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Proof step 1 from proof lift-cont (conclusion) (back to tree)

⊦ mon(lf) → cont(lift(lf))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof lift-cont (previous step) (back to tree)

mon(lf), ¬ cont(lift(lf)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:cont-def, seq: ⊦ cont(f) ↔ mon(f) ∧ pbd(f), subst:[f] → [lift(lf)]; [A, B] → [A, B], lhs:cont(lift(lf)), <all paths>

Additional information

Used terms: lift(lf)

Proof step 3 from proof lift-cont (previous step) (back to tree)

⊦ cont(f) ↔ mon(f) ∧ pbd(f)

Lemma cont-def

Proof step 4 from proof lift-cont (previous step) (back to tree)

mon(lf), ¬ mon(lift(lf)) ∨ ¬ pbd(lift(lf)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: mon-def: ⊦ mon(f) ↔ (∀ α1, α2. α1 ⊑ α2 → f(α1) ⊑ f(α2));

Proof step 5 from proof lift-cont (previous step) (back to tree)

mon(lf), ¬(α1 ⊑ α2 → lift(lf)(α1) ⊑ lift(lf)(α2)) ∨ ¬ pbd(lift(lf)) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 2

Additional information

Proof step 6 from proof lift-cont (previous step) (back to tree)

mon(lf), α1 ⊑ α2, ¬ lift(lf)(α1) ⊑ lift(lf)(α2) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: α2

Additional information

Used simplifier rules: Stream[A] freely generated by ⌝, ⌝;

Proof step 7 from proof lift-cont (previous step) (back to tree)

α2 = ⌜al⌝ , mon(lf), α1 ⊑ α2, ¬ lift(lf)(α1) ⊑ lift(lf)(α2) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-fin: ⊦ lift(lf)(⌜al⌝ ) = ⌜lf(al)⌝ ;

Proof step 8 from proof lift-cont (previous step) (back to tree)

mon(lf), α1 ⊑ ⌜al⌝ , ¬ lift(lf)(α1) ⊑ ⌜lf(al)⌝ ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: α1

Additional information

Used simplifier rules: Stream[A] freely generated by ⌝, ⌝; ⊦ ¬ ⌜nf⌝ ⊑ ⌜x⌝ ; (s)

Proof step 9 from proof lift-cont (previous step) (back to tree)

α1 = ⌜al0⌝ , mon(lf), α1 ⊑ ⌜al⌝ , ¬ lift(lf)(α1) ⊑ ⌜lf(al)⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-fin: ⊦ lift(lf)(⌜al⌝ ) = ⌜lf(al)⌝ ; ⊦ ⌜x⌝ ⊑ ⌜y⌝ ↔ x ⊑ y; (s)

Proof step 10 from proof lift-cont (previous step) (back to tree)

mon(lf), al0 ⊑ al, ¬ lf(al0) ⊑ lf(al) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:monl-def, seq: ⊦ mon(lf) ↔ (∀ al1, al2. al1 ⊑ al2 → lf(al1) ⊑ lf(al2)), subst:[lf] → [lf]; [A, B] → [A, B], lhs:mon(lf), <all paths>

Additional information

Used terms: lf

Proof step 11 from proof lift-cont (previous step) (back to tree)

⊦ mon(lf) ↔ (∀ al1, al2. al1 ⊑ al2 → lf(al1) ⊑ lf(al2))

Lemma monl-def

Proof step 12 from proof lift-cont (previous step) (back to tree)

al0 ⊑ al, ¬ lf(al0) ⊑ lf(al), ∀ al1, al2. al1 ⊑ al2 → lf(al1) ⊑ lf(al2) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 3 terms: al0, al

Additional information

Used terms: al0, al

Proof step 13 from proof lift-cont (previous step) (back to tree)

α2 = ⌜af⌝ , mon(lf), α1 ⊑ α2, ¬ lift(lf)(α1) ⊑ lift(lf)(α2) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-inf: ⊦ lift(lf)(⌜af⌝ ) = lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ );

