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Proof step 1 from proof finfrag-repla-sametrace (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)))((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)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

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

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: # pcsl = # pal

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ frag*(IOAut(start, step, ext))(s ' + sl, al) → (# sl = # al ↔ true); (ss) Used formulas: # pcsl = # pal

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

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

User applied rule structural induction

And obtained the following new goals

Rule argument

term: pcsl

Additional information

Used simplifier rules: repla-base: ⊦ repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ', [], ⌜as '⌝ + aσ, aα) = []; List('a) freely generated by [], +;; ⊦ # [] = 0; (s) ⊦ # x = 0 ↔ x = []; (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 ⊦ ext !([]) = []; (s)

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

#(pcs + pcsl0) = # pal, Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (pcs + pcsl0), pal, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (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, (s, cs) ' + (pcs + pcsl0), pal))⌝ ⊑ aext !(aα), ∀ pal, ext, aext, ren, aren, step, cstep, cren, cext, s, cs, as, aσ, aα, astart, astep, start, cstart. ¬ ( # pcsl0 = # pal ∧ Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl0, pal, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + 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)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ frag*(IOAut(start, step, ext))(s ' + sl, al) → (# sl = # al ↔ true); (ss) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a + x = a ' + x; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

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

# pcsl0 + 1 = # pal, Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcs ' + pcsl0, pal, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + 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, (s, cs) ' + pcs ' + pcsl0, pal))⌝ ⊑ aext !(aα), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: pcs

Additional information

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

# pcsl0 + 1 = # pal, pcs = (s1, cs1), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcs ' + pcsl0, pal, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + 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, (s, cs) ' + pcs ' + pcsl0, pal))⌝ ⊑ aext !(aα), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

# pcsl0 + 1 = # pal, Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, pal, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (s1, cs1) ' + 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) ' + (s1, cs1) ' + pcsl0, pal))⌝ ⊑ aext !(aα), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: pal

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ # [] = 0; (s) ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws) ⊦ # x = 0 ↔ x = []; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

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

# pcsl0 + 1 = # pal, pal = pa + pal1, Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, pal, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (s1, cs1) ' + 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) ' + (s1, cs1) ' + pcsl0, pal))⌝ ⊑ aext !(aα), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

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 ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + n = m + n0 ↔ n = n0; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

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

# pcsl0 = # pal1, Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pa ') + Π(ext, cext, ren, cren) !(pal1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1), 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) ' + (s1, cs1) ' + pcsl0, pa ' + pal1))⌝ ⊑ aext !(aα), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:finfrag-rec, seq:⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al), subst:[a, al, step, s, start, ext, sl] → [pa, pal1, Π(step, cstep, ren, cren, ext, cext), (s, cs), Π(start, cstart), Π(ext, cext, ren, cren), (s1, cs1) ' + pcsl0]; ['s, 'a, 'e] → [('s, 'cs), 'pa, 'e], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: pal1, (s, cs), Π(ext, cext, ren, cren), (s1, cs1) ' + pcsl0, pa, Π(start, cstart), Π(step, cstep, ren, cren, ext, cext)

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

⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al)

A spec-lemma

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

# pcsl0 = # pal1, Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pa ') + Π(ext, cext, ren, cren) !(pal1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), pa, ((s1, cs1) ' + pcsl0).head), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1))⌝ ⊑ aext !(aα), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

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 ' + x).head = a; (s)

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

# pcsl0 = # pal1, Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pa ') + Π(ext, cext, ren, cren) !(pal1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), Π(step, cstep, ren, cren, ext, cext)((s, cs), pa, (s1, cs1)), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1))⌝ ⊑ aext !(aα), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

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] → [pa, s1, cext, step, cs1, 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, pa, cs, cstep, ren, ext, cren, cs1, step, s, s1

Proof step 13 from proof finfrag-repla-sametrace (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 14 from proof finfrag-repla-sametrace (previous step) (back to tree)

