section ‹Basic theory of vector spaces, using locales› theory VectorSpace imports Main "HOL-Algebra.Module" "HOL-Algebra.Coset" RingModuleFacts MonoidSums LinearCombinations SumSpaces begin subsection ‹Basic definitions and facts carried over from modules› text ‹A ‹vectorspace› is a module where the ring is a field. Note that we switch notation from $(R, M)$ to $(K, V)$.› locale vectorspace = module?: module K V + field?: field K for K and V (* Use sets for bases, and functions from the sets to carrier K represent the coefficients. *) text ‹A ‹subspace› of a vectorspace is a nonempty subset that is closed under addition and scalar multiplication. These properties have already been defined in submodule. Caution: W is a set, while V is a module record. To get W as a vectorspace, write vs W.› locale subspace = fixes K and W and V (structure) assumes vs: "vectorspace K V" and submod: "submodule K W V" lemma (in vectorspace) is_module[simp]: "subspace K W V⟹submodule K W V" by (unfold subspace_def, auto) text ‹We introduce some basic facts and definitions copied from module. We introduce some abbreviations, to match convention.› abbreviation (in vectorspace) vs::"'c set ⇒ ('a, 'c, 'd) module_scheme" where "vs W ≡ V⦇carrier :=W⦈" lemma (in vectorspace) carrier_vs_is_self [simp]: "carrier (vs W) = W" by auto lemma (in vectorspace) subspace_is_vs: fixes W::"'c set" assumes 0: "subspace K W V" shows "vectorspace K (vs W)" proof - from 0 show ?thesis apply (unfold vectorspace_def subspace_def, auto) by (intro submodule_is_module, auto) qed abbreviation (in module) subspace_sum:: "['c set, 'c set] ⇒ 'c set" where "subspace_sum W1 W2 ≡submodule_sum W1 W2" lemma (in vectorspace) vs_zero_lin_dep: assumes h2: "S⊆carrier V" and h3: "lin_indpt S" shows "𝟬⇘_{V⇙}∉ S" proof - have vs: "vectorspace K V".. from vs have nonzero: "carrier K ≠{𝟬⇘_{K⇙}}" by (metis one_zeroI zero_not_one) from h2 h3 nonzero show ?thesis by (rule zero_nin_lin_indpt) qed text ‹A ‹linear_map› is a module homomorphism between 2 vectorspaces over the same field.› locale linear_map = V?: vectorspace K V + W?: vectorspace K W + mod_hom?: mod_hom K V W T for K and V and W and T context linear_map begin lemmas T_hom = f_hom lemmas T_add = f_add lemmas T_smult = f_smult lemmas T_im = f_im lemmas T_neg = f_neg lemmas T_minus = f_minus lemmas T_ker = f_ker abbreviation imT:: "'e set" where "imT ≡ mod_hom.im" abbreviation kerT:: "'c set" where "kerT ≡ mod_hom.ker" lemmas T0_is_0[simp] = f0_is_0 lemma kerT_is_subspace: "subspace K ker V" proof - have vs: "vectorspace K V".. from vs show ?thesis apply (unfold subspace_def, auto) by (rule ker_is_submodule) qed lemma imT_is_subspace: "subspace K imT W" proof - have vs: "vectorspace K W".. from vs show ?thesis apply (unfold subspace_def, auto) by (rule im_is_submodule) qed end lemma vs_criteria: fixes K and V assumes field: "field K" and zero: "𝟬⇘_{V⇙}∈ carrier V" and add: "∀v w. v∈carrier V ∧ w∈carrier V⟶ v⊕⇘_{V⇙}w∈ carrier V" and neg: "∀v∈carrier V. (∃neg_v∈carrier V. v⊕⇘_{V⇙}neg_v=𝟬⇘_{V⇙})" and smult: "∀c v. c∈ carrier K ∧ v∈carrier V⟶ c⊙⇘_{V⇙}v ∈ carrier V" and comm: "∀v w. v∈carrier V ∧ w∈carrier V⟶ v⊕⇘_{V⇙}w=w⊕⇘_{V⇙}v" and assoc: "∀v w x. v∈carrier V ∧ w∈carrier V ∧ x∈carrier V⟶ (v⊕⇘_{V⇙}w)⊕⇘_{V⇙}x= v⊕⇘_{V⇙}(w⊕⇘_{V⇙}x)" and add_id: "∀v∈carrier V. (v⊕⇘_{V⇙}𝟬⇘_{V⇙}=v)" and compat: "∀a b v. a∈ carrier K ∧ b∈ carrier K ∧ v∈carrier V⟶ (a⊗⇘_{K⇙}b)⊙⇘_{V⇙}v =a⊙⇘_{V⇙}(b⊙⇘_{V⇙}v)" and smult_id: "∀v∈carrier V. (𝟭⇘_{K⇙}⊙⇘_{V⇙}v =v)" and dist_f: "∀a b v. a∈ carrier K ∧ b∈ carrier K ∧ v∈carrier V⟶ (a⊕⇘_{K⇙}b)⊙⇘_{V⇙}v =(a⊙⇘_{V⇙}v) ⊕⇘_{V⇙}(b⊙⇘_{V⇙}v)" and dist_add: "∀a v w. a∈ carrier K ∧ v∈carrier V ∧ w∈carrier V⟶ a⊙⇘_{V⇙}(v⊕⇘_{V⇙}w) =(a⊙⇘_{V⇙}v) ⊕⇘_{V⇙}(a⊙⇘_{V⇙}w)" shows "vectorspace K V" proof - from field have 1: "cring K" by (unfold field_def domain_def, auto) from assms 1 have 2: "module K V" by (intro module_criteria, auto) from field 2 show ?thesis by (unfold vectorspace_def module_def, auto) qed text ‹For any set $S$, the space of functions $S\to K$ forms a vector space.› lemma (in vectorspace) func_space_is_vs: fixes S shows "vectorspace K (func_space S)" proof - have 0: "field K".. have 1: "module K (func_space S)" by (rule func_space_is_module) from 0 1 show ?thesis by (unfold vectorspace_def module_def, auto) qed lemma direct_sum_is_vs: fixes K V1 V2 assumes h1: "vectorspace K V1" and h2: "vectorspace K V2" shows "vectorspace K (direct_sum V1 V2)" proof - from h1 h2 have mod: "module K (direct_sum V1 V2)" by (unfold vectorspace_def, intro direct_sum_is_module, auto) from mod h1 show ?thesis by (unfold vectorspace_def, auto) qed lemma inj1_linear: fixes K V1 V2 assumes h1: "vectorspace K V1" and h2: "vectorspace K V2" shows "linear_map K V1 (direct_sum V1 V2) (inj1 V1 V2)" proof - from h1 h2 have mod: "mod_hom K V1 (direct_sum V1 V2) (inj1 V1 V2)" by (unfold vectorspace_def, intro inj1_hom, auto) from mod h1 h2 show ?thesis by (unfold linear_map_def vectorspace_def , auto, intro direct_sum_is_module, auto) qed lemma inj2_linear: fixes K V1 V2 assumes h1: "vectorspace K V1" and h2: "vectorspace K V2" shows "linear_map K V2 (direct_sum V1 V2) (inj2 V1 V2)" proof - from h1 h2 have mod: "mod_hom K V2 (direct_sum V1 V2) (inj2 V1 V2)" by (unfold vectorspace_def, intro inj2_hom, auto) from mod h1 h2 show ?thesis by (unfold linear_map_def vectorspace_def , auto, intro direct_sum_is_module, auto) qed text ‹For subspaces $V_1,V_2\subseteq V$, the map $V_1\oplus V_2\to V$ given by $(v_1,v_2)\mapsto v_1+v_2$ is linear.› lemma (in vectorspace) sum_map_linear: fixes V1 V2 assumes h1: "subspace K V1 V" and h2: "subspace K V2 V" shows "linear_map K (direct_sum (vs V1) (vs V2)) V (λ v. (fst v) ⊕⇘_{V⇙}(snd v))" proof - from h1 h2 have mod: "mod_hom K (direct_sum (vs V1) (vs V2)) V (λ v. (fst v) ⊕⇘_{V⇙}(snd v))" by ( intro sum_map_hom, unfold subspace_def, auto) from mod h1 h2 show ?thesis apply (unfold linear_map_def, auto) apply (intro direct_sum_is_vs subspace_is_vs, auto).. qed lemma (in vectorspace) sum_is_subspace: fixes W1 W2 assumes h1: "subspace K W1 V" and h2: "subspace K W2 V" shows "subspace K (subspace_sum W1 W2) V" proof - from h1 h2 have mod: "submodule K (submodule_sum W1 W2) V" by ( intro sum_is_submodule, unfold subspace_def, auto) from mod h1 h2 show ?thesis by (unfold subspace_def, auto) qed text ‹If $W_1,W_2\subseteq V$ are subspaces, $W_1\subseteq W_1 + W_2$› lemma (in vectorspace) in_sum_vs: fixes W1 W2 assumes h1: "subspace K W1 V" and h2: "subspace K W2 V" shows "W1 ⊆ subspace_sum W1 W2" proof - from h1 h2 show ?thesis by (intro in_sum, unfold subspace_def, auto) qed lemma (in vectorspace) vsum_comm: fixes W1 W2 assumes h1: "subspace K W1 V" and h2: "subspace K W2 V" shows "(subspace_sum W1 W2) = (subspace_sum W2 W1)" proof - from h1 h2 show ?thesis by (intro msum_comm, unfold subspace_def, auto) qed text ‹If $W_1,W_2\subseteq V$ are subspaces, then $W_1+W_2$ is the minimal subspace such that both $W_1\subseteq W$ and $W_2\subseteq W$.› lemma (in vectorspace) vsum_is_minimal: fixes W W1 W2 assumes h1: "subspace K W1 V" and h2: "subspace K W2 V" and h3: "subspace K W V" shows "(subspace_sum W1 W2) ⊆ W ⟷ W1 ⊆ W ∧ W2 ⊆ W" proof - from h1 h2 h3 show ?thesis by (intro sum_is_minimal, unfold subspace_def, auto) qed lemma (in vectorspace) span_is_subspace: fixes S assumes h2: "S⊆carrier V" shows "subspace K (span S) V" proof - have 0: "vectorspace K V".. from h2 have 1: "submodule K (span S) V" by (rule span_is_submodule) from 0 1 show ?thesis by (unfold subspace_def mod_hom_def linear_map_def, auto) qed subsubsection ‹Facts specific to vector spaces› text ‹If $av = w$ and $a\neq 0$, $v=a^{-1}w$.› lemma (in vectorspace) mult_inverse: assumes h1: "a∈carrier K" and h2: "v∈carrier V" and h3: "a ⊙⇘_{V⇙}v = w" and h4: "a≠𝟬⇘_{K⇙}" shows "v = (inv⇘_{K⇙}a )⊙⇘_{V⇙}w" proof - from h1 h2 h3 have 1: "w∈carrier V" by auto from h3 1 have 2: "(inv⇘_{K⇙}a )⊙⇘_{V⇙}(a ⊙⇘_{V⇙}v) =(inv⇘_{K⇙}a )⊙⇘_{V⇙}w" by auto from h1 h4 have 3: "inv⇘_{K⇙}a∈carrier K" by auto interpret g: group "(units_group K)" by (rule units_form_group) have f: "field K".. from f h1 h4 have 4: "a∈Units K" by (unfold field_def field_axioms_def, simp) from 4 h1 h4 have 5: "((inv⇘_{K⇙}a) ⊗⇘_{K⇙}a) = 𝟭⇘_{K⇙}" by (intro Units_l_inv, auto) from 5 have 6: "(inv⇘_{K⇙}a )⊙⇘_{V⇙}(a ⊙⇘_{V⇙}v) = v" proof - from h1 h2 h4 have 7: "(inv⇘_{K⇙}a )⊙⇘_{V⇙}(a ⊙⇘_{V⇙}v) =(inv⇘_{K⇙}a ⊗⇘_{K⇙}a) ⊙⇘_{V⇙}v" by (auto simp add: smult_assoc1) from 5 h2 have 8: "(inv⇘_{K⇙}a ⊗⇘_{K⇙}a) ⊙⇘_{V⇙}v = v" by auto from 7 8 show ?thesis by auto qed from 2 6 show ?thesis by auto qed text ‹If $w\in S$ and $\sum_{w\in S} a_ww=0$, we have $v=\sum_{w\not\in S}a_v^{-1}a_ww$› lemma (in vectorspace) lincomb_isolate: fixes A v assumes h1: "finite A" and h2: "A⊆carrier V" and h3: "a∈A→carrier K" and h4: "v∈A" and h5: "a v ≠ 𝟬⇘_{K⇙}" and h6: "lincomb a A=𝟬⇘_{V⇙}" shows "v=lincomb (λw. ⊖⇘_{K⇙}(inv⇘_{K⇙}(a v)) ⊗⇘_{K⇙}a w) (A-{v})" and "v∈ span (A-{v})" proof - from h1 h2 h3 h4 have 1: "lincomb a A = ((a v) ⊙⇘_{V⇙}v) ⊕⇘_{V⇙}lincomb a (A-{v})" by (rule lincomb_del2) from 1 have 2: "𝟬⇘_{V⇙}= ((a v) ⊙⇘_{V⇙}v) ⊕⇘_{V⇙}lincomb a (A-{v})" by (simp add: h6) from h1 h2 h3 have 5: "lincomb a (A-{v}) ∈carrier V" by auto (*intro lincomb_closed*) from 2 h1 h2 h3 h4 have 3: " ⊖⇘_{V⇙}lincomb a (A-{v}) = ((a v) ⊙⇘_{V⇙}v)" by (auto intro!: M.minus_equality) have 6: "v = (⊖⇘_{K⇙}(inv⇘_{K⇙}(a v))) ⊙⇘_{V⇙}lincomb a (A-{v})" proof - from h2 h3 h4 h5 3 have 7: "v = inv⇘_{K⇙}(a v) ⊙⇘_{V⇙}(⊖⇘_{V⇙}lincomb a (A-{v}))" by (intro mult_inverse, auto) from assms have 8: "inv⇘_{K⇙}(a v)∈carrier K" by auto from assms 5 8 have 9: "inv⇘_{K⇙}(a v) ⊙⇘_{V⇙}(⊖⇘_{V⇙}lincomb a (A-{v})) = (⊖⇘_{K⇙}(inv⇘_{K⇙}(a v))) ⊙⇘_{V⇙}lincomb a (A-{v})" by (simp add: smult_assoc_simp smult_minus_1_back r_minus) from 7 9 show ?thesis by auto qed from h1 have 10: "finite (A-{v})" by auto from assms have 11 : "(⊖⇘_{K⇙}(inv⇘_{K⇙}(a v)))∈ carrier K" by auto from assms have 12: "lincomb (λw. ⊖⇘_{K⇙}(inv⇘_{K⇙}(a v)) ⊗⇘_{K⇙}a w) (A-{v}) = (⊖⇘_{K⇙}(inv⇘_{K⇙}(a v))) ⊙⇘_{V⇙}lincomb a (A-{v})" by (intro lincomb_smult, auto) from 6 12 show "v=lincomb (λw. ⊖⇘_{K⇙}(inv⇘_{K⇙}(a v)) ⊗⇘_{K⇙}a w) (A-{v})" by auto with assms show "v∈ span (A-{v})" unfolding span_def by (force simp add: 11 ring_subset_carrier) qed text ‹The map $(S\to K)\mapsto V$ given by $(a_v)_{v\in S}\mapsto \sum_{v\in S} a_v v$ is linear.› lemma (in vectorspace) lincomb_is_linear: fixes S assumes h: "finite S" and h2: "S⊆carrier V" shows "linear_map K (func_space S) V (λa. lincomb a S)" proof - have 0: "vectorspace K V".. from h h2 have 1: "mod_hom K (func_space S) V (λa. lincomb a S)" by (rule lincomb_is_mod_hom) from 0 1 show ?thesis by (unfold vectorspace_def mod_hom_def linear_map_def, auto) qed subsection ‹Basic facts about span and linear independence› text ‹If $S$ is linearly independent, then $v\in \text{span}S$ iff $S\cup \{v\}$ is linearly dependent.