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Proof step 1 from proof finfrag-repls-head (conclusion) (back to tree)

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

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof finfrag-repls-head (previous step) (back to tree)

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), Πok(ext, cext, ren, cren), cext !(⌜πra(step, cstep, ren, cren, (s, cs) ' + pcsl, pal)⌝ ) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα).head = (s, as)

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: # pcsl = # pal

Additional information

Used simplifier rules: ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ frag(ioa)(σ, ⌜al⌝ ) ↔ frag*(ioa)(σ.l, al) ∧ finp(σ); (s) ⊦ (⌜x⌝ + ⌜y⌝ ).l = x + y; (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))(s ' + sl, al) → (# sl = # al ↔ true); (ss) Used formulas: # pcsl = # pal

Proof step 3 from proof finfrag-repls-head (previous step) (back to tree)

# pcsl = # pal, 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), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(⌜πra(step, cstep, ren, cren, (s, cs) ' + pcsl, pal)⌝ ) ⊑ aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα).head = (s, as)

User applied rule structural induction

And obtained the following new goals

Rule argument

term: pcsl

Additional information

Used simplifier rules: repls-base: ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ', [], ⌜as '⌝ + aσ, aα) = (s, as) '; List('a) freely generated by [], +;; ⊦ # [] = 0; (s) ⊦ # x = 0 ↔ x = []; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ σ + [] = σ; (s) ⊦ frag(IOAut(start, step, ext))(⌜s '⌝ , α) ↔ α = []; (s) ⊦ ⌜x⌝ = [] ↔ x = []; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ πra(lstep, rstep, lren, rren, (ls, rs) ', []) = []; (s) ⊦ ext !([]) = []; (s) ⊦ ⌜[]⌝ = []; (s) ⊦ [] ⊑ σ; (s) ⊦ a ' ≠ []; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ a '.head = a; (s)

Proof step 4 from proof finfrag-repls-head (previous step) (back to tree)

#(pcs + pcsl0) = # 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), Πok(ext, cext, ren, cren), cext !(⌜πra(step, cstep, ren, cren, (s, cs) ' + (pcs + pcsl0), pal)⌝ ) ⊑ aext !(aα), ∀ pal, start, cstart, step, cstep, ren, cren, ext, cext, s, cs, astart, astep, aext, as, aσ, aα, aren. # pcsl0 = # pal ∧ frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(⌜(s, cs) '⌝ + ⌜pcsl0⌝ , ⌜pal⌝ ) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ cext !(⌜πra(step, cstep, ren, cren, (s, cs) ' + pcsl0, pal)⌝ ) ⊑ aext !(aα) → repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl0, pal, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsl0, pal, ⌜as '⌝ + aσ, aα).head = (s, as) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (pcs + pcsl0), pal, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (pcs + pcsl0), pal, ⌜as '⌝ + aσ, aα).head = (s, as)

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 ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ a + x = a ' + x; (s) ⊦ frag*(IOAut(start, step, ext))(s ' + sl, al) → (# sl = # al ↔ true); (ss) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (⌜x⌝ + ⌜y⌝ ).l = x + y; (s) ⊦ frag(ioa)(σ, ⌜al⌝ ) ↔ frag*(ioa)(σ.l, al) ∧ finp(σ); (s) ⊦ ⌜x⌝ + ⌜y⌝ + ⌜z⌝ = ⌜x0⌝ ↔ x + y + z = x0; (s) ⊦ finp(σ) ↔ (∃ x. ⌜x⌝ = σ); (as) ⊦ (⌜x⌝ + ⌜y⌝ + ⌜y0⌝ ).l = x + y + y0; (s) ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 5 from proof finfrag-repls-head (previous step) (back to tree)

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

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 6 from proof finfrag-repls-head (previous step) (back to tree)

# pcsl0 + 1 = # pal, pal = pa + 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) ' + pcs ' + pcsl0, pal), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + pcs ' + pcsl0, pal))⌝ ⊑ aext !(aα), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcs ' + pcsl0, pal, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcs ' + pcsl0, pal, ⌜as '⌝ + aσ, aα).head = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (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) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 7 from proof finfrag-repls-head (previous step) (back to tree)

# pcsl0 = # 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) ' + pcs ' + pcsl0, pa ' + pal1), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + pcs ' + pcsl0, pa ' + pal1))⌝ ⊑ aext !(aα), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcs ' + pcsl0, pa ' + pal1, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcs ' + pcsl0, pa ' + pal1, ⌜as '⌝ + aσ, aα).head = (s, as)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: pcs

