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

⊦ ext !(α) = ⌜el⌝ + β ↔ (∃ al, α0. α = ⌜al⌝ + α0 ∧ ext !(⌜al⌝ ) = ⌜el⌝ ∧ ext !(α0) = β)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Used simplifier rules: finremtau-def: ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; remtau-append: ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream

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

ext !(α) = ⌜el⌝ + β ⊦ ∃ al, α0. α = ⌜al⌝ + α0 ∧ ext !(⌜al⌝ ) = ⌜el⌝ ∧ ext !(α0) = β

User applied rule apply rewrite 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:[ext, α, el, β] → [ext, α, el, β]; ['e, 'a] → ['e, 'a], <all paths>

Additional information

Used terms: ext, α, el, β

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

ext !(α) = ⌜el⌝ + β, el = [] ⊦ ∃ al, α0. α = ⌜al⌝ + α0 ∧ ext !(⌜al⌝ ) = ⌜el⌝ ∧ ext !(α0) = 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)⌝ ; remtau-append: ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜x⌝ = [] ↔ x = []; (s) ⊦ ⌜[]⌝ = []; (s) ⊦ [] + σ = σ; (s)

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

⊦ ∃ al, α0. α = ⌜al⌝ + α0 ∧ ext !(al) = [] ∧ ext !(α0) = ⌜ext !(al)⌝ + ext !(α0)

User applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: [], α abort: F discard: T computed: F

Additional information

Used simplifier rules: finremtau-empty: ⊦ ext !([]) = []; ⊦ ⌜[]⌝ = []; (s) ⊦ [] + σ = σ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: [], α

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

el ≠ [], ∃ al, α0. α = ⌜al⌝ + α0 ∧ al ≠ [] ∧ ext !(⌜al⌝ ) = ⌜el⌝ ∧ Some(el.last) = ext(al.last) ∧ ext !(α0) = β ⊦ ∃ al, α0. α = ⌜al⌝ + α0 ∧ ext !(⌜al⌝ ) = ⌜el⌝ ∧ ext !(α0) = 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 7 from proof remtau-split-weak (previous step) (back to tree)

Some(ext !(al).last) = ext(al.last), ext !(al) ≠ [], al ≠ [] ⊦ ∃ al0, α1. ⌜al⌝ + α0 = ⌜al0⌝ + α1 ∧ ext !(al0) = ext !(al) ∧ ext !(α1) = ext !(α0)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: al, α0

Additional information

Used terms: al, α0

Proof Tree for remtau-split-weak (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