Proof step 1 from proof multmx-column (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  #c mx1 = #r mx2, (mx1 * mx2)[c]c ≠ mx1 * mx2[c]c, c < #c mx2 ⊦

Rule argument

Additional information

Proof step 2 from proof multmx-column (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 3:

  ⊦ ar1 = ar2 ↔ # ar1 = # ar2 ∧ (∀ n. n < # ar1 → ar1[n] = ar2[n])

• Proof step 4:

  #c mx1 = #r mx2, c < #c mx2, # (mx1 * mx2)[c]c ≠ #(mx1 * mx2[c]c) ∨ (∃ n. ¬(n < # (mx1 * mx2)[c]c → (mx1 * mx2)[c]c[n] = (mx1 * mx2[c]c)[n])) ⊦

Rule argument

Additional information

Used terms: (mx1 * mx2)[c]c, mx1 * mx2[c]c

Proof step 3 from proof multmx-column (previous step) (back to tree)

A spec-lemma

Proof step 4 from proof multmx-column (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 5:

  ⊦ # mx[c]c = #r mx

• Proof step 6:

  #c mx1 = #r mx2, c < #c mx2, #r(mx1 * mx2) ≠ #(mx1 * mx2[c]c) ∨ (∃ n. ¬(n < #r(mx1 * mx2) → (mx1 * mx2)[c]c[n] = (mx1 * mx2[c]c)[n])) ⊦

Rule argument

Additional information

Used terms: mx1 * mx2, c

Proof step 5 from proof multmx-column (previous step) (back to tree)

A spec-lemma

Proof step 6 from proof multmx-column (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 7:

  ⊦ #c mx = # vec → #(mx * vec) = #r mx

• Proof step 8:

  #c mx1 = #r mx2, #c mx1 = # mx2[c]c, c < #c mx2, #r(mx1 * mx2) ≠ #r mx1 ∨ (∃ n. ¬(n < #r(mx1 * mx2) → (mx1 * mx2)[c]c[n] = (mx1 * mx2[c]c)[n])) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ # mx[c]c = #r mx; (s) Used terms: mx1, mx2[c]c

Proof step 7 from proof multmx-column (previous step) (back to tree)

Lemma multmxvec-size

Proof step 8 from proof multmx-column (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 9:

  ⊦ #c mx1 = #r mx2 → #r(mx1 * mx2) = #r mx1

• Proof step 10:

  #c mx1 = # mx2[c]c, #c mx1 = #r mx2, c < #c mx2, #r mx1 ≠ #r mx1 ∨ (∃ n. ¬(n < #r mx1 → (mx1 * mx2)[c]c[n] = (mx1 * mx2[c]c)[n])) ⊦

Rule argument

Additional information

Used terms: mx1, mx2

Proof step 9 from proof multmx-column (previous step) (back to tree)

Lemma multmx-rowno

Proof step 10 from proof multmx-column (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

  #c mx1 = #r mx2, (mx1 * mx2)[c]c[n] ≠ (mx1 * mx2[c]c)[n], n < #r mx1, c < #c mx2 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ # mx[c]c = #r mx; (s)

Proof step 11 from proof multmx-column (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 12:

  ⊦ mx[c]c[r] = mx[r, c]

• Proof step 13:

  #c mx1 = #r mx2, (mx1 * mx2)[n, c] ≠ (mx1 * mx2[c]c)[n], c < #c mx2, n < #r mx1 ⊦

Rule argument

Additional information

Used terms: mx1 * mx2, c, n

Proof step 12 from proof multmx-column (previous step) (back to tree)

A spec-lemma

Proof step 13 from proof multmx-column (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 14:

  #c mx1 = #r mx2, (mx1 * mx2)[n, c] ≠ (mx1 * mx2[c]c)[n], n < #r mx1, c < #c mx2 ⊦

Rule argument

Additional information

Proof step 14 from proof multmx-column (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 15:

  ⊦ #c mx = # vec ∧ r < #r mx → (mx * vec)[r] = Σ(0 to # vec, λ n. mx[r, n] * vec[n])

• Proof step 16:

  #c mx1 = #r mx2, #c mx1 = # mx2[c]c, (mx1 * mx2)[n, c] ≠ Σ(0 to # mx2[c]c, λ n0. mx1[n, n0] * mx2[c]c[n0]), c < #c mx2, n < #r mx1 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ # mx[c]c = #r mx; (s) Used terms: mx1, mx2[c]c, n

Proof step 15 from proof multmx-column (previous step) (back to tree)

Lemma multmxvec-get

Proof step 16 from proof multmx-column (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 17:

  #c mx1 = #r mx2, Σ(0 to #r mx2, λ n0. mx1[n, n0] * mx2[n0, c]) ≠ Σ(0 to #r mx2, λ n0. mx1[n, n0] * mx2[c]c[n0]), n < #r mx1, c < #c mx2 ⊦

Rule argument

Additional information

Used simplifier rules: multmx-get: ⊦ #c mx1 = #r mx2 ∧ r < #r mx1 ∧ c < #c mx2 → (mx1 * mx2)[r, c] = Σ(0 to #c mx1, λ n. mx1[r, n] * mx2[n, c]); ⊦ # mx[c]c = #r mx; (s)

Proof step 17 from proof multmx-column (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 18:

  ⊦ (∀ n. n ∈ ns → f1(n) = f2(n)) → Σ(ns, f1) = Σ(ns, f2)

• Proof step 19:

  #c mx1 = #r mx2, Σ(0 to #r mx2, λ n0. mx1[n, n0] * mx2[n0, c]) ≠ Σ(0 to #r mx2, λ n0. mx1[n, n0] * mx2[c]c[n0]), c < #c mx2, n < #r mx1, (∀ n0. n0 ∈ 0 to #r mx2 → mx1[n, n0] * mx2[n0, c] = mx1[n, n0] * mx2[c]c[n0]) → Σ(0 to #r mx2, λ n0. mx1[n, n0] * mx2[n0, c]) = Σ(0 to #r mx2, λ n0. mx1[n, n0] * mx2[c]c[n0]) ⊦

Rule argument

Additional information

Used terms: 0 to #r mx2, λ n0. mx1[n, n0] * mx2[n0, c], λ n0. mx1[n, n0] * mx2[c]c[n0]

Proof step 18 from proof multmx-column (previous step) (back to tree)

Lemma sum-red

Proof step 19 from proof multmx-column (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ mx[c]c[r] = mx[r, c]; (s)