(n0), σ0.drop(# aal), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ , ∀ la, ra. ¬(step(s, la, s0) ∧ ext(la) ≠ None ∧ cren(ca) = ren(la) ∧ cstep(cs, ra, cs0) ∧ cext(ra) ≠ None ∧ cren(ca) = cren(ra)), cs ≠ cs0 ∨ (∀ la. ¬(step(s, la, s0) ∧ cren(ca) = ren(la) ∧ ¬ cod(cren)(cren(ca)))), s ≠ s0 ∨ (∀ ra. ¬(cstep(cs, ra, cs0) ∧ cren(ca) = cren(ra) ∧ ¬ cod(ren)(cren(ca)))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ cod(f)(b) ↔ (∃ a. f(a) = b); (as) ⊦ ¬ cod(f)(b) → b ≠ f(a); (as) ⊦ Πok(lext, rext, lren, rren) ∧ lren(la) = rren(ra) → (rext(ra) = None ↔ false); (as) ⊦ Πok(lext, rext, lren, rren) → (rren(ra1) = rren(ra2) ↔ ra1 = ra2); (s) ⊦ Πok(lext, rext, lren, rren) ∧ lren(la) = rren(ra) → (lext(la) = None ↔ false); (as) ⊦ cod(f)(f(a)); (s)

Proof step 32 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), ren(a0) = cren(ca), lim (⌜λ n0. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n0), (cren(ca) ' + paf).take(n0), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n0. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n0), paf.take(n0), σ0.drop(# aal), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), ⌜as '⌝ + σ0, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ , ∀ la. ¬ step(s, la, s0) ∨ cren(ca) ≠ ren(la) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 15 terms: a

Additional information

Used terms: a

Proof step 33 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n + 1), (cren(ca) ' + paf).take(n + 1), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + nf).take(n + 1) = a ' + nf.take(n); (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 34 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: (λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ) = (λ n. ⌜aal.map(aren) + aren(aa) '⌝ + ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα) ⌝ )

Additional information

Used formulas: (λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ) = (λ n. ⌜aal.map(aren) + aren(aa) '⌝ + ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα) ⌝ )

Proof step 35 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], (λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα)⌝ ) = (λ n. ⌜aal.map(aren) + aren(aa) '⌝ + ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ), cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 2, flags: discard

Additional information

Proof step 36 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. ⌜aal.map(aren) + aren(aa) '⌝ + ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα) ⊦

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] → [aal.map(aren) + aren(aa) ', λ n0. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n0), paf.take(n0), aσ.drop(# aal + 1), aα) ⌝ ]; ['a] → ['pa], <all paths>

Additional information

Used simplifier rules: ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # x = # y → (⌜x⌝ + σ = ⌜y⌝ + σ0 ↔ x = y ∧ σ = σ0); (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: aal.map(aren) + aren(aa) ', λ n0. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n0), paf.take(n0), aσ.drop(# aal + 1), aα) ⌝

Proof step 37 from proof repla-infrec-addboth (previous step) (back to tree)

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

A spec-lemma

Proof step 38 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. ⌜aal.map(aren) + aren(aa) '⌝ + ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ mon(⌜λ n0. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n0), paf.take(n0), aσ.drop(# aal + 1), 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 a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s)

Proof step 39 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. (⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ ) + ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ¬ mon(⌜λ n0. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n0), paf.take(n0), aσ.drop(# aal + 1), 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 n0. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n0), paf.take(n0), aσ.drop(# aal + 1), aα) ⌝ ⌝ ]; ['a] → ['pa], <all paths>

Additional information

Used terms: ⌜λ n0. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n0), paf.take(n0), aσ.drop(# aal + 1), aα) ⌝ ⌝

Proof step 40 from proof repla-infrec-addboth (previous step) (back to tree)

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

A spec-lemma

Proof step 41 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. (⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ ) + ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ∃ n. ¬ ( ⌜n⌝ + ⌜1⌝ < # ⌜λ n0. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n0), paf.take(n0), aσ.drop(# aal + 1), aα)⌝ ⌝ → ⌜λ n0. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n0), paf.take(n0), aσ.drop(# aal + 1), aα)⌝ ⌝ [n] ⊑ ⌜λ n0. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n0), paf.take(n0), aσ.drop(# aal + 1), aα)⌝ ⌝ [n + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 42 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. (⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ ) + ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), aσ.drop(# aal + 1), aα)⌝ ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:repla-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) → ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⊑ ⌜repla (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, cs0, cext, aσ[# aal + 1], start, aσ.drop(# aal + 2), ext, ren, astart, cstart, n, 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 terms: aren, ren, cstep, cext, s0, aσ[# aal + 1], aext, aα, ext, astart, cstart, start, cren, astep, paf, cs0, n, aσ.drop(# aal + 2), step, pcsf

Proof step 43 from proof repla-infrec-addboth (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) → ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) + pcsf.take(n + 1), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝

