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Proof step 1 from proof finfrag-repla-sametrace-1 (conclusion) (back to tree)

⊦ nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(aσ, aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ ⊑ aext !(aα) → Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) = Π(ext, cext, ren, cren) !(pal)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof finfrag-repla-sametrace-1 (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), frag(IOAut(astart, astep, aext))(aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ ⊑ aext !(aα) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

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

Additional information

Used terms: aren, ren, ext, pcsl.head._1, hd(aσ), astep, astart, repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα), step, aext, cstart, cren, cstep, aα, pcsl.tail, start, tl(aσ), cext, pcsl.head._2

Proof step 3 from proof finfrag-repla-sametrace-1 (previous step) (back to tree)

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

Lemma finfrag-repla-sametrace

Proof step 4 from proof finfrag-repla-sametrace-1 (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), frag(IOAut(astart, astep, aext))(aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ ⊑ aext !(aα), nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsl.head._1, pcsl.head._2) ' + pcsl.tail, repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ ⌜cext ! (πra(step, cstep, ren, cren, (pcsl.head._1, pcsl.head._2) ' + pcsl.tail, repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα))) ⌝ ⊑ aext !(aα) → Π(ext, aext, ren, aren) ! (repla (step, cstep, cren, ren, aren, cext, aext, (pcsl.head._1, pcsl.head._2) ' + pcsl.tail, repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα), ⌜hd(aσ) '⌝ + tl(aσ), aα)) = Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 9

Additional information

Proof step 5 from proof finfrag-repla-sametrace-1 (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), frag(IOAut(astart, astep, aext))(aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ ⊑ aext !(aα) ⊦ nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsl.head._1, pcsl.head._2) ' + pcsl.tail, repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ ⌜cext ! (πra(step, cstep, ren, cren, (pcsl.head._1, pcsl.head._2) ' + pcsl.tail, repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα))) ⌝ ⊑ aext !(aα)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 6 from proof finfrag-repla-sametrace-1 (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), frag(IOAut(astart, astep, aext))(aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ ⊑ aext !(aα) ⊦ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsl.head._1, pcsl.head._2) ' + pcsl.tail, repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ ⌜cext ! (πra(step, cstep, ren, cren, (pcsl.head._1, pcsl.head._2) ' + pcsl.tail, repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα))) ⌝ ⊑ aext !(aα)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-list-c, seq:x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y, subst:[x, a, y] → [pcsl, pcs, pcsl0]; ['a] → [('s, 'cs)]

Additional information

Used simplifier rules: ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s) x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y; (elim) Used terms: pcsl, pcs, pcsl0

Proof step 7 from proof finfrag-repla-sametrace-1 (previous step) (back to tree)

pcsl = pcs + pcsl0, Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), pcsl ≠ [], frag(IOAut(astart, astep, aext))(aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ ⊑ aext !(aα) ⊦ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcs._1, pcs._2) ' + pcsl0, repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, (pcs._1, pcs._2) ' + pcsl0, repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)))⌝ ⊑ aext !(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: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 8 from proof finfrag-repla-sametrace-1 (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), 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 ' + pcsl0, pal), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, pcs ' + pcsl0, pal))⌝ ⊑ aext !(aα) ⊦ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcs._1, pcs._2) ' + pcsl0, repla(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, aσ, aα)) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, (pcs._1, pcs._2) ' + pcsl0, repla(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, aσ, aα)))⌝ ⊑ aext !(aα)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: pcs

Additional information

Proof step 9 from proof finfrag-repla-sametrace-1 (previous step) (back to tree)

pcs = (s, cs), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), 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 ' + pcsl0, pal), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, pcs ' + pcsl0, pal))⌝ ⊑ aext !(aα) ⊦ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcs._1, pcs._2) ' + pcsl0, repla(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, aσ, aα)) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, (pcs._1, pcs._2) ' + pcsl0, repla(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, aσ, aα)))⌝ ⊑ aext !(aα)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 10 from proof finfrag-repla-sametrace-1 (previous step) (back to tree)

Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl0, pal, aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), frag(IOAut(astart, astep, aext))(aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + pcsl0, pal), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + pcsl0, pal))⌝ ⊑ aext !(aα) ⊦ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((s, cs) ' + pcsl0, repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl0, pal, aσ, aα)) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + pcsl0, repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl0, pal, aσ, aα)))⌝ ⊑ aext !(aα)

