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Proof step 1 from proof proj-prefix (conclusion) (back to tree)

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

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof proj-prefix (previous step) (back to tree)

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: projaction-inf-def, seq:⊦ πra(lstep, rstep, lren, rren, ⌜psf⌝ , ⌜paf⌝ ) = lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), subst:[psf, lstep, paf, rren, lren, rstep] → [psf, lstep, paf, rren, lren, rstep]; ['ls, 'ra, 'la, 'pa, 'rs] → ['ls, 'ra, 'la, 'pa, 'rs], <all paths>

Additional information

Used terms: rstep, psf, lren, rren, lstep, paf

Proof step 3 from proof proj-prefix (previous step) (back to tree)

⊦ πra(lstep, rstep, lren, rren, ⌜psf⌝ , ⌜paf⌝ ) = lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )

Lemma projaction-inf-def

Proof step 4 from proof proj-prefix (previous step) (back to tree)

rext !(lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )) = β, fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), ¬ ⌜rext !(πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n)))⌝ ⊑ β ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: streamseq-mon-lim, name:lim-def, seq:⊦ mon(σσ) ∧ σσ ≠ [] → (∀ n. ⌜n⌝ < # σσ → σσ[n] ⊑ lim(σσ)) ∧ (∀ m. ¬ # lim(σσ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # σσ ∧ ⌜lim(σσ).take(m)⌝ ⊑ σσ[n])), subst:[σσ] → [⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ]; ['a] → ['ra]

Additional information

Used terms: ⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝

Proof step 5 from proof proj-prefix (previous step) (back to tree)

⊦ mon(σσ) ∧ σσ ≠ [] → (∀ n. ⌜n⌝ < # σσ → σσ[n] ⊑ lim(σσ)) ∧ (∀ m. ¬ # lim(σσ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # σσ ∧ ⌜lim(σσ).take(m)⌝ ⊑ σσ[n]))

A spec-lemma

Proof step 6 from proof proj-prefix (previous step) (back to tree)

rext !(lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )) = β, fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), ¬ ⌜rext !(πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n)))⌝ ⊑ β, mon(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) ∧ ⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ≠ [] → (∀ n. ⌜n⌝ < # ⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ → ⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ [n] ⊑ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )) ∧ (∀ m. ¬ # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # ⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ∧ ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ [n])) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ ⊑ ⌜y⌝ ↔ x ⊑ y; (s) ⊦ ⌜nf⌝ [n] = nf(n); (s) ⊦ ⌜m⌝ < ∞; (s) ⊦ # ⌜nf⌝ = ∞; (s) ⊦ [] ≠ ⌜nf⌝ ; (s)

Proof step 7 from proof proj-prefix (previous step) (back to tree)

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), ¬ ⌜rext !(πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n)))⌝ ⊑ rext !(lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )), mon(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) → (∀ n0. ⌜πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ ⊑ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )) ∧ (∀ m. ¬ # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n0. lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m) ⊑ πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)))) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 3

Additional information

Proof step 8 from proof proj-prefix (previous step) (back to tree)

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), ¬ mon(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ¬ ⌜rext !(πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n)))⌝ ⊑ rext !(lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

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

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) Used terms: rstart, rstep, psf, lren, rren, lext, rext, lstep, lstart, paf

Proof step 9 from proof proj-prefix (previous step) (back to tree)

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

Lemma mon-pia

Proof step 10 from proof proj-prefix (previous step) (back to tree)

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), ¬ ⌜rext !(πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n)))⌝ ⊑ rext !(lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )), ∀ n0. ⌜πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ ⊑ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n0. lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m) ⊑ πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: n abort: F discard: T computed: F

Additional information

Used terms: n

Proof step 11 from proof proj-prefix (previous step) (back to tree)

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ¬ ⌜rext !(πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n)))⌝ ⊑ rext !(lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )), ∀ m. ¬ # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n0. lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m) ⊑ πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))) ⊦

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] → [⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ , lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )]; ['a] → ['ra], <all paths>

Additional information

Used simplifier rules: ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s) ⊦ σ1 ⊑ σ1 + σ2; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ , lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )

Proof step 12 from proof proj-prefix (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof Tree for proj-prefix (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