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Proof step 1 from proof two-mon-size (conclusion) (back to tree)

mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ ⊦ # lim(⌜σf⌝ ) = # lim(⌜σg⌝ ) + ⌜m⌝

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof two-mon-size (previous step) (back to tree)

# lim(⌜σf⌝ ) ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, 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] → [⌜σf⌝ ]; [A] → [A]

Additional information

Used terms: ⌜σf⌝

Proof step 3 from proof two-mon-size (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)))

Lemma lim-def

Proof step 4 from proof two-mon-size (previous step) (back to tree)

# lim(⌜σf⌝ ) ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , mon(⌜σf⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ) ∧ ⌜σf⌝ ≠ [] → (∀ n. ⌜n⌝ < # ⌜σf⌝ → ⌜σf⌝ (n) ⊑ lim(⌜σf⌝ )) ∧ (∀ m. ¬ # lim(⌜σf⌝ ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # ⌜σf⌝ ∧ ⌜lim(⌜σf⌝ ).take(m)⌝ ⊑ ⌜σf⌝ (n))), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 5 from proof two-mon-size (previous step) (back to tree)

# lim(⌜σf⌝ ) ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, 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] → [⌜σg⌝ ]; [A] → [B]

Additional information

Used terms: ⌜σg⌝

Proof step 6 from proof two-mon-size (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)))

Lemma lim-def

Proof step 7 from proof two-mon-size (previous step) (back to tree)

# lim(⌜σf⌝ ) ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), mon(⌜σg⌝ ) ∧ ⌜σg⌝ ≠ [] → (∀ n. ⌜n⌝ < # ⌜σg⌝ → ⌜σg⌝ (n) ⊑ lim(⌜σg⌝ )) ∧ (∀ m. ¬ # lim(⌜σg⌝ ) < ⌜m⌝ → (∃ n. ⌜n⌝ < # ⌜σg⌝ ∧ ⌜lim(⌜σg⌝ ).take(m)⌝ ⊑ ⌜σg⌝ (n))), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 8 from proof two-mon-size (previous step) (back to tree)

# lim(⌜σf⌝ ) ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: # lim(⌜σf⌝ ) < # lim(⌜σg⌝ ) + ⌜m⌝

Additional information

Used formulas: # lim(⌜σf⌝ ) < # lim(⌜σg⌝ ) + ⌜m⌝

Proof step 9 from proof two-mon-size (previous step) (back to tree)

# lim(⌜σf⌝ ) ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , # lim(⌜σf⌝ ) < # lim(⌜σg⌝ ) + ⌜m⌝ , mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: # lim(⌜σf⌝ )

Additional information

Used simplifier rules: Natinf freely generated by ∞, ⌝; ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ¬ ∞ < n∞; (s)

Proof step 10 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σf⌝ ), # lim(⌜σf⌝ ) ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , # lim(⌜σf⌝ ) < # lim(⌜σg⌝ ) + ⌜m⌝ , mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 11 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , ⌜m0⌝ < # lim(⌜σg⌝ ) + ⌜m⌝ , mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: m ≤ m0

Additional information

Used formulas: m ≤ m0

Proof step 12 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m ≤ m0, ⌜m0⌝ < # lim(⌜σg⌝ ) + ⌜m⌝ , mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 8 terms: (m0 - m) + 1 abort: F discard: T computed: F

Additional information

Used terms: (m0 - m) + 1

Proof step 13 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m ≤ m0, ⌜m0⌝ < # lim(⌜σg⌝ ) + ⌜m⌝ , mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ¬ # lim(⌜σg⌝ ) < ⌜(m0 - m) + 1⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take((m0 - m) + 1)⌝ ⊑ σg(n)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 14 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m ≤ m0, ⌜m0⌝ < # lim(⌜σg⌝ ) + ⌜m⌝ , mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ⌜m0 - m⌝ < # lim(⌜σg⌝ ) → ⌜lim(⌜σg⌝ ).take((m0 - m) + 1)⌝ ⊑ σg(n), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-sub, seq:n ≤ m ⊦ n0 = m - n ↔ m = n0 + n, subst:[m, n, n0] → [m0, m, n0]

Additional information

Used simplifier rules: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: m0, m, n0

Proof step 15 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σf⌝ ), m0 = n0 + m, ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m ≤ m0, ⌜m0⌝ < # lim(⌜σg⌝ ) + ⌜m⌝ , mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ⌜n0⌝ < # lim(⌜σg⌝ ) → ⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ ⊑ σg(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m ≤ m + n; (s) ⊦ ⌜n⌝ + n∞1 < ⌜n⌝ + n∞2 ↔ n∞1 < n∞2; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ ⌜n⌝ + n∞1 = ⌜n⌝ + n∞2 ↔ n∞1 = n∞2; (s)

Proof step 16 from proof two-mon-size (previous step) (back to tree)

⌜n0⌝ + ⌜m⌝ = # lim(⌜σf⌝ ), ⌜n0⌝ ≠ # lim(⌜σg⌝ ), ⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ ⊑ σg(n), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ⌜n0⌝ < # lim(⌜σg⌝ ), ∀ m0. ¬ n0 + m < m0 → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ ⊦

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] → [⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ , σg(n)]; [A] → [B], <all paths>

Additional information

Used terms: ⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ , σg(n)

Proof step 17 from proof two-mon-size (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 18 from proof two-mon-size (previous step) (back to tree)

⌜n0⌝ + ⌜m⌝ = # lim(⌜σf⌝ ), ⌜n0⌝ ≠ # lim(⌜σg⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ⌜n0⌝ < # lim(⌜σg⌝ ), ∀ m0. ¬ n0 + m < m0 → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∃ x_σ. ⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ = σg(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 19 from proof two-mon-size (previous step) (back to tree)

⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ = σg(n), ⌜n0⌝ + ⌜m⌝ = # lim(⌜σf⌝ ), ⌜n0⌝ ≠ # lim(⌜σg⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ⌜n0⌝ < # lim(⌜σg⌝ ), ∀ m0. ¬ n0 + m < m0 → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: extensionality, seq: ⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0(n) = σ1(n)), subst:[σ0, σ1] → [⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ, σg(n)]; [A] → [B], <all paths>

Additional information

Used terms: ⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ, σg(n)

Proof step 20 from proof two-mon-size (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0(n) = σ1(n))

A spec-lemma

Proof step 21 from proof two-mon-size (previous step) (back to tree)

#(⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ) = # σg(n), ⌜n0⌝ + ⌜m⌝ = # lim(⌜σf⌝ ), ⌜n0⌝ ≠ # lim(⌜σg⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ⌜n0⌝ < # lim(⌜σg⌝ ), ∀ m0. ¬ n0 + m < m0 → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n1. ⌜n1⌝ < #(⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ) → (⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ)(n1) = σg(n)(n1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) ⊦ # ⌜σ.take(n)⌝ = ⌜# σ.take(n)⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 22 from proof two-mon-size (previous step) (back to tree)

⌜n0⌝ + ⌜m⌝ = # lim(⌜σf⌝ ), (⌜n0⌝ + ⌜1⌝ ) + # x_σ = # σg(n), ⌜n0⌝ ≠ # lim(⌜σg⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ⌜n0⌝ < # lim(⌜σg⌝ ), ∀ m0. ¬ n0 + m < m0 → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n1. ¬ ⌜n0⌝ + # x_σ < ⌜n1⌝ → (⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ)(n1) = σg(n)(n1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 10 terms: n abort: F discard: T computed: F

Additional information

Used terms: n

Proof step 23 from proof two-mon-size (previous step) (back to tree)

# σf(n) = # σg(n) + ⌜m⌝ , (⌜n0⌝ + ⌜1⌝ ) + # x_σ = # σg(n), ⌜n0⌝ + ⌜m⌝ = # lim(⌜σf⌝ ), ⌜n0⌝ ≠ # lim(⌜σg⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ⌜n0⌝ < # lim(⌜σg⌝ ), ∀ m0. ¬ n0 + m < m0 → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n1. ¬ ⌜n0⌝ + # x_σ < ⌜n1⌝ → (⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ)(n1) = σg(n)(n1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 9 terms: n abort: F discard: T computed: F

Additional information

Used terms: n

Proof step 24 from proof two-mon-size (previous step) (back to tree)

# σf(n) = # σg(n) + ⌜m⌝ , ⌜n0⌝ + ⌜m⌝ = # lim(⌜σf⌝ ), (⌜n0⌝ + ⌜1⌝ ) + # x_σ = # σg(n), ⌜n0⌝ ≠ # lim(⌜σg⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ⌜n0⌝ < # lim(⌜σg⌝ ), σf(n) ⊑ lim(⌜σf⌝ ), ∀ m0. ¬ n0 + m < m0 → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n1. ¬ ⌜n0⌝ + # x_σ < ⌜n1⌝ → (⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ)(n1) = σg(n)(n1) ⊦

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] → [σf(n), lim(⌜σf⌝ )]; [A] → [A], <all paths>

Additional information

Used terms: σf(n), lim(⌜σf⌝ )

Proof step 25 from proof two-mon-size (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 26 from proof two-mon-size (previous step) (back to tree)

# σf(n) = # σg(n) + ⌜m⌝ , (⌜n0⌝ + ⌜1⌝ ) + # x_σ = # σg(n), ⌜n0⌝ + ⌜m⌝ = # lim(⌜σf⌝ ), ⌜n0⌝ ≠ # lim(⌜σg⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ⌜n0⌝ < # lim(⌜σg⌝ ), ∀ m0. ¬ n0 + m < m0 → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n1. ¬ ⌜n0⌝ + # x_σ < ⌜n1⌝ → (⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ)(n1) = σg(n)(n1), ∃ σ1. σf(n) + σ1 = lim(⌜σf⌝ ) ⊦

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 ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 27 from proof two-mon-size (previous step) (back to tree)

# σf(n) = # σg(n) + ⌜m⌝ , (⌜n0⌝ + ⌜1⌝ ) + # x_σ = # σg(n), ⌜n0⌝ + ⌜m⌝ = # σf(n) + # σ1, σf(n) + σ1 = lim(⌜σf⌝ ), ⌜n0⌝ ≠ # lim(⌜σg⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ⌜n0⌝ < # lim(⌜σg⌝ ), ∀ m0. ¬ n0 + m < m0 → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n1. ¬ ⌜n0⌝ + # x_σ < ⌜n1⌝ → (⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ)(n1) = σg(n)(n1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: extensionality, seq: ⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0(n) = σ1(n)), subst:[σ0, σ1] → [σf(n) + σ1, lim(⌜σf⌝ )]; [A] → [A], <all paths>

Additional information

Used terms: σf(n) + σ1, lim(⌜σf⌝ )

Proof step 28 from proof two-mon-size (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0(n) = σ1(n))

A spec-lemma

Proof step 29 from proof two-mon-size (previous step) (back to tree)

