feat(theories/analysis): use new homomorphism names from algebra
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Rob Lewis
Leonardo de Moura
parent
89de67f4c3
commit
194cd89000
2 changed files with 14 additions and 71 deletions
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@ -66,62 +66,6 @@ section left_module
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by rewrite [sub_eq_add_neg, smul_right_distrib, neg_smul]
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end left_module
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/- linear maps -/
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structure is_linear_map [class] (R : Type) {M₁ M₂ : Type}
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[smul₁ : has_scalar R M₁] [smul₂ : has_scalar R M₂]
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[add₁ : has_add M₁] [add₂ : has_add M₂]
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(T : M₁ → M₂) :=
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(additive : ∀ u v : M₁, T (u + v) = T u + T v)
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(homogeneous : ∀ a : R, ∀ u : M₁, T (a • u) = a • T u)
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proposition linear_map_additive (R : Type) {M₁ M₂ : Type}
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[smul₁ : has_scalar R M₁] [smul₂ : has_scalar R M₂]
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[add₁ : has_add M₁] [add₂ : has_add M₂]
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(T : M₁ → M₂) [linT : is_linear_map R T] (u v : M₁) :
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T (u + v) = T u + T v :=
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is_linear_map.additive smul₁ smul₂ _ _ T u v
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proposition linear_map_homogeneous {R M₁ M₂ : Type}
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[smul₁ : has_scalar R M₁] [smul₂ : has_scalar R M₂]
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[add₁ : has_add M₁] [add₂ : has_add M₂]
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(T : M₁ → M₂) [linT : is_linear_map R T] (a : R) (u : M₁) :
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T (a • u) = a • T u :=
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is_linear_map.homogeneous smul₁ smul₂ _ _ T a u
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proposition is_linear_map_id [instance] (R : Type) {M : Type}
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[smulRM : has_scalar R M] [has_addM : has_add M] :
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is_linear_map R (id : M → M) :=
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is_linear_map.mk (take u v, rfl) (take a u, rfl)
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section is_linear_map
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variables {R M₁ M₂ : Type}
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variable [ringR : ring R]
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variable [moduleRM₁ : left_module R M₁]
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variable [moduleRM₂ : left_module R M₂]
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include ringR moduleRM₁ moduleRM₂
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variable T : M₁ → M₂
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variable [is_linear_mapT : is_linear_map R T]
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include is_linear_mapT
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proposition linear_map_zero : T 0 = 0 :=
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calc
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T 0 = T ((0 : R) • 0) : zero_smul
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... = (0 : R) • T 0 : linear_map_homogeneous T
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... = 0 : zero_smul
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proposition linear_map_neg (u : M₁) : T (-u) = -(T u) :=
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by rewrite [-neg_one_smul, linear_map_homogeneous T, neg_one_smul]
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proposition linear_map_smul_add_smul (a b : R) (u v : M₁) :
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T (a • u + b • v) = a • T u + b • T v :=
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by rewrite [linear_map_additive R T, *linear_map_homogeneous T]
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proposition linear_map_sub (u v : M₁) : T (u - v) = T u - T v :=
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by rewrite [*sub_eq_add_neg, linear_map_additive R T, linear_map_neg]
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end is_linear_map
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/- vector spaces -/
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structure vector_space [class] (F V : Type) [fieldF : field F]
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@ -5,7 +5,7 @@ Author: Robert Y. Lewis
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Bounded linear operators
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-/
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import .normed_space .real_limit algebra.module
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import .normed_space .real_limit algebra.module algebra.homomorphism
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open real nat classical
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noncomputable theory
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@ -15,7 +15,7 @@ namespace analysis
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section bdd_lin_op
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set_option pp.universes true
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structure is_bdd_linear_map [class] {V W : Type} [normed_vector_space V] [normed_vector_space W] (f : V → W)
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extends is_linear_map ℝ f :=
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extends is_module_hom ℝ f :=
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(op_norm : ℝ) (op_norm_pos : op_norm > 0) (op_norm_bound : ∀ v : V, ∥f v∥ ≤ op_norm * ∥v∥)
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theorem is_bdd_linear_map_id (V : Type) [normed_vector_space V] : is_bdd_linear_map (λ x : V, x) :=
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@ -48,10 +48,10 @@ theorem is_bdd_linear_map_add [instance] {V W : Type} [normed_vector_space V] [n
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begin
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fapply is_bdd_linear_map.mk,
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{intros,
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rewrite [linear_map_additive ℝ f, linear_map_additive ℝ g],
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rewrite [hom_add f, hom_add g],-- (this takes 4 seconds!!)rewrite [2 hom_add],
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simp},
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{intros,
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rewrite [linear_map_homogeneous f, linear_map_homogeneous g, smul_left_distrib]},
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rewrite [hom_smul f, hom_smul g, smul_left_distrib]},
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{exact is_bdd_linear_map.op_norm _ _ f + is_bdd_linear_map.op_norm _ _ g},
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{apply add_pos,
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repeat apply is_bdd_linear_map.op_norm_pos},
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@ -76,9 +76,10 @@ theorem is_bdd_linear_map_smul [instance] {V W : Type} [normed_vector_space V] [
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intro Hcnz,
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fapply is_bdd_linear_map.mk,
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{intros,
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rewrite [linear_map_additive ℝ f, smul_left_distrib]},
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rewrite [hom_add f, smul_left_distrib]},--rewrite [linear_map_additive ℝ f, smul_left_distrib]},
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{intros,
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rewrite [linear_map_homogeneous f, -*mul_smul, {c * a}mul.comm]},
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rewrite [hom_smul f, -*mul_smul, {c*r}mul.comm]},
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--rewrite [linear_map_homogeneous f, -*mul_smul, {c * a}mul.comm]},
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{exact (abs c) * is_bdd_linear_map.op_norm _ _ f},
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{have Hpos : abs c > 0, from abs_pos_of_ne_zero Hcnz,
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apply mul_pos,
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@ -106,9 +107,9 @@ theorem is_bdd_linear_map_comp {U V W : Type} [normed_vector_space U] [normed_ve
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begin
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fapply is_bdd_linear_map.mk,
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{intros,
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rewrite [linear_map_additive ℝ g, linear_map_additive ℝ f]},
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rewrite [hom_add g, hom_add f]},--rewrite [linear_map_additive ℝ g, linear_map_additive ℝ f]},
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{intros,
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rewrite [linear_map_homogeneous g, linear_map_homogeneous f]},
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rewrite [hom_smul g, hom_smul f]},--rewrite [linear_map_homogeneous g, linear_map_homogeneous f]},
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{exact is_bdd_linear_map.op_norm _ _ f * is_bdd_linear_map.op_norm _ _ g},
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{apply mul_pos, repeat apply is_bdd_linear_map.op_norm_pos},
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{intros,
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@ -144,7 +145,7 @@ theorem bounded_linear_operator_continuous : continuous f :=
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split,
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apply div_pos_of_pos_of_pos Hε !op_norm_pos,
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intro x' Hx',
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rewrite [-linear_map_sub f],
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rewrite [-hom_sub f],--[-linear_map_sub f],
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apply lt_of_le_of_lt,
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apply op_norm_bound f,
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rewrite [-@mul_div_cancel' _ _ ε (op_norm f) (ne_of_gt !op_norm_pos)],
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@ -375,13 +376,11 @@ theorem continuous_at_of_diffable_at [Hf : frechet_diffable_at f x] : continuous
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apply op_norm_bound (!frechet_deriv_at),
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rewrite [-add_halves ε at {2}],
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apply add_lt_add,
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exact calc
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ε * ∥x' - x∥ < ε * min δ ((ε / 2) / (ε + op_norm (frechet_deriv_at f x))) : mul_lt_mul_of_pos_left Hx' Hε
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... ≤ ε * ((ε / 2) / (ε + op_norm (frechet_deriv_at f x))) :
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mul_le_mul_of_nonneg_left !min_le_right (le_of_lt Hε)
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... < ε / 2 : mul_div_self_add_lt (div_pos_of_pos_of_pos Hε two_pos) Hε !op_norm_pos,
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let on := op_norm (frechet_deriv_at f x),
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exact calc
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ε * ∥x' - x∥ < ε * min δ ((ε / 2) / (ε + on)) : mul_lt_mul_of_pos_left Hx' Hε
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... ≤ ε * ((ε / 2) / (ε + on)) : mul_le_mul_of_nonneg_left !min_le_right (le_of_lt Hε)
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... < ε / 2 : mul_div_self_add_lt (div_pos_of_pos_of_pos Hε two_pos) Hε !op_norm_pos,
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exact calc
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on * ∥x' - x∥ < on * min δ ((ε / 2) / (ε + on)) : mul_lt_mul_of_pos_left Hx' !op_norm_pos
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... ≤ on * ((ε / 2) / (ε + on)) : mul_le_mul_of_nonneg_left !min_le_right (le_of_lt !op_norm_pos)
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