[Imprint][Data Privacy Statement][E-Mail]



View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

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

⊦ ext !(al) = el + el0 ↔ (∃ al1, al0. al = al1 + al0 ∧ ext !(al1) = el ∧ ext !(al0) = el0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: finremtau-append: ⊦ ext !(al + al0) = ext !(al) + ext !(al0); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws)

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

ext !(al) = el + el0 ⊦ ∃ al1, al0. al = al1 + al0 ∧ ext !(al1) = el ∧ ext !(al0) = el0

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:remtau-split-weak, seq:⊦ ext !(α) = ⌜el⌝ + β ↔ (∃ al, α0. α = ⌜al⌝ + α0 ∧ ext !(⌜al⌝ ) = ⌜el⌝ ∧ ext !(α0) = β), subst:[ext, α, el, β] → [ext, ⌜al⌝ , el, ⌜el0⌝ ]; ['a, 'e] → ['a, 'e]

Additional information

Used terms: ext, ⌜al⌝ , el, ⌜el0⌝

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

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

Lemma remtau-split-weak

Proof step 4 from proof finremtau-split-weak (previous step) (back to tree)

ext !(al) = el + el0, ext !(⌜al⌝ ) = ⌜el⌝ + ⌜el0⌝ ↔ (∃ al0, α0. ⌜al⌝ = ⌜al0⌝ + α0 ∧ ext !(⌜al0⌝ ) = ⌜el⌝ ∧ ext !(α0) = ⌜el0⌝ ) ⊦ ∃ al1, al0. al = al1 + al0 ∧ ext !(al1) = el ∧ ext !(al0) = el0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

ext !(α0) = ⌜el0⌝ , ext !(al) = ext !(al0) + el0, ⌜al⌝ = ⌜al0⌝ + α0 ⊦ ∃ al1, al2. al = al1 + al2 ∧ ext !(al1) = ext !(al0) ∧ ext !(al2) = el0

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 ⌜ ⌝, ⌜ ⌝;; ⊦ ⌜x⌝ ≠ σ + ⌜nf0⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream

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

ext !(al) = ext !(al0) + el0, ext !(α0) = ⌜el0⌝ , α0 = ⌜al3⌝ , ⌜al⌝ = ⌜al0⌝ + α0 ⊦ ∃ al1, al2. al = al1 + al2 ∧ ext !(al1) = ext !(al0) ∧ ext !(al2) = el0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: finremtau-def: ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; finremtau-append: ⊦ ext !(al + al0) = ext !(al) + ext !(al0); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜x⌝ + ⌜y⌝ = ⌜z⌝ ↔ x + y = z; (s)

Proof step 7 from proof finremtau-split-weak (previous step) (back to tree)

⊦ ∃ al1, al2. al0 + al3 = al1 + al2 ∧ ext !(al1) = ext !(al0) ∧ ext !(al2) = ext !(al3)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: al0, al3

Additional information

Used terms: al0, al3

Proof Tree for finremtau-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