# pcsl0 = # pal1, Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pa ') + Π(ext, cext, ren, cren) !(pal1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), 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) ' + (s1, cs1) ' + pcsl0, pa ' + pal1))⌝ ⊑ aext !(aα), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0), cs = cs1 ∧ (∃ la. step(s, la, s1) ∧ pa = ren(la) ∧ ¬ cod(cren)(pa)) ∨ s = s1 ∧ (∃ ra. cstep(cs, ra, cs1) ∧ pa = cren(ra) ∧ ¬ cod(ren)(pa)) ∨ (∃ la, ra. step(s, la, s1) ∧ ext(la) ≠ None ∧ pa = ren(la) ∧ cstep(cs, ra, cs1) ∧ cext(ra) ≠ None ∧ pa = cren(ra)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

# pcsl0 = # pal1, Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pa ') + Π(ext, cext, ren, cren) !(pal1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), 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) ' + (s1, cs1) ' + pcsl0, pa ' + pal1))⌝ ⊑ aext !(aα), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0), cs = cs1 ∧ step(s, a, s1) ∧ pa = ren(a) ∧ ¬ cod(cren)(ren(a)) ∨ s = s1 ∧ cstep(cs, ca, cs1) ∧ pa = cren(ca) ∧ ¬ cod(ren)(cren(ca)) ∨ step(s, a0, s1) ∧ ext(a0) ≠ None ∧ cren(ca0) = ren(a0) ∧ cstep(cs, ca0, cs1) ∧ cext(ca0) ≠ None ∧ pa = cren(ca0) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 11

Additional information

Proof step 16 from proof finfrag-repla-sametrace (previous step) (back to tree)

# pcsl0 = # pal1, cs = cs1, pa = ren(a), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pa ') + Π(ext, cext, ren, cren) !(pal1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), step(s, a, s1), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1))⌝ ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: repla-rec-addS: ⊦ step(s, a, s0) ∧ ¬ cod(cren)(ren(a)) → repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs) ' + pcsl, ren(a) ' + pal, ⌜as '⌝ + aσ, aα) = ren(a) ' + repla (step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # psl = # pal ∧ lstep(ls, la, ls0) ∧ ¬ cod(rren)(lren(la)) → πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psl, lren(la) ' + pal) = πra(lstep, rstep, lren, rren, (ls0, rs) ' + psl, pal); (s) ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream

Proof step 17 from proof finfrag-repla-sametrace (previous step) (back to tree)

# pcsl0 = # pal1, Π(ext, aext, ren, aren) !(ren(a) ') + Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(ren(a) ') + Π(ext, cext, ren, cren) !(pal1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα), ¬ cod(cren)(ren(a)), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 12 terms: pal1, ext, aext, ren, aren, step, cstep, cren, cext, s1, cs1, as, aσ, aα, astart, astep, start, cstart abort: F discard: T computed: T

Additional information

Used simplifier rules: same-concrete-abstract: ⊦ Πok(ext, aext, ren, aren) ∧ Πok(ext, cext, ren, cren) → (Π(ext, aext, ren, aren) !(ren(a) ') = Π(ext, cext, ren, cren) !(ren(a) ') ↔ true); ⊦ # y = # y0 → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: pal1, ext, aext, ren, aren, step, cstep, cren, cext, s1, cs1, as, aσ, aα, astart, astep, start, cstart

Proof step 18 from proof finfrag-repla-sametrace (previous step) (back to tree)

# pcsl0 = # pal1, pa = cren(ca), s = s1, Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pa ') + Π(ext, cext, ren, cren) !(pal1), cstep(cs, ca, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), 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) ' + (s1, cs1) ' + pcsl0, pa ' + pal1))⌝ ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: repla-rec-skip: ⊦ cstep(cs, ca, cs0) ∧ ¬ cod(ren)(cren(ca)) → repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s, cs0) ' + pcsl, cren(ca) ' + pal, ⌜as '⌝ + aσ, aα) = repla (step, cstep, cren, ren, aren, cext, aext, (s, cs0) ' + pcsl, pal, ⌜as '⌝ + aσ, aα); nounsharedextR-notcodren-project: ⊦ ¬ cod(ren)(cren(ca)) ∧ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) → Π(ext, cext, ren, cren) !(cren(ca) ') = []; nounsharedextR-notcodren: ⊦ ¬ cod(ren)(cren(ca)) ∧ nounsharedextR(cren, cext, ren) → cext !(ca ') = []; a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ [] + x = x; (s) ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s) ⊦ a + x = a ' + x; (s) ⊦ # psl = # pal ∧ rstep(rs, ra, rs0) ∧ ¬ cod(lren)(rren(ra)) → πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls, rs0) ' + psl, rren(ra) ' + pal) = ra + πra(lstep, rstep, lren, rren, (ls, rs0) + psl, pal); (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream

Proof step 19 from proof finfrag-repla-sametrace (previous step) (back to tree)

# pcsl0 = # pal1, Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pal1), cstep(cs, ca, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα), ¬ cod(ren)(cren(ca)), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 12 terms: pal1, ext, aext, ren, aren, step, cstep, cren, cext, s1, cs1, as, aσ, aα, astart, astep, start, cstart abort: F discard: T computed: T

Additional information

Used terms: pal1, ext, aext, ren, aren, step, cstep, cren, cext, s1, cs1, as, aσ, aα, astart, astep, start, cstart

Proof step 20 from proof finfrag-repla-sametrace (previous step) (back to tree)

# pcsl0 = # pal1, cren(ca0) = ren(a0), pa = cren(ca0), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(pa ') + Π(ext, cext, ren, cren) !(pal1), cext(ca0) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), step(s, a0, s1), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1))⌝ ⊑ aext !(aα), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

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 ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) ⊦ ext(a) ≠ None → ext !(a ' + al) = ext(a).get ' + ext !(al); (s) ⊦ a + x = a ' + x; (s) pa = lren(la) ⊦ # psl = # pal ∧ rstep(rs, ra, rs0) ∧ rren(ra) = lren(la) ∧ lstep(ls, la, ls0) → πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs0) ' + psl, pa ' + pal) = ra + πra(lstep, rstep, lren, rren, (ls0, rs0) ' + psl, pal); (s)

Proof step 21 from proof finfrag-repla-sametrace (previous step) (back to tree)

# pcsl0 = # pal1, cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), cext(ca0) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a0, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜cext(ca0).get '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: option, name:elim-Some, seq: ⊦ opt ≠ None → (get = opt.get ↔ opt = Some(get)), subst:[opt, get] → [cext(ca0), e]; ['a] → ['e]

Additional information

Used simplifier rules: ⊦ opt ≠ None → (get = opt.get ↔ opt = Some(get)); (elim) Used terms: cext(ca0), e

Proof step 22 from proof finfrag-repla-sametrace (previous step) (back to tree)

# pcsl0 = # pal1, cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), cext(ca0) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), step(s, a0, s1), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ⌜e '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 23 from proof finfrag-repla-sametrace (previous step) (back to tree)

# pcsl0 = # pal1, cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a0, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ⌜e '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: prefix-def, seq: ⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0), subst:[σ, σ0] → [⌜e '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ , aext !(aα)]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜e '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ , aext !(aα)

Proof step 24 from proof finfrag-repla-sametrace (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 25 from proof finfrag-repla-sametrace (previous step) (back to tree)

# pcsl0 = # pal1, cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), step(s, a0, s1), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0), ∃ σ1. (⌜e '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ) + σ1 = aext !(aα) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 26 from proof finfrag-repla-sametrace (previous step) (back to tree)

# pcsl0 = # pal1, (⌜e '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ) + β = aext !(aα), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a0, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name:remtau-split-one, seq:⊦ ext !(α) = ⌜e '⌝ + β ↔ (∃ al, a, α0. α = ⌜al⌝ + ⌜a '⌝ + α0 ∧ ext(a) = Some(e) ∧ ext !(⌜al⌝ ) = [] ∧ ext !(α0) = β), subst:[ext, α, e, β] → [aext, aα, e, ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β]; ['e, 'a] → ['e, 'aa], <all paths>

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: aext, aα, e, ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β

Proof step 27 from proof finfrag-repla-sametrace (previous step) (back to tree)

⊦ ext !(α) = ⌜e '⌝ + β ↔ (∃ al, a, α0. α = ⌜al⌝ + ⌜a '⌝ + α0 ∧ ext(a) = Some(e) ∧ ext !(⌜al⌝ ) = [] ∧ ext !(α0) = β)

A spec-lemma

Proof step 28 from proof finfrag-repla-sametrace (previous step) (back to tree)

