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

⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:mon-repla, seq:⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ fragω (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((s, cs) ' + pcsf, paf) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, (s, cs) + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ), subst:[astep, cs, cext, as, start, aσ, ext, ren, astart, cstart, cstep, aren, aext, step, aα, s, pcsf, cren, paf] → [astep, pcsf(0)._2, cext, as, start, aσ, ext, ren, astart, cstart, cstep, aren, aext, step, aα, pcsf(0)._1, λ n. pcsf(n + 1), cren, paf]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used terms: aren, ren, ext, pcsf(0)._1, as, astep, paf, astart, step, aext, cstart, cren, cstep, aα, start, aσ, λ n. pcsf(n + 1), cext, pcsf(0)._2

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

⊦ nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + pcsf, paf) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, (s, cs) + ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (s, cs) ' + pcsf.take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ )

Lemma mon-repla

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

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), nounsharedextR(cren, cext, ren) ∧ Πok(ext, cext, ren, cren) ∧ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, (pcsf(0)._1, pcsf(0)._2) + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcsf(0)._1, pcsf(0)._2) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα) → mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 7

Additional information

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

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcsf(0)._1, pcsf(0)._2) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: intrace-expand, seq: ⊦ ⌜nf⌝ = nf(0) + ⌜λ n. nf(n + 1)⌝ , subst:[nf] → [pcsf]; ['a] → [('s, 'cs)], <all paths>

Additional information

Used terms: pcsf

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

⊦ ⌜nf⌝ = nf(0) + ⌜λ n. nf(n + 1)⌝

A spec-lemma

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

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, pcsf(0) + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcsf(0)._1, pcsf(0)._2) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf(0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcsf(0)._1, pcsf(0)._2) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα)

User applied rule structural induction

And obtained the following new goals

Rule argument

term: pcsf(0)

Additional information

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

(s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf(0) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) ((pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)), paf) ∧ cext !(πra(step, cstep, ren, cren, ⌜(pcsf(0)._1, pcsf(0)._2) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

(s, cs) = pcsf(0), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (λ n. pcsf(n + 1)), paf), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: pcsf = (s, cs) ' + (λ n. pcsf(n + 1))

Additional information

Used formulas: pcsf = (s, cs) ' + (λ n. pcsf(n + 1))

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

(s, cs) = pcsf(0), pcsf ≠ (s, cs) ' + (λ n. pcsf(n + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))((s, cs) ' + (λ n. pcsf(n + 1)), paf), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:add-list-fun-def, seq: ⊦ x + nf = (λ n. (n < # x ⊃ x[n];nf(n - # x))), subst:[x, nf] → [(s, cs) ', λ n. pcsf(n + 1)]; ['a] → [('s, 'cs)], <all paths>

Additional information

Used terms: (s, cs) ', λ n. pcsf(n + 1)

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

⊦ x + nf = (λ n. (n < # x ⊃ x[n];nf(n - # x)))

A spec-lemma

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

(s, cs) = pcsf(0), pcsf ≠ (λ n. (n < # (s, cs) ' ⊃ (s, cs) '[n];pcsf((n - # (s, cs) ') + 1))), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (λ n. (n < # (s, cs) ' ⊃ (s, cs) '[n];pcsf((n - # (s, cs) ') + 1)), paf), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # a ' = 0 +1; (s) ⊦ a '[0] = a; (s) ⊦ m < 1 ↔ m = 0; (s)

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

(s, cs) = pcsf(0), pcsf(n) ≠ (n = 0 ⊃ (s, cs);pcsf((n - 1) + 1)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(λ n0. (n0 = 0 ⊃ (s, cs);pcsf((n0 - 1) + 1)), paf), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 2

Additional information

Proof step 16 from proof mon-repla-1 (previous step) (back to tree)

