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

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

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IO-Automaton, name: isIOAut-def, seq: ⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext)), subst:[start, step, ext] → [astart, astep, aext]; ['s, 'a, 'e] → ['as, 'aa, 'e], <all paths>

Additional information

Used terms: astart, astep, aext

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

⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext))

A spec-lemma

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ IOAut(start, step, ext).start = start; (s) ⊦ exec(ioa)(σ, α) ↔ frag(ioa)(σ, α) ∧ ioa.start(σ[0]); (s) ⊦ Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren) = IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)); (s)

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

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), astart(as), astart(aσ[0]), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(aext), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)) ∧ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0])

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IO-Automaton, name: isIOAut-def, seq: ⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext)), subst:[start, step, ext] → [start, step, ext]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used terms: start, step, ext

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

⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext))

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), IOAut?(IOAut(cstart, cstep, cext)), astart(aσ[0]), astart(as), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(astart, astep, aext))(aσ, aα), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(aext), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∃ s. start(s) ∧ oinj(ext) ⊦ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)) ∧ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), IOAut?(IOAut(cstart, cstep, cext)), astart(as), astart(aσ[0]), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), oinj(aext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcσ[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)) ∧ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0])

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IO-Automaton, name: isIOAut-def, seq: ⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext)), subst:[start, step, ext] → [cstart, cstep, cext]; ['s, 'a, 'e] → ['cs, 'ca, 'e], <all paths>

Additional information

Used terms: cstart, cstep, cext

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

⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext))

A spec-lemma

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

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), astart(aσ[0]), astart(as), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(astart, astep, aext))(aσ, aα), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(aext), oinj(ext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∃ cs. cstart(cs) ∧ oinj(cext) ⊦ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)) ∧ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 11 from proof repl-exec (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), astart(as), astart(aσ[0]), cstart(cs), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(cext), oinj(ext), oinj(aext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcσ[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)) ∧ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0])

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:repl-frag, seq:nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), oinj(cext), oinj(aext), oinj(ext), sameshared(cren, aren, ren), frag (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (pcσ, pα), frag(IOAut(astart, astep, aext))(aσ, aα), cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα) ⊦ frag (IOAut (Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), subst:[astep, cext, aα, start, pcσ, aσ, cren, ren, astart, cstart, cstep, aren, aext, step, ext, pα] → [astep, cext, aα, start, pcσ, aσ, cren, ren, astart, cstart, cstep, aren, aext, step, ext, pα]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used terms: aren, pα, ren, ext, astep, pcσ, astart, step, aext, cstart, cren, cstep, aα, start, aσ, cext

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

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

Lemma repl-frag

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

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), astart(aσ[0]), astart(as), cstart(cs), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(aext), oinj(ext), oinj(cext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren) ⊦ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)) ∧ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), astart(as), astart(aσ[0]), cstart(cs), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(cext), oinj(ext), oinj(aext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcσ[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:frag-repls-head, seq:nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), frag (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (⌜(s, cs) '⌝ + pcσ, pα), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + pcσ, pα)) ⊑ aext !(aα) ⊦ repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + pcσ, pα, ⌜as '⌝ + aσ, aα) ≠ [] ∧ hd (repls (step, cstep, cren, ren, aren, cext, aext, ⌜(s, cs) '⌝ + pcσ, pα, ⌜as '⌝ + aσ, aα)) = (s, as), subst:[astep, cs, cext, as, start, pcσ, aσ, cren, ren, astart, cstart, cstep, aren, aext, step, aα, s, ext, pα] → [astep, hd(pcσ)._2, cext, hd(aσ), start, tl(pcσ), tl(aσ), cren, ren, astart, cstart, cstep, aren, aext, step, aα, hd(pcσ)._1, ext, pα]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used terms: aren, pα, ren, ext, hd(pcσ)._1, hd(aσ), astep, tl(pcσ), astart, step, aext, cstart, cren, cstep, aα, start, tl(aσ), cext, hd(pcσ)._2

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

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

Lemma frag-repls-head

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

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), astart(aσ[0]), astart(as), cstart(cs), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(aext), oinj(ext), oinj(cext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]) ⊦ frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜(hd(pcσ)._1, hd(pcσ)._2) '⌝ + tl(pcσ), pα) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ cext !(πra(step, cstep, ren, cren, ⌜(hd(pcσ)._1, hd(pcσ)._2) '⌝ + tl(pcσ), pα)) ⊑ aext !(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] → [pcσ, pcs, pcσ0]; ['a] → [('s, 'cs)]

