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Proof step 12 from proof congmod-inverse-pos (previous step)

gcd(i, j) = 1, i < j, 0 < j, 1 < j, i ≤ 1, 0 ≤ i, 0 ≤ j, ≡mod(i * i0, 1, j), i0 < 0 ⊦ ∃ i0. ≡mod(i * i0, 1, j) ∧ 0 ≤ i0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-abs inst: name: le-rec seq: ⊦ i ≠ j → (i ≤ j ↔ i ≤ j - 1) subst: [j, i] → [1, i] flags: rewrite

Additional information

Used simplifier rules: anti: ⊦ i ≤ j → (j ≤ i ↔ i = j); from specification int-abs%gcd-zero-04: ⊦ gcd(i, 0) = abs(i); from specification int-gcd%abs: ⊦ 1 < i → abs(i) = i; from specification int-abs%gcd-com: ⊦ gcd(i, j) = gcd(j, i); from specification int-gcd%Mult-zero: ⊦ i * 0 = 0; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista] Used terms: 1, i