#(σf(n) + σ1) = # lim(⌜σf⌝ ), # σf(n) = # σg(n) + ⌜m⌝ , ⌜n0⌝ + ⌜m⌝ = # σf(n) + # σ1, (⌜n0⌝ + ⌜1⌝ ) + # x_σ = # σg(n), ⌜n0⌝ ≠ # lim(⌜σg⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ⌜n0⌝ < # lim(⌜σg⌝ ), ∀ m0. ¬ n0 + m < m0 → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n0. ⌜n0⌝ < #(σf(n) + σ1) → (σf(n) + σ1)(n0) = lim(⌜σf⌝ )(n0), ∀ n1. ¬ ⌜n0⌝ + # x_σ < ⌜n1⌝ → (⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ)(n1) = σg(n)(n1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ n∞ = ⌜m⌝ + ⌜n0⌝ → (⌜n⌝ < n∞ ↔ n < m + n0); (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 30 from proof two-mon-size (previous step) (back to tree)

# σf(n) = # σg(n) + ⌜m⌝ , (⌜n0⌝ + ⌜1⌝ ) + # x_σ = # σg(n), ⌜n0⌝ + ⌜m⌝ = # σf(n) + # σ1, ⌜n0⌝ + ⌜m⌝ = # lim(⌜σf⌝ ), ⌜n0⌝ ≠ # lim(⌜σg⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ⌜n0⌝ < # lim(⌜σg⌝ ), ∀ m0. ¬ n0 + m < m0 → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n1. ¬ ⌜n0⌝ + # x_σ < ⌜n1⌝ → (⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ)(n1) = σg(n)(n1), ∀ n1. n1 < n0 + m → (σf(n) + σ1)(n1) = lim(⌜σf⌝ )(n1) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: σ1

Additional information

Used simplifier rules: Stream[A] freely generated by ⌝, ⌝; ⊦ # ⌜nf⌝ = ∞; (s) ⊦ n∞ + ∞ = ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ ⌜n⌝ ≠ ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf

Proof step 31 from proof two-mon-size (previous step) (back to tree)

# σf(n) = # σg(n) + ⌜m⌝ , ⌜n0⌝ + ⌜m⌝ = # lim(⌜σf⌝ ), ⌜n0⌝ + ⌜m⌝ = # σf(n) + # σ1, (⌜n0⌝ + ⌜1⌝ ) + # x_σ = # σg(n), σ1 = ⌜x⌝ , ⌜n0⌝ ≠ # lim(⌜σg⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ⌜n0⌝ < # lim(⌜σg⌝ ), ∀ m0. ¬ n0 + m < m0 → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n1. n1 < n0 + m → (σf(n) + σ1)(n1) = lim(⌜σf⌝ )(n1), ∀ n1. ¬ ⌜n0⌝ + # x_σ < ⌜n1⌝ → (⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ)(n1) = σg(n)(n1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 32 from proof two-mon-size (previous step) (back to tree)

# σf(n) = # σg(n) + ⌜m⌝ , ⌜n0⌝ + ⌜m⌝ = # lim(⌜σf⌝ ), (⌜n0⌝ + ⌜1⌝ ) + # x_σ = # σg(n), ⌜n0⌝ + ⌜m⌝ = # σf(n) + ⌜# x⌝ , ⌜n0⌝ ≠ # lim(⌜σg⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ⌜n0⌝ < # lim(⌜σg⌝ ), ∀ m0. ¬ n0 + m < m0 → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n1. ¬ ⌜n0⌝ + # x_σ < ⌜n1⌝ → (⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ)(n1) = σg(n)(n1), ∀ n1. n1 < n0 + m → (σf(n) + ⌜x⌝ )(n1) = lim(⌜σf⌝ )(n1) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags:

Additional information

Proof step 33 from proof two-mon-size (previous step) (back to tree)

# σf(n) = # σg(n) + ⌜m⌝ , ⌜n0⌝ + ⌜m⌝ = (# σg(n) + ⌜m⌝ ) + ⌜# x⌝ , (⌜n0⌝ + ⌜1⌝ ) + # x_σ = # σg(n), ⌜n0⌝ + ⌜m⌝ = # lim(⌜σf⌝ ), ⌜n0⌝ ≠ # lim(⌜σg⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ⌜n0⌝ < # lim(⌜σg⌝ ), ∀ m0. ¬ n0 + m < m0 → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n1. n1 < n0 + m → (σf(n) + ⌜x⌝ )(n1) = lim(⌜σf⌝ )(n1), ∀ n1. ¬ ⌜n0⌝ + # x_σ < ⌜n1⌝ → (⌜lim(⌜σg⌝ ).take(n0 + 1)⌝ + x_σ)(n1) = σg(n)(n1) ⊦

User applied rule subst equation

And closed the goal!

Rule argument

pos: left 3, flags: rotate, discard

Additional information

Used simplifier rules: ⊦ ⌜n⌝ + n∞1 = ⌜n⌝ + n∞2 ↔ n∞1 = n∞2; (s) ⊦ ⌜n⌝ + n∞ = ⌜n⌝ ↔ n∞ = ⌜0⌝ ; (s) ⊦ ⌜n⌝ + n∞ = ⌜0⌝ ↔ n∞ = ⌜0⌝ ∧ n = 0; (ws) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf

Proof step 34 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , ⌜m0⌝ < # lim(⌜σg⌝ ) + ⌜m⌝ , mon(⌜σg⌝ ), mon(⌜σf⌝ ), ¬ m ≤ m0, ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ m < n → ⌜m⌝ < ⌜n⌝ + n∞0; (s) ⊦ m ≤ n ↔ ¬ n < m; (ss)

Proof step 35 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m0 < m, mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: m0 abort: F discard: T computed: F

Additional information

Used terms: m0

Proof step 36 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m0 < m, mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ n < n; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ # σ = ⌜n⌝ → σ.take(n) = σ.l; (s)

