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Proof step 17 from proof diff-primes (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: intnat inst: name: num-i2n-ls-02 seq: 0 ≤ i ⊦ (* m) < i→n(i) ↔ n→i(m) < i subst: [i, m] → [1 + (j - 1) / 2, n] flags: rewrite

Additional information

Used simplifier rules: numle-02: ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0; from specification int-abs%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%mone-06: ⊦ ~1 < i ↔ 0 ≤ i; from specification int-abs%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%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%succ-02: ⊦ i < j + 1 ↔ i ≤ j; from specification int-abs%lels-03: ⊦ i < j → (¬ ¬ j ≤ i ↔ false); from specification int-abs%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%div-ls-mone: 0 < j ⊦ i / (* j) < ~1 ↔ i + 2 * j ≤ 0; from specification int-div%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%sub-add-01: ⊦ i + (j - i0) = i + j - i0; from specification int-abs%numsub: ⊦ (* i) + i0 - (* j) = (i - j) + i0; from specification int-abs%numle-03: ⊦ i + (* j) ≤ (* j0) ↔ i ≤ j0 - j; from specification int-abs%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%forward-le-03: ⊦ j ≤ i ∧ i < 0 → j < 0; (forward) from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs Used terms: 1 + (j - 1) / 2, n