statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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is_compact.div_closed_ball (hs : is_compact s) (hδ : 0 ≤ δ) (x : E) :
s / closed_ball x δ = x⁻¹ • cthickening δ s | by simp [div_eq_mul_inv, mul_comm, hs.mul_closed_ball hδ] | lemma | is_compact.div_closed_ball | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"div_eq_mul_inv",
"is_compact",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.closed_ball_mul (hs : is_compact s) (hδ : 0 ≤ δ) (x : E) :
closed_ball x δ * s = x • cthickening δ s | by rw [mul_comm, hs.mul_closed_ball hδ] | lemma | is_compact.closed_ball_mul | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"is_compact",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.closed_ball_div (hs : is_compact s) (hδ : 0 ≤ δ) (x : E) :
closed_ball x δ * s = x • cthickening δ s | by simp [div_eq_mul_inv, mul_comm, hs.closed_ball_mul hδ] | lemma | is_compact.closed_ball_div | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"div_eq_mul_inv",
"is_compact",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_on_quotient (S : add_subgroup M) : has_norm (M ⧸ S) | { norm := λ x, Inf (norm '' {m | mk' S m = x}) } | instance | norm_on_quotient | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"has_norm",
"mk'"
] | The definition of the norm on the quotient by an additive subgroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_subgroup.quotient_norm_eq {S : add_subgroup M} (x : M ⧸ S) :
‖x‖ = Inf (norm '' {m : M | (m : M ⧸ S) = x}) | rfl | lemma | add_subgroup.quotient_norm_eq | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_norm_nonempty {S : add_subgroup M} :
∀ x : M ⧸ S, (norm '' {m | mk' S m = x}).nonempty | begin
rintro ⟨m⟩,
rw set.nonempty_image_iff,
use m,
change mk' S m = _,
refl
end | lemma | image_norm_nonempty | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"mk'",
"set.nonempty_image_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_below_image_norm (s : set M) : bdd_below (norm '' s) | begin
use 0,
rintro _ ⟨x, hx, rfl⟩,
apply norm_nonneg
end | lemma | bdd_below_image_norm | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_norm_neg {S : add_subgroup M} (x : M ⧸ S) : ‖-x‖ = ‖x‖ | begin
suffices : norm '' {m | mk' S m = x} = norm '' {m | mk' S m = -x},
by simp only [this, norm],
ext r,
split,
{ rintros ⟨m, rfl : mk' S m = x, rfl⟩,
rw ← norm_neg,
exact ⟨-m, by simp only [(mk' S).map_neg, set.mem_set_of_eq], rfl⟩ },
{ rintros ⟨m, hm : mk' S m = -x, rfl⟩,
exact ⟨-m, by sim... | lemma | quotient_norm_neg | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"mk'"
] | The norm on the quotient satisfies `‖-x‖ = ‖x‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_norm_sub_rev {S : add_subgroup M} (x y : M ⧸ S) : ‖x - y‖ = ‖y - x‖ | by rw [show x - y = -(y - x), by abel, quotient_norm_neg] | lemma | quotient_norm_sub_rev | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"quotient_norm_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_norm_mk_le (S : add_subgroup M) (m : M) :
‖mk' S m‖ ≤ ‖m‖ | begin
apply cInf_le,
use 0,
{ rintros _ ⟨n, h, rfl⟩,
apply norm_nonneg },
{ apply set.mem_image_of_mem,
rw set.mem_set_of_eq }
end | lemma | quotient_norm_mk_le | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"cInf_le",
"set.mem_image_of_mem"
] | The norm of the projection is smaller or equal to the norm of the original element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_norm_mk_le' (S : add_subgroup M) (m : M) :
‖(m : M ⧸ S)‖ ≤ ‖m‖ | quotient_norm_mk_le S m | lemma | quotient_norm_mk_le' | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"quotient_norm_mk_le"
] | The norm of the projection is smaller or equal to the norm of the original element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_norm_mk_eq (S : add_subgroup M) (m : M) :
‖mk' S m‖ = Inf ((λ x, ‖m + x‖) '' S) | begin
change Inf _ = _,
congr' 1,
ext r,
simp_rw [coe_mk', eq_iff_sub_mem],
split,
{ rintros ⟨y, h, rfl⟩,
use [y - m, h],
simp },
{ rintros ⟨y, h, rfl⟩,
use m + y,
simpa using h },
end | lemma | quotient_norm_mk_eq | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup"
] | The norm of the image under the natural morphism to the quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_norm_nonneg (S : add_subgroup M) : ∀ x : M ⧸ S, 0 ≤ ‖x‖ | begin
rintros ⟨m⟩,
change 0 ≤ ‖mk' S m‖,
apply le_cInf (image_norm_nonempty _),
rintros _ ⟨n, h, rfl⟩,
apply norm_nonneg
end | lemma | quotient_norm_nonneg | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"image_norm_nonempty",