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: pcσ, pcs, pcσ0

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

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), pcσ = pcs + pcσ0, pcσ ≠ [], astart(as), astart(aσ[0]), cstart(cs), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(cext), oinj(ext), oinj(aext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcσ[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]) ⊦ frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜(pcs._1, pcs._2) '⌝ + pcσ0, pα) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcs._1, pcs._2) '⌝ + pcσ0, pα)) ⊑ aext !(aα)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + σ = ⌜a '⌝ + σ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (⌜a '⌝ + σ)[0] = a; (s) ⊦ ⌜a '⌝ + σ ≠ []; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcs '⌝ + pcσ0, pα)) = aext !(aα), astart(aσ[0]), astart(as), cstart(cs), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜pcs '⌝ + pcσ0, pα), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + pcσ0, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + pcσ0, pα, aσ, aα)), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(aext), oinj(ext), oinj(cext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcs), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + pcσ0, pα, aσ, aα)[0]) ⊦ frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜(pcs._1, pcs._2) '⌝ + pcσ0, pα) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcs._1, pcs._2) '⌝ + pcσ0, pα)) ⊑ aext !(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, as0, aσ0]; ['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, as0, aσ0

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

cext !(πra(step, cstep, ren, cren, ⌜pcs '⌝ + pcσ0, pα)) = aext !(aα), aσ = as0 + aσ0, aσ ≠ [], astart(as), astart(aσ[0]), cstart(cs), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + pcσ0, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + pcσ0, pα, aσ, aα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜pcs '⌝ + pcσ0, pα), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(cext), oinj(ext), oinj(aext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcs), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + pcσ0, pα, aσ, aα)[0]) ⊦ frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜(pcs._1, pcs._2) '⌝ + pcσ0, pα) ∧ frag(IOAut(astart, astep, aext))(⌜as0 '⌝ + aσ0, aα) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcs._1, pcs._2) '⌝ + pcσ0, pα)) ⊑ aext !(aα)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + σ = ⌜a '⌝ + σ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (⌜a '⌝ + σ)[0] = a; (s) ⊦ ⌜a '⌝ + σ ≠ []; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcs '⌝ + pcσ0, pα)) = aext !(aα), astart(as), astart(as0), cstart(cs), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜pcs '⌝ + pcσ0, pα), frag(IOAut(astart, astep, aext))(⌜as0 '⌝ + aσ0, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + pcσ0, pα, ⌜as0 '⌝ + aσ0, aα), repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + pcσ0, pα, ⌜as0 '⌝ + aσ0, aα)), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(aext), oinj(ext), oinj(cext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcs), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + pcσ0, pα, ⌜as0 '⌝ + aσ0, aα)[0]) ⊦ frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜(pcs._1, pcs._2) '⌝ + pcσ0, pα) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcs._1, pcs._2) '⌝ + pcσ0, pα)) ⊑ aext !(aα)

User applied rule apply elim lemma

And closed the goal!

Rule argument

term: pcs._1

Additional information

Used simplifier rules: ⊦ Π(lstart, rstart)((ls, rs)) ↔ lstart(ls) ∧ rstart(rs); (s) ⊦ σ ⊑ σ; (s) Used terms: s0, cs0

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

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), hd(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(hd(pcσ)._1, hd(pcσ)._2) '⌝ + tl(pcσ), pα, ⌜hd(aσ) '⌝ + tl(aσ), aα)) = (hd(pcσ)._1, hd(aσ)), repls(step, cstep, cren, ren, aren, cext, aext, ⌜(hd(pcσ)._1, hd(pcσ)._2) '⌝ + tl(pcσ), pα, ⌜hd(aσ) '⌝ + tl(aσ), aα) ≠ [], astart(aσ[0]), astart(as), cstart(cs), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(aext), oinj(ext), oinj(cext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-hd-tl, seq: ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0), subst:[σ, a, σ0] → [repls (step, cstep, cren, ren, aren, cext, aext, ⌜(hd(pcσ)._1, hd(pcσ)._2) '⌝ + tl(pcσ), pα, ⌜hd(aσ) '⌝ + tl(aσ), aα), pas, paσ]; ['a] → [('s, 'as)]

Additional information

Used simplifier rules: ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0); (elim) Used terms: repls (step, cstep, cren, ren, aren, cext, aext, ⌜(hd(pcσ)._1, hd(pcσ)._2) '⌝ + tl(pcσ), pα, ⌜hd(aσ) '⌝ + tl(aσ), aα), pas, paσ