› theorem (in vectorspace) lin_dep_iff_in_span: fixes A v S assumes h1: "S ⊆ carrier V" and h2: "lin_indpt S" and h3: "v∈ carrier V" and h4: "v∉S" shows "v∈ span S ⟷ lin_dep (S ∪ {v})" proof - let ?T = "S ∪ {v}" have 0: "v∈?T " by auto from h1 h3 have h1_1: "?T ⊆ carrier V" by auto have a1:"lin_dep ?T ⟹ v∈ span S" proof - assume a11: "lin_dep ?T" from a11 obtain a w A where a: "(finite A ∧ A⊆?T ∧ (a∈ (A→carrier K)) ∧ (lincomb a A = 𝟬⇘_{V⇙}) ∧ (w∈A) ∧ (a w≠ 𝟬⇘_{K⇙}))" by (metis lin_dep_def) from assms a have nz2: "∃v∈A-S. a v≠𝟬⇘_{K⇙}" by (intro lincomb_must_include[where ?v="w" and ?T="S∪{v}"], auto) from a nz2 have singleton: "{v}=A-S" by auto from singleton nz2 have nz3: "a v≠𝟬⇘_{K⇙}" by auto (*Can modularize this whole section out. "solving for one variable"*) let ?b="(λw. ⊖⇘_{K⇙}(inv⇘_{K⇙}(a v)) ⊗⇘_{K⇙}(a w))" from singleton have Ains: "(A∩S) = A-{v}" by auto from assms a singleton nz3 have a31: "v= lincomb ?b (A∩S)" apply (subst Ains) by (intro lincomb_isolate(1), auto) from a a31 nz3 singleton show ?thesis apply (unfold span_def, auto) apply (rule_tac x="?b" in exI) apply (rule_tac x="A∩S" in exI) by (auto intro!: m_closed) qed have a2: "v∈ (span S) ⟹ lin_dep ?T" proof - assume inspan: "v∈ (span S)" from inspan obtain a A where a: "A⊆S ∧ finite A ∧ (v = lincomb a A)∧ a∈A→carrier K" by (simp add: span_def, auto) let ?b = "λ w. if (w=v) then (⊖⇘_{K⇙}𝟭⇘_{K⇙}) else a w" (*consider -v + \sum a_w w*) have lc0: " lincomb ?b (A∪{v})=𝟬⇘_{V⇙}" proof - from assms a have lc_ins: "lincomb ?b (A∪{v}) = ((?b v) ⊙⇘_{V⇙}v) ⊕⇘_{V⇙}lincomb ?b A" by (intro lincomb_insert, auto) from assms a have lc_elim: "lincomb ?b A=lincomb a A" by (intro lincomb_elim_if, auto) from assms lc_ins lc_elim a show ?thesis by (simp add: M.l_neg smult_minus_1) qed from a lc0 show ?thesis apply (unfold lin_dep_def) apply (rule_tac x="A∪{v}" in exI) apply (rule_tac x="?b" in exI) apply (rule_tac x="v" in exI) by auto qed from a1 a2 show ?thesis by auto qed text ‹If $v\in \text{span} A$ then $\text{span}A =\text{span}(A\cup \{v\})$› lemma (in vectorspace) already_in_span: fixes v A assumes inC: "A⊆carrier V" and inspan: "v∈span A" shows "span A= span (A∪{v})" proof - from inC inspan have dir1: "span A ⊆ span (A∪{v})" by (intro span_is_monotone, auto) from inC have inown: "A⊆span A" by (rule in_own_span) from inC have subm: "submodule K (span A) V" by (rule span_is_submodule) from inown inspan subm have dir2: "span (A ∪ {v}) ⊆ span A" by (intro span_is_subset, auto) from dir1 dir2 show ?thesis by auto qed subsection ‹The Replacement Theorem› text ‹If $A,B\subseteq V$ are finite, $A$ is linearly independent, $B$ generates $W$, and $A\subseteq W$, then there exists $C\subseteq V$ disjoint from $A$ such that $\text{span}(A\cup C) = W$ and $|C|\le |B|-|A|$. In other words, we can complete any linearly independent set to a generating set of $W$ by adding at most $|B|-|A|$ more elements.› theorem (in vectorspace) replacement: fixes A B (*A B are lists of vectors (colloquially we refer to them as sets)*) assumes h1: "finite A" and h2: "finite B" and h3: "B⊆carrier V" and h4: "lin_indpt A" (*A is linearly independent*) and h5: "A⊆span B" (*All entries of A are in K*) shows "∃C. finite C ∧ C⊆carrier V ∧ C⊆span B ∧ C∩A={} ∧ int (card C) ≤ (int (card B)) - (int (card A)) ∧ (span (A ∪ C) = span B)" (is "∃C. ?P A B C") (*There is a set C of cardinality |B| - |A| such that A∪C generates V*) using h1 h2 h3 h4 h5 proof (induct "card A" arbitrary: A B) case 0 from "0.prems"(1) "0.hyps" have a0: "A={}" by auto from "0.prems"(3) have a3: "B⊆span B" by (rule in_own_span) from a0 a3 "0.prems" show ?case by (rule_tac x="B" in exI, auto) next case (Suc m) let ?W="span B" from Suc.prems(3) have BinC: "span B⊆carrier V" by (rule span_is_subset2) (*everything you want to know about A*) from Suc.prems Suc.hyps BinC have A: "finite A" "lin_indpt A" "A⊆span B" "Suc m = card A" "A⊆carrier V" by auto (*everything you want to know about B*) from Suc.prems have B: "finite B" "B⊆carrier V" by auto (*A B are lists of vectors (colloquially we refer to them as sets)*) from Suc.hyps(2) obtain v where v: "v∈A" by fastforce let ?A'="A-{v}" (*?A' is linearly independent because it is the subset of a linearly independent set, A.*) from A(2) have liA': "lin_indpt ?A'" apply (intro subset_li_is_li[of "A" "?A'"]) by auto from v liA' Suc.prems Suc.hyps(2) have "∃C'. ?P ?A' B C'" apply (intro Suc.hyps(1)) by auto from this obtain C' where C': "?P ?A' B C'" by auto show ?case proof (cases "v∈ C'") case True have vinC': "v∈C'" by fact from vinC' v have seteq: "A - {v} ∪ C' = A ∪ (C' - {v})" by auto from C' seteq have spaneq: "span (A ∪ (C' - {v})) = span (B)" by algebra from Suc.prems Suc.hyps C' vinC' v spaneq show ?thesis apply (rule_tac x="C'-{v}" in exI) apply (subgoal_tac "card C' >0") by auto next case False have f: "v∉C'" by fact from A v C' have "∃a. a∈(?A'∪C')→carrier K ∧ lincomb a (?A' ∪ C') =v" by (intro finite_in_span, auto) from this obtain a where a: "a∈(?A'∪C')→carrier K ∧ v= lincomb a (?A' ∪ C')" by metis let ?b="(λ w. if (w=v) then ⊖⇘_{K⇙}𝟭⇘_{K⇙}else a w)" from a have b: "?b∈A∪C'→carrier K" by auto from v have rewrite_ins: "A∪C'=(?A'∪C')∪{v}" by auto from f have "v∉?A'∪C'" by auto from this A C' v a f have lcb: "lincomb ?b (A ∪ C') = 𝟬⇘_{V⇙}" apply (subst rewrite_ins) apply (subst lincomb_insert) apply (simp_all add: ring_subset_carrier coeff_in_ring) apply (auto split: if_split_asm) apply (subst lincomb_elim_if) by (auto simp add: smult_minus_1 l_neg ring_subset_carrier) (*NOTE: l_neg got deleted from the simp rules, but it is very useful.*) from C' f have rewrite_minus: "C'=(A∪C')-A" by auto from A C' b lcb v have exw: "∃w∈ C'. ?b w≠𝟬⇘_{K⇙}" apply (subst rewrite_minus) apply (intro lincomb_must_include[where ?T="A∪C'" and ?v="v"]) by auto from exw obtain w where w: "w∈ C'" "?b w≠𝟬⇘_{K⇙}" by auto from A C' w f b lcb have w_in: "w∈span ((A∪ C') -{w})" apply (intro lincomb_isolate[where a="?b"]) by auto have spaneq2: "span (A∪(C'-{w})) = span B" proof - have 1: "span (?A'∪C') = span (A∪ C')" (*adding v doesn't change the span*) proof - from A C' v have m1: "span (?A'∪C') = span ((?A'∪ C')∪{v})" apply (intro already_in_span) by auto from f m1 show ?thesis by (metis rewrite_ins) qed have 2: "span (A∪ (C'-{w})) = span (A∪ C')" (*removing w doesn't change the span*) proof - from C' w(1) f have b60: "A∪ (C'-{w}) = (A∪ C') -{w}" by auto from w(1) have b61: "A∪ C'= (A∪ C' -{w})∪{w}" by auto from A C' w_in show ?thesis apply (subst b61) apply (subst b60) apply (intro already_in_span) by auto qed from C' 1 2 show ?thesis by auto qed(* "span (A∪(C'-{w})) = span B"*) from A C' w f v spaneq2 show ?thesis apply (rule_tac x="C'-{w}" in exI) apply (subgoal_tac "card C' >0") by auto qed qed subsection ‹Defining dimension and bases.