"le_cInf"
] | The quotient norm is nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_mk_nonneg (S : add_subgroup M) (m : M) : 0 ≤ ‖mk' S m‖ | quotient_norm_nonneg S _ | lemma | norm_mk_nonneg | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"quotient_norm_nonneg"
] | The quotient norm is nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_norm_eq_zero_iff (S : add_subgroup M) (m : M) :
‖mk' S m‖ = 0 ↔ m ∈ closure (S : set M) | begin
have : 0 ≤ ‖mk' S m‖ := norm_mk_nonneg S m,
rw [← this.le_iff_eq, quotient_norm_mk_eq, real.Inf_le_iff],
simp_rw [zero_add],
{ calc (∀ ε > (0 : ℝ), ∃ r ∈ (λ x, ‖m + x‖) '' (S : set M), r < ε) ↔
(∀ ε > 0, (∃ x ∈ S, ‖m + x‖ < ε)) : by simp [set.bex_image_iff]
... ↔ ∀ ε > 0, (∃ x ∈ S, ‖m + -x‖ <... | lemma | quotient_norm_eq_zero_iff | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"closure",
"dist_comm",
"forall₂_congr",
"metric.ball",
"metric.mem_closure_iff",
"norm_mk_nonneg",
"quotient_norm_mk_eq",
"real.Inf_le_iff",
"set.bex_image_iff",
"set.nonempty_image_iff"
] | The norm of the image of `m : M` in the quotient by `S` is zero if and only if `m` belongs
to the closure of `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_mk_lt {S : add_subgroup M} (x : M ⧸ S) {ε : ℝ} (hε : 0 < ε) :
∃ (m : M), mk' S m = x ∧ ‖m‖ < ‖x‖ + ε | begin
obtain ⟨_, ⟨m : M, H : mk' S m = x, rfl⟩, hnorm : ‖m‖ < ‖x‖ + ε⟩ :=
real.lt_Inf_add_pos (image_norm_nonempty x) hε,
subst H,
exact ⟨m, rfl, hnorm⟩,
end | lemma | norm_mk_lt | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"image_norm_nonempty",
"mk'",
"real.lt_Inf_add_pos"
] | For any `x : M ⧸ S` and any `0 < ε`, there is `m : M` such that `mk' S m = x`
and `‖m‖ < ‖x‖ + ε`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_mk_lt' (S : add_subgroup M) (m : M) {ε : ℝ} (hε : 0 < ε) :
∃ s ∈ S, ‖m + s‖ < ‖mk' S m‖ + ε | begin
obtain ⟨n : M, hn : mk' S n = mk' S m, hn' : ‖n‖ < ‖mk' S m‖ + ε⟩ :=
norm_mk_lt (quotient_add_group.mk' S m) hε,
erw [eq_comm, quotient_add_group.eq] at hn,
use [- m + n, hn],
rwa [add_neg_cancel_left]
end | lemma | norm_mk_lt' | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"mk'",
"norm_mk_lt"
] | For any `m : M` and any `0 < ε`, there is `s ∈ S` such that `‖m + s‖ < ‖mk' S m‖ + ε`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_norm_add_le (S : add_subgroup M) (x y : M ⧸ S) : ‖x + y‖ ≤ ‖x‖ + ‖y‖ | begin
refine le_of_forall_pos_le_add (λ ε hε, _),
replace hε := half_pos hε,
obtain ⟨m, rfl, hm : ‖m‖ < ‖mk' S m‖ + ε / 2⟩ := norm_mk_lt x hε,
obtain ⟨n, rfl, hn : ‖n‖ < ‖mk' S n‖ + ε / 2⟩ := norm_mk_lt y hε,
calc ‖mk' S m + mk' S n‖ = ‖mk' S (m + n)‖ : by rw (mk' S).map_add
... ≤ ‖m + n‖ : quotient_norm_m... | lemma | quotient_norm_add_le | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"half_pos",
"mk'",
"norm_mk_lt",
"quotient_norm_mk_le"
] | The quotient norm satisfies the triangle inequality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_mk_zero (S : add_subgroup M) : ‖(0 : M ⧸ S)‖ = 0 | begin
erw quotient_norm_eq_zero_iff,
exact subset_closure S.zero_mem
end | lemma | norm_mk_zero | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"quotient_norm_eq_zero_iff",
"subset_closure"
] | The quotient norm of `0` is `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_zero_eq_zero (S : add_subgroup M) (hS : is_closed (S : set M)) (m : M)
(h : ‖mk' S m‖ = 0) : m ∈ S | by rwa [quotient_norm_eq_zero_iff, hS.closure_eq] at h | lemma | norm_zero_eq_zero | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"is_closed",
"quotient_norm_eq_zero_iff"
] | If `(m : M)` has norm equal to `0` in `M ⧸ S` for a closed subgroup `S` of `M`, then
`m ∈ S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_nhd_basis (S : add_subgroup M) :
(𝓝 (0 : M ⧸ S)).has_basis (λ ε : ℝ, 0 < ε) (λ ε, {x | ‖x‖ < ε}) | ⟨begin
intros U,
split,
{ intros U_in,
rw ← (mk' S).map_zero at U_in,
have := preimage_nhds_coinduced U_in,
rcases metric.mem_nhds_iff.mp this with ⟨ε, ε_pos, H⟩,
use [ε/2, half_pos ε_pos],
intros x x_in,
dsimp at x_in,
rcases norm_mk_lt x (half_pos ε_pos) with ⟨y, rfl, ry⟩,
apply ... | lemma | quotient_nhd_basis | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"filter.mem_of_superset",
"half_pos",
"is_open",
"is_open.mem_nhds",
"is_open_Union",
"mk'",
"norm_mk_lt",
"preimage_nhds_coinduced",
"quotient_norm_mk_le",
"set.subset.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_subgroup.seminormed_add_comm_group_quotient (S : add_subgroup M) :
