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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symm_symm {f₀ f₁ : C(X, Y)} (F : homotopy f₀ f₁) : F.symm.symm = F | by { ext, simp } | lemma | continuous_map.homotopy.symm_symm | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans {f₀ f₁ f₂ : C(X, Y)} (F : homotopy f₀ f₁) (G : homotopy f₁ f₂) :
homotopy f₀ f₂ | { to_fun := λ x, if (x.1 : ℝ) ≤ 1/2 then F.extend (2 * x.1) x.2 else G.extend (2 * x.1 - 1) x.2,
continuous_to_fun := begin
refine continuous_if_le (continuous_induced_dom.comp continuous_fst) continuous_const
(F.continuous.comp (by continuity)).continuous_on
(G.continuous.comp (by continuity)).contin... | def | continuous_map.homotopy.trans | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"continuity",
"continuous_const",
"continuous_fst",
"continuous_if_le",
"continuous_on",
"homotopy"
] | Given `homotopy f₀ f₁` and `homotopy f₁ f₂`, we can define a `homotopy f₀ f₂` by putting the first
homotopy on `[0, 1/2]` and the second on `[1/2, 1]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_apply {f₀ f₁ f₂ : C(X, Y)} (F : homotopy f₀ f₁) (G : homotopy f₁ f₂)
(x : I × X) : (F.trans G) x =
if h : (x.1 : ℝ) ≤ 1/2 then
F (⟨2 * x.1, (unit_interval.mul_pos_mem_iff zero_lt_two).2 ⟨x.1.2.1, h⟩⟩, x.2)
else
G (⟨2 * x.1 - 1, unit_interval.two_mul_sub_one_mem_iff.2 ⟨(not_le.1 h).le, x.1.2.2⟩⟩, x.2... | show ite _ _ _ = _,
by split_ifs; { rw [extend, continuous_map.coe_Icc_extend, set.Icc_extend_of_mem], refl } | lemma | continuous_map.homotopy.trans_apply | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"continuous_map.coe_Icc_extend",
"extend",
"homotopy",
"set.Icc_extend_of_mem",
"unit_interval.mul_pos_mem_iff",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans {f₀ f₁ f₂ : C(X, Y)} (F : homotopy f₀ f₁) (G : homotopy f₁ f₂) :
(F.trans G).symm = G.symm.trans F.symm | begin
ext x,
simp only [symm_apply, trans_apply],
split_ifs with h₁ h₂,
{ change (x.1 : ℝ) ≤ _ at h₂,
change (1 : ℝ) - x.1 ≤ _ at h₁,
have ht : (x.1 : ℝ) = 1/2,
{ linarith },
norm_num [ht] },
{ congr' 2,
apply subtype.ext,
simp only [unit_interval.coe_symm_eq, subtype.coe_mk],
lina... | lemma | continuous_map.homotopy.symm_trans | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy",
"subtype.coe_mk",
"subtype.ext",
"unit_interval.coe_symm_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast {f₀ f₁ g₀ g₁ : C(X, Y)} (F : homotopy f₀ f₁) (h₀ : f₀ = g₀) (h₁ : f₁ = g₁) :
homotopy g₀ g₁ | { to_fun := F,
map_zero_left' := by simp [←h₀],
map_one_left' := by simp [←h₁] } | def | continuous_map.homotopy.cast | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy"
] | Casting a `homotopy f₀ f₁` to a `homotopy g₀ g₁` where `f₀ = g₀` and `f₁ = g₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hcomp {f₀ f₁ : C(X, Y)} {g₀ g₁ : C(Y, Z)} (F : homotopy f₀ f₁) (G : homotopy g₀ g₁) :
homotopy (g₀.comp f₀) (g₁.comp f₁) | { to_fun := λ x, G (x.1, F x),
map_zero_left' := by simp,
map_one_left' := by simp } | def | continuous_map.homotopy.hcomp | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy"
] | If we have a `homotopy f₀ f₁` and a `homotopy g₀ g₁`, then we can compose them and get a
`homotopy (g₀.comp f₀) (g₁.comp f₁)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopic (f₀ f₁ : C(X, Y)) : Prop | nonempty (homotopy f₀ f₁) | def | continuous_map.homotopic | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopic",
"homotopy"
] | Given continuous maps `f₀` and `f₁`, we say `f₀` and `f₁` are homotopic if there exists a
`homotopy f₀ f₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl (f : C(X, Y)) : homotopic f f | ⟨homotopy.refl f⟩ | lemma | continuous_map.homotopic.refl | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm ⦃f g : C(X, Y)⦄ (h : homotopic f g) : homotopic g f | h.map homotopy.symm | lemma | continuous_map.homotopic.symm | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopic",