› text ‹Finite dimensional is defined as having a finite generating set.› definition (in vectorspace) fin_dim:: "bool" where "fin_dim = (∃ A. ((finite A) ∧ (A ⊆ carrier V) ∧ (gen_set A)))" text ‹The dimension is the size of the smallest generating set. For equivalent characterizations see below.› definition (in vectorspace) dim:: "nat" where "dim = (LEAST n. (∃ A. ((finite A) ∧ (card A = n) ∧ (A ⊆ carrier V) ∧ (gen_set A))))" text ‹A ‹basis› is a linearly independent generating set.› definition (in vectorspace) basis:: "'c set ⇒ bool" where "basis A = ((lin_indpt A) ∧ (gen_set A)∧ (A⊆carrier V))" text ‹From the replacement theorem, any linearly independent set is smaller than any generating set.› lemma (in vectorspace) li_smaller_than_gen: fixes A B assumes h1: "finite A" and h2: "finite B" and h3: "A⊆carrier V" and h4: "B⊆carrier V" and h5: "lin_indpt A" and h6: "gen_set B" shows "card A ≤ card B" proof - from h3 h6 have 1: "A⊆span B" by auto from h1 h2 h4 h5 1 obtain C where 2: "finite C ∧ C⊆carrier V ∧ C⊆span B ∧ C∩A={} ∧ int (card C) ≤ int (card B) - int (card A) ∧ (span (A ∪ C) = span B)" by (metis replacement) from 2 show ?thesis by arith qed text ‹The dimension is the cardinality of any basis. (In particular, all bases are the same size.)› lemma (in vectorspace) dim_basis: fixes A assumes fin: "finite A" and h2: "basis A" shows "dim = card A" proof - have 0: "⋀B m. ((finite B) ∧ (card B = m) ∧ (B ⊆ carrier V) ∧ (gen_set B)) ⟹ card A ≤ m" proof - fix B m assume 1: "((finite B) ∧ (card B = m) ∧ (B ⊆ carrier V) ∧ (gen_set B))" from 1 fin h2 have 2: "card A ≤ card B" apply (unfold basis_def) apply (intro li_smaller_than_gen) by auto from 1 2 show "?thesis B m" by auto qed from fin h2 0 show ?thesis apply (unfold dim_def basis_def) apply (intro Least_equality) apply (rule_tac x="A" in exI) by auto qed (*can define more generally with posets*) text ‹A ‹maximal› set with respect to $P$ is such that if $B\supseteq A$ and $P$ is also satisfied for $B$, then $B=A$.› definition maximal::"'a set ⇒ ('a set ⇒ bool) ⇒ bool" where "maximal A P = ((P A) ∧ (∀B. B⊇A ∧ P B ⟶ B = A))" text ‹A ‹minimal› set with respect to $P$ is such that if $B\subseteq A$ and $P$ is also satisfied for $B$, then $B=A$.› definition minimal::"'a set ⇒ ('a set ⇒ bool) ⇒ bool" where "minimal A P = ((P A) ∧ (∀B. B⊆A ∧ P B ⟶ B = A))" text ‹A maximal linearly independent set is a generating set.› lemma (in vectorspace) max_li_is_gen: fixes A assumes h1: "maximal A (λS. S⊆carrier V ∧ lin_indpt S)" shows "gen_set A" proof (rule ccontr) assume 0: "¬(gen_set A)" from h1 have 1: " A⊆ carrier V ∧ lin_indpt A" by (unfold maximal_def, auto) from 1 have 2: "span A ⊆ carrier V" by (intro span_is_subset2, auto) from 0 1 2 have 3: "∃v. v∈carrier V ∧ v ∉ (span A)" by auto from 3 obtain v where 4: "v∈carrier V ∧ v ∉ (span A)" by auto have 5: "v∉A" proof - from h1 1 have 51: "A⊆span A" apply (intro in_own_span) by auto from 4 51 show ?thesis by auto qed from lin_dep_iff_in_span have 6: "⋀S v. S ⊆ carrier V∧ lin_indpt S ∧ v∈ carrier V ∧ v∉S ∧ v∉ span S ⟹ (lin_indpt (S ∪ {v}))" by auto from 1 4 5 have 7: "lin_indpt (A ∪ {v})" apply (intro 6) by auto (* assumes h0:"finite S" and h1: "S ⊆ carrier V" and h2: "lin_indpt S" and h3: "v∈ carrier V" and h4: "v∉S" shows "v∈ span S ⟷ ¬ (lin_indpt (S ∪ {v}))"*) have 9: "¬(maximal A (λS. S⊆carrier V ∧ lin_indpt S))" proof - from 1 4 5 7 have 8: "(∃B. A ⊆ B ∧ B ⊆ carrier V ∧ lin_indpt B ∧ B ≠ A)" apply (rule_tac x="A∪{v}" in exI) by auto from 8 show ?thesis apply (unfold maximal_def) by simp qed from h1 9 show False by auto qed text ‹A minimal generating set is linearly independent.› lemma (in vectorspace) min_gen_is_li: fixes A assumes h1: "minimal A (λS. S⊆carrier V ∧ gen_set S)" shows "lin_indpt A" proof (rule ccontr) assume 0: "¬lin_indpt A" from h1 have 1: " A⊆ carrier V ∧ gen_set A" by (unfold minimal_def, auto) from 1 have 2: "span A = carrier V" by auto from 0 1 obtain a v A' where 3: "finite A' ∧ A'⊆A ∧ a ∈ A' → carrier K ∧ LinearCombinations.module.lincomb V a A' = 𝟬⇘_{V⇙}∧ v ∈ A' ∧ a v ≠ 𝟬⇘_{K⇙}" by (unfold lin_dep_def, auto) have 4: "gen_set (A-{v})" proof - from 1 3 have 5: "v∈span (A'-{v})" apply (intro lincomb_isolate[where a="a" and v="v"]) by auto from 3 5 have 51: "v∈span (A-{v})" apply (intro subsetD[where ?A="span (A'-{v})" and ?B="span (A-{v})" and ?c="v"]) by (intro span_is_monotone, auto) from 1 have 6: "A⊆span A" apply (intro in_own_span) by auto from 1 51 have 7: "span (A-{v}) = span ((A-{v})∪{v})" apply (intro already_in_span) by auto from 3 have 8: "A = ((A-{v})∪{v})" by auto from 2 7 8 have 9:"span (A-{v}) = carrier V" by auto (*can't use 3 directly :( *) from 9 show ?thesis by auto qed have 10: "¬(minimal A (λS. S⊆carrier V ∧ gen_set S))" proof - from 1 3 4 have 11: "(∃B. A ⊇ B ∧ B ⊆ carrier V ∧ gen_set B ∧ B ≠ A)" apply (rule_tac x="A-{v}" in exI) by auto from 11 show ?thesis apply (unfold minimal_def) by auto qed from h1 10 show False by auto qed text ‹Given that some finite set satisfies $P$, there is a minimal set that satisfies $P$.› lemma minimal_exists: fixes A P assumes h1: "finite A" and h2: "P A" shows "∃B. B⊆A ∧ minimal B P" using h1 h2 proof (induct "card A" arbitrary: A rule: less_induct) case (less A) show ?case proof (cases "card A = 0") case True from True less.hyps less.prems show ?thesis apply (rule_tac x="{}" in exI) apply (unfold minimal_def) by auto next case False show ?thesis proof (cases "minimal A P") case True then show ?thesis apply (rule_tac x="A" in exI) by auto next case False have 2: "¬minimal A P" by fact from less.prems 2 have 3: "∃B. P B ∧ B ⊆ A ∧ B≠A" apply (unfold minimal_def) by auto from 3 obtain B where 4: "P B ∧ B ⊂ A ∧ B≠A" by auto from 4 have 5: "card B < card A" by (metis less.prems(1) psubset_card_mono) from less.hyps less.prems 3 4 5 have 6: "∃C⊆B. minimal C P" apply (intro less.hyps) apply auto by (metis rev_finite_subset) from 6 obtain C where 7: "C⊆B ∧ minimal C P" by auto from 4 7 show ?thesis apply (rule_tac x="C" in exI) apply (unfold minimal_def) by auto qed qed qed text ‹If $V$ is finite-dimensional, then any linearly independent set is finite.