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Proof step 23 from proof min-skolemf (previous step)

mon<(nσ), # nσ ≠ ∞ ∧ nσ[(# nσ).n - 1] < nσ[m0] ∨ ⌜m0⌝ < # nσ ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ[m0] ≤ nσ[m] → m0 ≤ m), ∀ n0, nσ0. n0 < nσ[m0] +1 → mon<(nσ0) → nσ0 = [] ∨ # nσ0 ≠ ∞ ∧ nσ0[(# nσ0).n - 1] < n0 ∨ (∃ m0. ⌜m0⌝ < # nσ0 ∧ n0 ≤ nσ0[m0] ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n0 ≤ nσ0[m] → m0 ≤ m)) ⊦ nσ = [], # nσ ≠ ∞ ∧ ¬ nσ[m0] < nσ[(# nσ).n - 1], ⌜m0 + 1⌝ < # nσ ∧ nσ[m0] < nσ[m0 + 1] ∧ (∀ m1. ⌜m1⌝ < # nσ ∧ nσ[m0] < nσ[m1] → m0 + 1 ≤ m1), ∃ m. ⌜m⌝ < # nσ ∧ nσ[m0] < nσ[m] ∧ (∀ m1. ⌜m1⌝ < # nσ ∧ nσ[m0] < nσ[m1] → m ≤ m1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 24:
mon<(nσ), # nσ ≠ ∞ ∧ nσ[(# nσ).n - 1] < nσ[m0] ∨ ⌜m0⌝ < # nσ ∧ (∀ m. ⌜m⌝ < # nσ ∧ nσ[m0] ≤ nσ[m] → m0 ≤ m), ∀ n0, nσ0. n0 < nσ[m0] +1 → mon<(nσ0) → nσ0 = [] ∨ # nσ0 ≠ ∞ ∧ nσ0[(# nσ0).n - 1] < n0 ∨ (∃ m0. ⌜m0⌝ < # nσ0 ∧ n0 ≤ nσ0[m0] ∧ (∀ m. ⌜m⌝ < # nσ0 ∧ n0 ≤ nσ0[m] → m0 ≤ m)) ⊦ nσ = [], # nσ ≠ ∞ ∧ ¬ nσ[m0] < nσ[(# nσ).n - 1], ∃ m. ⌜m⌝ < # nσ ∧ nσ[m0] < nσ[m] ∧ (∀ m1. ⌜m1⌝ < # nσ ∧ nσ[m0] < nσ[m1] → m ≤ m1), ⌜m0⌝ + ⌜1⌝ < # nσ ∧ nσ[m0] < nσ[m0 + 1] ∧ (⌜m⌝ < # nσ ∧ nσ[m0] < nσ[m] → m0 < m)

Rule argument

Additional information

Used simplifier rules: succ-01: ⊦ m + 1 ≤ n ↔ m < n; from specification nat%c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%add-fin: ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; from specification nat-inf