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Proof step 13 from proof add (previous step)

(i→n(i1) + 1) + i→n(j) = i→n((i1 + 1) + j), 0 ≤ j, 0 ≤ i1, ∀ j0. 0 ≤ j0 → (i→n(i1) + 1) + i→n(j0) = i→n((i1 + 1) + j0) ⊦ i→n(i1) + i→n(j) = i→n(j + i1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ 0 < i → i→n(i) = i→n(i - 1) + 1

Rule argument

spec: <toplevel> name: Int2nat-pos seq: ⊦ 0 < i → i→n(i) = i→n(i - 1) + 1 subst: [i] → [(i1 + 1) + j] flags: <all paths>

Additional information

Used simplifier rules: ⊦ (* i) + i0 - (* j) = (i - j) + i0; (s)% ⊦ i + 0 = i; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ m + n = m + n0 ↔ n = n0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; (s)% ⊦ ~1 < i ↔ 0 ≤ i; (s)% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ 0 ≤ i ∧ 0 ≤ j → ¬ i + j < 0; (s) Used terms: (i1 + 1) + j