Additional information

Proof step 8 from proof finfrag-repls-head (previous step) (back to tree)

# pcsl0 = # pal1, pcs = (s1, 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)))((s, cs) ' + pcs ' + pcsl0, pa ' + pal1), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + pcs ' + pcsl0, pa ' + pal1))⌝ ⊑ aext !(aα), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcs ' + pcsl0, pa ' + pal1, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcs ' + pcsl0, pa ' + pal1, ⌜as '⌝ + aσ, aα).head = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 9 from proof finfrag-repls-head (previous step) (back to tree)

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

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-repls-head (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-repls-head (previous step) (back to tree)

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

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-repls-head (previous step) (back to tree)

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

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-repls-head (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-repls-head (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 15 from proof finfrag-repls-head (previous step) (back to tree)

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

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 8

Additional information

Used simplifier rules: repls-rec-addS: ⊦ step(s, a, s0) ∧ ¬ cod(cren)(ren(a)) → repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs) ' + pcsl, ren(a) ' + pal, ⌜as '⌝ + aσ, aα) = (s, as) ' + repls (step, cstep, cren, ren, aren, cext, aext, (s0, cs) ' + pcsl, pal, ⌜as '⌝ + aσ, aα); ⊦ # 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) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (a ' + x).head = a; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 16 from proof finfrag-repls-head (previous step) (back to tree)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: nounsharedextR-notcodren: ⊦ ¬ cod(ren)(cren(ca)) ∧ nounsharedextR(cren, cext, ren) → cext !(ca ') = []; repls-rec-skip: ⊦ cstep(cs, ca, cs0) ∧ ¬ cod(ren)(cren(ca)) → repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s, cs0) ' + pcsl, cren(ca) ' + pal, ⌜as '⌝ + aσ, aα) = repls (step, cstep, cren, ren, aren, cext, aext, (s, cs0) ' + pcsl, pal, ⌜as '⌝ + aσ, aα); a: ⊦ (x + y) + z = x + y + z; from specification list append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ [] + 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)

Proof step 17 from proof finfrag-repls-head (previous step) (back to tree)

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

User applied rule all left

And closed the goal!

Rule argument

pos: left 9 terms: pal1, start, cstart, step, cstep, ren, cren, ext, cext, s1, cs1, astart, astep, aext, as, aσ, aα, aren abort: F discard: T computed: T

Additional information

Used simplifier rules: ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss) ⊦ σ0 + ⌜x⌝ = ⌜nf0⌝ ↔ σ0 = ⌜nf0⌝ ; (s) ⊦ ⌜x⌝ ≠ ⌜nf⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: pal1, start, cstart, step, cstep, ren, cren, ext, cext, s1, cs1, astart, astep, aext, as, aσ, aα, aren

Proof step 18 from proof finfrag-repls-head (previous step) (back to tree)

# pcsl0 = # pal1, cren(ca0) = ren(a0), pa = cren(ca0), 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), step(s, a0, s1), Πok(ext, cext, ren, cren), ⌜cext !(πra(step, cstep, ren, cren, (s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1))⌝ ⊑ aext !(aα), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, pa ' + pal1, ⌜as '⌝ + aσ, aα).head = (s, as)

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 19 from proof finfrag-repls-head (previous step) (back to tree)

# pcsl0 = # pal1, cren(ca0) = ren(a0), 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), step(s, a0, s1), Πok(ext, cext, ren, cren), ⌜cext(ca0).get '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα).head = (s, as)

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 20 from proof finfrag-repls-head (previous step) (back to tree)

# pcsl0 = # pal1, cext(ca0) = Some(e), cren(ca0) = ren(a0), 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), step(s, a0, s1), Πok(ext, cext, ren, cren), ⌜e '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα).head = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 21 from proof finfrag-repls-head (previous step) (back to tree)

# pcsl0 = # pal1, cext(ca0) = Some(e), cren(ca0) = ren(a0), 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), step(s, a0, s1), Πok(ext, cext, ren, cren), ⌜e '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ⊑ aext !(aα), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα).head = (s, as)

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 22 from proof finfrag-repls-head (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 23 from proof finfrag-repls-head (previous step) (back to tree)