Proof step 37 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , ⌜lim(⌜σf⌝ ).l⌝ ⊑ σf(n), m0 < m, mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ ⊦

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] → [⌜lim(⌜σf⌝ ).l⌝ , σf(n)]; [A] → [A], <all paths>

Additional information

Used terms: ⌜lim(⌜σf⌝ ).l⌝ , σf(n)

Proof step 38 from proof two-mon-size (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 39 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m0 < m, mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∃ σ1. ⌜lim(⌜σf⌝ ).l⌝ + σ1 = σf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 40 from proof two-mon-size (previous step) (back to tree)

⌜lim(⌜σf⌝ ).l⌝ + σ1 = σf(n), ⌜m0⌝ = # lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m0 < m, mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 9 terms: n abort: F discard: F computed: F

Additional information

Used terms: n

Proof step 41 from proof two-mon-size (previous step) (back to tree)

⌜lim(⌜σf⌝ ).l⌝ + σ1 = σf(n), ⌜m0⌝ = # lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m0 < m, mon(⌜σg⌝ ), mon(⌜σf⌝ ), σf(n) ⊑ lim(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ) ⊦

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] → [σf(n), lim(⌜σf⌝ )]; [A] → [A], <all paths>

Additional information

Used terms: σf(n), lim(⌜σf⌝ )

Proof step 42 from proof two-mon-size (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 43 from proof two-mon-size (previous step) (back to tree)

⌜lim(⌜σf⌝ ).l⌝ + σ1 = σf(n), ⌜m0⌝ = # lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m0 < m, mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∃ σ1. σf(n) + σ1 = lim(⌜σf⌝ ) ⊦

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 ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 44 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # σf(n) + # σ, σf(n) + σ = lim(⌜σf⌝ ), ⌜(σf(n) + σ).l⌝ + σ1 = σf(n), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m0 < m, mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: σ

Additional information

Used simplifier rules: Stream[A] freely generated by ⌝, ⌝; ⊦ # ⌜nf⌝ = ∞; (s) ⊦ n∞ + ∞ = ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s)

Proof step 45 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # σf(n) + # σ, σ = ⌜x⌝ , ⌜(σf(n) + σ).l⌝ + σ1 = σf(n), σf(n) + σ = lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m0 < m, mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 46 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # σf(n) + ⌜# x⌝ , σf(n) + ⌜x⌝ = lim(⌜σf⌝ ), ⌜(σf(n) + ⌜x⌝ ).l⌝ + σ1 = σf(n), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m0 < m, mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: extensionality, seq: ⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0(n) = σ1(n)), subst:[σ0, σ1] → [⌜(σf(n) + ⌜x⌝ ).l⌝ + σ1, σf(n)]; [A] → [A], <all paths>

Additional information

Used terms: ⌜(σf(n) + ⌜x⌝ ).l⌝ + σ1, σf(n)

Proof step 47 from proof two-mon-size (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0(n) = σ1(n))

A spec-lemma

Proof step 48 from proof two-mon-size (previous step) (back to tree)

#(⌜(σf(n) + ⌜x⌝ ).l⌝ + σ1) = # σf(n), ⌜m0⌝ = # σf(n) + ⌜# x⌝ , σf(n) + ⌜x⌝ = lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m0 < m, mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n0. ⌜n0⌝ < #(⌜(σf(n) + ⌜x⌝ ).l⌝ + σ1) → (⌜(σf(n) + ⌜x⌝ ).l⌝ + σ1)(n0) = σf(n)(n0) ⊦

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⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 49 from proof two-mon-size (previous step) (back to tree)

⌜# (σf(n) + ⌜x⌝ ).l⌝ + # σ1 = # σf(n), ⌜m0⌝ = # σf(n) + ⌜# x⌝ , σf(n) + ⌜x⌝ = lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m0 < m, mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n0. ⌜n0⌝ < # σf(n) → (⌜(σf(n) + ⌜x⌝ ).l⌝ + σ1)(n0) = σf(n)(n0) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: σ1

Additional information

Used simplifier rules: Stream[A] freely generated by ⌝, ⌝; ⊦ # ⌜nf⌝ = ∞; (s) ⊦ # σ = ∞ → σ + σ0 = σ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜m⌝ + ∞ = ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s)

Proof step 50 from proof two-mon-size (previous step) (back to tree)

⌜# (σf(n) + ⌜x⌝ ).l⌝ + # σ1 = # σf(n), ⌜m0⌝ = # σf(n) + ⌜# x⌝ , σ1 = ⌜x0⌝ , σf(n) + ⌜x⌝ = lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m0 < m, mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n0. ⌜n0⌝ < # σf(n) → (⌜(σf(n) + ⌜x⌝ ).l⌝ + σ1)(n0) = σf(n)(n0) ⊦

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⌝ + ⌜x0⌝ )(m) = (x + x0)(m); (s) ⊦ n∞ = ⌜m⌝ + ⌜n0⌝ → (⌜n⌝ < n∞ ↔ n < m + n0); (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 51 from proof two-mon-size (previous step) (back to tree)

⌜# (σf(n) + ⌜x⌝ ).l⌝ + ⌜# x0⌝ = # σf(n), ⌜m0⌝ = # σf(n) + ⌜# x⌝ , σf(n) + ⌜x⌝ = lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m0 < m, mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n0. n0 < # (σf(n) + ⌜x⌝ ).l + # x0 → ((σf(n) + ⌜x⌝ ).l + x0)(n0) = σf(n)(n0) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: σf(n)

Additional information

Used simplifier rules: Stream[A] freely generated by ⌝, ⌝; ⊦ ⌜nf⌝ + σ = ⌜nf⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # ⌜nf⌝ = ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ ⌜n⌝ ≠ ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf

Proof step 52 from proof two-mon-size (previous step) (back to tree)

⌜# (σf(n) + ⌜x⌝ ).l⌝ + ⌜# x0⌝ = # σf(n), ⌜m0⌝ = # σf(n) + ⌜# x⌝ , ⌜x1⌝ = σf(n), σf(n) + ⌜x⌝ = lim(⌜σf⌝ ), ⌜m0⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , m0 < m, mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n0. n0 < # (σf(n) + ⌜x⌝ ).l + # x0 → ((σf(n) + ⌜x⌝ ).l + x0)(n0) = σf(n)(n0) ⊦

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 ⊦ σ + [] = σ; (s) ⊦ ⌜[]⌝ = []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + [] = x; (s) ⊦ [] + x = x; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ n + 0 = n; (s) ⊦ # [] = 0; (s) lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic ⊦ ⌜x⌝ (n) = x(n); (s) ⊦ (⌜x⌝ + ⌜y⌝ ).l = x + y; (s) ⊦ #(x + y) = # x + # y; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n + m = n0; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ # x = 0 ↔ x = []; (s) ⊦ ⌜n⌝ = ⌜n0⌝ ↔ n = n0; (s) ⊦ ⌜n⌝ + n∞ = ⌜0⌝ ↔ n∞ = ⌜0⌝ ∧ n = 0; (ws) ⊦ ⌜n⌝ + n∞ = ⌜n⌝ ↔ n∞ = ⌜0⌝ ; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 53 from proof two-mon-size (previous step) (back to tree)

⌜x1⌝ = σf(n), ⌜x1⌝ = lim(⌜σf⌝ ), ⌜# x1⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , # x1 < m, mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n0. σf(n0) ⊑ ⌜x1⌝ ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 9 terms: n abort: F discard: F computed: F

Additional information

Used terms: n

Proof step 54 from proof two-mon-size (previous step) (back to tree)

# σf(n) = # σg(n) + ⌜m⌝ , ⌜x1⌝ = lim(⌜σf⌝ ), ⌜x1⌝ = σf(n), ⌜# x1⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , # x1 < m, mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n0. σf(n0) ⊑ ⌜x1⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 55 from proof two-mon-size (previous step) (back to tree)

⌜# x1⌝ = # σg(n) + ⌜m⌝ , ⌜x1⌝ = lim(⌜σf⌝ ), ⌜x1⌝ = σf(n), ⌜# x1⌝ ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , # x1 < m, mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n0. σf(n0) ⊑ ⌜x1⌝ ⊦

User applied rule structural induction

And closed the goal!

Rule argument

term: σg(n)

Additional information

Used simplifier rules: Stream[A] freely generated by ⌝, ⌝; ⊦ # ⌜nf⌝ = ∞; (s) ⊦ ⌜m⌝ + ∞ = ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n + m = n0; (s) lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic ⊦ ¬ m + n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 56 from proof two-mon-size (previous step) (back to tree)

# lim(⌜σf⌝ ) ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , mon(⌜σf⌝ ), mon(⌜σg⌝ ), ¬ # lim(⌜σf⌝ ) < # lim(⌜σg⌝ ) + ⌜m⌝ , ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: nat-inf, name: lele-refl, seq: ⊦ ¬ n∞1 < n∞ → (n∞ < n∞1 ↔ n∞ ≠ n∞1), subst:[n∞, n∞1] → [# lim(⌜σf⌝ ), # lim(⌜σg⌝ ) + ⌜m⌝ ], <all paths>

Additional information

Used terms: # lim(⌜σf⌝ ), # lim(⌜σg⌝ ) + ⌜m⌝

Proof step 57 from proof two-mon-size (previous step) (back to tree)

⊦ ¬ n∞1 < n∞ → (n∞ < n∞1 ↔ n∞ ≠ n∞1)

A spec-lemma

Proof step 58 from proof two-mon-size (previous step) (back to tree)

# lim(⌜σf⌝ ) ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , # lim(⌜σg⌝ ) + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ¬ # lim(⌜σf⌝ ) < # lim(⌜σg⌝ ) + ⌜m⌝ , ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ n∞ < n∞0 → ¬ n∞0 < n∞; (s)

Proof step 59 from proof two-mon-size (previous step) (back to tree)

# lim(⌜σf⌝ ) ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , # lim(⌜σg⌝ ) + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: # lim(⌜σg⌝ )

Additional information

Used simplifier rules: Natinf freely generated by ∞, ⌝; ⊦ ⌜m⌝ + ∞ = ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ¬ ∞ < n∞; (s)

Proof step 60 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σg⌝ ), # lim(⌜σf⌝ ) ≠ # lim(⌜σg⌝ ) + ⌜m⌝ , # lim(⌜σg⌝ ) + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 61 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σg⌝ ), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ m0. ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0)⌝ ⊑ σf(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 7 terms: m0 + m + 1 abort: F discard: T computed: F

Additional information

Used terms: m0 + m + 1

Proof step 62 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σg⌝ ), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ¬ # lim(⌜σf⌝ ) < ⌜m0 + m + 1⌝ → (∃ n. ⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ ⊑ σf(n)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 63 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σg⌝ ), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ + ⌜m⌝ + ⌜1⌝ → ⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ ⊑ σf(n) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 10

Additional information

Proof step 64 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σg⌝ ), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , # lim(⌜σf⌝ ) < ⌜m0⌝ + ⌜m⌝ + ⌜1⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ) ⊦

User applied rule structural induction

And closed the goal!

