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Proof step 23 from proof coprimes-min (previous step)

filterFun(λ j. gcd(i, j) = 1, count-i2j(1, i)) ≠ [] → 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

User applied rule case distinction

And obtained the following new goals

Proof step 24:
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 ⊦ filterFun(λ j. gcd(i, j) = 1, count-i2j(1, i)) ≠ [], min(filterFun(λ j. gcd(i, j) = 1, count-i2j(1, i))) = 1
Proof step 31:
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

Rule argument

pos: left 1

Proof step 23 from proof coprimes-min (previous step)

User applied rule case distinction

And obtained the following new goals

Rule argument

Additional information