seminormed_add_comm_group (M ⧸ S) | { dist := λ x y, ‖x - y‖,
dist_self := λ x, by simp only [norm_mk_zero, sub_self],
dist_comm := quotient_norm_sub_rev,
dist_triangle := λ x y z,
begin
unfold dist,
have : x - z = (x - y) + (y - z) := by abel,
rw this,
exact quotient_norm_add_le S (x - y) (y -... | instance | add_subgroup.seminormed_add_comm_group_quotient | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"dist_comm",
"dist_self",
"dist_triangle",
"filter.has_basis_binfi_principal",
"norm_mk_zero",
"quotient_nhd_basis",
"quotient_norm_add_le",
"quotient_norm_sub_rev",
"seminormed_add_comm_group",
"set.nonempty_Ioi"
] | The seminormed group structure on the quotient by an additive subgroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_subgroup.normed_add_comm_group_quotient (S : add_subgroup M) [is_closed (S : set M)] :
normed_add_comm_group (M ⧸ S) | { eq_of_dist_eq_zero :=
begin
rintros ⟨m⟩ ⟨m'⟩ (h : ‖mk' S m - mk' S m'‖ = 0),
erw [← (mk' S).map_sub, quotient_norm_eq_zero_iff, ‹is_closed _›.closure_eq,
← quotient_add_group.eq_iff_sub_mem] at h,
exact h
end,
.. add_subgroup.seminormed_add_comm_group_quotient S } | instance | add_subgroup.normed_add_comm_group_quotient | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"add_subgroup.seminormed_add_comm_group_quotient",
"eq_of_dist_eq_zero",
"is_closed",
"mk'",
"normed_add_comm_group",
"quotient_norm_eq_zero_iff"
] | The quotient in the category of normed groups. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_mk (S : add_subgroup M) : normed_add_group_hom M (M ⧸ S) | { bound' := ⟨1, λ m, by simpa [one_mul] using quotient_norm_mk_le _ m⟩,
.. quotient_add_group.mk' S } | def | add_subgroup.normed_mk | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"bound'",
"normed_add_group_hom",
"one_mul",
"quotient_norm_mk_le"
] | The morphism from a seminormed group to the quotient by a subgroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_mk.apply (S : add_subgroup M) (m : M) : normed_mk S m = quotient_add_group.mk' S m | rfl | lemma | add_subgroup.normed_mk.apply | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup"
] | `S.normed_mk` agrees with `quotient_add_group.mk' S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
surjective_normed_mk (S : add_subgroup M) : function.surjective (normed_mk S) | surjective_quot_mk _ | lemma | add_subgroup.surjective_normed_mk | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"surjective_quot_mk"
] | `S.normed_mk` is surjective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ker_normed_mk (S : add_subgroup M) : S.normed_mk.ker = S | quotient_add_group.ker_mk _ | lemma | add_subgroup.ker_normed_mk | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup"
] | The kernel of `S.normed_mk` is `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_normed_mk_le (S : add_subgroup M) : ‖S.normed_mk‖ ≤ 1 | normed_add_group_hom.op_norm_le_bound _ zero_le_one (λ m, by simp [quotient_norm_mk_le']) | lemma | add_subgroup.norm_normed_mk_le | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"normed_add_group_hom.op_norm_le_bound",
"quotient_norm_mk_le'",
"zero_le_one"
] | The operator norm of the projection is at most `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_normed_mk (S : add_subgroup M) (h : (S.topological_closure : set M) ≠ univ) :
‖S.normed_mk‖ = 1 | begin
obtain ⟨x, hx⟩ := set.nonempty_compl.2 h,
let y := S.normed_mk x,
have hy : ‖y‖ ≠ 0,
{ intro h0,
exact set.not_mem_of_mem_compl hx ((quotient_norm_eq_zero_iff S x).1 h0) },
refine le_antisymm (norm_normed_mk_le S) (le_of_forall_pos_le_add (λ ε hε, _)),
suffices : 1 ≤ ‖S.normed_mk‖ + min ε ((1 : ℝ)... | lemma | add_subgroup.norm_normed_mk | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"div_lt_div_right",
"div_pos",
"div_self",
"inv_pos_lt_iff_one_lt_mul",
"lt_one_add",
"norm_mk_lt",
"one_half_lt_one",
"one_half_pos",
"quotient_norm_eq_zero_iff",
"quotient_norm_mk_le",
"ring",
"set.not_mem_of_mem_compl"
] | The operator norm of the projection is `1` if the subspace is not dense. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_trivial_quotient_mk (S : add_subgroup M)
(h : (S.topological_closure : set M) = set.univ) : ‖S.normed_mk‖ = 0 | begin
refine le_antisymm (op_norm_le_bound _ le_rfl (λ x, _)) (norm_nonneg _),
have hker : x ∈ (S.normed_mk).ker.topological_closure,
{ rw [S.ker_normed_mk],
exact set.mem_of_eq_of_mem h trivial },
rw [ker_normed_mk] at hker,