Rule argument

term: # lim(⌜σf⌝ )

Additional information

Used simplifier rules: Natinf freely generated by ∞, ⌝; ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n + m = n0; (s) ⊦ ⌜n⌝ + n∞1 < ⌜n⌝ + n∞2 ↔ n∞1 < n∞2; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ ¬ n < n; (s) ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ ¬ ∞ < n∞; (s)

Proof step 65 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σg⌝ ), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ ⊑ σf(n), ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ) ⊦

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] → [⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ , σf(n)]; [A] → [A], <all paths>

Additional information

Used terms: ⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ , σf(n)

Proof step 66 from proof two-mon-size (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 67 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σg⌝ ), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∃ σ1. ⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1 = σf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 68 from proof two-mon-size (previous step) (back to tree)

⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1 = σf(n), ⌜m0⌝ = # lim(⌜σg⌝ ), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: extensionality, seq: ⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0(n) = σ1(n)), subst:[σ0, σ1] → [⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1, σf(n)]; [A] → [A], <all paths>

Additional information

Used terms: ⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1, σf(n)

Proof step 69 from proof two-mon-size (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0(n) = σ1(n))

A spec-lemma

Proof step 70 from proof two-mon-size (previous step) (back to tree)

#(⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1) = # σf(n), ⌜m0⌝ = # lim(⌜σg⌝ ), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n0. ⌜n0⌝ < #(⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1) → (⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1)(n0) = σf(n)(n0) ⊦

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 ⊦ # ⌜σ.take(n)⌝ = ⌜# σ.take(n)⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 71 from proof two-mon-size (previous step) (back to tree)

⌜# lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + # σ1 = # σf(n), ⌜m0⌝ = # lim(⌜σg⌝ ), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n0. ⌜n0⌝ < # σf(n) → (⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1)(n0) = σf(n)(n0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: size-to, seq:¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n, subst:[σ, n] → [lim(⌜σf⌝ ), m0 + m + 1]; [A] → [A], <all paths>

Additional information

Used terms: lim(⌜σf⌝ ), m0 + m + 1

Proof step 72 from proof two-mon-size (previous step) (back to tree)

¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n

A spec-lemma

Proof step 73 from proof two-mon-size (previous step) (back to tree)

⌜# lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + # σ1 = # σf(n), ⌜m0⌝ = # lim(⌜σg⌝ ), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , # lim(⌜σf⌝ ) < ⌜m0 + m + 1⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n0. ⌜n0⌝ < # σf(n) → (⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1)(n0) = σf(n)(n0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ¬ n < n; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) ⊦ ⌜n⌝ + n∞1 < ⌜n⌝ + n∞2 ↔ n∞1 < n∞2; (s) ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 74 from proof two-mon-size (previous step) (back to tree)

⌜# lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + # σ1 = # σf(n), ⌜m0⌝ = # lim(⌜σg⌝ ), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , # lim(⌜σf⌝ ) < ⌜m0⌝ + ⌜m⌝ + ⌜1⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n0. ⌜n0⌝ < # σf(n) → (⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1)(n0) = σf(n)(n0) ⊦

User applied rule structural induction

And closed the goal!

Rule argument

term: # lim(⌜σf⌝ )

Additional information

Used simplifier rules: Natinf freely generated by ∞, ⌝; ⊦ ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n + m = n0; (s) ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ ¬ ∞ < n∞; (s)

Proof step 75 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σg⌝ ), ⌜m0 + m + 1⌝ + # σ1 = # σf(n), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ¬ # lim(⌜σf⌝ ) < ⌜m0 + m + 1⌝ , ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n0. ⌜n0⌝ < # σf(n) → (⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1)(n0) = σf(n)(n0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf

Proof step 76 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σg⌝ ), (⌜m0⌝ + ⌜m⌝ + ⌜1⌝ ) + # σ1 = # σf(n), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ + ⌜m⌝ + ⌜1⌝ , ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. σg(n) ⊑ lim(⌜σg⌝ ), ∀ n0. ⌜n0⌝ < # σf(n) → (⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1)(n0) = σf(n)(n0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 11 terms: n abort: F discard: T computed: F

Additional information

Used terms: n

Proof step 77 from proof two-mon-size (previous step) (back to tree)

(⌜m0⌝ + ⌜m⌝ + ⌜1⌝ ) + # σ1 = # σf(n), ⌜m0⌝ = # lim(⌜σg⌝ ), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), σg(n) ⊑ lim(⌜σg⌝ ), ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ + ⌜m⌝ + ⌜1⌝ , ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n0. ⌜n0⌝ < # σf(n) → (⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1)(n0) = σf(n)(n0) ⊦

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] → [σg(n), lim(⌜σg⌝ )]; [A] → [B], <all paths>

Additional information

Used terms: σg(n), lim(⌜σg⌝ )

Proof step 78 from proof two-mon-size (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 79 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σg⌝ ), (⌜m0⌝ + ⌜m⌝ + ⌜1⌝ ) + # σ1 = # σf(n), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ + ⌜m⌝ + ⌜1⌝ , ∀ m0. ¬ # lim(⌜σg⌝ ) < ⌜m0⌝ → (∃ n. ⌜lim(⌜σg⌝ ).take(m0)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n0. ⌜n0⌝ < # σf(n) → (⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1)(n0) = σf(n)(n0), ∃ x_σ. σg(n) + x_σ = lim(⌜σg⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 80 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # σg(n) + # x_σ, (⌜m0⌝ + ⌜m⌝ + ⌜1⌝ ) + # σ1 = # σf(n), σg(n) + x_σ = lim(⌜σg⌝ ), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ + ⌜m⌝ + ⌜1⌝ , ∀ m1. ¬ m0 < m1 → (∃ n. ⌜lim(⌜σg⌝ ).take(m1)⌝ ⊑ σg(n)), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n0. ⌜n0⌝ < # σf(n) → (⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1)(n0) = σf(n)(n0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: extensionality, seq: ⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0(n) = σ1(n)), subst:[σ0, σ1] → [σg(n) + x_σ, lim(⌜σg⌝ )]; [A] → [B], <all paths>