# pcsl0 = # pal1, cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), step(s, a0, s1), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0), ∃ al, aa, α0. aα = ⌜al⌝ + ⌜aa '⌝ + α0 ∧ aext(aa) = Some(e) ∧ aext !(⌜al⌝ ) = [] ∧ aext !(α0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ = [] ↔ x = []; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 29 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, aext !(aal) = [], # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a0, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

User applied rule apply rewrite 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] → [ca0, a0, aal, aa, ⌜as '⌝ + aσ, cren, ren, # aal, cstep, aren, pcsl0, aext, step, pal1, cs, cext, s1, cs1, aα0, s]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as], <all paths>

Additional information

Used simplifier rules: ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ ⌜x⌝ = [] ↔ x = []; (s) Used terms: aren, ren, cstep, cext, aa, s, aext, aα0, ca0, a0, pal1, cren, aal, pcsl0, cs1, s1, cs, # aal, ⌜as '⌝ + aσ, step

Proof step 30 from proof finfrag-repla-sametrace (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 step 31 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, aext !(⌜aal⌝ ) = [], # pcsl0 = # pal1, # aal = # aal, aext(aa) = Some(e), cext(ca0) = Some(e), cext(ca0) = aext(aa), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) ! (aal.map(aren) + aren(aa) ' + repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, (⌜as '⌝ + aσ).drop(# aal + 1), aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), step(s, a0, s1), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

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 ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (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) ⊦ ⌜x⌝ = [] ↔ x = []; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 32 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(aal.map(aren)) + Π(ext, aext, ren, aren) !(aren(aa) ') + Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, aσ.drop(# aal), aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), step(s, a0, s1), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ pal0, ext0, aext0, ren0, aren0, step0, cstep0, cren0, cext0, s0, cs0, as0, aσ0, aα0, astart0, astep0, start0, cstart0. Π(ext0, aext0, ren0, aren0) !(repla(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0)) = Π(ext0, cext0, ren0, cren0) !(pal0) ∨ ¬ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∨ ¬ frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∨ ¬ nounsharedextR(aren0, aext0, ren0) ∨ ¬ nounsharedextR(cren0, cext0, ren0) ∨ ¬ Πok(ext0, aext0, ren0, aren0) ∨ ¬ Πok(ext0, cext0, ren0, cren0) ∨ ¬ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 18 terms: pal1, ext, aext, ren, aren, step, cstep, cren, cext, s1, cs1, aσ[# aal], aσ.drop(# aal + 1), aα0, astart, astep, start, cstart abort: F discard: T computed: F

Additional information

Used terms: pal1, ext, aext, ren, aren, step, cstep, cren, cext, s1, cs1, aσ[# aal], aσ.drop(# aal + 1), aα0, astart, astep, start, cstart

Proof step 33 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, aext !(aal) = [], # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(aal.map(aren)) + Π(ext, aext, ren, aren) !(aren(aa) ') + Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, aσ.drop(# aal), aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a0, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) = Π(ext, cext, ren, cren) !(pal1) ∨ ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ∨ ¬ frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1) ∨ ¬ nounsharedextR(aren, aext, ren) ∨ ¬ nounsharedextR(cren, cext, ren) ∨ ¬ Πok(ext, aext, ren, aren) ∨ ¬ Πok(ext, cext, ren, cren) ∨ ¬ ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 34 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(aal.map(aren)) + Π(ext, aext, ren, aren) !(aren(aa) ') + Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, aσ.drop(# aal), aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), step(s, a0, s1), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) = Π(ext, cext, ren, cren) !(pal1) ∨ ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ∨ ¬ ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα0) ⊦

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]; ['a] → ['as], <all paths>

Additional information

Used terms: aσ, # aal

Proof step 35 from proof finfrag-repla-sametrace (previous step) (back to tree)

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

A spec-lemma

Proof step 36 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, aext !(aal) = [], # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(aal.map(aren)) + Π(ext, aext, ren, aren) !(aren(aa) ') + Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, aσ.drop(# aal), aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a0, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ ⌜# aal⌝ < # aσ, Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) = Π(ext, cext, ren, cren) !(pal1) ∨ ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ∨ ¬ ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα0) ⊦