"homotopy.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans ⦃f g h : C(X, Y)⦄ (h₀ : homotopic f g) (h₁ : homotopic g h) : homotopic f h | h₀.map2 homotopy.trans h₁ | lemma | continuous_map.homotopic.trans | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopic",
"homotopy.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hcomp {f₀ f₁ : C(X, Y)} {g₀ g₁ : C(Y, Z)} (h₀ : homotopic f₀ f₁) (h₁ : homotopic g₀ g₁) :
homotopic (g₀.comp f₀) (g₁.comp f₁) | h₀.map2 homotopy.hcomp h₁ | lemma | continuous_map.homotopic.hcomp | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence : equivalence (@homotopic X Y _ _) | ⟨refl, symm, trans⟩ | lemma | continuous_map.homotopic.equivalence | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopy_with (f₀ f₁ : C(X, Y)) (P : C(X, Y) → Prop) extends homotopy f₀ f₁ | (prop' : ∀ t, P ⟨λ x, to_fun (t, x),
continuous.comp continuous_to_fun (continuous_const.prod_mk continuous_id')⟩) | structure | continuous_map.homotopy_with | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"continuous.comp",
"continuous_id'",
"homotopy"
] | The type of homotopies between `f₀ f₁ : C(X, Y)`, where the intermediate maps satisfy the predicate
`P : C(X, Y) → Prop` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_injective : @function.injective (homotopy_with f₀ f₁ P) (I × X → Y) coe_fn | begin
rintros ⟨⟨⟨F, _⟩, _⟩, _⟩ ⟨⟨⟨G, _⟩, _⟩, _⟩ h,
congr' 3,
end | lemma | continuous_map.homotopy_with.coe_fn_injective | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {F G : homotopy_with f₀ f₁ P} (h : ∀ x, F x = G x) : F = G | coe_fn_injective $ funext h | lemma | continuous_map.homotopy_with.ext | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simps.apply (F : homotopy_with f₀ f₁ P) : I × X → Y | F
initialize_simps_projections homotopy_with
(to_homotopy_to_continuous_map_to_fun -> apply, -to_homotopy_to_continuous_map) | def | continuous_map.homotopy_with.simps.apply | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous (F : homotopy_with f₀ f₁ P) : continuous F | F.continuous_to_fun | lemma | continuous_map.homotopy_with.continuous | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_zero (F : homotopy_with f₀ f₁ P) (x : X) : F (0, x) = f₀ x | F.map_zero_left' x | lemma | continuous_map.homotopy_with.apply_zero | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_one (F : homotopy_with f₀ f₁ P) (x : X) : F (1, x) = f₁ x | F.map_one_left' x | lemma | continuous_map.homotopy_with.apply_one | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_continuous_map (F : homotopy_with f₀ f₁ P) : ⇑F.to_continuous_map = F | rfl | lemma | continuous_map.homotopy_with.coe_to_continuous_map | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_homotopy (F : homotopy_with f₀ f₁ P) : ⇑F.to_homotopy = F | rfl | lemma | continuous_map.homotopy_with.coe_to_homotopy | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prop (F : homotopy_with f₀ f₁ P) (t : I) : P (F.to_homotopy.curry t) | F.prop' t | lemma | continuous_map.homotopy_with.prop | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_prop (F : homotopy_with f₀ f₁ P) (t : ℝ) : P (F.to_homotopy.extend t) | begin
by_cases ht₀ : 0 ≤ t,
{ by_cases ht₁ : t ≤ 1,
{ convert F.prop ⟨t, ht₀, ht₁⟩,
ext,
rw [F.to_homotopy.extend_apply_of_mem_I ⟨ht₀, ht₁⟩, F.to_homotopy.curry_apply] },
{ convert F.prop 1,
ext,
rw [F.to_homotopy.extend_apply_of_one_le (le_of_not_le ht₁), F.to_homotopy.curry_apply,
... | lemma | continuous_map.homotopy_with.extend_prop | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl (f : C(X, Y)) (hf : P f) : homotopy_with f f P | { prop' := λ t, by { convert hf, cases f, refl },