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

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), pas = (hd(pcσ)._1, hd(aσ)), repls(step, cstep, cren, ren, aren, cext, aext, ⌜(hd(pcσ)._1, hd(pcσ)._2) '⌝ + tl(pcσ), pα, ⌜hd(aσ) '⌝ + tl(aσ), aα) = pas + paσ, repls(step, cstep, cren, ren, aren, cext, aext, ⌜(hd(pcσ)._1, hd(pcσ)._2) '⌝ + tl(pcσ), pα, ⌜hd(aσ) '⌝ + tl(aσ), aα) ≠ [], astart(as), astart(aσ[0]), cstart(cs), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(cext), oinj(ext), oinj(aext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcσ[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), repls(step, cstep, cren, ren, aren, cext, aext, ⌜(hd(pcσ)._1, hd(pcσ)._2) '⌝ + tl(pcσ), pα, ⌜hd(aσ) '⌝ + tl(aσ), aα) = ⌜(hd(pcσ)._1, hd(aσ)) '⌝ + paσ, astart(as), astart(aσ[0]), cstart(cs), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(cext), oinj(ext), oinj(aext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcσ[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]) ⊦

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] → [pcσ, pcs, pcσ0]; ['a] → [('s, 'cs)]

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: pcσ, pcs, pcσ0

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

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), pcσ = pcs + pcσ0, repls(step, cstep, cren, ren, aren, cext, aext, ⌜(pcs._1, pcs._2) '⌝ + pcσ0, pα, ⌜hd(aσ) '⌝ + tl(aσ), aα) = ⌜(pcs._1, hd(aσ)) '⌝ + paσ, pcσ ≠ [], astart(aσ[0]), astart(as), cstart(cs), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(aext), oinj(ext), oinj(cext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + σ = ⌜a '⌝ + σ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (⌜a '⌝ + σ)[0] = a; (s) ⊦ ⌜a '⌝ + σ ≠ []; (s)

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

cext !(πra(step, cstep, ren, cren, ⌜pcs '⌝ + pcσ0, pα)) = aext !(aα), repls(step, cstep, cren, ren, aren, cext, aext, ⌜(pcs._1, pcs._2) '⌝ + pcσ0, pα, ⌜hd(aσ) '⌝ + tl(aσ), aα) = ⌜(pcs._1, hd(aσ)) '⌝ + paσ, astart(as), astart(aσ[0]), cstart(cs), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + pcσ0, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + pcσ0, pα, aσ, aα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜pcs '⌝ + pcσ0, pα), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(cext), oinj(ext), oinj(aext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcs), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + pcσ0, pα, aσ, aα)[0]) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

term: pcs._1

Additional information

Used terms: s0, cs0

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

cext !(πra(step, cstep, ren, cren, ⌜pcs '⌝ + pcσ0, pα)) = aext !(aα), pcs = (s0, cs0), repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + pcσ0, pα, ⌜hd(aσ) '⌝ + tl(aσ), aα) = ⌜(s0, hd(aσ)) '⌝ + paσ, astart(aσ[0]), astart(as), cstart(cs), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜pcs '⌝ + pcσ0, pα), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + pcσ0, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + pcσ0, pα, aσ, aα)), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(aext), oinj(ext), oinj(cext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcs), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, ⌜pcs '⌝ + pcσ0, pα, aσ, aα)[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ Π(lstart, rstart)((ls, rs)) ↔ lstart(ls) ∧ rstart(rs); (s)

Proof step 27 from proof repl-exec (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, ⌜(s0, cs0) '⌝ + pcσ0, pα)) = aext !(aα), repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + pcσ0, pα, ⌜hd(aσ) '⌝ + tl(aσ), aα) = ⌜(s0, hd(aσ)) '⌝ + paσ, astart(as), astart(aσ[0]), cstart(cs), cstart(cs0), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + pcσ0, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + pcσ0, pα, aσ, aα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜(s0, cs0) '⌝ + pcσ0, pα), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(cext), oinj(ext), oinj(aext), sameshared(cren, aren, ren), start(s), start(s0), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, ⌜(s0, cs0) '⌝ + pcσ0, pα, aσ, aα)[0]) ⊦

User applied rule apply elim lemma

And closed the goal!

Rule argument

spec: stream, name:elim-hd-tl, seq: ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0), subst:[σ, a, σ0] → [aσ, as0, aσ0]; ['a] → ['as]

Additional information

Used simplifier rules: ⊦ ⌜a '⌝ + σ ≠ []; (s) ⊦ a + σ = ⌜a '⌝ + σ; (s) ⊦ (⌜a '⌝ + σ)[0] = a; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ Π(lstart, rstart)((ls, rs)) ↔ lstart(ls) ∧ rstart(rs); (s) ⊦ ¬ frag(IOAut(start, step, ext))([], α); (s) ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0); (elim) Used terms: aσ, as0, aσ0

Proof Tree for repl-exec (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