simp only [(quotient_norm_eq_zero_iff S x).mpr hker, normed_mk.apply, zero_mul],... | lemma | add_subgroup.norm_trivial_quotient_mk | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"le_rfl",
"quotient_norm_eq_zero_iff",
"set.mem_of_eq_of_mem",
"zero_mul"
] | The operator norm of the projection is `0` if the subspace is dense. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_quotient (f : normed_add_group_hom M N) : Prop | (surjective : function.surjective f)
(norm : ∀ x, ‖f x‖ = Inf ((λ m, ‖x + m‖) '' f.ker)) | structure | normed_add_group_hom.is_quotient | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"normed_add_group_hom"
] | `is_quotient f`, for `f : M ⟶ N` means that `N` is isomorphic to the quotient of `M`
by the kernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift {N : Type*} [seminormed_add_comm_group N] (S : add_subgroup M)
(f : normed_add_group_hom M N) (hf : ∀ s ∈ S, f s = 0) :
normed_add_group_hom (M ⧸ S) N | { bound' :=
begin
obtain ⟨c : ℝ, hcpos : (0 : ℝ) < c, hc : ∀ x, ‖f x‖ ≤ c * ‖x‖⟩ := f.bound,
refine ⟨c, λ mbar, le_of_forall_pos_le_add (λ ε hε, _)⟩,
obtain ⟨m : M, rfl : mk' S m = mbar, hmnorm : ‖m‖ < ‖mk' S m‖ + ε/c⟩ :=
norm_mk_lt mbar (div_pos hε hcpos),
calc ‖f m‖ ≤ c * ‖m‖ : hc m
... ≤ ... | def | normed_add_group_hom.lift | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"bound'",
"div_pos",
"lift",
"mk'",
"mul_div_cancel'",
"mul_lt_mul_left",
"norm_mk_lt",
"normed_add_group_hom",
"seminormed_add_comm_group"
] | Given `f : normed_add_group_hom M N` such that `f s = 0` for all `s ∈ S`, where,
`S : add_subgroup M` is closed, the induced morphism `normed_add_group_hom (M ⧸ S) N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_mk {N : Type*} [seminormed_add_comm_group N] (S : add_subgroup M)
(f : normed_add_group_hom M N) (hf : ∀ s ∈ S, f s = 0) (m : M) :
lift S f hf (S.normed_mk m) = f m | rfl | lemma | normed_add_group_hom.lift_mk | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"lift",
"lift_mk",
"normed_add_group_hom",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_unique {N : Type*} [seminormed_add_comm_group N] (S : add_subgroup M)
(f : normed_add_group_hom M N) (hf : ∀ s ∈ S, f s = 0)
(g : normed_add_group_hom (M ⧸ S) N) :
g.comp (S.normed_mk) = f → g = lift S f hf | begin
intro h,
ext,
rcases add_subgroup.surjective_normed_mk _ x with ⟨x,rfl⟩,
change (g.comp (S.normed_mk) x) = _,
simpa only [h]
end | lemma | normed_add_group_hom.lift_unique | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"add_subgroup.surjective_normed_mk",
"lift",
"lift_unique",
"normed_add_group_hom",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_quotient_quotient (S : add_subgroup M) : is_quotient (S.normed_mk) | ⟨S.surjective_normed_mk, λ m, by simpa [S.ker_normed_mk] using quotient_norm_mk_eq _ m⟩ | lemma | normed_add_group_hom.is_quotient_quotient | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"quotient_norm_mk_eq"
] | `S.normed_mk` satisfies `is_quotient`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_quotient.norm_lift {f : normed_add_group_hom M N} (hquot : is_quotient f) {ε : ℝ}
(hε : 0 < ε) (n : N) : ∃ (m : M), f m = n ∧ ‖m‖ < ‖n‖ + ε | begin
obtain ⟨m, rfl⟩ := hquot.surjective n,
have nonemp : ((λ m', ‖m + m'‖) '' f.ker).nonempty,
{ rw set.nonempty_image_iff,
exact ⟨0, f.ker.zero_mem⟩ },
rcases real.lt_Inf_add_pos nonemp hε with
⟨_, ⟨⟨x, hx, rfl⟩, H : ‖m + x‖ < Inf ((λ (m' : M), ‖m + m'‖) '' f.ker) + ε⟩⟩,
exact ⟨m+x, by rw [map_add,... | lemma | normed_add_group_hom.is_quotient.norm_lift | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"normed_add_group_hom",
"normed_add_group_hom.mem_ker",
"real.lt_Inf_add_pos",
"set.nonempty_image_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_quotient.norm_le {f : normed_add_group_hom M N} (hquot : is_quotient f) (m : M) :
‖f m‖ ≤ ‖m‖ | begin
rw hquot.norm,
apply cInf_le,
{ use 0,
rintros _ ⟨m', hm', rfl⟩,
apply norm_nonneg },
{ exact ⟨0, f.ker.zero_mem, by simp⟩ }
end | lemma | normed_add_group_hom.is_quotient.norm_le | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"cInf_le",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_norm_le {N : Type*} [seminormed_add_comm_group N] (S : add_subgroup M)
(f : normed_add_group_hom M N) (hf : ∀ s ∈ S, f s = 0)
{c : ℝ≥0} (fb : ‖f‖ ≤ c) :
‖lift S f hf‖ ≤ c | begin
apply op_norm_le_bound _ c.coe_nonneg,
intros x,
by_cases hc : c = 0,
{ simp only [hc, nnreal.coe_zero, zero_mul] at fb ⊢,
obtain ⟨x, rfl⟩ := surjective_quot_mk _ x,
show ‖f x‖ ≤ 0,
calc ‖f x‖ ≤ 0 * ‖x‖ : f.le_of_op_norm_le fb x
... = 0 : zero_mul _ },
