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Proof step 1 from proof projaction-infrec-skipL (conclusion) (back to tree)

⊦ Πok(lext, rext, lren, rren) ∧ lstep(ls, la, ls0) ∧ ¬ cod(rren)(lren(la)) ∧ fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf) → πra(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls0, rs) '⌝ + ⌜psf⌝ , lren(la) + ⌜paf⌝ ) = πra(lstep, rstep, lren, rren, ⌜(ls0, rs) '⌝ + ⌜psf⌝ , ⌜paf⌝ )

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof projaction-infrec-skipL (previous step) (back to tree)

πra(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls0, rs) '⌝ + ⌜psf⌝ , lren(la) + ⌜paf⌝ ) ≠ πra(lstep, rstep, lren, rren, ⌜(ls0, rs) '⌝ + ⌜psf⌝ , ⌜paf⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: pia-inf-simp: ⊦ πra(lstep, rstep, lren, rren, ⌜ps '⌝ + ⌜psf⌝ , ⌜paf⌝ ) = lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, ps ' + psf.take(n), paf.take(n))⌝ ⌝ ); append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ a + σ = ⌜a '⌝ + σ; (s)

Proof step 3 from proof projaction-infrec-skipL (previous step) (back to tree)

πra(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls0, rs) '⌝ + ⌜psf⌝ , ⌜lren(la) '⌝ + ⌜paf⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:pia-inf-simp, seq:⊦ πra(lstep, rstep, lren, rren, ⌜ps '⌝ + ⌜psf⌝ , ⌜paf⌝ ) = lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, ps ' + psf.take(n), paf.take(n))⌝ ⌝ ), subst:[psf, lstep, rren, lren, rstep, paf, ps] → [(⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f, lstep, rren, lren, rstep, (⌜lren(la) '⌝ + ⌜paf⌝ ).f, (ls, rs)]; ['ls, 'ra, 'la, 'pa, 'rs] → ['ls, 'ra, 'la, 'pa, 'rs]

Additional information

Used terms: (⌜lren(la) '⌝ + ⌜paf⌝ ).f, rstep, (ls, rs), lren, rren, lstep, (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f

Proof step 4 from proof projaction-infrec-skipL (previous step) (back to tree)

⊦ πra(lstep, rstep, lren, rren, ⌜ps '⌝ + ⌜psf⌝ , ⌜paf⌝ ) = lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, ps ' + psf.take(n), paf.take(n))⌝ ⌝ )

Lemma pia-inf-simp

Proof step 5 from proof projaction-infrec-skipL (previous step) (back to tree)

πra(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f⌝ , ⌜(⌜lren(la) '⌝ + ⌜paf⌝ ).f⌝ ) = lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ), πra(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls0, rs) '⌝ + ⌜psf⌝ , ⌜lren(la) '⌝ + ⌜paf⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)) ⊦

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 ⊦ ⌜(⌜x⌝ + ⌜nf⌝ ).f⌝ = ⌜x⌝ + ⌜nf⌝ ; (s)

Proof step 6 from proof projaction-infrec-skipL (previous step) (back to tree)

πra(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls0, rs) '⌝ + ⌜psf⌝ , ⌜lren(la) '⌝ + ⌜paf⌝ ) = lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ), πra(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls0, rs) '⌝ + ⌜psf⌝ , ⌜lren(la) '⌝ + ⌜paf⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags: discard

Additional information

Proof step 7 from proof projaction-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: streamseq-mon-lim, name:lim-postfix, seq: ⊦ mon(⌜σf⌝ ) → lim(⌜σf⌝ ) = lim(⌜λ n. σf(n + m)⌝ ), subst:[σf, m] → [λ n. ⌜πra (lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n)) ⌝ , 1]; ['a] → ['ra]

Additional information

Used terms: λ n. ⌜πra (lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n)) ⌝ , 1

Proof step 8 from proof projaction-infrec-skipL (previous step) (back to tree)

⊦ mon(⌜σf⌝ ) → lim(⌜σf⌝ ) = lim(⌜λ n. σf(n + m)⌝ )