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] → [⌜as '⌝ + aσ, astep, ⌜aal⌝ + ⌜aa '⌝ + aα0, astart, aext]; ['s, 'a, 'e] → ['as, 'aa, 'e], <all paths>

Additional information

Used terms: aext, ⌜aal⌝ + ⌜aa '⌝ + aα0, astart, ⌜as '⌝ + aσ, astep

Proof step 37 from proof finfrag-repla-sametrace (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 38 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, #(⌜as '⌝ + aσ) = #(⌜aal⌝ + ⌜aa '⌝ + aα0) + ⌜1⌝ , # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(aal.map(aren)) + Π(ext, aext, ren, aren) !(aren(aa) ') + Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, aσ.drop(# aal), aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs1), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), step(s, a0, s1), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ ⌜# aal⌝ < # aσ, ∀ n. ⌜n⌝ < #(⌜aal⌝ + ⌜aa '⌝ + aα0) → astep((⌜as '⌝ + aσ)[n], (⌜aal⌝ + ⌜aa '⌝ + aα0)[n], (⌜as '⌝ + aσ)[n +1]), Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) = Π(ext, cext, ren, cren) !(pal1) ∨ ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ∨ ¬ ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα0) ⊦

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)[n + 1] = σ[n]; (s) ⊦ 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) ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ n < m → (⌜(* n)⌝ + n∞1 = ⌜(* m)⌝ + n∞2 ↔ n∞1 = ⌜m - n⌝ + n∞2); (s) ⊦ ⌜(* n)⌝ + ⌜(* m)⌝ = ⌜n + m⌝ ; (s)

Proof step 39 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, aext !(aal) = [], # pcsl0 = # pal1, # aσ = ⌜1⌝ + ⌜# aal⌝ + # aα0, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(aal.map(aren)) + Π(ext, aext, ren, aren) !(aren(aa) ') + Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, aσ.drop(# aal), aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, cstep(cs, ca0, cs1), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a0, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ ⌜# aal⌝ < # aσ, ∀ n. ¬ ⌜# aal⌝ + # aα0 < ⌜n⌝ → astep((⌜as '⌝ + aσ)[n], (⌜aal⌝ + ⌜aa '⌝ + aα0)[n], aσ[n]), Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) = Π(ext, cext, ren, cren) !(pal1) ∨ ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ∨ ¬ ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα0) ⊦

User applied rule subst equation

And closed the goal!

Rule argument

pos: left 4, flags: discard

Additional information

Used simplifier rules: ⊦ ⌜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 40 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, aext !(aal) = [], # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(aal.map(aren)) + Π(ext, aext, ren, aren) !(aren(aa) ') + Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, aσ[# aal] + aσ.drop(# aal + 1), aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a0, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) = Π(ext, cext, ren, cren) !(pal1) ∨ ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ∨ ¬ ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα0) ⊦

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 ⊦ [] + x = x; (s) ⊦ a + σ = ⌜a '⌝ + σ; (s) ⊦ rext !(ral) = [] ∧ Πok(lext, rext, lren, rren) → Π(lext, rext, lren, rren) !(ral.map(rren)) = []; (s)

Proof step 41 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(aren(aa) ') + Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), step(s, a0, s1), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) = Π(ext, cext, ren, cren) !(pal1) ∨ ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ∨ ¬ ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα0) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 19

Additional information

Proof step 42 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, aext !(aal) = [], Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) = Π(ext, cext, ren, cren) !(pal1), # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(aren(aa) ') + Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a0, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦

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 ⊦ # y = # y0 → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws)

Proof step 43 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, aext !(aal) = [], Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) = Π(ext, cext, ren, cren) !(pal1), # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(aren(aa) ') ≠ Π(ext, cext, ren, cren) !(ren(a0) '), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a0, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 7, flags: rotate, discard

Additional information

Proof step 44 from proof finfrag-repla-sametrace (previous step) (back to tree)

Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) = Π(ext, cext, ren, cren) !(pal1), aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), Π(ext, aext, ren, aren) !(aren(aa) ') ≠ Π(ext, cext, ren, cren) !(cren(ca0) '), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), step(s, a0, s1), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name:finremtau-one, seq: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '), subst:[ext, a] → [Π(ext, cext, ren, cren), cren(ca0)]; ['a, 'e] → ['pa, 'e], <all paths>