..homotopy.refl f } | def | continuous_map.homotopy_with.refl | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy.refl"
] | Given a continuous function `f`, and a proof `h : P f`, we can define a `homotopy_with f f P` by
`F (t, x) = f x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm {f₀ f₁ : C(X, Y)} (F : homotopy_with f₀ f₁ P) : homotopy_with f₁ f₀ P | { prop' := λ t, by simpa using F.prop (σ t),
..F.to_homotopy.symm } | def | continuous_map.homotopy_with.symm | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | Given a `homotopy_with f₀ f₁ P`, we can define a `homotopy_with f₁ f₀ P` by reversing the homotopy. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_symm {f₀ f₁ : C(X, Y)} (F : homotopy_with f₀ f₁ P) : F.symm.symm = F | ext $ homotopy.congr_fun $ homotopy.symm_symm _ | lemma | continuous_map.homotopy_with.symm_symm | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans {f₀ f₁ f₂ : C(X, Y)} (F : homotopy_with f₀ f₁ P) (G : homotopy_with f₁ f₂ P) :
homotopy_with f₀ f₂ P | { prop' := λ t, begin
simp only [homotopy.trans],
change P ⟨λ _, ite ((t : ℝ) ≤ _) _ _, _⟩,
split_ifs,
{ exact F.extend_prop _ },
{ exact G.extend_prop _ }
end,
..F.to_homotopy.trans G.to_homotopy } | def | continuous_map.homotopy_with.trans | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy.trans"
] | Given `homotopy_with f₀ f₁ P` and `homotopy_with f₁ f₂ P`, we can define a `homotopy_with f₀ f₂ P`
by putting the first homotopy on `[0, 1/2]` and the second on `[1/2, 1]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_apply {f₀ f₁ f₂ : C(X, Y)} (F : homotopy_with f₀ f₁ P) (G : homotopy_with f₁ f₂ P)
(x : I × X) : (F.trans G) x =
if h : (x.1 : ℝ) ≤ 1/2 then
F (⟨2 * x.1, (unit_interval.mul_pos_mem_iff zero_lt_two).2 ⟨x.1.2.1, h⟩⟩, x.2)
else
G (⟨2 * x.1 - 1, unit_interval.two_mul_sub_one_mem_iff.2 ⟨(not_le.1 h).le, ... | homotopy.trans_apply _ _ _ | lemma | continuous_map.homotopy_with.trans_apply | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"unit_interval.mul_pos_mem_iff",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans {f₀ f₁ f₂ : C(X, Y)} (F : homotopy_with f₀ f₁ P) (G : homotopy_with f₁ f₂ P) :
(F.trans G).symm = G.symm.trans F.symm | ext $ homotopy.congr_fun $ homotopy.symm_trans _ _ | lemma | continuous_map.homotopy_with.symm_trans | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast {f₀ f₁ g₀ g₁ : C(X, Y)} (F : homotopy_with f₀ f₁ P) (h₀ : f₀ = g₀) (h₁ : f₁ = g₁) :
homotopy_with g₀ g₁ P | { prop' := F.prop,
..F.to_homotopy.cast h₀ h₁ } | def | continuous_map.homotopy_with.cast | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | Casting a `homotopy_with f₀ f₁ P` to a `homotopy_with g₀ g₁ P` where `f₀ = g₀` and `f₁ = g₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopic_with (f₀ f₁ : C(X, Y)) (P : C(X, Y) → Prop) : Prop | nonempty (homotopy_with f₀ f₁ P) | def | continuous_map.homotopic_with | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | Given continuous maps `f₀` and `f₁`, we say `f₀` and `f₁` are homotopic with respect to the
predicate `P` if there exists a `homotopy_with f₀ f₁ P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl (f : C(X, Y)) (hf : P f) : homotopic_with f f P | ⟨homotopy_with.refl f hf⟩ | lemma | continuous_map.homotopic_with.refl | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm ⦃f g : C(X, Y)⦄ (h : homotopic_with f g P) : homotopic_with g f P | ⟨h.some.symm⟩ | lemma | continuous_map.homotopic_with.symm | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans ⦃f g h : C(X, Y)⦄ (h₀ : homotopic_with f g P) (h₁ : homotopic_with g h P) :
