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Proof step 11 from proof max_exp-fac-add (previous step)

i0 < j, i0 < i, i0 < fac(i * j), i0 ≤ 0, prime(i), j0 < i, 0 < j, 0 ≤ j0, 0 < i, 0 ≤ j, 0 < fac(i * j), Π countFun(1, j0, λ j0. i * j + j0) ≤ 0, Π countFun(1, j0, λ j0. i * j + j0) < j, Π countFun(1, j0, λ j0. i * j + j0) < i, Π countFun(1, j0, λ j0. i * j + j0) < fac(i * j), i0 ∈ countFun(1, j0, λ j0. i * j + j0) ⊦ max_exp(i, fac(i * j) * Π countFun(1, j0, λ j0. i * j + j0)) = max_exp(i, fac(i * j))

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: intlist inst: name: countFun-in seq: ⊦ i0 ∈ countFun(i, j, intfun) ↔ (∃ j0. i ≤ j0 ∧ j0 ≤ j ∧ intfun(j0) = i0) subst: [intfun, j, i, i0] → [λ j0. i * j + j0, j0, 1, i0] flags: rewrite

Additional information

Used simplifier rules: leone: ⊦ 1 ≤ i ↔ 0 < i; from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%zero-ls: ⊦ 0 < i → (i * j < 0 ↔ j < 0); from specification int-mult%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: λ j0. i * j + j0, j0, 1, i0