Proof step 14 from proof lift-cont (previous step) (back to tree)

mon(lf), α1 ⊑ ⌜af⌝ , ¬ lift(lf)(α1) ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: α1

Additional information

Used simplifier rules: lift-inf: ⊦ lift(lf)(⌜af⌝ ) = lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ); Stream[A] freely generated by ⌝, ⌝; ⊦ ⌜nf⌝ ⊑ ⌜nf0⌝ ↔ nf = nf0; (s) ⊦ σ ⊑ σ; (s)

Proof step 15 from proof lift-cont (previous step) (back to tree)

α1 = ⌜al⌝ , mon(lf), α1 ⊑ ⌜af⌝ , ¬ lift(lf)(α1) ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-fin: ⊦ lift(lf)(⌜al⌝ ) = ⌜lf(al)⌝ ;

Proof step 16 from proof lift-cont (previous step) (back to tree)

mon(lf), ⌜al⌝ ⊑ ⌜af⌝ , ¬ ⌜lf(al)⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: streamseq-mon-lim, name:lim-def, seq:⊦ mon(σσ) ∧ σσ ≠ [] → (∀ n. ⌜n⌝ < # σσ → σσ(n) ⊑ lim(σσ)) ∧ (∀ m. ¬ # lim(σσ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # σσ ∧ ⌜lim(σσ).take(m)⌝ ⊑ σσ(n))), subst:[σσ] → [⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ]; [A] → [B]

Additional information

Used terms: ⌜λ n. ⌜lf(af.take(n))⌝ ⌝

Proof step 17 from proof lift-cont (previous step) (back to tree)

⊦ mon(σσ) ∧ σσ ≠ [] → (∀ n. ⌜n⌝ < # σσ → σσ(n) ⊑ lim(σσ)) ∧ (∀ m. ¬ # lim(σσ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # σσ ∧ ⌜lim(σσ).take(m)⌝ ⊑ σσ(n)))

A spec-lemma

Proof step 18 from proof lift-cont (previous step) (back to tree)

mon(lf), ⌜al⌝ ⊑ ⌜af⌝ , ¬ ⌜lf(al)⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ∧ ⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ≠ [] → (∀ n. ⌜n⌝ < # ⌜λ n. ⌜lf(af.take(n))⌝ ⌝ → ⌜λ n. ⌜lf(af.take(n))⌝ ⌝ (n) ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ )) ∧ (∀ m. ¬ # lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # ⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ∧ ⌜lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜λ n. ⌜lf(af.take(n))⌝ ⌝ (n))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ ⊑ ⌜y⌝ ↔ x ⊑ y; (s) ⊦ ⌜nf⌝ (n) = nf(n); (s) ⊦ ⌜m⌝ < ∞; (s) ⊦ # ⌜nf⌝ = ∞; (s) ⊦ [] ≠ ⌜nf⌝ ; (s)

Proof step 19 from proof lift-cont (previous step) (back to tree)

mon(lf), ⌜al⌝ ⊑ ⌜af⌝ , ¬ ⌜lf(al)⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) → (∀ n. ⌜lf(af.take(n))⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ )) ∧ (∀ m. ¬ # lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ).take(m) ⊑ lf(af.take(n)))) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:monl-equiv, seq: ⊦ mon(lf) ↔ (∀ af. mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ )), subst:[lf] → [lf]; [A, B] → [A, B], lhs:mon(lf), <all paths>

Additional information

Used terms: lf

Proof step 20 from proof lift-cont (previous step) (back to tree)

⊦ mon(lf) ↔ (∀ af. mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ))

Lemma monl-equiv

Proof step 21 from proof lift-cont (previous step) (back to tree)

⌜al⌝ ⊑ ⌜af⌝ , ¬ ⌜lf(al)⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) → (∀ n. ⌜lf(af.take(n))⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ )) ∧ (∀ m. ¬ # lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ).take(m) ⊑ lf(af.take(n)))), ∀ af. mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 3

