specification using natset, actualize matrix+array with ring by morphism :array → vector; mkarray → mkvector; # → #; [ ] → [ ]; [ ] → [ ]; [ ]r → [ ]r; [ ]c → [ ]c; end actualize functions O- . : matrix → matrix; O- . : vector → vector; Σ : natset × (nat → elem) → elem; variables mx : matrix; vec : vector; a : elem; r : nat; c : nat; m : nat; n : nat; f : nat → elem; ns : natset; g : nat × nat → elem; partial functions . + . : matrix × matrix → matrix with λ mx1, mx2. #r mx1 = #r mx2 ∧ #c mx2 = #c mx2; . + . : vector × vector → vector with λ vec1, vec2. # vec1 = # vec2; . * . : matrix × matrix → matrix prio 10 right with λ mx1, mx2. #c mx1 = #r mx2; . * . : matrix × vector → vector prio 10 right with λ mx, vec. #r mx = # vec; . * . : vector × matrix → vector prio 10 right with λ vec, mx. #c mx = # vec; . * . : vector × vector → elem prio 10 right with λ vec1, vec2. # vec1 = # vec2; axioms addinvvec-size: # O- vec = # vec; used for: s, ls; addinvvec-get: n < # vec → (O- vec)[n] = O- vec[n]; used for: s, ls; addinvmx-rowno: #r O- mx = #r mx; used for: s, ls; addinvmx-colno: #c O- mx = #c mx; used for: s, ls; addinvmx-get: c < #c mx ∧ c < #c mx → (O- mx)[r, c] = O- mx[r, c]; used for: s, ls; addvec-size: #(vec1 + vec2) = # vec1; used for: s, ls; addmvec-get: n < # vec1 → (vec1 + vec2)[n] = vec1[n] + vec2[n]; used for: s, ls; addmx-rowno: #r(mx1 + mx2) = #r mx1; used for: s, ls; addmx-colno: #c(mx1 + mx2) = #c mx1; used for: s, ls; addmx-get: r < #r mx1 ∧ c < #c mx1 → (mx1 + mx2)[r, c] = mx1[r, c] + mx2[r, c]; used for: s, ls; sum-base: Σ(∅, f) = O; used for: s, ls; sum-rec: ¬ n ∈ ns → Σ(ns ++ n, f) = f(n) + Σ(ns, f); used for: s, ls; multmx-rowno: #c mx1 = #r mx2 → #r(mx1 * mx2) = #r mx1; used for: s, ls; multmx-colno: #c mx1 = #r mx2 → #c(mx1 * mx2) = #c mx2; used for: s, ls; 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]); used for: s, ls; multmxvec-size: #c mx = # vec → #(mx * vec) = #r mx; used for: s, ls; multmxvec-get: #c mx = # vec ∧ r < #r mx → (mx * vec)[r] = Σ(0 to # vec, λ n. mx[r, n] * vec[n]); used for: s, ls; multvecmx-size: #r mx = # vec → #(vec * mx) = #c mx; used for: s, ls; multvecmx-get: #r mx = # vec ∧ c < #c mx → (vec * mx)[c] = Σ(0 to # vec, λ n. vec[n] * mx[n, c]); used for: s, ls; end specification