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Proof step 1 from proof congmod-inverse-smallest (conclusion)

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

User applied rule insert lemma

And obtained the following new goals

Proof step 2:
gcd(i, j) = 1, 0 ≤ i, 0 ≤ j ⊦ ∃ i0. ≡mod(i * i0, 1, j) ∧ 0 ≤ i0
Proof step 3:
∃ i0. ≡mod(i * i0, 1, j) ∧ 0 ≤ i0, gcd(i, j) = 1, 0 ≤ i, 0 < j ⊦ ∃ i0. ≡mod(i * i0, 1, j) ∧ 0 ≤ i0 ∧ i0 < j

Rule argument

spec: inst: name: congmod-inverse-pos seq: gcd(i, j) = 1, 0 ≤ i, 0 ≤ j ⊦ ∃ i0. ≡mod(i * i0, 1, j) ∧ 0 ≤ i0 subst: [i, j] → [i, j] flags: current, []

Additional information

Used simplifier rules: lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: i, j