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Proof step 1 from proof pcont-pis (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. πrs(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-pis (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. πrs(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-pis (previous step) (back to tree)

⊦ lift(frag*(ioa)) = frag(ioa)

A spec-lemma

Proof step 4 from proof pcont-pis (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. πrs(lstep, rstep, lren, rren, pσ, pα)) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:lifted-pis, seq:⊦ lifted (frag* (IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))) (λ psl, pal. πrs(lstep, rstep, lren, rren, psl, pal), λ pσ, pα. πrs(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-pis (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. πrs(lstep, rstep, lren, rren, psl, pal), λ pσ, pα. πrs(lstep, rstep, lren, rren, pσ, pα))

Lemma lifted-pis

Proof step 6 from proof pcont-pis (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. πrs(lstep, rstep, lren, rren, psl, pal), λ pσ, pα. πrs(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. πrs(lstep, rstep, lren, rren, pσ, pα)) ⊦

User applied rule apply 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:[trlp, trf, βf] → [frag* (IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))), λ psl, pal. πrs(lstep, rstep, lren, rren, psl, pal), λ pσ, pα. πrs(lstep, rstep, lren, rren, pσ, pα)]; ['s, 'a, 'b] → [('ls, 'rs), 'pa, 'rs]

Additional information

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

Proof step 7 from proof pcont-pis (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-pis (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. πrs(lstep, rstep, lren, rren, psl, pal), λ pσ, pα. πrs(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. πrs(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. πrs(lstep, rstep, lren, rren, psl, pal), λ pσ, pα. πrs(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. πrs(lstep, rstep, lren, rren, psl, pal)) → pcont(lift(frag*(IOAut(Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren))))) (λ pσ, pα. πrs(lstep, rstep, lren, rren, pσ, pα)) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 3

Additional information

Proof step 9 from proof pcont-pis (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. πrs(lstep, rstep, lren, rren, psl, pal), λ pσ, pα. πrs(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. πrs(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. πrs(lstep, rstep, lren, rren, psl, pal), λ pσ, pα. πrs(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. πrs(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 10 from proof pcont-pis (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. πrs(lstep, rstep, lren, rren, psl, pal), λ pσ, pα. πrs(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. πrs(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. πrs(lstep, rstep, lren, rren, psl, pal)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:pmon-pis, seq:⊦ pmon (frag* (IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)))) (λ psl, pal. πrs(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 11 from proof pcont-pis (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. πrs(lstep, rstep, lren, rren, psl, pal))

Lemma pmon-pis

Proof Tree for pcont-pis (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