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

⊦ pbd(ext !)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

¬ pbd(ext !) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: contfun-lim, name:pbd-def, seq: ⊦ pbd(f) ↔ (∀ af, bl. (∀ n. f(⌜af.take(n)⌝ ) ⊑ ⌜bl⌝ ) ↔ f(⌜af⌝ ) ⊑ ⌜bl⌝ ), subst:[f] → [ext !]; ['a, 'b] → ['a, 'e], <all paths>

Additional information

Used terms: ext !

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

⊦ pbd(f) ↔ (∀ af, bl. (∀ n. f(⌜af.take(n)⌝ ) ⊑ ⌜bl⌝ ) ↔ f(⌜af⌝ ) ⊑ ⌜bl⌝ )

A spec-lemma

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

∃ af, bl. ¬((∀ n. ext !(⌜af.take(n)⌝ ) ⊑ ⌜bl⌝ ) ↔ ext !(⌜af⌝ ) ⊑ ⌜bl⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ ⊑ ⌜y⌝ ↔ x ⊑ y; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

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

⊦ (∀ n. ext !(af.take(n)) ⊑ el) ↔ ext !(⌜af⌝ ) ⊑ ⌜el⌝

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

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

¬ ext !(⌜af⌝ ) ⊑ ⌜el⌝ , ∀ n. ext !(af.take(n)) ⊑ el ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

term: ext !(⌜af⌝ )

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;;

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

ext !(⌜af⌝ ) = ⌜el0⌝ , ¬ ext !(⌜af⌝ ) ⊑ ⌜el⌝ , ∀ n. ext !(af.take(n)) ⊑ el ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ ⊑ ⌜y⌝ ↔ x ⊑ y; (s)

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

ext !(⌜af⌝ ) = ⌜el0⌝ , ¬ el0 ⊑ el, ∀ n. ext !(af.take(n)) ⊑ el ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

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

Additional information

Used terms: ext, ⌜af⌝ , el0

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

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

A spec-lemma

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

¬ el0 ⊑ el, ∀ n. ext !(af.take(n)) ⊑ el, ∃ al, α0. ⌜af⌝ = ⌜al⌝ + α0 ∧ ext !(al) = el0 ∧ ext !(α0) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ .f = nf; (s) ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜x⌝ + σ = ⌜nf⌝ ↔ ¬ finp(σ) ∧ x + σ.f = nf; (as)

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

ext !(⌜af0⌝ ) = [], ¬ ext !(al) ⊑ el, ∀ n. ext !((al + af0).take(n)) ⊑ el ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: # al abort: F discard: T computed: F

Additional information

Used terms: # al

Proof step 12 from proof remtau-pbd (previous step) (back to tree)

ext !(⌜af0⌝ ) = [], ext !((al + af0).take(# al)) ⊑ el, ¬ ext !(al) ⊑ el ⊦

Heuristic elimination applied rule apply elim lemma

And closed the goal!

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [al + af0, # al, al0, nf0]; ['a] → ['a]

Additional information

Used simplifier rules: ⊦ (⌜x⌝ + ⌜nf⌝ ).f = x + nf; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # x = # x0 → (x + nf = x0 + nf0 ↔ x = x0 ∧ nf = nf0); (s) ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: al + af0, # al, al0, nf0

Proof step 13 from proof remtau-pbd (previous step) (back to tree)

ext !(⌜af⌝ ) = ⌜ef⌝ , ¬ ext !(⌜af⌝ ) ⊑ ⌜el⌝ , ∀ n. ext !(af.take(n)) ⊑ el ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ ⌜nf⌝ ⊑ ⌜x⌝ ; (s)

Proof step 14 from proof remtau-pbd (previous step) (back to tree)

ext !(⌜af⌝ ) = ⌜ef⌝ , ∀ n. ext !(af.take(n)) ⊑ el ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: remtau, 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, ⌜af⌝ , ef.take(# el + 1), ⌜ef.drop(# el + 1)⌝ ]; ['a, 'e] → ['a, 'e]

Additional information

Used terms: ext, ⌜af⌝ , ef.take(# el + 1), ⌜ef.drop(# el + 1)⌝

Proof step 15 from proof remtau-pbd (previous step) (back to tree)

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

A spec-lemma

Proof step 16 from proof remtau-pbd (previous step) (back to tree)

ext !(⌜af⌝ ) = ⌜ef⌝ , ext !(⌜af⌝ ) = ⌜ef.take(# el + 1)⌝ + ⌜ef.drop(# el + 1)⌝ ↔ (∃ al, α0. ⌜af⌝ = ⌜al⌝ + α0 ∧ ext !(⌜al⌝ ) = ⌜ef.take(# el + 1)⌝ ∧ ext !(α0) = ⌜ef.drop(# el + 1)⌝ ), ∀ n. ext !(af.take(n)) ⊑ el ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜x⌝ + ⌜nf⌝ = ⌜nf0⌝ ↔ x + nf = nf0; (s) ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 17 from proof remtau-pbd (previous step) (back to tree)

ext !(⌜af⌝ ) = ⌜ef⌝ , ef.take(# el + 1) + ef.drop(# el + 1) = ef ↔ (∃ al, α0. ⌜af⌝ = ⌜al⌝ + α0 ∧ ext !(al) = ef.take(# el + 1) ∧ ext !(α0) = ⌜ef.drop(# el + 1)⌝ ), ∀ n. ext !(af.take(n)) ⊑ el ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [ef, # el + 1, el0, nf0]; ['a] → ['e]