(s, cs) = pcsf(0), n ≠ 0, pcsf(n) ≠ pcsf((n - 1) + 1), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜(s, cs) '⌝ + ⌜λ n. pcsf(n + 1)⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(λ n0. (n0 = 0 ⊃ (s, cs);pcsf((n0 - 1) + 1)), paf), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

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] → [n, n1]

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: n, n1

Proof step 17 from proof mon-repla-1 (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), mon (⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: streamseq-mon-lim, name: mon-def, seq: ⊦ mon(σσ) ↔ (∀ n. ⌜n⌝ + ⌜1⌝ < # σσ → σσ[n] ⊑ σσ[n + 1]), subst:[σσ] → [⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝ ]; ['a] → ['pa], <all paths>

Additional information

Used terms: ⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝

Proof step 18 from proof mon-repla-1 (previous step) (back to tree)

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

A spec-lemma

Proof step 19 from proof mon-repla-1 (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ n. ⌜n⌝ + ⌜1⌝ < # ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ → ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ [n ] ⊑ ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ [n + 1 ] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 20 from proof mon-repla-1 (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ mon(⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ ), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

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

Additional information

Used terms: ⌜λ n. ⌜repla (step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα) ⌝ ⌝

Proof step 21 from proof mon-repla-1 (previous step) (back to tree)

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

A spec-lemma

Proof step 22 from proof mon-repla-1 (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝ , ∃ n. ¬ ( ⌜n⌝ + ⌜1⌝ < # ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ → ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ [n] ⊑ ⌜λ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⌝ [n + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ⌜nf⌝ [n] = nf(n); (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ n∞ < ∞ ↔ n∞ ≠ ∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 23 from proof mon-repla-1 (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(n + 1), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: to-succ-nf, seq: ⊦ nf.take(n + 1) = nf(0) + (λ n. nf(n + 1)).take(n), subst:[nf, n] → [pcsf, n]; ['a] → [('s, 'cs)], <all paths>

Additional information

Used terms: pcsf, n

Proof step 24 from proof mon-repla-1 (previous step) (back to tree)

⊦ nf.take(n + 1) = nf(0) + (λ n. nf(n + 1)).take(n)

A spec-lemma

Proof step 25 from proof mon-repla-1 (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf(0) + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 26 from proof mon-repla-1 (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf(0) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf.take(2 + n), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: to-succ2-nf, seq: ⊦ nf.take(n + 2) = nf(0) + (λ n. nf(n + 1)).take(n + 1), subst:[nf, n] → [pcsf, n]; ['a] → [('s, 'cs)], <all paths>

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic Used terms: pcsf, n

Proof step 27 from proof mon-repla-1 (previous step) (back to tree)

⊦ nf.take(n + 2) = nf(0) + (λ n. nf(n + 1)).take(n + 1)

A spec-lemma

Proof step 28 from proof mon-repla-1 (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf(0) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf(0) + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 29 from proof mon-repla-1 (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf(0) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf(0) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝ , ∀ n. ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝ ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 7 terms: n abort: F discard: T computed: F

Additional information

Used terms: n

Proof step 30 from proof mon-repla-1 (previous step) (back to tree)

frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα), fragω(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcsf, paf), nounsharedextR(cren, cext, ren), Πok(ext, cext, ren, cren), cext !(πra(step, cstep, ren, cren, ⌜pcsf⌝ , ⌜paf⌝ )) ⊑ aext !(aα), ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, (pcsf(0)._1, pcsf(0)._2) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝ , ¬ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf(0) ' + (λ n. pcsf(n + 1)).take(n), paf.take(n), ⌜as '⌝ + aσ, aα)⌝ ⊑ ⌜repla(step, cstep, cren, ren, aren, cext, aext, pcsf(0) ' + (λ n. pcsf(n + 1)).take(n + 1), paf.take(n + 1), ⌜as '⌝ + aσ, aα)⌝ ⊦

User applied rule structural induction

And closed the goal!

Rule argument

term: pcsf(0)

Additional information

Proof Tree for mon-repla-1 (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.

the proof tree