Lemma repla-mon

Proof step 44 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. (⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ ) + ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), aσ.drop(# aal + 1), 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)))((s0, cs0) ' + pcsf, paf) ∧ frag(IOAut(astart, astep, aext))(⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) ∧ cext !(πra(step, cstep, ren, cren, (s0, cs0) + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) + pcsf.take(n), paf.take(n), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα)⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) + pcsf.take(n + 1), paf.take(n + 1), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) ⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 45 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. (⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ ) + ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), aσ.drop(# aal + 1), aα)⌝ , frag(IOAut(astart, astep, aext))(⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) → ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα)⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) ⌝ ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:frag-from-1, seq:⊦ frag(IOAut(start, step, ext))(σ, α) ∧ ¬ # α < ⌜n⌝ → (frag(IOAut(start, step, ext))(σ.drop(n), α.drop(n)) ↔ true), subst:[step, α, start, ext, n, σ] → [astep, ⌜aal⌝ + ⌜aa '⌝ + aα, astart, aext, # aal + 1, aσ]; ['s, 'a, 'e] → ['as, 'aa, 'e]

Additional information

Used terms: aext, ⌜aal⌝ + ⌜aa '⌝ + aα, astart, aσ, # aal + 1, astep

Proof step 46 from proof repla-infrec-addboth (previous step) (back to tree)

⊦ frag(IOAut(start, step, ext))(σ, α) ∧ ¬ # α < ⌜n⌝ → (frag(IOAut(start, step, ext))(σ.drop(n), α.drop(n)) ↔ true)

A spec-lemma

Proof step 47 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. (⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ ) + ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), aσ.drop(# aal + 1), aα)⌝ , frag(IOAut(astart, astep, aext))(⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) → ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα)⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) ⌝ , frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ∧ ¬ #(⌜aal⌝ + ⌜aa '⌝ + aα) < ⌜# aal + 1⌝ → (frag(IOAut(astart, astep, aext))(aσ.drop(# aal + 1), (⌜aal⌝ + ⌜aa '⌝ + aα).drop(# aal + 1)) ↔ true) ⊦

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 ⊦ (⌜a '⌝ + σ).drop(1) = σ; (s) ⊦ (⌜x⌝ + σ).drop(m + # x) = σ.drop(m); (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ¬ n∞ + n∞1 < n∞1; (s) ⊦ n∞ + ⌜0⌝ = n∞; (s) ⊦ m ≤ n → (⌜(* m)⌝ + n∞2 < ⌜(* n)⌝ + n∞1 ↔ n∞2 < ⌜n - m⌝ + n∞1); (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 48 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. (⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ ) + ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), frag(IOAut(astart, astep, aext))(aσ.drop(# aal + 1), aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), aσ.drop(# aal + 1), aα)⌝ , frag(IOAut(astart, astep, aext))(⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) → ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα)⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) ⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: from-succ, seq: ⊦ ⌜n⌝ < # σ → σ.drop(n) = σ[n] + σ.drop(n + 1), subst:[σ, n] → [aσ, # aal + 1]; ['a] → ['as], <all paths>

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ a + σ = ⌜a '⌝ + σ; (s) Used terms: aσ, # aal + 1

Proof step 49 from proof repla-infrec-addboth (previous step) (back to tree)

⊦ ⌜n⌝ < # σ → σ.drop(n) = σ[n] + σ.drop(n + 1)

A spec-lemma

Proof step 50 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. (⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ ) + ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ.drop(# aal + 1), aα), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ ⌜# aal + 1⌝ < # aσ, ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), aσ.drop(# aal + 1), aα)⌝ , frag(IOAut(astart, astep, aext))(⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) → ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα)⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) ⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 51 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. (⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ ) + ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), frag(IOAut(astart, astep, aext))(aσ.drop(# aal + 1), aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ¬ ⌜# aal⌝ + ⌜1⌝ < # aσ, ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), aσ.drop(# aal + 1), aα)⌝ , frag(IOAut(astart, astep, aext))(⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) → ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα)⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) ⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name: frag, seq:⊦ frag(IOAut(start, step, ext))(σ, α) ↔ # σ = # α + ⌜1⌝ ∧ (∀ n. ⌜n⌝ < # α → step(σ[n], α[n], σ[n +1])), subst:[σ, step, α, start, ext] → [aσ, astep, ⌜aal⌝ + ⌜aa '⌝ + aα, astart, aext]; ['s, 'a, 'e] → ['as, 'aa, 'e], <all paths>

Additional information

Used terms: aext, ⌜aal⌝ + ⌜aa '⌝ + aα, astart, aσ, astep

Proof step 52 from proof repla-infrec-addboth (previous step) (back to tree)