Additional information

Used simplifier rules: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: ef, # el + 1, el0, nf0

Proof step 18 from proof remtau-pbd (previous step) (back to tree)

ext !(⌜af⌝ ) = ⌜ef⌝ , # el0 = # el + 1, ef = (⌜el0⌝ + ⌜nf0⌝ ).f, el0 + nf0 = ef ↔ (∃ al, α0. ⌜af⌝ = ⌜al⌝ + α0 ∧ ext !(al) = el0 ∧ ext !(α0) = ⌜nf0⌝ ), ∀ n. ext !(af.take(n)) ⊑ el ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ .f = nf; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # x = # y → (⌜x⌝ + σ = ⌜y⌝ + σ0 ↔ x = y ∧ σ = σ0); (s) ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s) ⊦ ⌜x + nf⌝ = ⌜x⌝ + ⌜nf⌝ ; (s) ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss) ⊦ ⌜x⌝ + σ = ⌜nf⌝ ↔ ¬ finp(σ) ∧ x + σ.f = nf; (as) ⊦ # x = # x0 → (x + nf = x0 + nf0 ↔ x = x0 ∧ nf = nf0); (s) ⊦ (⌜x⌝ + ⌜nf⌝ ).f = x + nf; (s)

Proof step 19 from proof remtau-pbd (previous step) (back to tree)

ext !(⌜af0⌝ ) = ⌜nf0⌝ , # ext !(al) = # el + 1, ∀ n. ext !((al + af0).take(n)) ⊑ el ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: # al abort: F discard: T computed: F

Additional information

Used terms: # al

Proof step 20 from proof remtau-pbd (previous step) (back to tree)

ext !(⌜af0⌝ ) = ⌜nf0⌝ , # ext !(al) = # el + 1, ext !((al + af0).take(# al)) ⊑ el ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [al + af0, # al, al0, af]; ['a] → ['a]

Additional information

Used simplifier rules: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: al + af0, # al, al0, af

Proof step 21 from proof remtau-pbd (previous step) (back to tree)

ext !(⌜af0⌝ ) = ⌜nf0⌝ , # ext !(al) = # el + 1, # al0 = # al, al + af0 = (⌜al0⌝ + ⌜af⌝ ).f, ext !(al0) ⊑ el ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = # x0 → (x + nf = x0 + nf0 ↔ x = x0 ∧ nf = nf0); (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (⌜x⌝ + ⌜nf⌝ ).f = x + nf; (s)

Proof step 22 from proof remtau-pbd (previous step) (back to tree)

ext !(⌜af0⌝ ) = ⌜nf0⌝ , # ext !(al0) = # el + 1, ext !(al0) ⊑ el ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last, name:prefix, seq: ⊦ x ⊑ y ↔ (∃ z. x + z = y), subst:[x, y] → [ext !(al0), el]; ['a] → ['e], <all paths>

Additional information

Used simplifier rules: ⊦ #(x + y) = # x + # y; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m = m + n ↔ n = 0; (s) ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws) ⊦ # x = 0 ↔ x = []; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: ext !(al0), el

Proof step 23 from proof remtau-pbd (previous step) (back to tree)

⊦ x ⊑ y ↔ (∃ z. x + z = y)

A spec-lemma

Proof step 24 from proof remtau-pbd (previous step) (back to tree)

ext !(⌜af⌝ ) ⊑ ⌜el⌝ ⊦ ∀ n. ext !(af.take(n)) ⊑ el

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 25 from proof remtau-pbd (previous step) (back to tree)

ext !(⌜af⌝ ) ⊑ ⌜el⌝ , ¬ ext !(af.take(n)) ⊑ el ⊦

User applied rule apply elim lemma

And closed the goal!

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [af, n, al, nf0]; ['a] → ['a]

Additional information

Used simplifier rules: ⊦ (⌜x⌝ + ⌜nf⌝ ).f = x + nf; (s) ⊦ ⌜x + nf⌝ = ⌜x⌝ + ⌜nf⌝ ; (s) ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s) ⊦ σ0 + σ ⊑ ⌜z⌝ ↔ finp(σ0) ∧ finp(σ) ∧ σ0.l + σ.l ⊑ z; (as) ⊦ finp(⌜x⌝ ); (s) ⊦ ⌜x⌝ .l = x; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ finp(σ) ↔ (∃ x. ⌜x⌝ = σ); (as) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + y ⊑ z → x ⊑ z; (s) ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: af, n, al, nf0

Proof Tree for remtau-pbd (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