A spec-lemma

Proof step 9 from proof projaction-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)), mon(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) → lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) = lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n + 1), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n + 1))⌝ ⌝ ) ⊦

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 '⌝ + ⌜nf⌝ ).f.take(n + 1) = a ' + nf.take(n); (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 10 from proof projaction-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)), mon(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) → lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) = lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n))⌝ ⌝ ) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 6

Additional information

Proof step 11 from proof projaction-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)), ¬ mon(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ⊦

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. ⌜πra (lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n)) ⌝ ⌝ ]; ['a] → ['ra], <all paths>

Additional information

Used terms: ⌜λ n. ⌜πra (lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n)) ⌝ ⌝

Proof step 12 from proof projaction-infrec-skipL (previous step) (back to tree)

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

A spec-lemma

Proof step 13 from proof projaction-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)), ∃ n. ¬ ( ⌜n⌝ + ⌜1⌝ < # ⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ → ⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ [n] ⊑ ⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ [n + 1]) ⊦

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 '⌝ + ⌜nf⌝ ).f.take(n + 1) = a ' + nf.take(n); (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ⌜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 14 from proof projaction-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)), ¬ ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n))⌝ ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: n

Additional information

Used simplifier rules: projaction-rec-skipL-one: ⊦ lstep(ls, la, ls0) ∧ ¬ cod(rren)(lren(la)) → πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ', lren(la) ') = πra(lstep, rstep, lren, rren, (ls0, rs) ', []); projaction-base: ⊦ πra(lstep, rstep, lren, rren, (ls, rs) ', []) = []; Nat freely generated by 0, +1;; ⊦ nf.take(0) = []; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ⌜[]⌝ = []; (s) ⊦ [] ⊑ σ; (s)

Proof step 15 from proof projaction-infrec-skipL (previous step) (back to tree)

n = n0 +1, lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)), ¬ ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n))⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) a: ⊦ (x + y) + z = x + y + z; from specification list 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 '⌝ + ⌜nf⌝ ).f.take(n + 1) = a ' + nf.take(n); (s)

Proof step 16 from proof projaction-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)), ¬ ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n0), lren(la) ' + paf.take(n0))⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n0 + 1), lren(la) ' + paf.take(n0 + 1))⌝ ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:pia-mon, seq:⊦ fragω(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren)) (psf, paf) → ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, psf.take(2 + n), paf.take(n + 1))⌝ , subst:[psf, lstart, n, rstart, rext, lstep, paf, rren, lren, lext, rstep] → [(ls0, rs) ' + psf, lstart, n0, rstart, rext, lstep, paf, rren, lren, lext, rstep]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['ls, 'ra, 'la, 'pa, 'rs, 'e]

Additional information

Used terms: rext, paf, n0, rstep, rstart, lren, rren, lstart, lstep, (ls0, rs) ' + psf, lext

Proof step 17 from proof projaction-infrec-skipL (previous step) (back to tree)

⊦ fragω(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren))(psf, paf) → ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, psf.take(2 + n), paf.take(n + 1))⌝

Lemma pia-mon

Proof step 18 from proof projaction-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)), ¬ ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n0), lren(la) ' + paf.take(n0))⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n0 + 1), lren(la) ' + paf.take(n0 + 1))⌝ , fragω(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren))((ls0, rs) ' + psf, paf) → ⌜πra(lstep, rstep, lren, rren, ((ls0, rs) ' + psf).take(n0 + 1), paf.take(n0))⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, ((ls0, rs) ' + psf).take(2 + n0), paf.take(n0 + 1))⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + nf).take(n + 2) = a ' + nf.take(n + 1); (s) ⊦ (a ' + nf).take(n + 1) = a ' + nf.take(n); (s) ⊦ Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren) = IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)); (s)