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

A spec-lemma

Proof step 53 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], # aσ = #(⌜aal⌝ + ⌜aa '⌝ + aα) + ⌜1⌝ , cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. (⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ ) + ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ.drop(# aal + 1), aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ ⌜# aal⌝ + ⌜1⌝ < # aσ, ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), aσ.drop(# aal + 1), aα)⌝ , frag(IOAut(astart, astep, aext))(⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) → ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα)⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) ⌝ , ∀ n. ⌜n⌝ < #(⌜aal⌝ + ⌜aa '⌝ + aα) → astep(aσ[n], (⌜aal⌝ + ⌜aa '⌝ + aα)[n], aσ[n +1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜(* n)⌝ + ⌜(* m)⌝ = ⌜n + m⌝ ; (s)

Proof step 54 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], # aσ = ⌜2⌝ + ⌜# aal⌝ + # aα, cext(ca) = aext(aa), ren(a) = cren(ca), lim (⌜λ n. (⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ ) + ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ.drop(# aal + 1), aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), ¬ ⌜# aal⌝ + ⌜1⌝ < # aσ, ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), aσ.drop(# aal + 1), aα)⌝ , frag(IOAut(astart, astep, aext))(⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) → ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα)⌝ ⊑ ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n + 1), paf.take(n + 1), ⌜aσ[# aal + 1] '⌝ + aσ.drop(# aal + 2), aα) ⌝ , ∀ n0. ¬ ⌜# aal⌝ + # aα < ⌜n0⌝ → astep(aσ[n0], (⌜aal⌝ + ⌜aa '⌝ + aα)[n0], aσ[n0 + 1]) ⊦

User applied rule subst equation

And closed the goal!

Rule argument

pos: left 2, flags: discard

Additional information

Used simplifier rules: ⊦ m ≤ n → (⌜(* m)⌝ + n∞2 < ⌜(* n)⌝ + n∞1 ↔ n∞2 < ⌜n - m⌝ + n∞1); (s) ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) ⊦ ¬ n∞ + n∞1 < n∞1; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf

Proof step 55 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), (λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα)⌝ ) ≠ (λ n. ⌜aal.map(aren) + aren(aa) '⌝ + ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ), lim (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜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⌝ + ⌜y⌝ + ⌜z⌝ = ⌜x0⌝ ↔ x + y + z = x0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s)

Proof step 56 from proof repla-infrec-addboth (previous step) (back to tree)

aext !(aal) = [], cext(ca) = aext(aa), ren(a) = cren(ca), aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα) ≠ repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), lim (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsf.take(n), cren(ca) ' + paf.take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα)⌝ ⌝ ) ≠ ⌜aal.map(aren)⌝ + ⌜aren(aa) '⌝ + lim(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsf.take(n), paf.take(n), aσ.drop(# aal + 1), aα)⌝ ⌝ ), cstep(cs, ca, cs0), frag(IOAut(astart, astep, aext))(aσ, ⌜aal⌝ + ⌜aa '⌝ + aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s0, cs0) ' + pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + ((s0, cs0) ' + pcsf).take(n), (cren(ca) ' + paf).take(n), aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) ⌝ ⌝ ), nounsharedextR(cren, cext, ren), step(s, a, s0), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜(s0, cs0) '⌝ + ⌜pcsf⌝ , ⌜cren(ca) '⌝ + ⌜paf⌝ )) ⊑ (aext(aa) = None ⊃ [];⌜aext(aa).get '⌝ ) + aext !(aα) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:repla-rec-addboth, seq:⊦ step(s, a, s0) ∧ cstep(cs, ca, cs0) ∧ cren(ca) = ren(a) ∧ aext !(⌜aal⌝ ) = [] ∧ cext(ca) = aext(aa) ∧ n = # aal → repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a) ' + pal, aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) = aal.map(aren) + aren(aa) ' + repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal, aσ.drop(n + 1), aα), subst:[ca, a, aal, aa, aσ, cren, ren, n, cstep, aren, pcsl, aext, step, pal, cs, cext, s0, cs0, aα, s] → [ca, a, aal, aa, aσ, cren, ren, # aal, cstep, aren, pcsf.take(n), aext, step, paf.take(n), cs, cext, s0, cs0, aα, s]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used simplifier rules: ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ ⌜x⌝ = [] ↔ x = []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: aren, ren, cstep, cext, aa, s, aext, aα, ca, a, paf.take(n), cren, aal, pcsf.take(n), cs0, s0, cs, # aal, aσ, step

Proof step 57 from proof repla-infrec-addboth (previous step) (back to tree)

⊦ step(s, a, s0) ∧ cstep(cs, ca, cs0) ∧ cren(ca) = ren(a) ∧ aext !(⌜aal⌝ ) = [] ∧ cext(ca) = aext(aa) ∧ n = # aal → repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a) ' + pal, aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) = aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s0, cs0) ' + pcsl, pal, aσ.drop(n + 1), aα)

Lemma repla-rec-addboth

Proof Tree for repla-infrec-addboth (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