homotopic_with f h P | ⟨h₀.some.trans h₁.some⟩ | lemma | continuous_map.homotopic_with.trans | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopy_rel (f₀ f₁ : C(X, Y)) (S : set X) | homotopy_with f₀ f₁ (λ f, ∀ x ∈ S, f x = f₀ x ∧ f x = f₁ x) | abbreviation | continuous_map.homotopy_rel | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | A `homotopy_rel f₀ f₁ S` is a homotopy between `f₀` and `f₁` which is fixed on the points in `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_fst (F : homotopy_rel f₀ f₁ S) (t : I) {x : X} (hx : x ∈ S) :
F (t, x) = f₀ x | (F.prop t x hx).1 | lemma | continuous_map.homotopy_rel.eq_fst | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_snd (F : homotopy_rel f₀ f₁ S) (t : I) {x : X} (hx : x ∈ S) :
F (t, x) = f₁ x | (F.prop t x hx).2 | lemma | continuous_map.homotopy_rel.eq_snd | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_eq_snd (F : homotopy_rel f₀ f₁ S) {x : X} (hx : x ∈ S) :
f₀ x = f₁ x | F.eq_fst 0 hx ▸ F.eq_snd 0 hx | lemma | continuous_map.homotopy_rel.fst_eq_snd | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl (f : C(X, Y)) (S : set X) : homotopy_rel f f S | homotopy_with.refl f (λ x hx, ⟨rfl, rfl⟩) | def | continuous_map.homotopy_rel.refl | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | Given a map `f : C(X, Y)` and a set `S`, we can define a `homotopy_rel f f S` by setting
`F (t, x) = f x` for all `t`. This is defined using `homotopy_with.refl`, but with the proof
filled in. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm (F : homotopy_rel f₀ f₁ S) : homotopy_rel f₁ f₀ S | { prop' := λ t x hx, by simp [F.eq_snd _ hx, F.fst_eq_snd hx],
..homotopy_with.symm F } | def | continuous_map.homotopy_rel.symm | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | Given a `homotopy_rel f₀ f₁ S`, we can define a `homotopy_rel f₁ f₀ S` by reversing the homotopy. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_symm (F : homotopy_rel f₀ f₁ S) : F.symm.symm = F | homotopy_with.symm_symm F | lemma | continuous_map.homotopy_rel.symm_symm | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (F : homotopy_rel f₀ f₁ S) (G : homotopy_rel f₁ f₂ S) : homotopy_rel f₀ f₂ S | { prop' := λ t, begin
intros x hx,
simp only [homotopy.trans],
change (⟨λ _, ite ((t : ℝ) ≤ _) _ _, _⟩ : C(X, Y)) _ = _ ∧ _ = _,
split_ifs,
{ simp [(homotopy_with.extend_prop F (2 * t) x hx).1, F.fst_eq_snd hx, G.fst_eq_snd hx] },
{ simp [(homotopy_with.extend_prop G (2 * t - 1) x hx).1, F.fst_e... | def | continuous_map.homotopy_rel.trans | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy.trans"
] | Given `homotopy_rel f₀ f₁ S` and `homotopy_rel f₁ f₂ S`, we can define a `homotopy_rel f₀ f₂ S`
by putting the first homotopy on `[0, 1/2]` and the second on `[1/2, 1]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_apply (F : homotopy_rel f₀ f₁ S) (G : homotopy_rel f₁ f₂ S)
(x : I × X) : (F.trans G) x =
if h : (x.1 : ℝ) ≤ 1/2 then
F (⟨2 * x.1, (unit_interval.mul_pos_mem_iff zero_lt_two).2 ⟨x.1.2.1, h⟩⟩, x.2)
else
G (⟨2 * x.1 - 1, unit_interval.two_mul_sub_one_mem_iff.2 ⟨(not_le.1 h).le, x.1.2.2⟩⟩, x.2) | homotopy.trans_apply _ _ _ | lemma | continuous_map.homotopy_rel.trans_apply | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"unit_interval.mul_pos_mem_iff",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans (F : homotopy_rel f₀ f₁ S) (G : homotopy_rel f₁ f₂ S) :
(F.trans G).symm = G.symm.trans F.symm | homotopy_with.ext $ homotopy.congr_fun $ homotopy.symm_trans _ _ | lemma | continuous_map.homotopy_rel.symm_trans | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast {f₀ f₁ g₀ g₁ : C(X, Y)} (F : homotopy_rel f₀ f₁ S) (h₀ : f₀ = g₀) (h₁ : f₁ = g₁) :
homotopy_rel g₀ g₁ S | { prop' := λ t x hx, by { simpa [←h₀, ←h₁] using F.prop t x hx },
..homotopy.cast F.to_homotopy h₀ h₁ } | def | continuous_map.homotopy_rel.cast | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | Casting a `homotopy_rel f₀ f₁ S` to a `homotopy_rel g₀ g₁ S` where `f₀ = g₀` and `f₁ = g₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopic_rel (f₀ f₁ : C(X, Y)) (S : set X) : Prop | nonempty (homotopy_rel f₀ f₁ S) | def | continuous_map.homotopic_rel | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | Given continuous maps `f₀` and `f₁`, we say `f₀` and `f₁` are homotopic relative to a set `S` if