› lemma (in vectorspace) fin_dim_li_fin: assumes fd: "fin_dim" and li: "lin_indpt A" and inC: "A⊆carrier V" shows fin: "finite A" proof (rule ccontr) assume A: "¬finite A" from fd obtain C where C: "finite C ∧ C⊆carrier V ∧ gen_set C" by (unfold fin_dim_def, auto) from A obtain B where B: "B⊆A∧ finite B ∧ card B = card C + 1" by (metis infinite_arbitrarily_large) from B li have liB: "lin_indpt B" by (intro subset_li_is_li[where ?A="A" and ?B="B"], auto) from B C liB inC have "card B ≤ card C" by (intro li_smaller_than_gen, auto) from this B show False by auto qed text ‹If $V$ is finite-dimensional (has a finite generating set), then a finite basis exists.› lemma (in vectorspace) finite_basis_exists: assumes h1: "fin_dim" shows "∃β. finite β ∧ basis β" proof - from h1 obtain A where 1: "finite A ∧ A⊆carrier V ∧ gen_set A" by (metis fin_dim_def) hence 2: "∃β. β⊆A ∧ minimal β (λS. S⊆carrier V ∧ gen_set S)" apply (intro minimal_exists) by auto then obtain β where 3: "β⊆A ∧ minimal β (λS. S⊆carrier V ∧ gen_set S)" by auto hence 4: "lin_indpt β" apply (intro min_gen_is_li) by auto moreover from 3 have 5: "gen_set β ∧ β⊆carrier V" apply (unfold minimal_def) by auto moreover from 1 3 have 6: "finite β" by (auto simp add: finite_subset) ultimately show ?thesis apply (unfold basis_def) by auto qed text‹ The proof is as follows. \begin{enumerate} \item Because $V$ is finite-dimensional, there is a finite generating set (we took this as our definition of finite-dimensional). \item Hence, there is a minimal $\beta \subseteq A$ such that $\beta$ generates $V$. \item $\beta$ is linearly independent because a minimal generating set is linearly independent. \end{enumerate} Finally, $\beta$ is a basis because it is both generating and linearly independent. › text ‹Any linearly independent set has cardinality at most equal to the dimension.› lemma (in vectorspace) li_le_dim: fixes A assumes fd: "fin_dim" and c: "A⊆carrier V" and l: "lin_indpt A" shows "finite A" "card A ≤ dim" proof - from fd c l show fa: "finite A" by (intro fin_dim_li_fin, auto) from fd obtain β where 1: "finite β ∧ basis β" by (metis finite_basis_exists) from assms fa 1 have 2: "card A ≤ card β" apply (intro li_smaller_than_gen, auto) by (unfold basis_def, auto) from assms 1 have 3: "dim = card β" by (intro dim_basis, auto) from 2 3 show "card A ≤ dim" by auto qed text ‹Any generating set has cardinality at least equal to the dimension.› lemma (in vectorspace) gen_ge_dim: fixes A assumes fa: "finite A" and c: "A⊆carrier V" and l: "gen_set A" shows "card A ≥ dim" proof - from assms have fd: "fin_dim" by (unfold fin_dim_def, auto) from fd obtain β where 1: "finite β ∧ basis β" by (metis finite_basis_exists) from assms 1 have 2: "card A ≥ card β" apply (intro li_smaller_than_gen, auto) by (unfold basis_def, auto) from assms 1 have 3: "dim = card β" by (intro dim_basis, auto) from 2 3 show ?thesis by auto qed text ‹If there is an upper bound on the cardinality of sets satisfying $P$, then there is a maximal set satisfying $P$.› lemma maximal_exists: fixes P B N assumes maxc: "⋀A. P A ⟹ finite A ∧ (card A ≤N)" and b: "P B" shows "∃A. finite A ∧ maximal A P" proof - (*take the maximal*) let ?S="{card A| A. P A}" let ?n="Max ?S" from maxc have 1:"finite ?S" apply (simp add: finite_nat_set_iff_bounded_le) by auto from 1 have 2: "?n∈?S" by (metis (mono_tags, lifting) Collect_empty_eq Max_in b) from assms 2 have 3: "∃A. P A ∧ finite A ∧ card A = ?n" by auto from 3 obtain A where 4: "P A ∧ finite A ∧ card A = ?n" by auto from 1 maxc have 5: "⋀A. P A ⟹ finite A ∧ (card A ≤?n)" by (metis (mono_tags, lifting) Max.coboundedI mem_Collect_eq) from 4 5 have 6: "maximal A P" apply (unfold maximal_def) by (metis card_seteq) from 4 6 show ?thesis by auto qed text ‹Any maximal linearly independent set is a basis.› lemma (in vectorspace) max_li_is_basis: fixes A assumes h1: "maximal A (λS. S⊆carrier V ∧ lin_indpt S)" shows "basis A" proof - from h1 have 1: "gen_set A" by (rule max_li_is_gen) from assms 1 show ?thesis by (unfold basis_def maximal_def, auto) qed text ‹Any minimal linearly independent set is a generating set.› lemma (in vectorspace) min_gen_is_basis: fixes A assumes h1: "minimal A (λS. S⊆carrier V ∧ gen_set S)" shows "basis A" proof - from h1 have 1: "lin_indpt A" by (rule min_gen_is_li) from assms 1 show ?thesis by (unfold basis_def minimal_def, auto) qed text ‹Any linearly independent set with cardinality at least the dimension is a basis.› lemma (in vectorspace) dim_li_is_basis: fixes A assumes fd: "fin_dim" and fa: "finite A" and ca: "A⊆carrier V" and li: "lin_indpt A" and d: "card A ≥ dim" (*≥*) shows "basis A" proof - from fd have 0: "⋀S. S⊆carrier V ∧ lin_indpt S ⟹ finite S ∧ card S ≤dim" by (auto intro: li_le_dim) (*fin_dim_li_fin*) from 0 assms have h1: "finite A ∧ maximal A (λS. S⊆carrier V ∧ lin_indpt S)" apply (unfold maximal_def) apply auto by (metis card_seteq eq_iff) from h1 show ?thesis by (auto intro: max_li_is_basis) qed text ‹Any generating set with cardinality at most the dimension is a basis.› lemma (in vectorspace) dim_gen_is_basis: fixes A assumes fa: "finite A" and ca: "A⊆carrier V" and li: "gen_set A" and d: "card A ≤ dim" shows "basis A" proof - have 0: "⋀S. finite S∧ S⊆carrier V ∧ gen_set S ⟹ card S ≥dim" by (intro gen_ge_dim, auto) (*li_le_dim*) from 0 assms have h1: "minimal A (λS. finite S ∧ S⊆carrier V ∧ gen_set S)" apply (unfold minimal_def) apply auto by (metis card_seteq eq_iff) (*slightly annoying: we have to get rid of "finite S" inside.