Additional information

Proof step 22 from proof lift-cont (previous step) (back to tree)

⌜al⌝ ⊑ ⌜af⌝ , ¬ mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ¬ ⌜lf(al)⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ af. mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: af

Additional information

Used terms: af

Proof step 23 from proof lift-cont (previous step) (back to tree)

⌜al⌝ ⊑ ⌜af⌝ , ¬ ⌜lf(al)⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ af. mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ n. ⌜lf(af.take(n))⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ).take(m) ⊑ lf(af.take(n))) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 4 terms: # al abort: F discard: F computed: F

Additional information

Used terms: # al

Proof step 24 from proof lift-cont (previous step) (back to tree)

⌜al⌝ ⊑ ⌜af⌝ , ⌜lf(af.take(# al))⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ¬ ⌜lf(al)⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ).take(m) ⊑ lf(af.take(n))), ∀ n. ⌜lf(af.take(n))⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ af. mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊦

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] → [af, # al, al0, nf0]; [A] → [A]

Additional information

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

Proof step 25 from proof lift-cont (previous step) (back to tree)

# al0 = # al, af = (⌜al0⌝ + ⌜nf0⌝ ).f, ⌜lf(al0)⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ⌜al⌝ ⊑ ⌜af⌝ , ¬ ⌜lf(al)⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ af. mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ n. ⌜lf(af.take(n))⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ).take(m) ⊑ lf(af.take(n))) ⊦

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⌝ ).f = x + nf; (s) ⊦ ⌜x + nf⌝ = ⌜x⌝ + ⌜nf⌝ ; (s)

Proof step 26 from proof lift-cont (previous step) (back to tree)

# al0 = # al, ⌜al⌝ ⊑ ⌜al0⌝ + ⌜nf0⌝ , ⌜lf(al0)⌝ ⊑ lim(⌜λ n. ⌜lf((al0 + nf0).take(n))⌝ ⌝ ), ¬ ⌜lf(al)⌝ ⊑ lim(⌜λ n. ⌜lf((al0 + nf0).take(n))⌝ ⌝ ), ∀ af. mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜lf((al0 + nf0).take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. lim(⌜λ n. ⌜lf((al0 + nf0).take(n))⌝ ⌝ ).take(m) ⊑ lf((al0 + nf0).take(n))), ∀ n. ⌜lf((al0 + nf0).take(n))⌝ ⊑ lim(⌜λ n. ⌜lf((al0 + nf0).take(n))⌝ ⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:prefix-def, seq: ⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0), subst:[σ, σ0] → [⌜al⌝ , ⌜al0⌝ + ⌜nf0⌝ ]; [A] → [A], lhs:⌜al⌝ ⊑ ⌜al0⌝ + ⌜nf0⌝ , <all paths>

Additional information

Used simplifier rules: ⊦ # x = # y → (⌜x⌝ + σ = ⌜y⌝ + σ0 ↔ x = y ∧ σ = σ0); (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: ⌜al⌝ , ⌜al0⌝ + ⌜nf0⌝

Proof step 27 from proof lift-cont (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 28 from proof lift-cont (previous step) (back to tree)

mon(lf), ¬ pbd(lift(lf)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:pbd-def, seq: ⊦ pbd(f) ↔ (∀ af, bl. (∀ n. f(⌜af.take(n)⌝ ) ⊑ ⌜bl⌝ ) ↔ f(⌜af⌝ ) ⊑ ⌜bl⌝ ), subst:[f] → [lift(lf)]; [A, B] → [A, B], lhs:pbd(lift(lf)), <all paths>

Additional information

Used terms: lift(lf)

Proof step 29 from proof lift-cont (previous step) (back to tree)