there exists a `homotopy_rel f₀ f₁ S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl (f : C(X, Y)) : homotopic_rel f f S | ⟨homotopy_rel.refl f S⟩ | lemma | continuous_map.homotopic_rel.refl | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm ⦃f g : C(X, Y)⦄ (h : homotopic_rel f g S) : homotopic_rel g f S | h.map homotopy_rel.symm | lemma | continuous_map.homotopic_rel.symm | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans ⦃f g h : C(X, Y)⦄ (h₀ : homotopic_rel f g S) (h₁ : homotopic_rel g h S) :
homotopic_rel f h S | h₀.map2 homotopy_rel.trans h₁ | lemma | continuous_map.homotopic_rel.trans | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence : equivalence (λ f g : C(X, Y), homotopic_rel f g S) | ⟨refl, symm, trans⟩ | lemma | continuous_map.homotopic_rel.equivalence | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nullhomotopic (f : C(X, Y)) : Prop | ∃ y : Y, homotopic f (continuous_map.const _ y) | def | continuous_map.nullhomotopic | topology.homotopy | src/topology/homotopy/contractible.lean | [
"topology.homotopy.path",
"topology.homotopy.equiv"
] | [
"continuous_map.const",
"homotopic"
] | A map is nullhomotopic if it is homotopic to a constant map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nullhomotopic_of_constant (y : Y) : nullhomotopic (continuous_map.const X y) | ⟨y, by refl⟩ | lemma | continuous_map.nullhomotopic_of_constant | topology.homotopy | src/topology/homotopy/contractible.lean | [
"topology.homotopy.path",
"topology.homotopy.equiv"
] | [
"continuous_map.const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nullhomotopic.comp_right {f : C(X, Y)} (hf : f.nullhomotopic) (g : C(Y, Z)) :
(g.comp f).nullhomotopic | by { cases hf with y hy, use g y, exact homotopic.hcomp hy (homotopic.refl g), } | lemma | continuous_map.nullhomotopic.comp_right | topology.homotopy | src/topology/homotopy/contractible.lean | [
"topology.homotopy.path",
"topology.homotopy.equiv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nullhomotopic.comp_left {f : C(Y, Z)} (hf : f.nullhomotopic) (g : C(X, Y)) :
(f.comp g).nullhomotopic | by { cases hf with y hy, use y, exact homotopic.hcomp (homotopic.refl g) hy, } | lemma | continuous_map.nullhomotopic.comp_left | topology.homotopy | src/topology/homotopy/contractible.lean | [
"topology.homotopy.path",
"topology.homotopy.equiv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contractible_space (X : Type*) [topological_space X] : Prop | (hequiv_unit [] : nonempty (X ≃ₕ unit)) | class | contractible_space | topology.homotopy | src/topology/homotopy/contractible.lean | [
"topology.homotopy.path",
"topology.homotopy.equiv"
] | [
"topological_space"
] | A contractible space is one that is homotopy equivalent to `unit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_nullhomotopic (X : Type*) [topological_space X] [contractible_space X] :
(continuous_map.id X).nullhomotopic | begin
obtain ⟨hv⟩ := contractible_space.hequiv_unit X,
use hv.inv_fun (),
convert hv.left_inv.symm,
ext, simp, congr,
end | lemma | id_nullhomotopic | topology.homotopy | src/topology/homotopy/contractible.lean | [
"topology.homotopy.path",
"topology.homotopy.equiv"
] | [
"continuous_map.id",
"contractible_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contractible_iff_id_nullhomotopic (Y : Type*) [topological_space Y] :
contractible_space Y ↔ (continuous_map.id Y).nullhomotopic | begin
split, { introI, apply id_nullhomotopic, },
rintro ⟨p, h⟩,
refine_struct { hequiv_unit := ⟨
{ to_fun := continuous_map.const _ (),
