[Imprint][Data Privacy Statement][E-Mail]



View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

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

execω(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren))(psf, paf) ⊦ exec(IOAut(rstart, rstep, rext))(πrs(lstep, rstep, lren, rren, ⌜psf⌝ , ⌜paf⌝ ), πra(lstep, rstep, lren, rren, ⌜psf⌝ , ⌜paf⌝ ))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

execω(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren))(psf, paf), ¬ exec(IOAut(rstart, rstep, rext))(πrs(lstep, rstep, lren, rren, ⌜psf⌝ , ⌜paf⌝ ), πra(lstep, rstep, lren, rren, ⌜psf⌝ , ⌜paf⌝ )) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: projstate-inf-def: ⊦ πrs(lstep, rstep, lren, rren, ⌜psf⌝ , ⌜paf⌝ ) = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ); projaction-inf-def: ⊦ πra(lstep, rstep, lren, rren, ⌜psf⌝ , ⌜paf⌝ ) = lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ); ⊦ IOAut(start, step, ext).start = start; (s) ⊦ exec(ioa)(σ, α) ↔ frag(ioa)(σ, α) ∧ ioa.start(σ[0]); (s) ⊦ execω(ioa)(sf, af) ↔ fragω(ioa)(sf, af) ∧ ioa.start(sf(0)); (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 3 from proof proj-exec-inf (previous step) (back to tree)

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)) ⊦ frag(IOAut(rstart, rstep, rext)) (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )) ∧ rstart(lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[0])

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. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ]; ['a] → ['rs]

Additional information

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

Proof step 4 from proof proj-exec-inf (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 5 from proof proj-exec-inf (previous step) (back to tree)

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), mon(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) ∧ ⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ≠ [] → (∀ n. ⌜n⌝ < # ⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ → ⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ [n] ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )) ∧ (∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # ⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ∧ ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ [n])) ⊦ frag(IOAut(rstart, rstep, rext)) (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )) ∧ rstart(lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[0])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), mon(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) → (∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )) ∧ (∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ )) ⊦ frag(IOAut(rstart, rstep, rext)) (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )) ∧ rstart(lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[0])

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 3

Additional information

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

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ mon(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) ⊦ frag(IOAut(rstart, rstep, rext)) (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )) ∧ rstart(lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[0])

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:mon-pis, seq:⊦ fragω(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren)) (psf, paf) → mon(⌜λ n. ⌜πrs(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]

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

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

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

Lemma mon-pis

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

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ) ⊦ frag(IOAut(rstart, rstep, rext)) (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )) ∧ rstart(lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[0])

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 10 from proof proj-exec-inf (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 11 from proof proj-exec-inf (previous step) (back to tree)

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), 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])), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) ⊦ frag(IOAut(rstart, rstep, rext)) (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )) ∧ rstart(lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[0])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), mon(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) → (∀ 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))⌝ ⌝ )) ∧ (∀ m. ¬ # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ )), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ) ⊦ frag(IOAut(rstart, rstep, rext)) (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )) ∧ rstart(lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[0])

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 3

Additional information

Proof step 13 from proof proj-exec-inf (previous step) (back to tree)

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ mon(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) ⊦ frag(IOAut(rstart, rstep, rext)) (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )) ∧ rstart(lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[0])

User applied rule apply 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]

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

Proof step 14 from proof proj-exec-inf (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 15 from proof proj-exec-inf (previous step) (back to tree)

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ 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))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ) ⊦ frag(IOAut(rstart, rstep, rext)) (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )) ∧ rstart(lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[0])

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 16 from proof proj-exec-inf (previous step) (back to tree)

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ frag(IOAut(rstart, rstep, rext)) (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )), ∀ m. ¬ # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(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: IOtrace, name: frag, seq:⊦ frag(IOAut(start, step, ext))(σ, α) ↔ # σ = # α + ⌜1⌝ ∧ (∀ n. ⌜n⌝ < # α → step(σ[n], α[n], σ[n +1])), subst:[σ, step, α, start, ext] → [lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), rstep, lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), rstart, rext]; ['s, 'a, 'e] → ['rs, 'ra, 'e], <all paths>

Additional information

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

Proof step 17 from proof proj-exec-inf (previous step) (back to tree)

⊦ frag(IOAut(start, step, ext))(σ, α) ↔ # σ = # α + ⌜1⌝ ∧ (∀ n. ⌜n⌝ < # α → step(σ[n], α[n], σ[n +1]))