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

Lemma pbd-def

Proof step 30 from proof lift-cont (previous step) (back to tree)

mon(lf), ∃ af, bl. ¬((∀ n. lift(lf)(⌜af.take(n)⌝ ) ⊑ ⌜bl⌝ ) ↔ lift(lf)(⌜af⌝ ) ⊑ ⌜bl⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lift-fin: ⊦ lift(lf)(⌜al⌝ ) = ⌜lf(al)⌝ ; lift-inf: ⊦ lift(lf)(⌜af⌝ ) = lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ); ⊦ ⌜x⌝ ⊑ ⌜y⌝ ↔ x ⊑ y; (s)

Proof step 31 from proof lift-cont (previous step) (back to tree)

mon(lf) ⊦ (∀ n. lf(af.take(n)) ⊑ bl) ↔ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊑ ⌜bl⌝

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: streamseq-mon-lim, name:lim-def, seq:⊦ mon(σσ) ∧ σσ ≠ [] → (∀ n. ⌜n⌝ < # σσ → σσ(n) ⊑ lim(σσ)) ∧ (∀ m. ¬ # lim(σσ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # σσ ∧ ⌜lim(σσ).take(m)⌝ ⊑ σσ(n))), subst:[σσ] → [⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ]; [A] → [B]

Additional information

Used terms: ⌜λ n. ⌜lf(af.take(n))⌝ ⌝

Proof step 32 from proof lift-cont (previous step) (back to tree)

⊦ mon(σσ) ∧ σσ ≠ [] → (∀ n. ⌜n⌝ < # σσ → σσ(n) ⊑ lim(σσ)) ∧ (∀ m. ¬ # lim(σσ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # σσ ∧ ⌜lim(σσ).take(m)⌝ ⊑ σσ(n)))

A spec-lemma

Proof step 33 from proof lift-cont (previous step) (back to tree)

mon(lf), mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ∧ ⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ≠ [] → (∀ n. ⌜n⌝ < # ⌜λ n. ⌜lf(af.take(n))⌝ ⌝ → ⌜λ n. ⌜lf(af.take(n))⌝ ⌝ (n) ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ )) ∧ (∀ m. ¬ # lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # ⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ∧ ⌜lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ).take(m)⌝ ⊑ ⌜λ n. ⌜lf(af.take(n))⌝ ⌝ (n))) ⊦ (∀ n. lf(af.take(n)) ⊑ bl) ↔ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊑ ⌜bl⌝

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ ⊑ ⌜y⌝ ↔ x ⊑ y; (s) ⊦ ⌜nf⌝ (n) = nf(n); (s) ⊦ ⌜m⌝ < ∞; (s) ⊦ # ⌜nf⌝ = ∞; (s) ⊦ [] ≠ ⌜nf⌝ ; (s)

Proof step 34 from proof lift-cont (previous step) (back to tree)

mon(lf), mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) → (∀ n. ⌜lf(af.take(n))⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ )) ∧ (∀ m. ¬ # lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ).take(m) ⊑ lf(af.take(n)))) ⊦ (∀ n. lf(af.take(n)) ⊑ bl) ↔ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊑ ⌜bl⌝

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 2

Additional information

Proof step 35 from proof lift-cont (previous step) (back to tree)

mon(lf), ¬ mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊦ (∀ n. lf(af.take(n)) ⊑ bl) ↔ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊑ ⌜bl⌝

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:monl-equiv, seq: ⊦ mon(lf) ↔ (∀ af. mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ )), subst:[lf] → [lf]; [A, B] → [A, B], lhs:mon(lf), <all paths>

Additional information

Used terms: lf

Proof step 36 from proof lift-cont (previous step) (back to tree)

⊦ mon(lf) ↔ (∀ af. mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ))

Lemma monl-equiv

Proof step 37 from proof lift-cont (previous step) (back to tree)

¬ mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ af. mon(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊦ (∀ n. lf(af.take(n)) ⊑ bl) ↔ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊑ ⌜bl⌝

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 2 terms: af

Additional information

Used terms: af

Proof step 38 from proof lift-cont (previous step) (back to tree)

mon(lf), ∀ n. ⌜lf(af.take(n))⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ).take(m) ⊑ lf(af.take(n))) ⊦ (∀ n. lf(af.take(n)) ⊑ bl) ↔ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊑ ⌜bl⌝

