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Proof step 1 from proof projstate-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) → πrs(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls0, rs) '⌝ + ⌜psf⌝ , lren(la) + ⌜paf⌝ ) = πrs(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 projstate-infrec-skipL (previous step) (back to tree)

πrs(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls0, rs) '⌝ + ⌜psf⌝ , lren(la) + ⌜paf⌝ ) ≠ πrs(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: pis-inf-simp: ⊦ πrs(lstep, rstep, lren, rren, ⌜ps '⌝ + ⌜psf⌝ , ⌜paf⌝ ) = lim(⌜λ n. ⌜πrs(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 projstate-infrec-skipL (previous step) (back to tree)

πrs(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls0, rs) '⌝ + ⌜psf⌝ , ⌜lren(la) '⌝ + ⌜paf⌝ ) ≠ lim(⌜λ n. ⌜πrs(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:pis-inf-simp, seq:⊦ πrs(lstep, rstep, lren, rren, ⌜ps '⌝ + ⌜psf⌝ , ⌜paf⌝ ) = lim(⌜λ n. ⌜πrs(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 projstate-infrec-skipL (previous step) (back to tree)

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

Lemma pis-inf-simp

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

πrs(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f⌝ , ⌜(⌜lren(la) '⌝ + ⌜paf⌝ ).f⌝ ) = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ), πrs(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls0, rs) '⌝ + ⌜psf⌝ , ⌜lren(la) '⌝ + ⌜paf⌝ ) ≠ lim(⌜λ n. ⌜πrs(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 projstate-infrec-skipL (previous step) (back to tree)

πrs(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls0, rs) '⌝ + ⌜psf⌝ , ⌜lren(la) '⌝ + ⌜paf⌝ ) = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ), πrs(lstep, rstep, lren, rren, ⌜(ls, rs) '⌝ + ⌜(ls0, rs) '⌝ + ⌜psf⌝ , ⌜lren(la) '⌝ + ⌜paf⌝ ) ≠ lim(⌜λ n. ⌜πrs(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 projstate-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πrs(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. ⌜πrs (lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n)) ⌝ , 1]; ['a] → ['rs]

Additional information

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

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

lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πrs(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. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) → lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) = lim(⌜λ n. ⌜πrs(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 projstate-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πrs(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. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) → lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) = lim(⌜λ n. ⌜πrs(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 projstate-infrec-skipL (previous step) (back to tree)

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

Additional information

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

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

lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πrs(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. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ → ⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ [n] ⊑ ⌜λ n. ⌜πrs(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 projstate-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πrs(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)), ¬ ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⊑ ⌜πrs(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: projstate-rec-skipL-one: ⊦ lstep(ls, la, ls0) ∧ ¬ cod(rren)(lren(la)) → πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ', lren(la) ') = πrs(lstep, rstep, lren, rren, (ls0, rs) ', []); projstate-base: ⊦ πrs(lstep, rstep, lren, rren, (ls, rs) ', []) = rs '; Nat freely generated by 0, +1;; ⊦ nf.take(0) = []; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ σ ⊑ σ; (s)

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

n = n0 +1, lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πrs(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)), ¬ ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⊑ ⌜πrs(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 projstate-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πrs(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)), ¬ ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n0), lren(la) ' + paf.take(n0))⌝ ⊑ ⌜πrs(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: projstate-rec-skipL, seq:⊦ # psl = # pal ∧ lstep(ls, la, ls0) ∧ ¬ cod(rren)(lren(la)) → πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psl, lren(la) ' + pal) = πrs(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 17 from proof projstate-infrec-skipL (previous step) (back to tree)

⊦ # psl = # pal ∧ lstep(ls, la, ls0) ∧ ¬ cod(rren)(lren(la)) → πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psl, lren(la) ' + pal) = πrs(lstep, rstep, lren, rren, (ls0, rs) ' + psl, pal)

Lemma projstate-rec-skipL

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

# psf.take(n0) = # paf.take(n0), lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πrs(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)), ¬ ⌜πrs(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n0), paf.take(n0))⌝ ⊑ ⌜πrs(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 19 from proof projstate-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πrs(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)), ¬ ⌜πrs(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n0), paf.take(n0))⌝ ⊑ ⌜πrs(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: projstate-rec-skipL, seq:⊦ # psl = # pal ∧ lstep(ls, la, ls0) ∧ ¬ cod(rren)(lren(la)) → πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psl, lren(la) ' + pal) = πrs(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 20 from proof projstate-infrec-skipL (previous step) (back to tree)

⊦ # psl = # pal ∧ lstep(ls, la, ls0) ∧ ¬ cod(rren)(lren(la)) → πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psl, lren(la) ' + pal) = πrs(lstep, rstep, lren, rren, (ls0, rs) ' + psl, pal)

Lemma projstate-rec-skipL

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

# psf.take(n0 + 1) = # paf.take(n0 + 1), lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πrs(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)), ¬ ⌜πrs(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n0), paf.take(n0))⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n0 + 1), 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 ⊦ m + n = m + n0 ↔ n = n0; (s) ⊦ # nf.take(n) = n; (s)

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

lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πrs(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)), ¬ ⌜πrs(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n0), paf.take(n0))⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n0 + 1), paf.take(n0 + 1))⌝ ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:pis-mon, seq:⊦ fragω(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren)) (psf, paf) → ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ ⌜πrs(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 simplifier rules: ⊦ 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) ⊦ (a ' + nf).take(n + 1) = a ' + nf.take(n); (s) ⊦ (a ' + nf).take(n + 2) = a ' + nf.take(n + 1); (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: rext, paf, n0, rstep, rstart, lren, rren, lstart, lstep, (ls0, rs) ' + psf, lext

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

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

Lemma pis-mon

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

lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n))⌝ ⌝ ), lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πrs(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 projstate-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n))⌝ ⌝ ), lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πrs(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 cut formula

And obtained the following new goals

Rule argument

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

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (⌜a '⌝ + ⌜nf⌝ ).f.take(n + 1) = a ' + nf.take(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 Used formulas: (λ n. ⌜πrs (lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n + 1), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n + 1)) ⌝ ) = (λ n. ⌜πrs(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ )

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

lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n))⌝ ⌝ ), (λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n + 1), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n + 1))⌝ ) ≠ (λ n. ⌜πrs(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ), lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πrs(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) 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 27 from proof projstate-infrec-skipL (previous step) (back to tree)

lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (⌜(ls0, rs) '⌝ + ⌜psf⌝ ).f.take(n), (⌜lren(la) '⌝ + ⌜paf⌝ ).f.take(n))⌝ ⌝ ) = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n))⌝ ⌝ ), lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n))⌝ ⌝ ) ≠ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, (ls0, rs) ' + psf.take(n), paf.take(n))⌝ ⌝ ), πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psf.take(n), lren(la) ' + paf.take(n)) ≠ πrs(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: projstate-rec-skipL, seq:⊦ # psl = # pal ∧ lstep(ls, la, ls0) ∧ ¬ cod(rren)(lren(la)) → πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psl, lren(la) ' + pal) = πrs(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 28 from proof projstate-infrec-skipL (previous step) (back to tree)

⊦ # psl = # pal ∧ lstep(ls, la, ls0) ∧ ¬ cod(rren)(lren(la)) → πrs(lstep, rstep, lren, rren, (ls, rs) ' + (ls0, rs) ' + psl, lren(la) ' + pal) = πrs(lstep, rstep, lren, rren, (ls0, rs) ' + psl, pal)

Lemma projstate-rec-skipL

Proof Tree for projstate-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