Additional information

Used terms: σg(n) + x_σ, lim(⌜σg⌝ )

Proof step 81 from proof two-mon-size (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0(n) = σ1(n))

A spec-lemma

Proof step 82 from proof two-mon-size (previous step) (back to tree)

#(σg(n) + x_σ) = # lim(⌜σg⌝ ), (⌜m0⌝ + ⌜m⌝ + ⌜1⌝ ) + # σ1 = # σf(n), ⌜m0⌝ = # σg(n) + # x_σ, # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ + ⌜m⌝ + ⌜1⌝ , ∀ m1. ¬ m0 < m1 → (∃ n. ⌜lim(⌜σg⌝ ).take(m1)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n0. ⌜n0⌝ < # σf(n) → (⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1)(n0) = σf(n)(n0), ∀ n0. ⌜n0⌝ < #(σg(n) + x_σ) → (σg(n) + x_σ)(n0) = lim(⌜σg⌝ )(n0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 83 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σg⌝ ), (⌜m0⌝ + ⌜m⌝ + ⌜1⌝ ) + # σ1 = # σf(n), ⌜m0⌝ = # σg(n) + # x_σ, # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ + ⌜m⌝ + ⌜1⌝ , ∀ m1. ¬ m0 < m1 → (∃ n. ⌜lim(⌜σg⌝ ).take(m1)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n0. ⌜n0⌝ < # σf(n) → (⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1)(n0) = σf(n)(n0), ∀ n0. n0 < m0 → (σg(n) + x_σ)(n0) = lim(⌜σg⌝ )(n0) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: x_σ

Additional information

Used simplifier rules: Stream[A] freely generated by ⌝, ⌝; ⊦ # ⌜nf⌝ = ∞; (s) ⊦ n∞ + ∞ = ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s)

Proof step 84 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # σg(n) + # x_σ, (⌜m0⌝ + ⌜m⌝ + ⌜1⌝ ) + # σ1 = # σf(n), ⌜m0⌝ = # lim(⌜σg⌝ ), x_σ = ⌜x_List<B>⌝ , # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ + ⌜m⌝ + ⌜1⌝ , ∀ m1. ¬ m0 < m1 → (∃ n. ⌜lim(⌜σg⌝ ).take(m1)⌝ ⊑ σg(n)), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n0. n0 < m0 → (σg(n) + x_σ)(n0) = lim(⌜σg⌝ )(n0), ∀ n0. ⌜n0⌝ < # σf(n) → (⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1)(n0) = σf(n)(n0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 85 from proof two-mon-size (previous step) (back to tree)

⌜m0⌝ = # lim(⌜σg⌝ ), (⌜m0⌝ + ⌜m⌝ + ⌜1⌝ ) + # σ1 = # σf(n), ⌜m0⌝ = # σg(n) + ⌜# x_List<B>⌝ , # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ + ⌜m⌝ + ⌜1⌝ , ∀ m1. ¬ m0 < m1 → (∃ n. ⌜lim(⌜σg⌝ ).take(m1)⌝ ⊑ σg(n)), ∀ n. # σf(n) = # σg(n) + ⌜m⌝ , ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n0. ⌜n0⌝ < # σf(n) → (⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1)(n0) = σf(n)(n0), ∀ n0. n0 < m0 → (σg(n) + ⌜x_List<B>⌝ )(n0) = lim(⌜σg⌝ )(n0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 10 terms: n abort: F discard: T computed: F

Additional information

Used terms: n

Proof step 86 from proof two-mon-size (previous step) (back to tree)

# σf(n) = # σg(n) + ⌜m⌝ , ⌜m0⌝ = # σg(n) + ⌜# x_List<B>⌝ , (⌜m0⌝ + ⌜m⌝ + ⌜1⌝ ) + # σ1 = # σf(n), ⌜m0⌝ = # lim(⌜σg⌝ ), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ + ⌜m⌝ + ⌜1⌝ , ∀ m1. ¬ m0 < m1 → (∃ n. ⌜lim(⌜σg⌝ ).take(m1)⌝ ⊑ σg(n)), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n0. n0 < m0 → (σg(n) + ⌜x_List<B>⌝ )(n0) = lim(⌜σg⌝ )(n0), ∀ n0. ⌜n0⌝ < # σf(n) → (⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1)(n0) = σf(n)(n0) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: σ1

Additional information

Used simplifier rules: Stream[A] freely generated by ⌝, ⌝; ⊦ # ⌜nf⌝ = ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ ⌜m⌝ < ∞; (s) ⊦ # σ = ∞ → σ + σ0 = σ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜m⌝ + ∞ = ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s)

Proof step 87 from proof two-mon-size (previous step) (back to tree)

# σf(n) = # σg(n) + ⌜m⌝ , ⌜m0⌝ = # lim(⌜σg⌝ ), (⌜m0⌝ + ⌜m⌝ + ⌜1⌝ ) + # σ1 = # σf(n), ⌜m0⌝ = # σg(n) + ⌜# x_List<B>⌝ , σ1 = ⌜x⌝ , # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ + ⌜m⌝ + ⌜1⌝ , ∀ m1. ¬ m0 < m1 → (∃ n. ⌜lim(⌜σg⌝ ).take(m1)⌝ ⊑ σg(n)), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n0. ⌜n0⌝ < # σf(n) → (⌜lim(⌜σf⌝ ).take(m0 + m + 1)⌝ + σ1)(n0) = σf(n)(n0), ∀ n0. n0 < m0 → (σg(n) + ⌜x_List<B>⌝ )(n0) = lim(⌜σg⌝ )(n0) ⊦