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 39 from proof lift-cont (previous step) (back to tree)

mon(lf), ¬ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊑ ⌜bl⌝ , ∀ m. ¬ # lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ).take(m) ⊑ lf(af.take(n))), ∀ n. ⌜lf(af.take(n))⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ n. lf(af.take(n)) ⊑ bl ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

term: lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ )

Additional information

Used simplifier rules: Stream[A] freely generated by ⌝, ⌝;

Proof step 40 from proof lift-cont (previous step) (back to tree)

lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) = ⌜bl0⌝ , mon(lf), ¬ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊑ ⌜bl⌝ , ∀ n. lf(af.take(n)) ⊑ bl, ∀ n. ⌜lf(af.take(n))⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ).take(m) ⊑ lf(af.take(n))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ .take(n) = x.take(n); (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜x⌝ ⊑ ⌜y⌝ ↔ x ⊑ y; (s)

Proof step 41 from proof lift-cont (previous step) (back to tree)

lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) = ⌜bl0⌝ , mon(lf), ¬ bl0 ⊑ bl, ∀ n. lf(af.take(n)) ⊑ bl, ∀ n. lf(af.take(n)) ⊑ bl0, ∀ m. ¬ # bl0 < m → (∃ n. bl0.take(m) ⊑ lf(af.take(n))) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: # bl0 abort: F discard: T computed: T

Additional information

Used terms: # bl0

Proof step 42 from proof lift-cont (previous step) (back to tree)

lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) = ⌜bl0⌝ , mon(lf), ¬ bl0 ⊑ bl, ¬ # bl0 < # bl0 → (∃ n. bl0.take(# bl0) ⊑ lf(af.take(n))), ∀ n. lf(af.take(n)) ⊑ bl0, ∀ n. lf(af.take(n)) ⊑ bl ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = n → x.take(n) = x; (s) ⊦ ¬ n < n; (s)

Proof step 43 from proof lift-cont (previous step) (back to tree)

lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) = ⌜bl0⌝ , mon(lf), bl0 ⊑ lf(af.take(n)), ¬ bl0 ⊑ bl, ∀ n. lf(af.take(n)) ⊑ bl, ∀ n. lf(af.take(n)) ⊑ bl0 ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 5 terms: n

Additional information

Used simplifier rules: prefix-trans: ⊦ x ⊑ y ∧ y ⊑ z → x ⊑ z; (forward) from specification list-last Used terms: n

Proof step 44 from proof lift-cont (previous step) (back to tree)

lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) = ⌜bf⌝ , mon(lf), ¬ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊑ ⌜bl⌝ , ∀ n. lf(af.take(n)) ⊑ bl, ∀ n. ⌜lf(af.take(n))⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ).take(m) ⊑ lf(af.take(n))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ¬ ∞ < n∞; (s) ⊦ # ⌜nf⌝ = ∞; (s) ⊦ ¬ ⌜nf⌝ ⊑ ⌜x⌝ ; (s)

Proof step 45 from proof lift-cont (previous step) (back to tree)

lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) = ⌜bf⌝ , mon(lf), ∀ n. lf(af.take(n)) ⊑ bl, ∀ n. ⌜lf(af.take(n))⌝ ⊑ ⌜bf⌝ , ∀ m. ∃ n. bf.take(m) ⊑ lf(af.take(n)) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 5 terms: # bl + 1 abort: F discard: T computed: F

Additional information

Used terms: # bl + 1

Proof step 46 from proof lift-cont (previous step) (back to tree)

lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) = ⌜bf⌝ , mon(lf), ∀ n. ⌜lf(af.take(n))⌝ ⊑ ⌜bf⌝ , ∀ n. lf(af.take(n)) ⊑ bl, ∃ n. bf.take(# bl + 1) ⊑ lf(af.take(n)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 47 from proof lift-cont (previous step) (back to tree)

lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) = ⌜bf⌝ , mon(lf), bf.take(# bl + 1) ⊑ lf(af.take(n)), ∀ n. lf(af.take(n)) ⊑ bl, ∀ n. ⌜lf(af.take(n))⌝ ⊑ ⌜bf⌝ ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 4 terms: n abort: F discard: T computed: F

