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Proof step 1 from proof remtau-split-inf (conclusion) (back to tree)

⊦ el ≠ [] → (ext !(⌜af⌝ ) = ⌜el⌝ + β ↔ (∃ al, af0. ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ ∧ al ≠ [] ∧ ext !(al) = el ∧ Some(el.last) = ext(al.last) ∧ ext !(⌜af0⌝ ) = β))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof remtau-split-inf (previous step) (back to tree)

el ≠ [] ⊦ ext !(⌜af⌝ ) = ⌜el⌝ + β ↔ (∃ al, af0. ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ ∧ al ≠ [] ∧ ext !(al) = el ∧ Some(el.last) = ext(al.last) ∧ ext !(⌜af0⌝ ) = β)

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:remtau-split, seq:⊦ el ≠ [] → ( ext !(α) = ⌜el⌝ + β ↔ (∃ al, α0. α = ⌜al⌝ + α0 ∧ al ≠ [] ∧ ext !(⌜al⌝ ) = ⌜el⌝ ∧ Some(el.last) = ext(al.last) ∧ ext !(α0) = β)), subst:[el, ext, α, β] → [el, ext, ⌜af⌝ , β]; ['e, 'a] → ['e, 'a]

Additional information

Used terms: el, ext, ⌜af⌝ , β

Proof step 3 from proof remtau-split-inf (previous step) (back to tree)

⊦ el ≠ [] → (ext !(α) = ⌜el⌝ + β ↔ (∃ al, α0. α = ⌜al⌝ + α0 ∧ al ≠ [] ∧ ext !(⌜al⌝ ) = ⌜el⌝ ∧ Some(el.last) = ext(al.last) ∧ ext !(α0) = β))

Lemma remtau-split

Proof step 4 from proof remtau-split-inf (previous step) (back to tree)

el ≠ [], el ≠ [] → (ext !(⌜af⌝ ) = ⌜el⌝ + β ↔ (∃ al, α0. ⌜af⌝ = ⌜al⌝ + α0 ∧ al ≠ [] ∧ ext !(⌜al⌝ ) = ⌜el⌝ ∧ Some(el.last) = ext(al.last) ∧ ext !(α0) = β)) ⊦ ext !(⌜af⌝ ) = ⌜el⌝ + β ↔ (∃ al, af0. ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ ∧ al ≠ [] ∧ ext !(al) = el ∧ Some(el.last) = ext(al.last) ∧ ext !(⌜af0⌝ ) = β)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: finremtau-def: ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s)

Proof step 5 from proof remtau-split-inf (previous step) (back to tree)

el ≠ [], ext !(⌜af⌝ ) = ⌜el⌝ + β ↔ (∃ al, α0. ⌜af⌝ = ⌜al⌝ + α0 ∧ al ≠ [] ∧ ext !(al) = el ∧ Some(ext !(al).last) = ext(al.last) ∧ ext !(α0) = β) ⊦ ext !(⌜af⌝ ) = ⌜el⌝ + β ↔ (∃ al, af0. ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ ∧ al ≠ [] ∧ ext !(al) = el ∧ Some(ext !(al).last) = ext(al.last) ∧ ext !(⌜af0⌝ ) = β)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 6 from proof remtau-split-inf (previous step) (back to tree)

ext !(⌜af⌝ ) = ⌜el⌝ + β, el ≠ [], ext !(⌜af⌝ ) = ⌜el⌝ + β ↔ (∃ al, α0. ⌜af⌝ = ⌜al⌝ + α0 ∧ al ≠ [] ∧ ext !(al) = el ∧ Some(ext !(al).last) = ext(al.last) ∧ ext !(α0) = β) ⊦ ∃ al, af0. ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ ∧ al ≠ [] ∧ ext !(al) = el ∧ Some(ext !(al).last) = ext(al.last) ∧ ext !(⌜af0⌝ ) = β

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 7 from proof remtau-split-inf (previous step) (back to tree)

ext !(⌜af⌝ ) = ⌜ext !(al)⌝ + ext !(α0), Some(ext !(al).last) = ext(al.last), ⌜af⌝ = ⌜al⌝ + α0, ext !(al) ≠ [], al ≠ [] ⊦ ∃ al0, af0. ⌜af⌝ = ⌜al0⌝ + ⌜af0⌝ ∧ al0 ≠ [] ∧ ext !(al0) = ext !(al) ∧ Some(ext !(al0).last) = ext(al0.last) ∧ ext !(⌜af0⌝ ) = ext !(α0)

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: α0

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;; ⊦ σ0 + ⌜x⌝ = ⌜nf0⌝ ↔ σ0 = ⌜nf0⌝ ; (s) ⊦ ⌜x⌝ ≠ ⌜nf⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream

Proof step 8 from proof remtau-split-inf (previous step) (back to tree)

ext !(⌜af⌝ ) = ⌜ext !(al)⌝ + ext !(α0), α0 = ⌜af1⌝ , ⌜af⌝ = ⌜al⌝ + α0, Some(ext !(al).last) = ext(al.last), ext !(al) ≠ [], al ≠ [] ⊦ ∃ al0, af0. ⌜af⌝ = ⌜al0⌝ + ⌜af0⌝ ∧ al0 ≠ [] ∧ ext !(al0) = ext !(al) ∧ Some(ext !(al0).last) = ext(al0.last) ∧ ext !(⌜af0⌝ ) = ext !(α0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 9 from proof remtau-split-inf (previous step) (back to tree)

ext !(⌜af⌝ ) = ⌜ext !(al)⌝ + ext !(⌜af1⌝ ), Some(ext !(al).last) = ext(al.last), ⌜af⌝ = ⌜al⌝ + ⌜af1⌝ , ext !(al) ≠ [], al ≠ [] ⊦ ∃ al0, af0. ⌜af⌝ = ⌜al0⌝ + ⌜af0⌝ ∧ al0 ≠ [] ∧ ext !(al0) = ext !(al) ∧ Some(ext !(al0).last) = ext(al0.last) ∧ ext !(⌜af0⌝ ) = ext !(⌜af1⌝ )

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: al, af1

Additional information

Used terms: al, af1

Proof step 10 from proof remtau-split-inf (previous step) (back to tree)

ext !(⌜af⌝ ) ≠ ⌜el⌝ + β, el ≠ [], ext !(⌜af⌝ ) = ⌜el⌝ + β ↔ (∃ al, α0. ⌜af⌝ = ⌜al⌝ + α0 ∧ al ≠ [] ∧ ext !(al) = el ∧ Some(ext !(al).last) = ext(al.last) ∧ ext !(α0) = β), ∃ al, af0. ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ ∧ al ≠ [] ∧ ext !(al) = el ∧ Some(ext !(al).last) = ext(al.last) ∧ ext !(⌜af0⌝ ) = β ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 11 from proof remtau-split-inf (previous step) (back to tree)

⌜af⌝ = ⌜al⌝ + ⌜af0⌝ , Some(ext !(al).last) = ext(al.last), ext !(⌜af⌝ ) ≠ ⌜ext !(al)⌝ + ext !(⌜af0⌝ ), ext !(al) ≠ [], al ≠ [] ⊦ ∃ al0, α0. ⌜af⌝ = ⌜al0⌝ + α0 ∧ al0 ≠ [] ∧ ext !(al0) = ext !(al) ∧ Some(ext !(al0).last) = ext(al0.last) ∧ ext !(α0) = ext !(⌜af0⌝ )

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: al, ⌜af0⌝

Additional information

Used terms: al, ⌜af0⌝

Proof Tree for remtau-split-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