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⌝ + ⌜x0⌝ )(m) = (x + x0)(m); (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s)

Proof step 88 from proof two-mon-size (previous step) (back to tree)

# σf(n) = # σg(n) + ⌜m⌝ , ⌜m0⌝ = # lim(⌜σg⌝ ), ⌜m0⌝ = # σg(n) + ⌜# x_List<B>⌝ , ⌜m0⌝ + ⌜m⌝ + ⌜1⌝ + ⌜# x⌝ = # σf(n), # lim(⌜σf⌝ ) ≠ ⌜m0⌝ + ⌜m⌝ , ⌜m0⌝ + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ¬ # lim(⌜σf⌝ ) < ⌜m0⌝ + ⌜m⌝ + ⌜1⌝ , ∀ m1. ¬ m0 < m1 → (∃ n. ⌜lim(⌜σg⌝ ).take(m1)⌝ ⊑ σg(n)), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n0. n0 < m0 → (σg(n) + ⌜x_List<B>⌝ )(n0) = lim(⌜σg⌝ )(n0), ∀ n0. ⌜n0⌝ < # σf(n) → (lim(⌜σf⌝ ).take(m0 + m + 1) + x)(n0) = σf(n)(n0) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 3, flags:

Additional information

Proof step 89 from proof two-mon-size (previous step) (back to tree)

(# σg(n) + ⌜# x_List<B>⌝ ) + ⌜m⌝ + ⌜1⌝ + ⌜# x⌝ = # σf(n), # σg(n) + ⌜# x_List<B>⌝ = # lim(⌜σg⌝ ), # σf(n) = # σg(n) + ⌜m⌝ , ⌜m0⌝ = # σg(n) + ⌜# x_List<B>⌝ , # lim(⌜σf⌝ ) ≠ (# σg(n) + ⌜# x_List<B>⌝ ) + ⌜m⌝ , (# σg(n) + ⌜# x_List<B>⌝ ) + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σg⌝ ), mon(⌜σf⌝ ), ¬ # lim(⌜σf⌝ ) < (# σg(n) + ⌜# x_List<B>⌝ ) + ⌜m⌝ + ⌜1⌝ , ∀ m1. ¬ m0 < m1 → (∃ n. ⌜lim(⌜σg⌝ ).take(m1)⌝ ⊑ σg(n)), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n0. ⌜n0⌝ < # σf(n) → (lim(⌜σf⌝ ).take(m0 + m + 1) + x)(n0) = σf(n)(n0), ∀ n0. n0 < m0 → (σg(n) + ⌜x_List<B>⌝ )(n0) = lim(⌜σg⌝ )(n0) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 3, flags:

Additional information

Proof step 90 from proof two-mon-size (previous step) (back to tree)

# σf(n) = # σg(n) + ⌜m⌝ , # σg(n) + ⌜# x_List<B>⌝ = # lim(⌜σg⌝ ), (# σg(n) + ⌜# x_List<B>⌝ ) + ⌜m⌝ + ⌜1⌝ + ⌜# x⌝ = # σg(n) + ⌜m⌝ , ⌜m0⌝ = # σg(n) + ⌜# x_List<B>⌝ , # lim(⌜σf⌝ ) ≠ (# σg(n) + ⌜# x_List<B>⌝ ) + ⌜m⌝ , (# σg(n) + ⌜# x_List<B>⌝ ) + ⌜m⌝ < # lim(⌜σf⌝ ), mon(⌜σf⌝ ), mon(⌜σg⌝ ), ¬ # lim(⌜σf⌝ ) < (# σg(n) + ⌜# x_List<B>⌝ ) + ⌜m⌝ + ⌜1⌝ , ∀ m1. ¬ m0 < m1 → (∃ n. ⌜lim(⌜σg⌝ ).take(m1)⌝ ⊑ σg(n)), ∀ n. σf(n) ⊑ lim(⌜σf⌝ ), ∀ n0. n0 < m0 → (σg(n) + ⌜x_List<B>⌝ )(n0) = lim(⌜σg⌝ )(n0), ∀ n0. ⌜n0⌝ < # σg(n) + ⌜m⌝ → (lim(⌜σf⌝ ).take(m0 + m + 1) + x)(n0) = σf(n)(n0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: nat-inf, name: add-assoc, seq: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2, subst:[n∞, n∞1, n∞2] → [# σg(n), ⌜# x_List<B>⌝ , ⌜m⌝ + ⌜1⌝ + ⌜# x⌝ ], <all paths>

Additional information

Used simplifier rules: ⊦ ⌜n⌝ + n∞1 = ⌜n⌝ + n∞2 ↔ n∞1 = n∞2; (s) ⊦ n∞1 = n∞1 + n∞2 ↔ n∞1 = ∞ ∨ n∞2 = ⌜0⌝ ; (s) ⊦ n < m → (⌜(* n)⌝ = ⌜(* m)⌝ + n∞2 ↔ false); (s) ⊦ # σ = ∞ → σ + σ0 = σ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜m⌝ + ∞ = ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s) Used terms: # σg(n), ⌜# x_List<B>⌝ , ⌜m⌝ + ⌜1⌝ + ⌜# x⌝

Proof step 91 from proof two-mon-size (previous step) (back to tree)

⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2

A spec-lemma

Proof Tree for two-mon-size (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