Additional information

Used terms: Π(ext, cext, ren, cren), cren(ca0)

Proof step 45 from proof finfrag-repla-sametrace (previous step) (back to tree)

⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get ')

A spec-lemma

Proof step 46 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, aext !(aal) = [], Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) = Π(ext, cext, ren, cren) !(pal1), # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), Π(ext, aext, ren, aren) !(aren(aa) ') ≠ (Π(ext, cext, ren, cren)(cren(ca0)) = None ⊃ [];Π(ext, cext, ren, cren)(cren(ca0)).get '), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a0, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ Πok(lext, rext, lren, rren) → Π(lext, rext, lren, rren)(rren(ra)) = rext(ra); (s)

Proof step 47 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, aext !(aal) = [], Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) = Π(ext, cext, ren, cren) !(pal1), # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), Π(ext, aext, ren, aren) !(aren(aa) ') ≠ Some(e).get ', Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a0, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name:finremtau-one, seq: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '), subst:[ext, a] → [Π(ext, aext, ren, aren), aren(aa)]; ['a, 'e] → ['pa, 'e], <all paths>

Additional information

Used simplifier rules: ⊦ Πok(lext, rext, lren, rren) → Π(lext, rext, lren, rren)(rren(ra)) = rext(ra); (s) ⊦ a ' = a0 ' ↔ a = a0; (s) Used terms: Π(ext, aext, ren, aren), aren(aa)

Proof step 48 from proof finfrag-repla-sametrace (previous step) (back to tree)

⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get ')

A spec-lemma

Proof step 49 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, aext !(aal) = [], # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(aren(aa) ') + Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a0, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ⊦

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 '⌝ + aα0, astart, aext, # aal + 1, ⌜as '⌝ + aσ]; ['s, 'a, 'e] → ['as, 'aa, 'e]

Additional information

Used terms: aext, ⌜aal⌝ + ⌜aa '⌝ + aα0, astart, ⌜as '⌝ + aσ, # aal + 1, astep

Proof step 50 from proof finfrag-repla-sametrace (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 51 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(aren(aa) ') + Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), step(s, a0, s1), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0) ∧ ¬ #(⌜aal⌝ + ⌜aa '⌝ + aα0) < ⌜# aal + 1⌝ → (frag(IOAut(astart, astep, aext))((⌜as '⌝ + aσ).drop(# aal + 1), (⌜aal⌝ + ⌜aa '⌝ + aα0).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) ⊦ n + 0 = n; (s) ⊦ n ≠ 0 → (⌜a '⌝ + σ).drop((* n) + m) = σ.drop((n - 1) + 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 52 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, aext !(aal) = [], # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(aren(aa) ') + Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag(IOAut(astart, astep, aext))(aσ.drop(# aal), aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a0, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ frag(IOAut(astart, astep, aext))(⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0) ⊦

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]; ['a] → ['as], <all paths>

Additional information

Used simplifier rules: ⊦ a + σ = ⌜a '⌝ + σ; (s) Used terms: aσ, # aal

Proof step 53 from proof finfrag-repla-sametrace (previous step) (back to tree)

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

A spec-lemma

Proof step 54 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, aext !(aal) = [], # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(aren(aa) ') + Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), step(s, a0, s1), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: prefix-def, seq: ⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0), subst:[σ, σ0] → [⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ , aext !(aα0)]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ , aext !(aα0)

Proof step 55 from proof finfrag-repla-sametrace (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 56 from proof finfrag-repla-sametrace (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, # pcsl0 = # pal1, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Π(ext, aext, ren, aren) !(aren(aa) ') + Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, ⌜aσ[# aal] '⌝ + aσ.drop(# aal + 1), aα0)) ≠ Π(ext, cext, ren, cren) !(ren(a0) ') + Π(ext, cext, ren, cren) !(pal1), Some(e) ≠ None, ext(a0) ≠ None, ⌜# aal⌝ < # aσ, cstep(cs, ca0, cs1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0), frag*(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s1, cs1) ' + pcsl0, pal1), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), step(s, a0, s1), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ σ1. ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + σ1 ≠ aext !(aα0) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 19 terms: β

Additional information

Used terms: β

Proof Tree for finfrag-repla-sametrace (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