specs/matrixops/proofs/1623070846000multmx-column ⊦ #c mx1 = #r mx2 ∧ c < #c mx2 → (mx1 * mx2)[c]c = mx1 * mx2[c]csum-redmultmx-getmultmxvec-getmultmx-rownomultmxvec-size712multmx-column-proofmultmx-column-proof-infoslssum-add ⊦ Σ(ns, f1) + Σ(ns, f2) = Σ(ns, λ n. f1(n) + f2(n))sum-recsum-base37sum-add-proofsum-add-proof-infomultvecmx-get ⊦ #r mx = # vec ∧ c < #c mx → (vec * mx)[c] = Σ(0 to # vec, λ n. vec[n] * mx[n, c])00multvecmx-get-proofmultvecmx-get-proof-infoslsaddmx-assoc ⊦ #r mx1 = #r mx2 ∧ #r mx2 = #r mx3 ∧ #c mx1 = #c mx2 ∧ #c mx2 = #c mx3 → (mx1 + mx2) + mx3 = mx1 + mx2 + mx3addmx-getaddmx-colnoaddmx-rowno24addmx-assoc-proofaddmx-assoc-proof-infoaddvec-com ⊦ # vec1 = # vec2 → vec1 + vec2 = vec2 + vec1addmvec-getaddvec-size12addvec-com-proofaddvec-com-proof-infoaddmvec-get ⊦ n < # vec1 → (vec1 + vec2)[n] = vec1[n] + vec2[n]00addmvec-get-proofaddmvec-get-proof-infoslssum-base ⊦ Σ(∅, f) = O00sum-base-proofsum-base-proof-infoslsmultmxvec-get ⊦ #c mx = # vec ∧ r < #r mx → (mx * vec)[r] = Σ(0 to # vec, λ n. mx[r, n] * vec[n])00multmxvec-get-proofmultmxvec-get-proof-infoslsmultmx-colno ⊦ #c mx1 = #r mx2 → #c(mx1 * mx2) = #c mx200multmx-colno-proofmultmx-colno-proof-infoslssum-rec ⊦ ¬ n ∈ ns → Σ(ns ++ n, f) = f(n) + Σ(ns, f)00sum-rec-proofsum-rec-proof-infoslsmultvecmx-size ⊦ #r mx = # vec → #(vec * mx) = #c mx00multvecmx-size-proofmultvecmx-size-proof-infoslsaddinvmx-colno ⊦ #c O- mx = #c mx00addinvmx-colno-proofaddinvmx-colno-proof-infoslsaddmx-colno ⊦ #c(mx1 + mx2) = #c mx100addmx-colno-proofaddmx-colno-proof-infoslsdistrmxvecvec ⊦ #c mx = # vec1 ∧ # vec1 = # vec2 → mx * (vec1 + vec2) = mx * vec1 + mx * vec2sum-redsum-addaddmvec-getmultmxvec-getmultmxvec-sizeaddvec-size918distrmxvecvec-proofdistrmxvecvec-proof-infoaddinvvec-size ⊦ # O- vec = # vec00addinvvec-size-proofaddinvvec-size-proof-infoslsaddvec-assoc ⊦ # vec1 = # vec2 ∧ # vec1 = # vec2 → (vec1 + vec2) + vec3 = vec1 + vec2 + vec3addmvec-getaddvec-size24addvec-assoc-proofaddvec-assoc-proof-infodistrmxmxvec ⊦ #r mx1 = #r mx2 ∧ #c mx1 = #c mx2 ∧ #c mx1 = # vec → (mx1 + mx2) * vec = mx1 * vec + mx2 * vecaddmx-getsum-redsum-addaddmvec-getmultmxvec-getaddmx-rownoaddmx-colnomultmxvec-sizeaddvec-size816distrmxmxvec-proofdistrmxmxvec-proof-infomult-assoc ⊦ #c mx1 = #r mx2 ∧ #c mx2 = #r mx3 → (mx1 * mx2) * mx3 = mx1 * mx2 * mx3sum-multrsum-multlsum-redsum-switchmultmx-getmultmx-colnomultmx-rowno1526mult-assoc-proofmult-assoc-proof-infomultmx-rowno ⊦ #c mx1 = #r mx2 → #r(mx1 * mx2) = #r mx100multmx-rowno-proofmultmx-rowno-proof-infoslsmultmx-row ⊦ #c mx1 = #r mx2 ∧ r < #r mx1 → (mx1 * mx2)[r]r = mx1[r]r * mx2multvecmx-getmultmx-getmultvecmx-sizemultmx-colno48multmx-row-proofmultmx-row-proof-infoslsaddinvmx-rowno ⊦ #r O- mx = #r mx00addinvmx-rowno-proofaddinvmx-rowno-proof-infoslsaddmx-rowno ⊦ #r(mx1 + mx2) = #r mx100addmx-rowno-proofaddmx-rowno-proof-infoslssum-union ⊦ Σ(ns1 ∪ ns2, f) = Σ(ns1, f) + Σ(ns2 \ ns1, f)sum-recsum-base611sum-union-proofsum-union-proof-infodistrvecmxmx ⊦ #r mx1 = # vec ∧ #r mx1 = #r mx2 ∧ #c mx1 = #c mx2 → vec * (mx1 + mx2) = vec * mx1 + vec * mx2addmx-getsum-redsum-addaddmvec-getmultvecmx-getaddmx-colnoaddmx-rownomultvecmx-sizeaddvec-size1020distrvecmxmx-proofdistrvecmxmx-proof-infosingletonmxvec-get ⊦ r < m ∧ c < # vec ∧ n < m → (mkmatrix(m, # vec, O)[r, c, a] * vec)[n] = (n = r ⊃ a * vec[c];O)singletonmxvec-get-sum-auxmultmxvec-get24singletonmxvec-get-proofsingletonmxvec-get-proof-infoaddvec-size ⊦ #(vec1 + vec2) = # vec100addvec-size-proofaddvec-size-proof-infoslsaddinvmx-get ⊦ c < #c mx ∧ c < #c mx → (O- mx)[r, c] = O- mx[r, c]00addinvmx-get-proofaddinvmx-get-proof-infoslssum-multl ⊦ a * Σ(ns, f) = Σ(ns, λ n. a * f(n))sum-recsum-base48sum-multl-proofsum-multl-proof-infosum-switch ⊦ Σ(ns1, λ m. Σ(ns2, λ n. g(m, n))) = Σ(ns2, λ n. Σ(ns1, λ m. g(m, n)))sum-addsum-recsum-base712sum-switch-proofsum-switch-proof-infoaddmx-get ⊦ r < #r mx1 ∧ c < #c mx1 → (mx1 + mx2)[r, c] = mx1[r, c] + mx2[r, c]00addmx-get-proofaddmx-get-proof-infoslssingletonmxvec-get-sum-aux⊦ n < m ∧ c < # vec ∧ r < m ∧ n1 ≤ # vec → Σ(0 to n1, λ n0. mkmatrix(m, # vec, O)[r, c, a][n, n0] * vec[n0]) = (n = r ∧ c < n1 ⊃ a * vec[c];O)sum-recsum-base714singletonmxvec-get-sum-aux-proofsingletonmxvec-get-sum-aux-proof-infodistrvevecmx ⊦ #r mx = # vec1 ∧ # vec1 = # vec2 → (vec1 + vec2) * mx = vec1 * mx + vec2 * mxsum-redsum-addaddmvec-getmultvecmx-getmultvecmx-sizeaddvec-size1019distrvevecmx-proofdistrvevecmx-proof-infoaddmx-com ⊦ #r mx1 = #r mx2 ∧ #c mx1 = #c mx2 → mx1 + mx2 = mx2 + mx1addmx-getaddmx-colnoaddmx-rowno12addmx-com-proofaddmx-com-proof-infosum-multr ⊦ Σ(ns, f) * a = Σ(ns, λ n. f(n) * a)sum-recsum-base48sum-multr-proofsum-multr-proof-infoaddinvvec-get ⊦ n < # vec → (O- vec)[n] = O- vec[n]00addinvvec-get-proofaddinvvec-get-proof-infoslsdistrmxmx-l ⊦ #r mx1 = #r mx2 ∧ #c mx1 = #c mx2 ∧ #c mx1 = #r mx → (mx1 + mx2) * mx = mx1 * mx + mx2 * mxsum-redsum-addaddmx-getmultmx-getmultmx-colnoaddmx-colnomultmx-rownoaddmx-rowno918distrmxmx-l-proofdistrmxmx-l-proof-infomultmx-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])00multmx-get-proofmultmx-get-proof-infoslssum-red ⊦ (∀ n. n ∈ ns → f1(n) = f2(n)) → Σ(ns, f1) = Σ(ns, f2)sum-recsum-base511sum-red-proofsum-red-proof-infodistrmxmx-r ⊦ #r mx1 = #r mx2 ∧ #c mx1 = #c mx2 ∧ #c mx = #r mx1 → mx * (mx1 + mx2) = mx * mx1 + mx * mx2sum-redsum-addaddmx-getmultmx-getmultmx-colnomultmx-rownoaddmx-colnoaddmx-rowno1121distrmxmx-r-proofdistrmxmx-r-proof-infomultmxvec-size ⊦ #c mx = # vec → #(mx * vec) = #r mx00multmxvec-size-proofmultmxvec-size-proof-infosls0