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Proof step 28 from proof correctness (previous step)

A.cols = B.rows, m2 = 0, C1 = A.rows × B.cols, m ≤ C2.rows, C2.rows = A.rows, C2.cols = B.cols, ∀ m0, n0. m0 < C2.rows ∧ n0 < C2.cols → C2[m0, n0] = C1[m0, n0] + (m0 < m ∧ n0 < C2.cols ⊃ Σ(B.rows, λ k. A[m0, k] * B[k, n0]);0), C2 = C0, m = m1, m < C2.rows, k1 = 0, k ≤ B.rows, C4.rows = A.rows, C4.cols = B.cols, ∀ m0, n0. m0 < C4.rows ∧ n0 < C4.cols → C4[m0, n0] = C2[m0, n0] + (m0 = m ∧ n0 < C4.cols ⊃ Σ(k, λ k. A[m0, k] * B[k, n0]);0), k = k0, k < B.rows, n2 = 0, n ≤ C.cols, C.rows = A.rows, C.cols = B.cols, ∀ m0, n0. m0 < C.rows ∧ n0 < C.cols → C[m0, n0] = C4[m0, n0] + (m0 = m ∧ n0 < n ⊃ A[m0, k] * B[k, n0];0), ¬ n < C.cols, ( ¬ k + 1 ≤ B.rows ∨ C.rows ≠ A.rows ∨ C.cols ≠ B.cols ∨ (∃ m0, n0. ¬(m0 < C.rows ∧ n0 < C.cols → C[m0, n0] = C2[m0, n0] + (m0 = m ∧ n0 < C.cols ⊃ Σ(k + 1, λ k. A[m0, k] * B[k, n0]);0)))) ∨ ¬ B.rows - k + 1 < B.rows - k0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 29:
C.cols = B.cols, C.rows = A.rows, C4.cols = B.cols, C4.rows = A.rows, C2.cols = B.cols, C2.rows = A.rows, A.cols = B.rows, n0 < B.cols, m0 < A.rows, k0 < B.rows, m < A.rows, ∀ m0, n0. m0 < A.rows ∧ n0 < B.cols → C4[m0, n0] = C2[m0, n0] + (m0 = m ⊃ Σ(k0, λ k. A[m, k] * B[k, n0]);0), ∀ m1, n. m1 < A.rows ∧ n < B.cols → C[m1, n] = C4[m1, n] + (m1 = m ⊃ A[m, k0] * B[k0, n];0), ∀ m1, n. m1 < A.rows ∧ n < B.cols → C2[m1, n] = (m1 < m ⊃ Σ(B.rows, λ k. A[m1, k] * B[k, n]);0) ⊦ B.cols = 0, A.rows = 0, B.rows = 0, C[m0, n0] = C2[m0, n0] + (m0 = m ⊃ A[m, k0] * B[k0, n0] + Σ(k0, λ k. A[m, k] * B[k, n0]);0)

Rule argument

Additional information

Used simplifier rules: add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2%at-base: ⊦ m0 < m ∧ n0 < n → (m × n)[m0, n0] = 0; from specification matrix%ler: ⊦ m ≤ m; from specification nat%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic1%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1%ref-1: ⊦ ¬ n < n; from specification nat-basic1%s: ⊦ m < n + 1 ↔ ¬ n < m; from specification nat-basic2%prop-07: ⊦ n < m → (m - n) + n = m; from specification nat%ls-01: n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; from specification nat%ls-le-simp: ⊦ m < n → (n ≤ m ↔ false); from specification nat%sum-rec: ⊦ Σ(n + 1, nf) = nf(n) + Σ(n, nf); from specification matrix-dot%c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%succ-01: ⊦ m + 1 ≤ n ↔ m < n; from specification nat%anti: ⊦ m ≤ n → (n ≤ m ↔ m = n); from specification nat%lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat%lels-03: ⊦ m < n → m ≤ n; from specification nat