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Proof step 31 from proof coprimes-min (previous step)

min(count-i2j(1, i)) ≤ min(filterFun(λ j. gcd(i, j) = 1, count-i2j(1, i))), 0 < i, min(filterFun(λ j. gcd(i, j) = 1, count-i2j(1, i))) ≤ 0, min(filterFun(λ j. gcd(i, j) = 1, count-i2j(1, i))) < i, ∀ i0. i0 ∈ filterFun(λ j. gcd(i, j) = 1, count-i2j(1, i)) → min(filterFun(λ j. gcd(i, j) = 1, count-i2j(1, i))) ≤ i0 ⊦ min(filterFun(λ j. gcd(i, j) = 1, count-i2j(1, i))) = 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 32:
min(count-i2j(1, i)) ≤ min(filterFun(λ j. gcd(i, j) = 1, count-i2j(1, i))), 0 < i, min(filterFun(λ j. gcd(i, j) = 1, count-i2j(1, i))) ≤ 0, min(filterFun(λ j. gcd(i, j) = 1, count-i2j(1, i))) < i, min(count-i2j(1, i)) ≤ 0, min(count-i2j(1, i)) < i, ∀ i0. i0 ∈ filterFun(λ j. gcd(i, j) = 1, count-i2j(1, i)) → min(filterFun(λ j. gcd(i, j) = 1, count-i2j(1, i))) ≤ i0 ⊦ min(filterFun(λ j. gcd(i, j) = 1, count-i2j(1, i))) = 1

Rule argument

Additional information

Used simplifier rules: forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs