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Proof step 1 from proof lifted-pia (conclusion) (back to tree)

⊦ lifted(frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))) (λ psl, pal. πra(lstep, rstep, lren, rren, psl, pal), λ pσ, pα. πra(lstep, rstep, lren, rren, pσ, pα))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof lifted-pia (previous step) (back to tree)

¬ lifted(frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))) (λ psl, pal. πra(lstep, rstep, lren, rren, psl, pal), λ pσ, pα. πra(lstep, rstep, lren, rren, pσ, pα)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: contfun-lim, name:liftedtr-def, seq: ⊦ lifted(trlp)(trf, βf) ↔ (∀ σ, α. lift(trlp)(σ, α) → lift(trf)(σ, α) = βf(σ, α)), subst:[trlp, trf, βf] → [frag* (IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))), λ psl, pal. πra(lstep, rstep, lren, rren, psl, pal), λ pσ, pα. πra(lstep, rstep, lren, rren, pσ, pα)]; ['s, 'a, 'b] → [('ls, 'rs), 'pa, 'ra], <all paths>

Additional information

Used terms: frag* (IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))), λ psl, pal. πra(lstep, rstep, lren, rren, psl, pal), λ pσ, pα. πra(lstep, rstep, lren, rren, pσ, pα)

Proof step 3 from proof lifted-pia (previous step) (back to tree)

⊦ lifted(trlp)(trf, βf) ↔ (∀ σ, α. lift(trlp)(σ, α) → lift(trf)(σ, α) = βf(σ, α))

A spec-lemma

Proof step 4 from proof lifted-pia (previous step) (back to tree)

∃ σ, α. ¬ ( lift(frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))))(σ, α) → lift(λ psl, pal. πra(lstep, rstep, lren, rren, psl, pal))(σ, α) = πra(lstep, rstep, lren, rren, σ, α)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 5 from proof lifted-pia (previous step) (back to tree)

lift(λ psl, pal. πra(lstep, rstep, lren, rren, psl, pal))(pσ, pα) ≠ πra(lstep, rstep, lren, rren, pσ, pα), lift(frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))))(pσ, pα) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace-lift, name:frag-is-lift-fragstar, seq: ⊦ lift(frag*(ioa)) = frag(ioa), subst:[ioa] → [IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))]; ['s, 'a, 'e] → [('ls, 'rs), 'pa, 'e], <all paths>

Additional information

Used terms: IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))

Proof step 6 from proof lifted-pia (previous step) (back to tree)

⊦ lift(frag*(ioa)) = frag(ioa)

A spec-lemma

Proof step 7 from proof lifted-pia (previous step) (back to tree)

lift(λ psl, pal. πra(lstep, rstep, lren, rren, psl, pal))(pσ, pα) ≠ πra(lstep, rstep, lren, rren, pσ, pα), frag(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(pσ, pα) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: pσ

Additional information

Used simplifier rules: 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))⌝ ⌝ ); Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;; ⊦ frag(ioa)(⌜sf⌝ , α) ↔ fragω(ioa)(sf, α.f) ∧ ¬ finp(α); (s) ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss) ⊦ lift(trf)(⌜sf⌝ , ⌜af⌝ ) = lim(⌜λ n. ⌜trf(sf.take(n + 1), af.take(n))⌝ ⌝ ); (s)

Proof step 8 from proof lifted-pia (previous step) (back to tree)

pσ = ⌜psl0⌝ , lift(λ psl, pal. πra(lstep, rstep, lren, rren, psl, pal))(pσ, pα) ≠ πra(lstep, rstep, lren, rren, pσ, pα), frag(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(pσ, pα) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ .l = x; (s) ⊦ lift(trf)(⌜sl⌝ , ⌜al⌝ ) = ⌜trf(sl, al)⌝ ; (s) ⊦ finp(σ) ↔ (∃ x. ⌜x⌝ = σ); (as) ⊦ frag(ioa)(⌜sl⌝ , α) ↔ frag*(ioa)(sl, α.l) ∧ finp(α); (s)

Proof step 9 from proof lifted-pia (previous step) (back to tree)

⌜πra(lstep, rstep, lren, rren, psl0, pal)⌝ ≠ πra(lstep, rstep, lren, rren, ⌜psl0⌝ , ⌜pal⌝ ), frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))(psl0, pal) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:projaction-fin-def, seq:⊦ # psl = # pal + 1 → πra(lstep, rstep, lren, rren, ⌜psl⌝ , ⌜pal⌝ ) = ⌜πra(lstep, rstep, lren, rren, psl, pal)⌝ , subst:[pal, lstep, lren, rstep, rren, psl] → [pal, lstep, lren, rstep, rren, psl0]; ['ls, 'ra, 'la, 'pa, 'rs] → ['ls, 'ra, 'la, 'pa, 'rs], <all paths>

Additional information

Used simplifier rules: ⊦ frag*(IOAut(start, step, ext))(sl, al) → (# sl = # al + 1 ↔ true); (ss) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: psl0, rstep, pal, lren, rren, lstep

Proof step 10 from proof lifted-pia (previous step) (back to tree)

⊦ # psl = # pal + 1 → πra(lstep, rstep, lren, rren, ⌜psl⌝ , ⌜pal⌝ ) = ⌜πra(lstep, rstep, lren, rren, psl, pal)⌝

Lemma projaction-fin-def

Proof Tree for lifted-pia (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