*) from h1 have h: "⋀B. B ⊆ A ∧ B ⊆ carrier V ∧ LinearCombinations.module.gen_set K V B ⟹ B = A" proof - fix B assume asm: "B ⊆ A ∧ B ⊆ carrier V ∧ LinearCombinations.module.gen_set K V B" from asm h1 have "finite B" apply (unfold minimal_def) apply (intro finite_subset[where ?A="B" and ?B="A"]) by auto from h1 asm this show "?thesis B" apply (unfold minimal_def) by simp qed from h1 h have h2: "minimal A (λS. S⊆carrier V ∧ gen_set S)" apply (unfold minimal_def) by presburger from h2 show ?thesis by (rule min_gen_is_basis) qed text ‹$\beta$ is a basis iff for all $v\in V$, there exists a unique $(a_v)_{v\in S}$ such that $\sum_{v\in S} a_v v=v$.› lemma (in vectorspace) basis_criterion: assumes A_fin: "finite A" and AinC: "A⊆carrier V" shows "basis A ⟷ (∀v. v∈ carrier V ⟶(∃! a. a∈A →⇩_{E}carrier K ∧ lincomb a A = v))" proof - have 1: "¬(∀v. v∈ carrier V ⟶(∃! a. a∈A →⇩_{E}carrier K ∧ lincomb a A = v)) ⟹ ¬basis A" proof - assume "¬(∀v. v∈ carrier V ⟶(∃! a. a∈A →⇩_{E}carrier K ∧ lincomb a A = v))" then obtain v where v: "v∈ carrier V ∧ ¬(∃! a. a∈A →⇩_{E}carrier K ∧ lincomb a A = v)" by metis (*either there is more than 1 rep, or no reps*) from v have vinC: "v∈carrier V" by auto from v have "¬(∃ a. a∈A →⇩_{E}carrier K ∧ lincomb a A = v) ∨ (∃ a b. a∈A →⇩_{E}carrier K ∧ lincomb a A = v ∧ b∈A →⇩_{E}carrier K ∧ lincomb b A = v ∧ a≠b)" by metis then show ?thesis proof assume a: "¬(∃ a. a∈A →⇩_{E}carrier K ∧ lincomb a A = v)" from A_fin AinC have "⋀a. a∈A → carrier K ⟹ lincomb a A = lincomb (restrict a A) A" unfolding lincomb_def restrict_def by (simp cong: finsum_cong add: ring_subset_carrier coeff_in_ring) with a have "¬(∃ a. a∈A → carrier K ∧ lincomb a A = v)" by auto with A_fin AinC have "v∉span A" using finite_in_span by blast with AinC v show "¬(basis A)" by (unfold basis_def, auto) next assume a2: "(∃ a b. a∈A →⇩_{E}carrier K ∧ lincomb a A = v ∧ b∈A →⇩_{E}carrier K ∧ lincomb b A = v ∧ a≠b)" then obtain a b where ab: "a∈A →⇩_{E}carrier K ∧ lincomb a A = v ∧ b∈A →⇩_{E}carrier K ∧ lincomb b A = v ∧ a≠b" by metis from ab obtain w where w: "w∈A ∧ a w ≠ b w" apply (unfold PiE_def, auto) by (metis extensionalityI) let ?c="λ x. (if x∈A then ((a x) ⊖⇘_{K⇙}(b x)) else undefined)" from ab have a_fun: "a∈A → carrier K" and b_fun: "b∈A → carrier K" by (unfold PiE_def, auto) from w a_fun b_fun have abinC: "a w ∈carrier K" "b w ∈carrier K" by auto from abinC w have nz: "a w ⊖⇘_{K⇙}b w ≠ 𝟬⇘_{K⇙}" by auto (*uses M.minus_other_side*) from A_fin AinC a_fun b_fun ab vinC have a_b: "LinearCombinations.module.lincomb V (λx. if x ∈ A then a x ⊖⇘_{K⇙}b x else undefined) A = 𝟬⇘_{V⇙}" by (simp cong: lincomb_cong add: coeff_in_ring lincomb_diff) from A_fin AinC ab w v nz a_b have "lin_dep A" apply (intro lin_dep_crit[where ?A="A" and ?a="?c" and ?v="w"]) apply (auto simp add: PiE_def) by auto thus "¬basis A" by (unfold basis_def, auto) qed qed have 2: "(∀v. v∈ carrier V ⟶(∃! a. a∈A →⇩_{E}carrier K ∧ lincomb a A = v)) ⟹ basis A" proof - assume b1: "(∀v. v∈ carrier V ⟶(∃! a. a∈A →⇩_{E}carrier K ∧ lincomb a A = v))" (is "(∀v. v∈ carrier V ⟶(∃! a. ?Q a v))") from b1 have b2: "(∀v. v∈ carrier V ⟶(∃ a. a∈A → carrier K ∧ lincomb a A = v))" apply (unfold PiE_def) by blast from A_fin AinC b2 have "gen_set A" apply (unfold span_def) by blast from b1 have A_li: "lin_indpt A" proof - let ?z="λ x. (if (x∈A) then 𝟬⇘_{K⇙}else undefined)" from A_fin AinC have zero: "?Q ?z 𝟬⇘_{V⇙}" by (unfold PiE_def extensional_def lincomb_def, auto simp add: ring_subset_carrier) (*uses finsum_all0*) from A_fin AinC show ?thesis proof (rule finite_lin_indpt2) fix a assume a_fun: "a ∈ A → carrier K" and lc_a: "LinearCombinations.module.lincomb V a A = 𝟬⇘_{V⇙}" from a_fun have a_res: "restrict a A ∈ A →⇩_{E}carrier K" by auto from a_fun A_fin AinC lc_a have lc_a_res: "LinearCombinations.module.lincomb V (restrict a A) A = 𝟬⇘_{V⇙}" apply (unfold lincomb_def restrict_def) by (simp cong: finsum_cong2 add: coeff_in_ring ring_subset_carrier) from a_fun a_res lc_a_res zero b1 have "restrict a A = ?z" by auto from this show "∀v∈A. a v = 𝟬⇘_{K⇙}" apply (unfold restrict_def) by meson qed qed have A_gen: "gen_set A" proof - from AinC have dir1: "span A⊆carrier V" by (rule span_is_subset2) have dir2: "carrier V⊆span A" proof (auto) fix v assume v: "v∈carrier V" from v b2 obtain a where "a∈A → carrier K ∧ lincomb a A = v" by auto from this A_fin AinC show "v∈span A" by (subst finite_span, auto) qed from dir1 dir2 show ?thesis by auto qed from A_li A_gen AinC show "basis A" by (unfold basis_def, auto) qed from 1 2 show ?thesis by satx qed lemma (in linear_map) surj_imp_imT_carrier: assumes surj: "T` (carrier V) = carrier W" shows "(imT) = carrier W" by (simp add: surj im_def) subsection ‹The rank-nullity (dimension) theorem› text ‹If $V$ is finite-dimensional and $T:V\to W$ is a linear map, then $\text{dim}(\text{im}(T))+ \text{dim}(\text{ker}(T)) = \text{dim} V$. Moreover, we prove that if $T$ is surjective linear-map between $V$ and $W$, where $V$ is finite-dimensional, then also $W$ is finite-dimensional.› theorem (in linear_map) rank_nullity_main: assumes fd: "V.fin_dim" shows "(vectorspace.dim K (W.vs imT)) + (vectorspace.dim K (V.vs kerT)) = V.dim" "T ` (carrier V) = carrier W ⟹ W.fin_dim" proof - ― ‹First interpret kerT, imT as vectorspaces› have subs_ker: "subspace K kerT V" by (intro kerT_is_subspace) from subs_ker have vs_ker: "vectorspace K (V.vs kerT)" by (rule V.subspace_is_vs) from vs_ker interpret ker: vectorspace K "(V.vs kerT)" by auto have kerInC: "kerT⊆carrier V" by (unfold ker_def, auto) have subs_im: "subspace K imT W" by (intro imT_is_subspace) from subs_im have vs_im: "vectorspace K (W.vs imT)" by (rule W.subspace_is_vs) from vs_im interpret im: vectorspace K "(W.vs imT)" by auto have imInC: "imT⊆carrier W" by (unfold im_def, auto) (* obvious fact *) have zero_same[simp]: "𝟬⇘_{V.vs kerT⇙}= 𝟬⇘_{V⇙}" apply (unfold ker_def) by auto ― ‹Show ker T has a finite basis. This is not obvious. Show that any linearly independent set has size at most that of V. There exists a maximal linearly independent set, which is the basis.