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Proof step 5 from proof coef-gcd (previous step)

gcd(i, j) = 1, j ≤ i, 0 ≤ j, 0 ≤ i, 0 < j → i * binom_coef(i - 1, j - 1) = j * binom_coef(i, j) ⊦ i | binom_coef(i, j)

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 6:
⊦ gcd(i, j) = 1 ∧ i * i0 = j * j0 → i | j0

Rule argument

spec: int-gcd inst: name: coprime-divides seq: ⊦ gcd(i, j) = 1 ∧ i * i0 = j * j0 → i | j0 subst: [i, j, i0, j0] → [i, j, binom_coef(i - 1, j - 1), binom_coef(i, j)] flags:

Additional information

Used simplifier rules: coef-neg: ⊦ j < 0 → binom_coef(i, j) = 0;%coef-zero: ⊦ 0 ≤ i → binom_coef(i, 0) = 1;%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%anti: ⊦ i ≤ j → (j ≤ i ↔ i = j); from specification int-abs%gcd-zero-04: ⊦ gcd(i, 0) = abs(i); from specification int-gcd%absle: ⊦ 0 ≤ i → abs(i) = i; from specification int-abs%sub-eq-01: ⊦ i = j → i - j = 0; from specification int-abs%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 Used terms: i, j, binom_coef(i - 1, j - 1), binom_coef(i, j)