A spec-lemma

Proof step 18 from proof proj-exec-inf (previous step) (back to tree)

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ 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))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), # lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ ∨ (∃ n. ¬ ( ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) → rstep (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n +1]))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)

Proof step 19 from proof proj-exec-inf (previous step) (back to tree)

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ∀ m. ¬ # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), # lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ ∨ ¬ ( ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) → rstep (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1])) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:inffrag-lim-size-pis-pia, seq:fragω(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren))(psf, paf) ⊦ # lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , 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]

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

Proof step 20 from proof proj-exec-inf (previous step) (back to tree)

fragω(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren))(psf, paf) ⊦ # lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝

Lemma inffrag-lim-size-pis-pia

Proof step 21 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ 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))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), # lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ ∨ ¬ ( ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) → rstep (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1])) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 22 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ∀ m. ¬ # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: n + 1 abort: F discard: T computed: F

Additional information

Used terms: n + 1

Proof step 23 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ¬ # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜n + 1⌝ → (∃ n0. ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n + 1)⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ 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))⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: 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 ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 24 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n + 1)⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , ¬ rstep (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(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: stream, name: prefix-def, seq: ⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0), subst:[σ, σ0] → [⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) .take (n + 1 ) ⌝ , ⌜πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ ]; ['a] → ['ra], <all paths>

Additional information

Used terms: ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n + 1)⌝ , ⌜πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝

Proof step 25 from proof proj-exec-inf (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 26 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ 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))⌝ ⌝ ), ∃ σ1. ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n + 1)⌝ + σ1 = ⌜πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 27 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n + 1)⌝ + rα = ⌜πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(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: stream, name: extensionality, seq: ⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0[n] = σ1[n]), subst:[σ0, σ1] → [ ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) .take (n + 1 ) ⌝ + rα, ⌜πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ ]; ['a] → ['ra], <all paths>

Additional information

Used terms: ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n + 1) ⌝ + rα, ⌜πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝

Proof step 28 from proof proj-exec-inf (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0[n] = σ1[n])

A spec-lemma

Proof step 29 from proof proj-exec-inf (previous step) (back to tree)

#(⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n + 1)⌝ + rα) = # ⌜πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , # lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ 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))⌝ ⌝ ), ∀ n1. ⌜n1⌝ < #(⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n + 1)⌝ + rα) → (⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n + 1)⌝ + rα)[n1] = ⌜πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ [n1] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ [n] = x[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⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (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 ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) ⊦ # ⌜σ.take(n)⌝ = ⌜# σ.take(n)⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 30 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n1. ¬ ⌜n⌝ + # rα < ⌜n1⌝ → (⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n + 1)⌝ + rα)[n1] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n1] ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 10 terms: n abort: F discard: T computed: F

Additional information

Used terms: n

Proof step 31 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ¬ ⌜n⌝ + # rα < ⌜n⌝ → (⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n + 1)⌝ + rα)[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ 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))⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: 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∞ + n∞1 < n∞1; (s)

Proof step 32 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# 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))⌝ ⌝ ).take(n + 1)⌝ + rα)[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(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: stream, name: get-append-ls, seq: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m], subst:[σ, σ1, m] → [⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) .take (n + 1 ) ⌝ , rα, n]; ['a] → ['ra], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]; (s) ⊦ ⌜n⌝ ≠ ∞; (s) n∞ ≠ ∞ ⊦ (⌜m⌝ + n∞).n = m + n∞.n; (s) ⊦ ⌜n⌝ .n = n; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # ⌜σ.take(n)⌝ = ⌜# σ.take(n)⌝ ; (s) ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ ¬ n < n; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf Used terms: ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n + 1)⌝ , rα, n

Proof step 33 from proof proj-exec-inf (previous step) (back to tree)

⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]

A spec-lemma

Proof step 34 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# 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))⌝ ⌝ ).take(n + 1)⌝ [n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ⌜n⌝ < # lim(⌜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))⌝ ⌝ ).take(n + 1)⌝ , fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ 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))⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: 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 < n; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) ⊦ # ⌜σ.take(n)⌝ = ⌜# σ.take(n)⌝ ; (s) m < n, ¬ # σ < ⌜n⌝ ⊦ σ.take(n)[m] = σ[m]; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ⌜x⌝ [n] = x[n]; (s)