› have every_li_small: "⋀A. (A ⊆ kerT)∧ ker.lin_indpt A ⟹ finite A ∧ card A ≤ V.dim" proof - fix A assume eli_asm: "(A ⊆ kerT)∧ ker.lin_indpt A" (*annoying: I can't just use subst V.module.span_li_not_depend(2) in the show ?thesis statement because it doesn't appear in the conclusion.*) note V.module.span_li_not_depend(2)[where ?N="kerT" and ?S="A"] from this subs_ker fd eli_asm kerInC show "?thesis A" apply (intro conjI) by (auto intro!: V.li_le_dim) qed from every_li_small have exA: "∃A. finite A ∧ maximal A (λS. S⊆carrier (V.vs kerT) ∧ ker.lin_indpt S)" apply (intro maximal_exists[where ?N="V.dim" and ?B="{}"]) apply auto by (unfold ker.lin_dep_def, auto) from exA obtain A where A:" finite A ∧ maximal A (λS. S⊆carrier (V.vs kerT) ∧ ker.lin_indpt S)" by blast hence finA: "finite A" and Ainker: "A⊆carrier (V.vs kerT)" and AinC: "A⊆carrier V" by (unfold maximal_def ker_def, auto) ― ‹We obtain the basis A of kerT. It is also linearly independent when considered in V rather than kerT› from A have Abasis: "ker.basis A" by (intro ker.max_li_is_basis, auto) from subs_ker Abasis have spanA: "V.module.span A = kerT" apply (unfold ker.basis_def) by (subst sym[OF V.module.span_li_not_depend(1)[where ?N="kerT"]], auto) from Abasis have Akerli: "ker.lin_indpt A" apply (unfold ker.basis_def) by auto from subs_ker Ainker Akerli have Ali: "V.module.lin_indpt A" by (auto simp add: V.module.span_li_not_depend(2)) txt‹Use the replacement theorem to find C such that $A\cup C$ is a basis of V.› from fd obtain B where B: "finite B∧ V.basis B" by (metis V.finite_basis_exists) from B have Bfin: "finite B" and Bbasis:"V.basis B" by auto from B have Bcard: "V.dim = card B" by (intro V.dim_basis, auto) from Bbasis have 62: "V.module.span B = carrier V" by (unfold V.basis_def, auto) from A Abasis Ali B vs_ker have "∃C. finite C ∧ C⊆carrier V ∧ C⊆ V.module.span B ∧ C∩A={} ∧ int (card C) ≤ (int (card B)) - (int (card A)) ∧ (V.module.span (A ∪ C) = V.module.span B)" apply (intro V.replacement) apply (unfold vectorspace.basis_def V.basis_def) by (unfold ker_def, auto) txt ‹From replacement we got $|C|\leq |B|-|A|$. Equality must actually hold, because no generating set can be smaller than $B$. Now $A\cup C$ is a maximal generating set, hence a basis; its cardinality equals the dimension.› txt ‹We claim that $T(C)$ is basis for $\text{im}(T)$.› then obtain C where C: "finite C ∧ C⊆carrier V ∧ C⊆ V.module.span B ∧ C∩A={} ∧ int (card C) ≤ (int (card B)) - (int (card A)) ∧ (V.module.span (A ∪ C) = V.module.span B)" by auto hence Cfin: "finite C" and CinC: "C⊆carrier V" and CinspanB: " C⊆V.module.span B" and CAdis: "C∩A={}" and Ccard: "int (card C) ≤ (int (card B)) - (int (card A))" and ACspanB: "(V.module.span (A ∪ C) = V.module.span B)" by auto from C have cardLe: "card A + card C ≤ card B" by auto from B C have ACgen: "V.module.gen_set (A∪C)" apply (unfold V.basis_def) by auto from finA C ACgen AinC B have cardGe: "card (A∪C) ≥ card B" by (intro V.li_smaller_than_gen, unfold V.basis_def, auto) from finA C have cardUn: "card (A∪C)≤ card A + card C" by (metis Int_commute card_Un_disjoint le_refl) from cardLe cardUn cardGe Bcard have cardEq: "card (A∪C) = card A + card C" "card (A∪C) = card B" "card (A∪C) = V.dim" by auto from Abasis C cardEq have disj: "A∩C={}" by auto from finA AinC C cardEq 62 have ACfin: "finite (A∪C)" and ACbasis: "V.basis (A∪C)" by (auto intro!: V.dim_gen_is_basis) have lm: "linear_map K V W T".. txt ‹Let $C'$ be the image of $C$ under $T$. We will show $C'$ is a basis for $\text{im}(T)$.› let ?C' = "T`C" from Cfin have C'fin: "finite ?C'" by auto from AinC C have cim: "?C'⊆imT" by (unfold im_def, auto) txt ‹"There is a subtle detail: we first have to show $T$ is injective on $C$.› txt ‹We establish that no nontrivial linear combination of $C$ can have image 0 under $T$, because that would mean it is a linear combination of $A$, giving that $A\cup C$ is linearly dependent, contradiction. We use this result in 2 ways: (1) if $T$ is not injective on $C$, then we obtain $v$, $w\in C$ such that $v-w$ is in the kernel, contradiction, (2) if $T(C)$ is linearly dependent, taking the inverse image of that linear combination gives a linear combination of $C$ in the kernel, contradiction. Hence $T$ is injective on $C$ and $T(C)$ is linearly independent.› have lc_in_ker: "⋀d D v. ⟦D⊆C; d∈D→carrier K; T (V.module.lincomb d D) = 𝟬⇘_{W⇙}; v∈D; d v ≠𝟬⇘_{K⇙}⟧⟹False" proof - fix d D v assume D: "D⊆C" and d: "d∈D→carrier K" and T0: "T (V.module.lincomb d D) = 𝟬⇘_{W⇙}" and v: "v∈D" and dvnz: "d v ≠𝟬⇘_{K⇙}" from D Cfin have Dfin: "finite D" by (auto intro: finite_subset) from D CinC have DinC: "D⊆carrier V" by auto from T0 d Dfin DinC have lc_d: "V.module.lincomb d D∈kerT" by (unfold ker_def, auto) from lc_d spanA AinC have "∃a' A'. A'⊆A ∧ a'∈A'→carrier K ∧ V.module.lincomb a' A'= V.module.lincomb d D" by (intro V.module.in_span, auto) then obtain a' A' where a': "A'⊆A ∧ a'∈A'→carrier K ∧ V.module.lincomb d D = V.module.lincomb a' A'" by metis hence A'sub: "A'⊆A" and a'fun: "a'∈A'→carrier K" and a'_lc:"V.module.lincomb d D = V.module.lincomb a' A'" by auto from a' finA Dfin have A'fin: "finite (A')" by (auto intro: finite_subset) from AinC A'sub have A'inC: "A'⊆carrier V" by auto let ?e = "(λv. if v ∈ A' then a' v else ⊖⇘_{K⇙}𝟭⇘_{K⇙}⊗⇘_{K⇙}d v)" from a'fun d have e_fun: "?e ∈ A' ∪ D → carrier K" apply (unfold Pi_def) by auto from