inv_fun := continuous_map.const _ p }⟩ },
{ exact h.symm, }, { convert homotopic.refl (continuous_map.id unit), ext, },
end | lemma | contractible_iff_id_nullhomotopic | topology.homotopy | src/topology/homotopy/contractible.lean | [
"topology.homotopy.path",
"topology.homotopy.equiv"
] | [
"continuous_map.const",
"continuous_map.id",
"contractible_space",
"id_nullhomotopic",
"inv_fun",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_map.homotopy_equiv.contractible_space [contractible_space Y]
(e : X ≃ₕ Y) :
contractible_space X | ⟨(contractible_space.hequiv_unit Y).map e.trans⟩ | lemma | continuous_map.homotopy_equiv.contractible_space | topology.homotopy | src/topology/homotopy/contractible.lean | [
"topology.homotopy.path",
"topology.homotopy.equiv"
] | [
"contractible_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_map.homotopy_equiv.contractible_space_iff (e : X ≃ₕ Y) :
contractible_space X ↔ contractible_space Y | ⟨by { introI h, exact e.symm.contractible_space }, by { introI h, exact e.contractible_space }⟩ | lemma | continuous_map.homotopy_equiv.contractible_space_iff | topology.homotopy | src/topology/homotopy/contractible.lean | [
"topology.homotopy.path",
"topology.homotopy.equiv"
] | [
"contractible_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph.contractible_space [contractible_space Y] (e : X ≃ₜ Y) :
contractible_space X | e.to_homotopy_equiv.contractible_space | lemma | homeomorph.contractible_space | topology.homotopy | src/topology/homotopy/contractible.lean | [
"topology.homotopy.path",
"topology.homotopy.equiv"
] | [
"contractible_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph.contractible_space_iff (e : X ≃ₜ Y) :
contractible_space X ↔ contractible_space Y | e.to_homotopy_equiv.contractible_space_iff | lemma | homeomorph.contractible_space_iff | topology.homotopy | src/topology/homotopy/contractible.lean | [
"topology.homotopy.path",
"topology.homotopy.equiv"
] | [
"contractible_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopy_equiv (X : Type u) (Y : Type v) [topological_space X] [topological_space Y] | (to_fun : C(X, Y))
(inv_fun : C(Y, X))
(left_inv : (inv_fun.comp to_fun).homotopic (continuous_map.id X))
(right_inv : (to_fun.comp inv_fun).homotopic (continuous_map.id Y)) | structure | continuous_map.homotopy_equiv | topology.homotopy | src/topology/homotopy/equiv.lean | [
"topology.homotopy.basic"
] | [
"continuous_map.id",
"homotopic",
"homotopy_equiv",
"inv_fun",
"topological_space"
] | A homotopy equivalence between topological spaces `X` and `Y` are a pair of functions
`to_fun : C(X, Y)` and `inv_fun : C(Y, X)` such that `to_fun.comp inv_fun` and `inv_fun.comp to_fun`
are both homotopic to `id`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe (h : homotopy_equiv X Y) : (h.to_fun : X → Y) = h | rfl | lemma | continuous_map.homotopy_equiv.to_fun_eq_coe | topology.homotopy | src/topology/homotopy/equiv.lean | [
"topology.homotopy.basic"
] | [
"homotopy_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (h : homotopy_equiv X Y) : continuous h | h.to_fun.continuous | lemma | continuous_map.homotopy_equiv.continuous | topology.homotopy | src/topology/homotopy/equiv.lean | [
"topology.homotopy.basic"
] | [
"continuous",
"homotopy_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_homotopy_equiv (h : X ≃ₜ Y) : X ≃ₕ Y | { to_fun := ⟨h⟩,
inv_fun := ⟨h.symm⟩,
left_inv := by { convert continuous_map.homotopic.refl _, ext, simp },
right_inv := by { convert continuous_map.homotopic.refl _, ext, simp } } | def | homeomorph.to_homotopy_equiv | topology.homotopy | src/topology/homotopy/equiv.lean | [
"topology.homotopy.basic"
] | [
"continuous_map.homotopic.refl",
"inv_fun"