Proof step 35 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# 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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 10 terms: n0 abort: F discard: F computed: F

Additional information

Used terms: n0

Proof step 36 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# 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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜n0⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ 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))⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 37 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# 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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜n0⌝ → ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n1 + 1), paf.take(n1))⌝ , ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:inffrag-size-pis-pia, 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)) = # πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n)) + 1, subst:[psf, lstart, n, rstart, rext, lstep, paf, rren, lren, lext, rstep] → [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) Used terms: rext, paf, n0, rstep, rstart, lren, rren, lstart, lstep, psf, lext

Proof step 38 from proof proj-exec-inf (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)) = # πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n)) + 1

Lemma inffrag-size-pis-pia

Proof step 39 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , # πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) = # πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) + 1, (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# 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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜n0⌝ → ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n1 + 1), paf.take(n1))⌝ , ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ 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))⌝ ⌝ ) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 11 terms: n0 abort: F discard: F computed: F

Additional information

Used terms: n0

Proof step 40 from proof proj-exec-inf (previous step) (back to tree)

# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) = # πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) + 1, # lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# 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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ¬ rstep (lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜n0⌝ → ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n1 + 1), paf.take(n1))⌝ , ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(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: stream, name: prefix-def, seq: ⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0), subst:[σ, σ0] → [⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )]; ['a] → ['rs], <all paths>

Additional information

Used terms: ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )

Proof step 41 from proof proj-exec-inf (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 42 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πrs(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))⌝ ⌝ ) + ⌜1⌝ , # πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) = # πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) + 1, (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# 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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n + 1]), ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜n0⌝ → ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n1 + 1), paf.take(n1))⌝ , ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ 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))⌝ ⌝ ), ∃ σ1. ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + σ1 = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) ⊦

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⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 43 from proof proj-exec-inf (previous step) (back to tree)

# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) = # πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) + 1, (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ = # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) + ⌜1⌝ , lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep ((⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n + 1]), ¬ ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ < ⌜n0⌝ → ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n1 + 1), paf.take(n1))⌝ , ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ, ∀ 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))⌝ ⌝ ), ∀ m. ¬ ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ < ⌜m⌝ → (∃ n. ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(m)⌝ ⊑ ⌜πrs(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: stream, name: extensionality, seq: ⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0[n] = σ1[n]), subst:[σ0, σ1] → [⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ, lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )]; ['a] → ['rs], <all paths>

Additional information

Used terms: ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ, lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )

Proof step 44 from proof proj-exec-inf (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0[n] = σ1[n])

A spec-lemma

Proof step 45 from proof proj-exec-inf (previous step) (back to tree)

#(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ) = # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), # πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) = # πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) + 1, ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ = # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# 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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep ((⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n + 1]), ¬ ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ < ⌜n0⌝ → ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n1 + 1), paf.take(n1))⌝ , ∀ m. ¬ ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ < ⌜m⌝ → (∃ n. ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ, ∀ n. ⌜n⌝ < #(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ) → (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n] = lim(⌜λ n. ⌜πrs(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: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 46 from proof proj-exec-inf (previous step) (back to tree)

# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) = # πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) + 1, (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ = # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) + ⌜1⌝ , ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ = # lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep ((⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n + 1]), ¬ ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ < ⌜n0⌝ → ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n1 + 1), paf.take(n1))⌝ , ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ, ∀ 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))⌝ ⌝ ), ∀ m. ¬ ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ < ⌜m⌝ → (∃ n. ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n2. ⌜n2⌝ < ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ → (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n2] = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n2] ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 14 terms: n abort: F discard: F computed: F

Additional information

Used terms: n

Proof step 47 from proof proj-exec-inf (previous step) (back to tree)

# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) = # πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) + 1, ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ = # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ = # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# 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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep ((⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n + 1]), ¬ ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ < ⌜n0⌝ → ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n1 + 1), paf.take(n1))⌝ , ⌜n⌝ < ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ → (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n] = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n], ∀ n2. ⌜n2⌝ < ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ → (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n2] = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n2], ∀ m. ¬ ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ < ⌜m⌝ → (∃ n. ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ n∞ = n∞0 + ⌜1⌝ ∧ ⌜n⌝ < n∞0 → (⌜n⌝ < n∞ ↔ true); (ss) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf

Proof step 48 from proof proj-exec-inf (previous step) (back to tree)

# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) = # πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) + 1, (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ = # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) + ⌜1⌝ , ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ = # lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n] = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n], ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep ((⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n + 1]), ¬ ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ < ⌜n0⌝ → ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n1 + 1), paf.take(n1))⌝ , ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ, ∀ 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))⌝ ⌝ ), ∀ m. ¬ ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ < ⌜m⌝ → (∃ n. ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n2. ⌜n2⌝ < ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ → (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n2] = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n2] ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: rα

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;; ⊦ # ⌜nf⌝ = ∞; (s) ⊦ ⌜m⌝ + ∞ = ∞; (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)

Proof step 49 from proof proj-exec-inf (previous step) (back to tree)

# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) = # πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) + 1, ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ = # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ = # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) + ⌜1⌝ , (⌜n⌝ + ⌜1⌝ ) + # rα = ⌜# πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n] = lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], rα = ⌜ral⌝ , ⌜n⌝ < # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep ((⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n + 1]), ¬ ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ < ⌜n0⌝ → ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n1 + 1), paf.take(n1))⌝ , ∀ n2. ⌜n2⌝ < ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ → (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n2] = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n2], ∀ m. ¬ ⌜# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + # rσ < ⌜m⌝ → (∃ n. ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: 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∞ + n∞1 < n∞1; (s) ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ n < m → (⌜(* n)⌝ + n∞1 = ⌜(* m)⌝ + n∞2 ↔ n∞1 = ⌜m - n⌝ + n∞2); (s) ⊦ ⌜n1⌝ + ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n1 + n + m = n0; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 50 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# ral⌝ + # rσ, # πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) = 2 + n + # ral, (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ = # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), n + 1 + # ral = # 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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n] = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n], fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep ((⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n + 1]), ¬ (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ < ⌜n0⌝ → ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n1 + 1), paf.take(n1))⌝ , ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ < ⌜m⌝ → (∃ n. ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n2. ⌜n2⌝ < (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ → (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n2] = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n2], ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ, ∀ 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))⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: get-append-ls, seq: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m], subst:[σ, σ1, m] → [⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , rσ, n]; ['a] → ['rs], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]; (s) ⊦ ⌜n⌝ .n = n; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m + m0 ≤ m ↔ m0 = 0; (s) ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , rσ, n

Proof step 51 from proof proj-exec-inf (previous step) (back to tree)

⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]

A spec-lemma

Proof step 52 from proof proj-exec-inf (previous step) (back to tree)

# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) = 2 + n + # ral, # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# ral⌝ + # rσ, n + 1 + # ral = # πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)), (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ = # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ [n] = lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ⌜n⌝ < # ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ [n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n + 1]), ¬ (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ < ⌜n0⌝ → ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n1 + 1), paf.take(n1))⌝ , ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ, ∀ n2. ⌜n2⌝ < (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ → (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n2] = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n2], ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ < ⌜m⌝ → (∃ n. ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ) ⊦

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 n < n0 ⊦ (* n) ≠ m0 + (* n0); (s) ⊦ m < m + n ↔ n ≠ 0; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ ⌜x⌝ [n] = x[n]; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 53 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# ral⌝ + # rσ, # πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) = 2 + n + # ral, (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ = # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), n + 1 + # ral = # 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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n] = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n], fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n + 1]), ¬ (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ < ⌜n0⌝ → ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n1 + 1), paf.take(n1))⌝ , ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ < ⌜m⌝ → (∃ n. ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n2. ⌜n2⌝ < (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ → (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n2] = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n2], ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ, ∀ 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))⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: get-append-ls, seq: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m], subst:[σ, σ1, m] → [⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , rσ, n + 1]; ['a] → ['rs], <all paths>

Additional information

Used simplifier rules: ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s) ⊦ m ≠ 0 → (m + n ≤ 1 ↔ m = 1 ∧ n = 0); (ws) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , rσ, n + 1

Proof step 54 from proof proj-exec-inf (previous step) (back to tree)

⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]

A spec-lemma

Proof step 55 from proof proj-exec-inf (previous step) (back to tree)

# πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) = 2 + n + # ral, # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# ral⌝ + # rσ, n + 1 + # ral = # πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)), (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ = # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n] = lim(⌜λ n. ⌜πrs(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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ⌜n + 1⌝ < # ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ , fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ [n + 1]), ¬ (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ < ⌜n0⌝ → ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n1 + 1), paf.take(n1))⌝ , ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ, ∀ n2. ⌜n2⌝ < (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ → (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n2] = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n2], ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ < ⌜m⌝ → (∃ n. ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ) ⊦

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 n < n0 ⊦ (* n) ≠ m0 + (* n0); (s) ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s) ⊦ m + n < m + n0 ↔ n < n0; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ ⌜x⌝ [n] = x[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⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 56 from proof proj-exec-inf (previous step) (back to tree)

# lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) = ⌜1⌝ + ⌜n⌝ + ⌜# ral⌝ + # rσ, # πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0)) = 2 + n + # ral, (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ = # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), n + 1 + # ral = # 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))⌝ ⌝ )[n] = πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n] = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n], fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstep (πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], πra(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n], πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))[n + 1]), ¬ (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ < ⌜n0⌝ → ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(n0)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n1 + 1), paf.take(n1))⌝ , ∀ m. ¬ (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ < ⌜m⌝ → (∃ n. ⌜(⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n2. ⌜n2⌝ < (⌜2⌝ + ⌜n⌝ + ⌜# ral⌝ ) + # rσ → (⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ)[n2] = lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[n2], ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n0 + 1), paf.take(n0))⌝ + rσ, ∀ 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))⌝ ⌝ ) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:finfrag-pi-step, seq:⊦ frag*(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren)) (psl, pal) ∧ n + 1 < # πrs(lstep, rstep, lren, rren, psl, pal) → rstep (πrs(lstep, rstep, lren, rren, psl, pal)[n], πra(lstep, rstep, lren, rren, psl, pal)[n], πrs(lstep, rstep, lren, rren, psl, pal)[n + 1]), subst:[lstart, n, pal, rstart, rext, lstep, lren, lext, rstep, rren, psl] → [lstart, n, paf.take(n0), rstart, rext, lstep, lren, lext, rstep, rren, psf.take(n0 + 1)]; ['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) ⊦ fragω(IOAut(start, step, ext))(sf, af) → (frag*(IOAut(start, step, ext))(sf.take(n + 1), af.take(n)) ↔ true); (ss) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s) ⊦ m ≠ 0 → (m + n ≤ 1 ↔ m = 1 ∧ n = 0); (ws) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: rext, psf.take(n0 + 1), n, paf.take(n0), rstep, rstart, lren, rren, lstart, lstep, lext

Proof step 57 from proof proj-exec-inf (previous step) (back to tree)

⊦ frag*(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren))(psl, pal) ∧ n + 1 < # πrs(lstep, rstep, lren, rren, psl, pal) → rstep(πrs(lstep, rstep, lren, rren, psl, pal)[n], πra(lstep, rstep, lren, rren, psl, pal)[n], πrs(lstep, rstep, lren, rren, psl, pal)[n + 1])

Lemma finfrag-pi-step

Proof step 58 from proof proj-exec-inf (previous step) (back to tree)

fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstart(lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[0]), ∀ m. ¬ # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:inffrag-pis-lim-init, seq:fragω(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren))(psf, paf) ⊦ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[0] = psf(0)._2, 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]

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

Proof step 59 from proof proj-exec-inf (previous step) (back to tree)

fragω(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren))(psf, paf) ⊦ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[0] = psf(0)._2

Lemma inffrag-pis-lim-init

Proof step 60 from proof proj-exec-inf (previous step) (back to tree)

lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[0] = psf(0)._2, fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstart(lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[0]), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ 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))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 61 from proof proj-exec-inf (previous step) (back to tree)

lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ )[0] = psf(0)._2, fragω(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psf, paf), Π(lstart, rstart)(psf(0)), ¬ rstart(psf(0)._2), ∀ m. ¬ # lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πra(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ), ∀ 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))⌝ ⌝ ), ∀ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⊑ lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜lim(⌜λ n. ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜πrs(lstep, rstep, lren, rren, psf.take(n + 1), paf.take(n))⌝ ) ⊦

User applied rule structural induction

And closed the goal!

Rule argument

term: psf(0)

Additional information

Used simplifier rules: ⊦ Π(lstart, rstart)((ls, rs)) ↔ lstart(ls) ∧ rstart(rs); (s)

Proof Tree for proj-exec-inf (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