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Proof step 1 from proof monge-boundedge-converges (conclusion) (back to tree)

⊦ bounded≥(f, r) ∧ mon≥(f) → converges(f)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof monge-boundedge-converges (previous step) (back to tree)

bounded≥(f, r), mon≥(f), ¬ converges(f) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: boundedge-neg-boundedle, seq: ⊦ bounded≥(f, r) ↔ bounded≤(λ n. ~ f(n), ~ r), subst:[f, r] → [f, r], <all paths>

Additional information

Used terms: f, r

Proof step 3 from proof monge-boundedge-converges (previous step) (back to tree)

⊦ bounded≥(f, r) ↔ bounded≤(λ n. ~ f(n), ~ r)

Lemma boundedge-neg-boundedle

Proof step 4 from proof monge-boundedge-converges (previous step) (back to tree)

bounded≤(λ n. ~ f(n), ~ r), mon≥(f), ¬ converges(f) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: monge-neg-monle, seq: ⊦ mon≥(f) ↔ mon≤(λ n. ~ f(n)), subst:[f] → [f], <all paths>

Additional information

Used terms: f

Proof step 5 from proof monge-boundedge-converges (previous step) (back to tree)

⊦ mon≥(f) ↔ mon≤(λ n. ~ f(n))

Lemma monge-neg-monle

Proof step 6 from proof monge-boundedge-converges (previous step) (back to tree)

bounded≤(λ n. ~ f(n), ~ r), mon≤(λ n. ~ f(n)), ¬ converges(f) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:monle-boundedle-converges, seq: ⊦ bounded≤(f, r) ∧ mon≤(f) → converges(f), subst:[f, r] → [λ n. ~ f(n), ~ r]

Additional information

Used terms: λ n. ~ f(n), ~ r

Proof step 7 from proof monge-boundedge-converges (previous step) (back to tree)

⊦ bounded≤(f, r) ∧ mon≤(f) → converges(f)

Lemma monle-boundedle-converges

Proof step 8 from proof monge-boundedge-converges (previous step) (back to tree)

bounded≤(λ n. ~ f(n), ~ r), mon≤(λ n. ~ f(n)), ¬ converges(f), bounded≤(λ n. ~ f(n), ~ r) ∧ mon≤(λ n. ~ f(n)) → converges(λ n. ~ f(n)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 9 from proof monge-boundedge-converges (previous step) (back to tree)

bounded≤(λ n. ~ f(n), ~ r), converges(λ n. ~ f(n)), mon≤(λ n. ~ f(n)), ¬ converges(f) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: converges-neg-converges, seq: ⊦ converges(f) ↔ converges(λ n. ~ f(n)), subst:[f] → [f], <all paths>

Additional information

Used terms: f

Proof step 10 from proof monge-boundedge-converges (previous step) (back to tree)

⊦ converges(f) ↔ converges(λ n. ~ f(n))

Lemma converges-neg-converges

Proof Tree for monge-boundedge-converges (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