{ replace hc : 0 < c := pos_i... | lemma | normed_add_group_hom.lift_norm_le | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"aux",
"div_pos",
"lift_mk",
"mul_div_cancel'",
"mul_le_mul_left",
"nnreal.coe_zero",
"normed_add_group_hom",
"seminormed_add_comm_group",
"surjective_quot_mk",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_norm_noninc {N : Type*} [seminormed_add_comm_group N] (S : add_subgroup M)
(f : normed_add_group_hom M N) (hf : ∀ s ∈ S, f s = 0)
(fb : f.norm_noninc) :
(lift S f hf).norm_noninc | λ x,
begin
have fb' : ‖f‖ ≤ (1 : ℝ≥0) := norm_noninc.norm_noninc_iff_norm_le_one.mp fb,
simpa using le_of_op_norm_le _ (f.lift_norm_le _ _ fb') _,
end | lemma | normed_add_group_hom.lift_norm_noninc | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup",
"lift",
"normed_add_group_hom",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.quotient.seminormed_add_comm_group :
seminormed_add_comm_group (M ⧸ S) | add_subgroup.seminormed_add_comm_group_quotient S.to_add_subgroup | instance | submodule.quotient.seminormed_add_comm_group | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup.seminormed_add_comm_group_quotient",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.quotient.normed_add_comm_group [hS : is_closed (S : set M)] :
normed_add_comm_group (M ⧸ S) | @add_subgroup.normed_add_comm_group_quotient _ _ S.to_add_subgroup hS | instance | submodule.quotient.normed_add_comm_group | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"add_subgroup.normed_add_comm_group_quotient",
"is_closed",
"normed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.quotient.complete_space [complete_space M] : complete_space (M ⧸ S) | quotient_add_group.complete_space M S.to_add_subgroup | instance | submodule.quotient.complete_space | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"complete_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.quotient.norm_mk_lt {S : submodule R M} (x : M ⧸ S) {ε : ℝ} (hε : 0 < ε) :
∃ m : M, submodule.quotient.mk m = x ∧ ‖m‖ < ‖x‖ + ε | norm_mk_lt x hε | lemma | submodule.quotient.norm_mk_lt | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"norm_mk_lt",
"submodule",
"submodule.quotient.mk"
] | For any `x : M ⧸ S` and any `0 < ε`, there is `m : M` such that `submodule.quotient.mk m = x`
and `‖m‖ < ‖x‖ + ε`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.quotient.norm_mk_le (m : M) :
‖(submodule.quotient.mk m : M ⧸ S)‖ ≤ ‖m‖ | quotient_norm_mk_le S.to_add_subgroup m | lemma | submodule.quotient.norm_mk_le | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"quotient_norm_mk_le",
"submodule.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.quotient.normed_space (𝕜 : Type*) [normed_field 𝕜] [normed_space 𝕜 M]
[has_smul 𝕜 R] [is_scalar_tower 𝕜 R M] : normed_space 𝕜 (M ⧸ S) | { norm_smul_le := λ k x, le_of_forall_pos_le_add $ λ ε hε,
begin
have := (nhds_basis_ball.tendsto_iff nhds_basis_ball).mp
((@real.uniform_continuous_const_mul (‖k‖)).continuous.tendsto (‖x‖)) ε hε,
simp only [mem_ball, exists_prop, dist, abs_sub_lt_iff] at this,
rcases this with ⟨δ, hδ, h⟩,
obta... | instance | submodule.quotient.normed_space | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"abs_sub_lt_iff",
"continuous.tendsto",
"exists_prop",
"has_smul",
"is_scalar_tower",
"norm_smul",
"norm_smul_le",
"normed_field",
"normed_space",
"quotient_norm_mk_le",
"real.uniform_continuous_const_mul",
"submodule.quotient.module'",
"submodule.quotient.norm_mk_le",
"submodule.quotient.... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.quotient.norm_mk_lt {I : ideal R} (x : R ⧸ I) {ε : ℝ} (hε : 0 < ε) :
∃ r : R, ideal.quotient.mk I r = x ∧ ‖r‖ < ‖x‖ + ε | norm_mk_lt x hε | lemma | ideal.quotient.norm_mk_lt | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ideal.quotient.mk",
"norm_mk_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.quotient.norm_mk_le (r : R) :
‖ideal.quotient.mk I r‖ ≤ ‖r‖ | quotient_norm_mk_le I.to_add_subgroup r | lemma | ideal.quotient.norm_mk_le | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"quotient_norm_mk_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.quotient.semi_normed_comm_ring : semi_normed_comm_ring (R ⧸ I) | { mul_comm := mul_comm,
norm_mul := λ x y, le_of_forall_pos_le_add $ λ ε hε,
begin
have := ((nhds_basis_ball.prod_nhds nhds_basis_ball).tendsto_iff nhds_basis_ball).mp
(real.continuous_mul.tendsto (‖x‖, ‖y‖)) ε hε,
simp only [set.mem_prod, mem_ball, and_imp, prod.forall, exists_prop, prod.exists] at t... | instance | ideal.quotient.semi_normed_comm_ring | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"abs_sub_lt_iff",
