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Proof step 5 from proof diff-primes-neg (previous step)

countFun(0, (j - 1) / 2, λ i. (~1 - i ^2) |%| j)[m] = countFun(0, (j - 1) / 2, λ i. (~1 - i ^2) |%| j)[n], n < # countFun(0, (j - 1) / 2, λ i. (~1 - i ^2) |%| j), m < # countFun(0, (j - 1) / 2, λ i. (~1 - i ^2) |%| j), prime(j) ⊦ m = n, j = 2, countFun(0, (j - 1) / 2, λ i. (~1 - i ^2) |%| j) = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ 0 ≤ i ∧ i ≤ j → # countFun(i, j, intfun) = i→n(1 + j - i)
Proof step 7:
0 ≤ 0 ∧ 0 ≤ (j - 1) / 2, countFun(0, (j - 1) / 2, λ i. (~1 - i ^2) |%| j)[m] = countFun(0, (j - 1) / 2, λ i. (~1 - i ^2) |%| j)[n], n < i→n(1 + (j - 1) / 2 - 0), m < i→n(1 + (j - 1) / 2 - 0), prime(j) ⊦ m = n, j = 2, countFun(0, (j - 1) / 2, λ i. (~1 - i ^2) |%| j) = []

Rule argument

spec: intlist inst: name: countFun-length seq: ⊦ 0 ≤ i ∧ i ≤ j → # countFun(i, j, intfun) = i→n(1 + j - i) subst: [intfun, j, i] → [λ i. (~1 - i ^2) |%| j, (j - 1) / 2, 0] flags: rewrite

Additional information

Used simplifier rules: div-two-zero-01: ⊦ 0 ≤ i / 2 ↔ ~1 ≤ i; from specification int-div%sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%prime-not-neg: ⊦ j < 0 → ¬ prime(j); from specification int-gcd Used terms: λ i. (~1 - i ^2) |%| j, (j - 1) / 2, 0