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, as, aσ0]; ['a] → ['as]

Additional information

Used simplifier rules: finfrag-repla-sametrace: ⊦ nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ frag* (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((s, cs) ' + pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + pcsl, pal))⌝ ⊑ aext !(aα) → Π(ext, aext, ren, aren) ! (repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(pal); ⊦ ⌜a '⌝ + σ ≠ []; (s) ⊦ a + σ = ⌜a '⌝ + σ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ¬ frag(IOAut(start, step, ext))([], α); (s) ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0); (elim) Used terms: aσ, as, aσ0

Proof step 11 from proof finfrag-repla-sametrace-1 (previous step) (back to tree)

Π(ext, aext, ren, aren) ! (repla (step, cstep, cren, ren, aren, cext, aext, (pcsl.head._1, pcsl.head._2) ' + pcsl.tail, repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα), ⌜hd(aσ) '⌝ + tl(aσ), aα)) = Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), frag(IOAut(astart, astep, aext))(aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ ⊑ aext !(aα) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-list-c, seq:x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y, subst:[x, a, y] → [pcsl, pcs, pcsl0]; ['a] → [('s, 'cs)]

Additional information

Used simplifier rules: ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s) x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y; (elim) Used terms: pcsl, pcs, pcsl0

Proof step 12 from proof finfrag-repla-sametrace-1 (previous step) (back to tree)

Π(ext, aext, ren, aren) ! (repla (step, cstep, cren, ren, aren, cext, aext, (pcs._1, pcs._2) ' + pcsl0, repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα), ⌜hd(aσ) '⌝ + tl(aσ), aα)) = Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)), pcsl = pcs + pcsl0, Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcsl, pal, aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), pcsl ≠ [], frag(IOAut(astart, astep, aext))(aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsl, pal), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, pcsl, pal))⌝ ⊑ aext !(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: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 13 from proof finfrag-repla-sametrace-1 (previous step) (back to tree)

Π(ext, aext, ren, aren) ! (repla (step, cstep, cren, ren, aren, cext, aext, (pcs._1, pcs._2) ' + pcsl0, repla(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, aσ, aα), ⌜hd(aσ) '⌝ + tl(aσ), aα)) = Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, aσ, aα)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), 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 ' + pcsl0, pal), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, pcs ' + pcsl0, pal))⌝ ⊑ aext !(aα) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: pcs

Additional information

Proof step 14 from proof finfrag-repla-sametrace-1 (previous step) (back to tree)

Π(ext, aext, ren, aren) ! (repla (step, cstep, cren, ren, aren, cext, aext, (pcs._1, pcs._2) ' + pcsl0, repla(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, aσ, aα), ⌜hd(aσ) '⌝ + tl(aσ), aα)) = Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, aσ, aα)), pcs = (s, cs), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcs ' + pcsl0, pal, aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), 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 ' + pcsl0, pal), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, pcs ' + pcsl0, pal))⌝ ⊑ aext !(aα) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 15 from proof finfrag-repla-sametrace-1 (previous step) (back to tree)

Π(ext, aext, ren, aren) ! (repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl0, repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl0, pal, aσ, aα), ⌜hd(aσ) '⌝ + tl(aσ), aα)) = Π(ext, cext, ren, cren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl0, pal, aσ, aα)), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl0, pal, aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), frag(IOAut(astart, astep, aext))(aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + pcsl0, pal), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + pcsl0, pal))⌝ ⊑ aext !(aα) ⊦

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, as, aσ0]; ['a] → ['as]

Additional information

Used simplifier rules: finfrag-repla-sametrace: ⊦ nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ frag* (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((s, cs) ' + pcsl, pal) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + pcsl, pal))⌝ ⊑ aext !(aα) → Π(ext, aext, ren, aren) ! (repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(pal); ⊦ ⌜a '⌝ + σ ≠ []; (s) ⊦ a + σ = ⌜a '⌝ + σ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ¬ frag(IOAut(start, step, ext))([], α); (s) ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0); (elim) Used terms: aσ, as, aσ0

Proof Tree for finfrag-repla-sametrace-1 (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