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Proof step 2 from proof sum-constant-countFun (previous step)

(∀ i1. i ≤ i1 ∧ i1 ≤ j → intfun(i1) = i0) ∧ i ≤ j ∧ 0 ≤ i, ∀ j0, i1, intfun, i0. i→n(j0) - i→n(i1) < i→n(j) - i→n(i) → (∀ i2. i1 ≤ i2 ∧ i2 ≤ j0 → intfun(i2) = i0) ∧ i1 ≤ j0 ∧ 0 ≤ i1 → Σ countFun(i1, j0, intfun) = (1 + j0 - i1) * i0 ⊦ Σ countFun(i, j, intfun) = (1 + j - i) * i0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ i ≤ j → countFun(i, j, intfun) = countFun(i, j - 1, intfun) + intfun(j) '
Proof step 4:
i ≤ j, (∀ i1. i ≤ i1 ∧ i1 ≤ j → intfun(i1) = i0) ∧ i ≤ j ∧ 0 ≤ i, ∀ j0, i1, intfun, i0. i→n(j0) - i→n(i1) < i→n(j) - i→n(i) → (∀ i2. i1 ≤ i2 ∧ i2 ≤ j0 → intfun(i2) = i0) ∧ i1 ≤ j0 ∧ 0 ≤ i1 → Σ countFun(i1, j0, intfun) = (1 + j0 - i1) * i0 ⊦ Σ(countFun(i, j - 1, intfun) + intfun(j) ') = (1 + j - i) * i0

Rule argument

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

Additional information

Used terms: intfun, j, i