"and_imp",
"exists_prop",
"ideal.quotient.norm_mk_le",
"ideal.quotient.norm_mk_lt",
"mul_comm",
"norm_mul",
"norm_mul_le",
"semi_normed_comm_ring",
"set.mem_prod",
"submodule.quotient.seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.quotient.normed_comm_ring [is_closed (I : set R)] :
normed_comm_ring (R ⧸ I) | { .. ideal.quotient.semi_normed_comm_ring I,
.. submodule.quotient.normed_add_comm_group I } | instance | ideal.quotient.normed_comm_ring | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"ideal.quotient.semi_normed_comm_ring",
"is_closed",
"normed_comm_ring",
"submodule.quotient.normed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.quotient.normed_algebra [normed_algebra 𝕜 R] :
normed_algebra 𝕜 (R ⧸ I) | { .. submodule.quotient.normed_space I 𝕜,
.. ideal.quotient.algebra 𝕜 } | instance | ideal.quotient.normed_algebra | analysis.normed.group | src/analysis/normed/group/quotient.lean | [
"analysis.normed_space.basic",
"analysis.normed.group.hom",
"ring_theory.ideal.quotient_operations"
] | [
"ideal.quotient.algebra",
"normed_algebra",
"submodule.quotient.normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_group_seminorm (G : Type*) [add_group G] extends zero_hom G ℝ | (add_le' : ∀ r s, to_fun (r + s) ≤ to_fun r + to_fun s)
(neg' : ∀ r, to_fun (-r) = to_fun r) | structure | add_group_seminorm | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"add_group",
"zero_hom"
] | A seminorm on an additive group `G` is a function `f : G → ℝ` that preserves zero, is
subadditive and such that `f (-x) = f x` for all `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group_seminorm (G : Type*) [group G] | (to_fun : G → ℝ)
(map_one' : to_fun 1 = 0)
(mul_le' : ∀ x y, to_fun (x * y) ≤ to_fun x + to_fun y)
(inv' : ∀ x, to_fun x⁻¹ = to_fun x) | structure | group_seminorm | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"group"
] | A seminorm on a group `G` is a function `f : G → ℝ` that sends one to zero, is submultiplicative
and such that `f x⁻¹ = f x` for all `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonarch_add_group_seminorm (G : Type*) [add_group G] extends zero_hom G ℝ | (add_le_max' : ∀ r s, to_fun (r + s) ≤ max (to_fun r) (to_fun s))
(neg' : ∀ r, to_fun (-r) = to_fun r) | structure | nonarch_add_group_seminorm | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"add_group",
"zero_hom"
] | A nonarchimedean seminorm on an additive group `G` is a function `f : G → ℝ` that preserves
zero, is nonarchimedean and such that `f (-x) = f x` for all `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_group_norm (G : Type*) [add_group G] extends add_group_seminorm G | (eq_zero_of_map_eq_zero' : ∀ x, to_fun x = 0 → x = 0) | structure | add_group_norm | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"add_group",
"add_group_seminorm"
] | A norm on an additive group `G` is a function `f : G → ℝ` that preserves zero, is subadditive
and such that `f (-x) = f x` and `f x = 0 → x = 0` for all `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group_norm (G : Type*) [group G] extends group_seminorm G | (eq_one_of_map_eq_zero' : ∀ x, to_fun x = 0 → x = 1) | structure | group_norm | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"group",
"group_seminorm"
] | A seminorm on a group `G` is a function `f : G → ℝ` that sends one to zero, is submultiplicative
and such that `f x⁻¹ = f x` and `f x = 0 → x = 1` for all `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonarch_add_group_norm (G : Type*) [add_group G] extends nonarch_add_group_seminorm G | (eq_zero_of_map_eq_zero' : ∀ x, to_fun x = 0 → x = 0) | structure | nonarch_add_group_norm | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"add_group",
"nonarch_add_group_seminorm"
] | A nonarchimedean norm on an additive group `G` is a function `f : G → ℝ` that preserves zero, is
nonarchimedean and such that `f (-x) = f x` and `f x = 0 → x = 0` for all `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonarch_add_group_seminorm_class (F : Type*) (α : out_param $ Type*) [add_group α]
extends nonarchimedean_hom_class F α ℝ | (map_zero (f : F) : f 0 = 0)
(map_neg_eq_map' (f : F) (a : α) : f (-a) = f a) | class | nonarch_add_group_seminorm_class | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"add_group",
"nonarchimedean_hom_class"
] | `nonarch_add_group_seminorm_class F α` states that `F` is a type of nonarchimedean seminorms on
the additive group `α`.