# pcsl0 = # pal1, cext(ca0) = Some(e), cren(ca0) = ren(a0), 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), step(s, a0, s1), Πok(ext, cext, ren, cren), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0), ∃ σ1. (⌜e '⌝ + ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ ) + σ1 = aext !(aα) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα).head = (s, as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 24 from proof finfrag-repls-head (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), 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), step(s, a0, s1), Πok(ext, cext, ren, cren), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα).head = (s, as)

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 25 from proof finfrag-repls-head (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 26 from proof finfrag-repls-head (previous step) (back to tree)

# pcsl0 = # pal1, cext(ca0) = Some(e), cren(ca0) = ren(a0), 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), step(s, a0, s1), Πok(ext, cext, ren, cren), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0), ∃ 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 ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, aα).head = (s, as)

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 27 from proof finfrag-repls-head (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), 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), step(s, a0, s1), Πok(ext, cext, ren, cren), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0) ⊦ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0) ≠ [] ∧ repls(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s1, cs1) ' + pcsl0, ren(a0) ' + pal1, ⌜as '⌝ + aσ, ⌜aal⌝ + ⌜aa '⌝ + aα0).head = (s, as)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: repls-rec-addboth, seq:⊦ step(s, a, s0) ∧ cstep(cs, ca, cs0) ∧ aext !(⌜aal⌝ ) = [] ∧ cext(ca) = aext(aa) ∧ cren(ca) = ren(a) ∧ n = # aal → repls (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + (s0, cs0) ' + pcsl, ren(a) ' + pal, aσ, ⌜aal⌝ + ⌜aa '⌝ + aα) = lift(aσ.take(n), s) + (s, aσ[n]) ' + repls (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 28 from proof finfrag-repls-head (previous step) (back to tree)

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

Lemma repls-rec-addboth

Proof step 29 from proof finfrag-repls-head (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), 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), step(s, a0, s1), Πok(ext, cext, ren, cren), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0) ⊦ lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, (⌜as '⌝ + aσ).drop(# aal + 1), aα0) ≠ [] ∧ ( lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' + repls(step, cstep, cren, ren, aren, cext, aext, (s1, cs1) ' + pcsl0, pal1, (⌜as '⌝ + aσ).drop(# aal + 1), aα0) ).head = (s, as)

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 + a ' + y).head = (x + a ').head; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ n + 0 = n; (s) ⊦ n ≠ 0 → (⌜a '⌝ + σ).drop((* n) + m) = σ.drop((n - 1) + m); (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ ⌜x⌝ = [] ↔ x = []; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 30 from proof finfrag-repls-head (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), (lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ').head ≠ (s, as), 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), step(s, a0, s1), Πok(ext, cext, ren, cren), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0) ⊦

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] → [lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ', pas, x0]; ['a] → [('s, 'as)]

Additional information

Used simplifier rules: ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list x ≠ [] ⊦ a = x.head ∧ y = x.tail ↔ x = a + y; (elim) Used terms: lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ', pas, x0

Proof step 31 from proof finfrag-repls-head (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, aext !(aal) = [], # pcsl0 = # pal1, lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' = pas + x0, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' ≠ [], Some(e) ≠ None, ext(a0) ≠ None, pas ≠ (s, as), 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), step(s, a0, s1), Πok(ext, cext, ren, cren), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0) ⊦

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 ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a + x = a ' + x; (s)

Proof step 32 from proof finfrag-repls-head (previous step) (back to tree)

aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, aext !(aal) = [], # pcsl0 = # pal1, lift((⌜as '⌝ + aσ).take(# aal), s) + (s, (⌜as '⌝ + aσ)[# aal]) ' = pas ' + x0, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, pas ≠ (s, as), 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), step(s, a0, s1), Πok(ext, cext, ren, cren), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0), (s, (⌜as '⌝ + aσ)[# aal]) = pas ∨ (s, (⌜as '⌝ + aσ)[# aal]) ∈ x0, pas ∈ lift((⌜as '⌝ + aσ).take(# aal), s) ∨ pas = (s, (⌜as '⌝ + aσ)[# aal]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: get-append-notls, seq: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n], subst:[σ, σ1, m] → [⌜as '⌝ , aσ, # aal]; ['a] → ['as], <all paths>