A'fin Dfin (*finiteness*) A'inC DinC (*in carrier*) a'fun d e_fun (*coefficients valid*) disj D A'sub (*A and C disjoint*) have lccomp1: "V.module.lincomb a' A' ⊕⇘_{V⇙}⊖⇘_{K⇙}𝟭⇘_{K⇙}⊙⇘_{V⇙}V.module.lincomb d D = V.module.lincomb (λv. if v∈A' then a' v else ⊖⇘_{K⇙}𝟭⇘_{K⇙}⊗⇘_{K⇙}d v) (A'∪D)" apply (subst sym[OF V.module.lincomb_smult]) apply (simp_all) apply (subst V.module.lincomb_union2) by (auto) from A'fin (*finiteness*) A'inC (*in carrier*) a'fun (*coefficients valid*) have lccomp2: "V.module.lincomb a' A' ⊕⇘_{V⇙}⊖⇘_{K⇙}𝟭⇘_{K⇙}⊙⇘_{V⇙}V.module.lincomb d D = 𝟬⇘_{V⇙}" by (simp add: a'_lc V.module.smult_minus_1 V.module.M.r_neg) from lccomp1 lccomp2 have lc0: "V.module.lincomb (λv. if v∈A' then a' v else ⊖⇘_{K⇙}𝟭⇘_{K⇙}⊗⇘_{K⇙}d v) (A'∪D) =𝟬⇘_{V⇙}" by auto from disj a' v D have v_nin: "v∉A'" by auto from A'fin Dfin (*finiteness*) A'inC DinC (*in carrier*) e_fun d (*coefficients valid*) A'sub D disj (*A' D are disjoint subsets*) v dvnz (*d v is nonzero coefficient*) lc0 have AC_ld: "V.module.lin_dep (A∪C)" apply (intro V.module.lin_dep_crit[where ?A="A'∪D" and ?S="A∪C" and ?a="λv. if v∈A' then a' v else ⊖⇘_{K⇙}𝟭⇘_{K⇙}⊗⇘_{K⇙}d v" and ?v="v"]) by (auto dest: integral) from AC_ld ACbasis show False by (unfold V.basis_def, auto) qed have C'_card: "inj_on T C" "card C = card ?C'" proof - show "inj_on T C" proof (rule ccontr) assume "¬inj_on T C" then obtain v w where "v∈C" "w∈C" "v≠w" "T v = T w" by (unfold inj_on_def, auto) from this CinC show False apply (intro lc_in_ker[where ?D1="{v,w}" and ?d1="λx. if x=v then 𝟭⇘_{K⇙}else ⊖⇘_{K⇙}𝟭⇘_{K⇙}" and ?v1="v"]) by (auto simp add: V.module.lincomb_def hom_sum ring_subset_carrier W.module.smult_minus_1 r_neg T_im) qed from this Cfin show "card C = card ?C'" by (metis card_image) qed let ?f="the_inv_into C T" have f: "⋀x. x∈C ⟹ ?f (T x) = x" "⋀y. y∈?C' ⟹ T (?f y) = y" apply (insert C'_card(1)) apply (metis the_inv_into_f_f) by (metis f_the_inv_into_f) (*We show C' is a basis for the image. First we show it is linearly independent.*) have C'_li: "im.lin_indpt ?C'" proof (rule ccontr) assume Cld: "¬im.lin_indpt ?C'" from Cld cim subs_im have CldW: "W.module.lin_dep ?C'" apply (subst sym[OF W.module.span_li_not_depend(2)[where ?S="T`C" and ?N="imT"]]) by auto from C CldW have "∃c' v'. (c'∈ (?C'→carrier K)) ∧ (W.module.lincomb c' ?C' = 𝟬⇘_{W⇙}) ∧ (v'∈?C') ∧ (c' v'≠ 𝟬⇘_{K⇙})" by (intro W.module.finite_lin_dep, auto) then obtain c' v' where c': "(c'∈ (?C'→carrier K)) ∧ (W.module.lincomb c' ?C' = 𝟬⇘_{W⇙}) ∧ (v'∈?C') ∧ (c' v'≠ 𝟬⇘_{K⇙})" by auto hence c'fun: "(c'∈ (?C'→carrier K))" and c'lc: "(W.module.lincomb c' ?C' = 𝟬⇘_{W⇙})" and v':"(v'∈?C')" and cvnz: "(c' v'≠ 𝟬⇘_{K⇙})" by auto (*can't get c' directly with metis/auto with W.module.finite_lin_dep*) txt ‹We take the inverse image of $C'$ under $T$ to get a linear combination of $C$ that is in the kernel and hence a linear combination of $A$. This contradicts $A\cup C$ being linearly independent.› let ?c="λv. c' (T v)" from c'fun have c_fun: "?c∈ C→carrier K" by auto from Cfin (*C finite*) c_fun c'fun (*coefficients valid*) C'_card (*bijective*) CinC (*C in carrier*) f (*inverse to T*) c'lc (*lc c' = 0*) have "T (V.module.lincomb ?c C) = 𝟬⇘_{W⇙}" apply (unfold V.module.lincomb_def W.module.lincomb_def) apply (subst hom_sum, auto) apply (simp cong: finsum_cong add: ring_subset_carrier coeff_in_ring) apply (subst finsum_reindex[where ?f="λw. c' w ⊙⇘_{W⇙}w" and ?h="T" and ?A="C", THEN sym]) by auto with f c'fun cvnz v' show False by (intro lc_in_ker[where ?D1="C" and ?d1="?c" and ?v1="?f v'"], auto) qed have C'_gen: "im.gen_set ?C'" proof - have C'_span: "span ?C' = imT" proof (rule equalityI) from cim subs_im show "W.module.span ?C' ⊆ imT" by (intro span_is_subset, unfold subspace_def, auto) next show "imT⊆W.module.span ?C'" proof (auto) fix w assume w: "w∈imT" from this finA Cfin AinC CinC obtain v where v_inC: "v∈carrier V" and w_eq_T_v: "w= T v" by (unfold im_def image_def, auto) from finA Cfin AinC CinC v_inC ACgen have "∃a. a ∈ A∪C → carrier K∧ V.module.lincomb a (A∪C) = v" by (intro V.module.finite_in_span, auto) then obtain a where a_fun: "a ∈ A∪C → carrier K" and lc_a_v: "v= V.module.lincomb a (A∪C)" by auto let ?a'="λv. a (?f v)" from finA Cfin AinC CinC a_fun disj Ainker f C'_card have Tv: "T v = W.module.lincomb ?a' ?C'" apply (subst lc_a_v) apply (subst V.module.lincomb_union, simp_all) (*Break up the union A∪C*) apply (unfold lincomb_def V.module.lincomb_def) apply (subst hom_sum, auto) (*Take T inside the sum over A*) apply (simp add: subsetD coeff_in_ring hom_sum (*Take T inside the sum over C*) T_ker (*all terms become 0 because the vectors are in the kernel.*) ) apply (subst finsum_reindex[where ?h="T" and ?f="λv. ?a' v⊙⇘_{W⇙}v"], auto) by (auto cong: finsum_cong simp add: coeff_in_ring ring_subset_carrier) from a_fun f have a'_fun: "?a'∈?C' → carrier K" by auto from C'fin CinC this w_eq_T_v a'_fun Tv show "w ∈ LinearCombinations.module.span K W (T ` C)" by (subst finite_span, auto) qed qed from this subs_im CinC show ?thesis apply (subst span_li_not_depend(1)) by (unfold im_def subspace_def, auto) qed from C'_li C'_gen C cim have C'_basis: "im.basis (T`C)" by (unfold im.basis_def, auto) have C_card_im: "card C = (vectorspace.dim K (W.vs imT))" using C'_basis C'_card(2) C'fin im.dim_basis by auto from finA Abasis have "ker.dim = card A" by (rule ker.dim_basis) note * = this C_card_im cardEq show "(vectorspace.dim K (W.vs imT)) + (vectorspace.dim K (V.vs kerT)) = V.dim" using * by auto assume "T` (carrier V) = carrier W" from * surj_imp_imT_carrier[OF this] show "W.fin_dim" using C'_basis C'fin unfolding W.fin_dim_def im.basis_def by auto qed theorem (in linear_map) rank_nullity: assumes fd: "V.fin_dim" shows "(vectorspace.dim K (W.vs imT)) + (vectorspace.dim K (V.vs kerT)) = V.dim" by (rule rank_nullity_main[OF fd]) end