You should extend this class when you extend `nonarch_add_group_seminorm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonarch_add_group_norm_class (F : Type*) (α : out_param $ Type*) [add_group α]
extends nonarch_add_group_seminorm_class F α | (eq_zero_of_map_eq_zero (f : F) {a : α} : f a = 0 → a = 0) | class | nonarch_add_group_norm_class | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"add_group",
"nonarch_add_group_seminorm_class"
] | `nonarch_add_group_norm_class F α` states that `F` is a type of nonarchimedean norms on the
additive group `α`.
You should extend this class when you extend `nonarch_add_group_norm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_sub_le_max : f (x - y) ≤ max (f x) (f y) | by { rw [sub_eq_add_neg, ← nonarch_add_group_seminorm_class.map_neg_eq_map' f y],
exact map_add_le_max _ _ _ } | lemma | map_sub_le_max | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonarch_add_group_seminorm_class.to_add_group_seminorm_class [add_group E]
[nonarch_add_group_seminorm_class F E] :
add_group_seminorm_class F E ℝ | { map_add_le_add := λ f x y, begin
have h_nonneg : ∀ a, 0 ≤ f a,
{ intro a,
rw [← nonarch_add_group_seminorm_class.map_zero f, ← sub_self a],
exact le_trans (map_sub_le_max _ _ _) (by rw max_self (f a)) },
exact le_trans (map_add_le_max _ _ _)
(max_le (le_add_of_nonneg_right (h_nonneg _)) ... | instance | nonarch_add_group_seminorm_class.to_add_group_seminorm_class | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"add_group",
"add_group_seminorm_class",
"map_sub_le_max",
"nonarch_add_group_seminorm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonarch_add_group_norm_class.to_add_group_norm_class [add_group E]
[nonarch_add_group_norm_class F E] :
add_group_norm_class F E ℝ | { map_add_le_add := map_add_le_add,
map_neg_eq_map := nonarch_add_group_seminorm_class.map_neg_eq_map',
..‹nonarch_add_group_norm_class F E› } | instance | nonarch_add_group_norm_class.to_add_group_norm_class | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"add_group",
"add_group_norm_class",
"nonarch_add_group_norm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group_seminorm_class : group_seminorm_class (group_seminorm E) E ℝ | { coe := λ f, f.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_one_eq_zero := λ f, f.map_one',
map_mul_le_add := λ f, f.mul_le',
map_inv_eq_map := λ f, f.inv' } | instance | group_seminorm.group_seminorm_class | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"group_seminorm",
"group_seminorm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe : p.to_fun = p | rfl | lemma | group_seminorm.to_fun_eq_coe | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext : (∀ x, p x = q x) → p = q | fun_like.ext p q | lemma | group_seminorm.ext | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_def : p ≤ q ↔ (p : E → ℝ) ≤ q | iff.rfl | lemma | group_seminorm.le_def | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_def : p < q ↔ (p : E → ℝ) < q | iff.rfl | lemma | group_seminorm.lt_def | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_le_coe : (p : E → ℝ) ≤ q ↔ p ≤ q | iff.rfl | lemma | group_seminorm.coe_le_coe | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lt_coe : (p : E → ℝ) < q ↔ p < q | iff.rfl | lemma | group_seminorm.coe_lt_coe | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ⇑(0 : group_seminorm E) = 0 | rfl | lemma | group_seminorm.coe_zero | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (x : E) : (0 : group_seminorm E) x = 0 | rfl | lemma | group_seminorm.zero_apply | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add : ⇑(p + q) = p + q | rfl | lemma | group_seminorm.coe_add | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_apply (x : E) : (p + q) x = p x + q x | rfl | lemma | group_seminorm.add_apply | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sup : ⇑(p ⊔ q) = p ⊔ q | rfl | lemma | group_seminorm.coe_sup | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_apply (x : E) : (p ⊔ q) x = p x ⊔ q x | rfl | lemma | group_seminorm.sup_apply | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (p : group_seminorm E) (f : F →* E) : group_seminorm F | { to_fun := λ x, p (f x),
map_one' := by rw [f.map_one, map_one_eq_zero p],
mul_le' := λ _ _, (congr_arg p $ f.map_mul _ _).trans_le $ map_mul_le_add p _ _,
inv' := λ x, by rw [map_inv, map_inv_eq_map p] } | def | group_seminorm.comp | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"group_seminorm",
"map_inv"
] | Composition of a group seminorm with a monoid homomorphism as a group seminorm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp : ⇑(p.comp f) = p ∘ f | rfl | lemma | group_seminorm.coe_comp | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (x : F) : (p.comp f) x = p (f x) | rfl | lemma | group_seminorm.comp_apply | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id : p.comp (monoid_hom.id _) = p | ext $ λ _, rfl | lemma | group_seminorm.comp_id | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"monoid_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_zero : p.comp (1 : F →* E) = 0 | ext $ λ _, map_one_eq_zero p | lemma | group_seminorm.comp_zero | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_comp : (0 : group_seminorm E).comp f = 0 | ext $ λ _, rfl | lemma | group_seminorm.zero_comp | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (g : F →* E) (f : G →* F) : p.comp (g.comp f) = (p.comp g).comp f | ext $ λ _, rfl | lemma | group_seminorm.comp_assoc | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comp (f : F →* E) : (p + q).comp f = p.comp f + q.comp f | ext $ λ _, rfl | lemma | group_seminorm.add_comp | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_mono (hp : p ≤ q) : p.comp f ≤ q.comp f | λ _, hp _ | lemma | group_seminorm.comp_mono | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_mul_le (f g : F →* E) : p.comp (f * g) ≤ p.comp f + p.comp g | λ _, map_mul_le_add p _ _ | lemma | group_seminorm.comp_mul_le | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_bdd_below_range_add {p q : group_seminorm E} {x : E} :