Additional information

Used simplifier rules: lift-base: ⊦ lift([], s) = []; ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]; (s) ⊦ ⌜x⌝ [n] = x[n]; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ # a ' = 0 +1; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ # x = 0 ↔ x = []; (s) in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ σ.take(0) = []; (s) ⊦ # [] = 0; (s) ⊦ a '[0] = a; (s) ⊦ [] + x = x; (s) ⊦ a ' = a0 ' + x ↔ a = a0 ∧ x = []; (ws) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: ⌜as '⌝ , aσ, # aal

Proof step 33 from proof finfrag-repls-head (previous step) (back to tree)

⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]

A spec-lemma

Proof step 34 from proof finfrag-repls-head (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, # pcsl0 = # pal1, lift((⌜as '⌝ + aσ).take(# aal), s) + (s, aσ[# aal - (# ⌜as '⌝ ).n]) ' = pas ' + x0, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, ext(a0) ≠ None, pas ≠ (s, as), 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), step(s, a0, s1), Πok(ext, cext, ren, cren), ¬ ⌜# aal⌝ < # ⌜as '⌝ , ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0), pas ∈ lift((⌜as '⌝ + aσ).take(# aal), s) ∨ pas = (s, aσ[# aal - (# ⌜as '⌝ ).n]), (s, aσ[# aal - (# ⌜as '⌝ ).n]) = pas ∨ (s, aσ[# aal - (# ⌜as '⌝ ).n]) ∈ x0 ⊦

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 ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list ⊦ ⌜n⌝ .n = n; (s) ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ # x = 0 ↔ x = []; (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)

Proof step 35 from proof finfrag-repls-head (previous step) (back to tree)

aext !(aal) = [], aext !(aα0) = ⌜cext !(πra(step, cstep, ren, cren, (s1, cs1) ' + pcsl0, pal1))⌝ + β, # pcsl0 = # pal1, lift((⌜as '⌝ + aσ).take(# aal), s) + (s, aσ[# aal - 1]) ' = pas ' + x0, aext(aa) = Some(e), cext(ca0) = Some(e), cren(ca0) = ren(a0), Some(e) ≠ None, aal ≠ [], ext(a0) ≠ None, pas ≠ (s, as), 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), step(s, a0, s1), Πok(ext, cext, ren, cren), ∀ pal0, start0, cstart0, step0, cstep0, ren0, cren0, ext0, cext0, s0, cs0, astart0, astep0, aext0, as0, aσ0, aα0, aren0. frag*(IOAut(Π(start0, cstart0), Π(step0, cstep0, ren0, cren0, ext0, cext0), Π(ext0, cext0, ren0, cren0)))((s0, cs0) ' + pcsl0, pal0) ∧ finp(⌜(s0, cs0) '⌝ + ⌜pcsl0⌝ ) ∧ frag(IOAut(astart0, astep0, aext0))(⌜as0 '⌝ + aσ0, aα0) ∧ nounsharedextR(cren0, cext0, ren0) ∧ Πok(ext0, cext0, ren0, cren0) ∧ ⌜cext0 !(πra(step0, cstep0, ren0, cren0, (s0, cs0) ' + pcsl0, pal0))⌝ ⊑ aext0 !(aα0) → repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0) ≠ [] ∧ repls(step0, cstep0, cren0, ren0, aren0, cext0, aext0, (s0, cs0) ' + pcsl0, pal0, ⌜as0 '⌝ + aσ0, aα0).head = (s0, as0), (s, aσ[# aal - 1]) = pas ∨ (s, aσ[# aal - 1]) ∈ x0, pas ∈ lift((⌜as '⌝ + aσ).take(# aal), s) ∨ pas = (s, aσ[# aal - 1]) ⊦

Heuristic elimination applied rule apply elim lemma

And closed the goal!

Rule argument

spec: nat, name:elim-pred-c, seq:n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1, subst:[n, m] → [# aal, n0]

Additional information

Used simplifier rules: lift-rec: ⊦ lift(as ' + asl, s) = (s, as) ' + lift(asl, s); ⊦ # x = 0 ↔ x = []; (s) in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ n ≠ 0 → (⌜a '⌝ + σ).take((* n) + m) = a ' + σ.take((n - 1) + m); (s) ⊦ n + 0 = n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ a ' + x = a0 ' + y ↔ a = a0 ∧ x = y; (ws) a: ⊦ (x + y) + z = x + y + z; from specification list n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: # aal, n0

Proof Tree for finfrag-repls-head (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