] | Any homeomorphism is a homotopy equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_homotopy_equiv (h : X ≃ₜ Y) : ⇑(h.to_homotopy_equiv) = h | rfl | lemma | homeomorph.coe_to_homotopy_equiv | topology.homotopy | src/topology/homotopy/equiv.lean | [
"topology.homotopy.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (h : X ≃ₕ Y) : Y ≃ₕ X | { to_fun := h.inv_fun,
inv_fun := h.to_fun,
left_inv := h.right_inv,
right_inv := h.left_inv } | def | continuous_map.homotopy_equiv.symm | topology.homotopy | src/topology/homotopy/equiv.lean | [
"topology.homotopy.basic"
] | [
"inv_fun"
] | If `X` is homotopy equivalent to `Y`, then `Y` is homotopy equivalent to `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_inv_fun (h : homotopy_equiv X Y) : (⇑h.inv_fun : Y → X) = ⇑h.symm | rfl | lemma | continuous_map.homotopy_equiv.coe_inv_fun | topology.homotopy | src/topology/homotopy/equiv.lean | [
"topology.homotopy.basic"
] | [
"homotopy_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simps.apply (h : X ≃ₕ Y) : X → Y | h | def | continuous_map.homotopy_equiv.simps.apply | topology.homotopy | src/topology/homotopy/equiv.lean | [
"topology.homotopy.basic"
] | [] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.symm_apply (h : X ≃ₕ Y) : Y → X | h.symm
initialize_simps_projections homotopy_equiv (to_fun_to_fun -> apply,
inv_fun_to_fun -> symm_apply, -to_fun, -inv_fun) | def | continuous_map.homotopy_equiv.simps.symm_apply | topology.homotopy | src/topology/homotopy/equiv.lean | [
"topology.homotopy.basic"
] | [
"homotopy_equiv",
"inv_fun"
] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl (X : Type u) [topological_space X] : X ≃ₕ X | (homeomorph.refl X).to_homotopy_equiv | def | continuous_map.homotopy_equiv.refl | topology.homotopy | src/topology/homotopy/equiv.lean | [
"topology.homotopy.basic"
] | [
"homeomorph.refl",
"topological_space"
] | Any topological space is homotopy equivalent to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans (h₁ : X ≃ₕ Y) (h₂ : Y ≃ₕ Z) : X ≃ₕ Z | { to_fun := h₂.to_fun.comp h₁.to_fun,
inv_fun := h₁.inv_fun.comp h₂.inv_fun,
left_inv := begin
refine homotopic.trans _ h₁.left_inv,
change ((h₁.inv_fun.comp h₂.inv_fun).comp (h₂.to_fun.comp h₁.to_fun)) with
h₁.inv_fun.comp ((h₂.inv_fun.comp h₂.to_fun).comp h₁.to_fun),
refine homotopic.hcomp _ (ho... | def | continuous_map.homotopy_equiv.trans | topology.homotopy | src/topology/homotopy/equiv.lean | [
"topology.homotopy.basic"
] | [
"continuous_map.id_comp",
"inv_fun"
] | If `X` is homotopy equivalent to `Y`, and `Y` is homotopy equivalent to `Z`, then `X` is homotopy
equivalent to `Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_trans (h₁ : X ≃ₕ Y) (h₂ : Y ≃ₕ Z) :
(h₁.trans h₂).symm = h₂.symm.trans h₁.symm | by ext; refl | lemma | continuous_map.homotopy_equiv.symm_trans | topology.homotopy | src/topology/homotopy/equiv.lean | [
"topology.homotopy.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_to_homotopy_equiv (X : Type u) [topological_space X] :
(homeomorph.refl X).to_homotopy_equiv = homotopy_equiv.refl X | rfl | lemma | homeomorph.refl_to_homotopy_equiv | topology.homotopy | src/topology/homotopy/equiv.lean | [
"topology.homotopy.basic"
] | [
"homeomorph.refl",
"homotopy_equiv.refl",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_to_homotopy_equiv (h : X ≃ₜ Y) :
h.symm.to_homotopy_equiv = h.to_homotopy_equiv.symm | rfl | lemma | homeomorph.symm_to_homotopy_equiv | topology.homotopy | src/topology/homotopy/equiv.lean | [
"topology.homotopy.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_to_homotopy_equiv (h₀ : X ≃ₜ Y) (h₁ : Y ≃ₜ Z) :
(h₀.trans h₁).to_homotopy_equiv = h₀.to_homotopy_equiv.trans h₁.to_homotopy_equiv | rfl | lemma | homeomorph.trans_to_homotopy_equiv | topology.homotopy | src/topology/homotopy/equiv.lean | [