Additional information

Used terms: n

Proof step 48 from proof lift-cont (previous step) (back to tree)

lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) = ⌜bf⌝ , mon(lf), bf.take(# bl + 1) ⊑ lf(af.take(n)), lf(af.take(n)) ⊑ bl, ∀ n. ⌜lf(af.take(n))⌝ ⊑ ⌜bf⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: prefix-trans: ⊦ x ⊑ y ∧ y ⊑ z → x ⊑ z; (forward) from specification list-last

Proof step 49 from proof lift-cont (previous step) (back to tree)

lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) = ⌜bf⌝ , mon(lf), lf(af.take(n)) ⊑ bl, bf.take(# bl + 1) ⊑ lf(af.take(n)), bf.take(# bl + 1) ⊑ bl, ∀ n. ⌜lf(af.take(n))⌝ ⊑ ⌜bf⌝ ⊦

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] → [bf.take(# bl + 1), bl]; [A] → [B], lhs:bf.take(# bl + 1) ⊑ bl, <all paths>

Additional information

Used terms: bf.take(# bl + 1), bl

Proof step 50 from proof lift-cont (previous step) (back to tree)

⊦ x ⊑ y ↔ (∃ z. x + z = y)

A spec-lemma

Proof step 51 from proof lift-cont (previous step) (back to tree)

lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) = ⌜bf⌝ , mon(lf), bf.take(# bl + 1) ⊑ lf(af.take(n)), lf(af.take(n)) ⊑ bl, ∀ n. ⌜lf(af.take(n))⌝ ⊑ ⌜bf⌝ , ∃ z. bf.take(# bl + 1) + z = bl ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 52 from proof lift-cont (previous step) (back to tree)

bf.take(# bl + 1) + bl0 = bl, lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) = ⌜bf⌝ , mon(lf), lf(af.take(n)) ⊑ bl, bf.take(# bl + 1) ⊑ lf(af.take(n)), ∀ n. ⌜lf(af.take(n))⌝ ⊑ ⌜bf⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-getput, name:extension, seq: ⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x(n) = y(n)), subst:[y, x] → [bf.take(# bl + 1) + bl0, bl]; [A] → [B], lhs:bf.take(# bl + 1) + bl0 = bl, <all paths>

Additional information

Used simplifier rules: ⊦ #(x + y) = # x + # y; (s) ⊦ # nf.take(n) = n; (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) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: bf.take(# bl + 1) + bl0, bl

Proof step 53 from proof lift-cont (previous step) (back to tree)

⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x(n) = y(n))

A spec-lemma

Proof step 54 from proof lift-cont (previous step) (back to tree)

mon(lf), lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊑ ⌜bl⌝ , ∀ m. ¬ # lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ).take(m) ⊑ lf(af.take(n))), ∀ n. ⌜lf(af.take(n))⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊦ ∀ n. lf(af.take(n)) ⊑ bl

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 55 from proof lift-cont (previous step) (back to tree)

mon(lf), lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) ⊑ ⌜bl⌝ , ¬ lf(af.take(n)) ⊑ bl, ∀ n. ⌜lf(af.take(n))⌝ ⊑ lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ), ∀ m. ¬ # lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ) < ⌜m⌝ → (∃ n. lim(⌜λ n. ⌜lf(af.take(n))⌝ ⌝ ).take(m) ⊑ lf(af.take(n))) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: n

Additional information

Used simplifier rules: leq-trans: ⊦ σ ⊑ σ1 ∧ σ1 ⊑ σ2 → σ ⊑ σ2; (forward) from specification stream ⊦ ⌜x⌝ ⊑ ⌜y⌝ ↔ x ⊑ y; (s) Used terms: n

Proof Tree for lift-cont (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