Proof step 19 from proof projaction-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n0), paf.take(n0))⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n0 + 1), paf.take(n0 + 1))⌝ , ¬ cod(rren)(lren(la)), ¬ ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n0), lren(la) ' + paf.take(n0))⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n0 + 1), lren(la) ' + paf.take(n0 + 1))⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: projaction-rec-skipL, seq:⊦ # 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), subst:[lstep, la, ls, rs, ls0, pal, lren, rstep, rren, psl] → [lstep, la, ls, rs, ls0, paf.take(n0), lren, rstep, rren, psf.take(n0)]; ['ls, 'ra, 'la, 'pa, 'rs] → ['ls, 'ra, 'la, 'pa, 'rs], <all paths>

Additional information

Used simplifier rules: ⊦ # nf.take(n) = n; (s) Used terms: psf.take(n0), rs, paf.take(n0), rstep, lren, rren, la, lstep, ls, ls0

Proof step 20 from proof projaction-infrec-skipL (previous step) (back to tree)

⊦ # 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)

Lemma projaction-rec-skipL

Proof step 21 from proof projaction-infrec-skipL (previous step) (back to tree)

# psf.take(n0) = # paf.take(n0), lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n0), paf.take(n0))⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n0 + 1), paf.take(n0 + 1))⌝ , ¬ cod(rren)(lren(la)), ¬ ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n0), paf.take(n0))⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n0 + 1), lren(la) ' + paf.take(n0 + 1))⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # nf.take(n) = n; (s)

Proof step 22 from proof projaction-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n0), paf.take(n0))⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n0 + 1), paf.take(n0 + 1))⌝ , ¬ cod(rren)(lren(la)), ¬ ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n0), paf.take(n0))⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n0 + 1), lren(la) ' + paf.take(n0 + 1))⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: projaction-rec-skipL, seq:⊦ # 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), subst:[lstep, la, ls, rs, ls0, pal, lren, rstep, rren, psl] → [lstep, la, ls, rs, ls0, paf.take(n0 + 1), lren, rstep, rren, psf.take(n0 + 1)]; ['ls, 'ra, 'la, 'pa, 'rs] → ['ls, 'ra, 'la, 'pa, 'rs], <all paths>

Additional information

Used simplifier rules: ⊦ # nf.take(n) = n; (s) ⊦ m + n = m + n0 ↔ n = n0; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: psf.take(n0 + 1), rs, paf.take(n0 + 1), rstep, lren, rren, la, lstep, ls, ls0

Proof step 23 from proof projaction-infrec-skipL (previous step) (back to tree)

⊦ # 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)

Lemma projaction-rec-skipL

Proof step 24 from proof projaction-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) = lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n))⌝ ⌝ ), lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 25 from proof projaction-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) = lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n))⌝ ⌝ ), lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)) ⊦

User applied rule weakening

And obtained the following new goals

Rule argument

poslist: left 1

Additional information

Proof step 26 from proof projaction-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)) ⊦

User applied rule reduce equation

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Proof step 27 from proof projaction-infrec-skipL (previous step) (back to tree)

⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n))⌝ ⌝ ≠ ⌜λ n. ⌜πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ , fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s) ⊦ ⌜nf⌝ = ⌜nf0⌝ ↔ nf = nf0; (s)

Proof step 28 from proof projaction-infrec-skipL (previous step) (back to tree)

πra(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n)) ≠ πra(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n)), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))((ls0, rs) ' + psf, paf), lstep(ls, la, ls0), Πok(lext, rext, lren, rren), ¬ cod(rren)(lren(la)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: projaction-rec-skipL, seq:⊦ # 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), subst:[lstep, la, ls, rs, ls0, pal, lren, rstep, rren, psl] → [lstep, la, ls, rs, ls0, paf.take(n), lren, rstep, rren, psf.take(n)]; ['ls, 'ra, 'la, 'pa, 'rs] → ['ls, 'ra, 'la, 'pa, 'rs], <all paths>

Additional information

Used simplifier rules: ⊦ # nf.take(n) = n; (s) Used terms: psf.take(n), rs, paf.take(n), rstep, lren, rren, la, lstep, ls, ls0

Proof step 29 from proof projaction-infrec-skipL (previous step) (back to tree)

⊦ # 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)

Lemma projaction-rec-skipL

Proof Tree for projaction-infrec-skipL (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