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Proof step 24 from proof exin2pot (previous step)

0 < i0, 0 < 2 ^ i, 0 < i0 * 2 + j1, i0 < 2 ^ i * 2, 2 ^ i ≤ i0, 0 ≤ i, 0 ≤ i0, j1 < 2, 0 ≤ j1, 1 < 2 ^ i * 2, ∀ j0. i→n(j0) < i→n(i0 * 2 + j1) → 0 < j0 → (∃ i. 0 ≤ i ∧ 2 ^ i ≤ j0 ∧ j0 < 2 ^ (i + 1)) ⊦ i0 * 2 + j1 = 0, i0 * 2 + j1 = 1, 2 ^ i * 2 ≤ i0 * 2 + j1

User applied rule insert spec-lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: modlep-02 seq: j * i + j0 ≤ j * i0, abs(j0) < j ⊦ i ≤ i0 subst: [j, i, j0, i0] → [2, i0, j1, 2 ^ i] flags:

Additional information

Used simplifier rules: anti: ⊦ i ≤ j → (j ≤ i ↔ i = j); from specification int-abs%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%lf-02: ⊦ i < j ∧ j = i0 → i < i0; (forward) from specification int-basic1%mods-ls-04: ⊦ j0 < j ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 ≤ j0 → (i * j + j0 < i0 * j ↔ i < i0); from specification int-mult%lels-04: ⊦ i ≤ j → (¬ ¬ j < i ↔ false); from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%absle: ⊦ 0 ≤ i → abs(i) = i; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: 2, i0, j1, 2 ^ i