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

0 < i + j, 0 ≤ i, 0 < j, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ i ∧ 0 ≤ j0 → fac(i + j0) = fac(i) * Π countFun(1, j0, λ j0. i + j0) ⊦ j = 0, (i + j) * fac(i + j - 1) = fac(i) * Π countFun(1, j, λ j0. i + j0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 12:
⊦ i ≤ j → countFun(i, j, intfun) = countFun(i, j - 1, intfun) + intfun(j) '
Proof step 13:
1 ≤ j, 0 < i + j, 0 ≤ i, 0 < j, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ i ∧ 0 ≤ j0 → fac(i + j0) = fac(i) * Π countFun(1, j0, λ j0. i + j0) ⊦ j = 0, (i + j) * fac(i + j - 1) = fac(i) * Π(countFun(1, j - 1, λ j0. i + j0) + (i + j)')

Rule argument

spec: intlist inst: name: countFun-last seq: ⊦ i ≤ j → countFun(i, j, intfun) = countFun(i, j - 1, intfun) + intfun(j) ' subst: [intfun, j, i] → [λ j0. i + j0, j, 1] flags: rewrite

Additional information

Used simplifier rules: leone: ⊦ 1 ≤ i ↔ 0 < i; from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: λ j0. i + j0, j, 1