bdd_below (range $ λ y, p y + q (x / y)) | ⟨0, by { rintro _ ⟨x, rfl⟩, dsimp, positivity }⟩ | lemma | group_seminorm.mul_bdd_below_range_add | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"bdd_below",
"group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_apply : (p ⊓ q) x = ⨅ y, p y + q (x / y) | rfl | lemma | group_seminorm.inf_apply | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_one [decidable_eq E] (x : E) :
(1 : add_group_seminorm E) x = if x = 0 then 0 else 1 | rfl | lemma | add_group_seminorm.apply_one | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"add_group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (r : R) (p : add_group_seminorm E) : ⇑(r • p) = r • p | rfl | lemma | add_group_seminorm.coe_smul | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"add_group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_apply (r : R) (p : add_group_seminorm E) (x : E) : (r • p) x = r • p x | rfl | lemma | add_group_seminorm.smul_apply | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"add_group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_sup (r : R) (p q : add_group_seminorm E) : r • (p ⊔ q) = r • p ⊔ r • q | have real.smul_max : ∀ x y : ℝ, r • max x y = max (r • x) (r • y),
from λ x y, by simpa only [←smul_eq_mul, ←nnreal.smul_def, smul_one_smul ℝ≥0 r (_ : ℝ)]
using mul_max_of_nonneg x y (r • 1 : ℝ≥0).coe_nonneg,
ext $ λ x, real.smul_max _ _ | lemma | add_group_seminorm.smul_sup | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"add_group_seminorm",
"mul_max_of_nonneg",
"smul_one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonarch_add_group_seminorm_class :
nonarch_add_group_seminorm_class (nonarch_add_group_seminorm E) E | { coe := λ f, f.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_add_le_max := λ f, f.add_le_max',
map_zero := λ f, f.map_zero',
map_neg_eq_map' := λ f, f.neg', } | instance | nonarch_add_group_seminorm.nonarch_add_group_seminorm_class | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"nonarch_add_group_seminorm",
"nonarch_add_group_seminorm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ⇑(0 : nonarch_add_group_seminorm E) = 0 | rfl | lemma | nonarch_add_group_seminorm.coe_zero | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"nonarch_add_group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (x : E) : (0 : nonarch_add_group_seminorm E) x = 0 | rfl | lemma | nonarch_add_group_seminorm.zero_apply | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"nonarch_add_group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_bdd_below_range_add {p q : nonarch_add_group_seminorm E} {x : E} :
bdd_below (range $ λ y, p y + q (x - y)) | ⟨0, by { rintro _ ⟨x, rfl⟩, dsimp, positivity }⟩ | lemma | nonarch_add_group_seminorm.add_bdd_below_range_add | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"bdd_below",
"nonarch_add_group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_one [decidable_eq E] (x : E) :
(1 : group_seminorm E) x = if x = 1 then 0 else 1 | rfl | lemma | group_seminorm.apply_one | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (r : R) (p : group_seminorm E) : ⇑(r • p) = r • p | rfl | lemma | group_seminorm.coe_smul | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_apply (r : R) (p : group_seminorm E) (x : E) : (r • p) x = r • p x | rfl | lemma | group_seminorm.smul_apply | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_sup (r : R) (p q : group_seminorm E) : r • (p ⊔ q) = r • p ⊔ r • q | have real.smul_max : ∀ x y : ℝ, r • max x y = max (r • x) (r • y),
from λ x y, by simpa only [←smul_eq_mul, ←nnreal.smul_def, smul_one_smul ℝ≥0 r (_ : ℝ)]
using mul_max_of_nonneg x y (r • 1 : ℝ≥0).coe_nonneg,
ext $ λ x, real.smul_max _ _ | lemma | group_seminorm.smul_sup | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"group_seminorm",
"mul_max_of_nonneg",
"smul_one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_one [decidable_eq E] (x : E) :
(1 : nonarch_add_group_seminorm E) x = if x = 0 then 0 else 1 | rfl | lemma | nonarch_add_group_seminorm.apply_one | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"nonarch_add_group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (r : R) (p : nonarch_add_group_seminorm E) : ⇑(r • p) = r • p | rfl | lemma | nonarch_add_group_seminorm.coe_smul | analysis.normed.group | src/analysis/normed/group/seminorm.lean | [
"tactic.positivity",
"data.real.nnreal"
] | [
"nonarch_add_group_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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