[Imprint][Data Privacy Statement][E-Mail]

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Proof step 9 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, 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, λ k. A[m0, k] * B[k, n0]);0), k = k0, k < B.rows, n = 0 ⊦ ⟨while n < C.cols do {/* (correctness invariant: n ≤ C.cols ∧ C.rows = A.rows ∧ C.cols = B.cols ∧ (∀ m0, n0. m0 < C.rows ∧ n0 < C.cols → C[m0, n0] = (C`)[m0, n0] + (m0 = m ∧ n0 < n ⊃ A[m0, k] * B[k, n0];0)) wfbound: C.cols - n) */; C[m, n] := C[m, n] + A[m, k] * B[k, n]; n := n + 1}⟩ ⟨k := k + 1⟩ ( ( k ≤ 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, λ k. A[m0, k] * B[k, n0]);0))) ∧ B.rows - k < B.rows - k0)

Heuristic symbolic execution applied rule invariant right

And obtained the following new goals

Proof step 10:
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), C = C3, n = n1, n < C.cols ⊦ ⟨C[m, n] := C[m, n] + A[m, k] * B[k, n]⟩ ⟨n := n + 1⟩ ( ( 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))) ∧ C.cols - n < C3.cols - n1)
Proof step 27:
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 := k + 1⟩ ( ( k ≤ 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, λ k. A[m0, k] * B[k, n0]);0))) ∧ B.rows - k < B.rows - k0)

Rule argument

bound: C.cols - n inv: n ≤ C.cols ∧ C.rows = A.rows ∧ C.cols = B.cols ∧ (∀ m0, n0. m0 < C.rows ∧ n0 < C.cols → C[m0, n0] = (C`)[m0, n0] + (m0 = m ∧ n0 < n ⊃ A[m0, k] * B[k, n0];0)) fmapos: right 1

Additional information

Used simplifier rules: lels-03: ⊦ m < n → m ≤ n; from specification nat%lez: ⊦ 0 ≤ m; from specification nat%zero-01: ⊦ ¬ m < 0; from specification nat-basic1%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used formulas: n ≤ C.cols ∧ C.rows = A.rows ∧ C.cols = B.cols ∧ (∀ m0, n0. m0 < C.rows ∧ n0 < C.cols → C[m0, n0] = (C`)[m0, n0] + (m0 = m ∧ n0 < n ⊃ A[m0, k] * B[k, n0];0)), C.cols - n = C.cols - n