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

⊦ pcont(frag(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))) (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 pcont-pia (previous step) (back to tree)

¬ pcont(frag(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))) (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: 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], <rotated> <all paths>

Additional information

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

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

⊦ lift(frag*(ioa)) = frag(ioa)

A spec-lemma

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

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

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:lifted-pia, seq:⊦ 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)), subst:[lstart, rstart, rext, lstep, rren, lren, lext, rstep] → [lstart, rstart, rext, lstep, rren, lren, lext, rstep]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['ls, 'ra, 'la, 'pa, 'rs, 'e]

Additional information

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

Proof step 5 from proof pcont-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))

Lemma lifted-pia

Proof step 6 from proof pcont-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)), ¬ pcont(lift(frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))))) (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:pconttr-lifted, seq: ⊦ lifted(trlp)(trf, βf) ∧ pmon(trlp)(trf) → pcont(lift(trlp))(βf), subst:[trf, trlp, βf] → [λ psl, pal. πra(lstep, rstep, lren, rren, psl, pal), frag* (IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))), λ pσ, pα. πra(lstep, rstep, lren, rren, pσ, pα)]; ['s, 'a, 'b] → [('ls, 'rs), 'pa, 'ra], <all paths>

Additional information

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

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

⊦ lifted(trlp)(trf, βf) ∧ pmon(trlp)(trf) → pcont(lift(trlp))(βf)

A spec-lemma

Proof step 8 from proof pcont-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)), ¬ pcont(lift(frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))))) (λ pσ, pα. πra(lstep, rstep, lren, rren, pσ, pα)) ⊦ 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)) ∧ pmon(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))

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 9 from proof pcont-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)), ¬ pcont(lift(frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))))) (λ pσ, pα. πra(lstep, rstep, lren, rren, pσ, pα)), ¬ pmon(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)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:pmon-pia, seq:⊦ pmon (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)), subst:[lstart, rstart, rext, lstep, rren, lren, lext, rstep] → [lstart, rstart, rext, lstep, rren, lren, lext, rstep]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['ls, 'ra, 'la, 'pa, 'rs, 'e], <all paths>

Additional information

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

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

⊦ pmon(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))

Lemma pmon-pia

Proof Tree for pcont-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