"topology.homotopy.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundary (N : Type*) : set (I^N) | {y | ∃ i, y i = 0 ∨ y i = 1} | def | cube.boundary | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [] | The points in a cube with at least one projection equal to 0 or 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_at (i : N) : (I^N) ≃ₜ I × I^{j // j ≠ i} | fun_split_at I i | def | cube.split_at | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [] | The forward direction of the homeomorphism
between the cube $I^N$ and $I × I^{N\setminus\{j\}}$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
insert_at (i : N) : I × (I^{j // j ≠ i}) ≃ₜ I^N | (fun_split_at I i).symm | def | cube.insert_at | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [] | The backward direction of the homeomorphism
between the cube $I^N$ and $I × I^{N\setminus\{j\}}$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
insert_at_boundary (i : N) {t₀ : I} {t} (H : (t₀ = 0 ∨ t₀ = 1) ∨ t ∈ boundary {j // j ≠ i}) :
insert_at i ⟨t₀, t⟩ ∈ boundary N | begin
obtain H | ⟨j, H⟩ := H,
{ use i, rwa [fun_split_at_symm_apply, dif_pos rfl] },
{ use j, rwa [fun_split_at_symm_apply, dif_neg j.prop, subtype.coe_eta] },
end | lemma | cube.insert_at_boundary | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [
"subtype.coe_eta"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
loop_space | path x x | def | loop_space | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [
"path"
] | The space of paths with both endpoints equal to a specified point `x : X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
loop_space.inhabited : inhabited (path x x) | ⟨path.refl x⟩ | instance | loop_space.inhabited | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gen_loop : set C(I^N, X) | {p | ∀ y ∈ cube.boundary N, p y = x} | def | gen_loop | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [
"cube.boundary"
] | The `n`-dimensional generalized loops based at `x` in a space `X` are
continuous functions `I^n → X` that sends the boundary to `x`.
We allow an arbitrary indexing type `N` in place of `fin n` here. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
copy (f : Ω^N X x) (g : (I^N) → X) (h : g = f) : Ω^N X x | ⟨⟨g, h.symm ▸ f.1.2⟩, by { convert f.2, ext1, simp_rw h, refl }⟩ | def | gen_loop.copy | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [] | Copy of a `gen_loop` with a new map from the unit cube equal to the old one.
Useful to fix definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : Ω^N X x) {g : (I^N) → X} (h : g = f) : ⇑(copy f g h) = g | rfl | lemma | gen_loop.coe_copy | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : Ω^N X x) {g : (I^N) → X} (h : g = f) : copy f g h = f | by { ext x, exact congr_fun h x } | lemma | gen_loop.copy_eq | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boundary (f : Ω^N X x) : ∀ y ∈ cube.boundary N, f y = x | f.2 | lemma | gen_loop.boundary | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [
"cube.boundary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fun_like : fun_like (Ω^N X x) (I^N) (λ _, X) | { coe := λ f, f.1,
coe_injective' := λ ⟨⟨f, _⟩, _⟩ ⟨⟨g, _⟩, _⟩ h, by { congr, exact h } } | instance | gen_loop.fun_like | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [
"fun_like"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (f g : Ω^N X x) (H : ∀ y, f y = g y) : f = g | fun_like.coe_injective' (funext H) | lemma | gen_loop.ext | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_apply (f : C(I^N, X)) (H y) : (⟨f, H⟩ : Ω^N X x) y = f y | rfl | lemma | gen_loop.mk_apply | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const : Ω^N X x | ⟨continuous_map.const _ x, λ _ _, rfl⟩ | def | gen_loop.const | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [] | The constant `gen_loop` at `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_apply {t} : (@const N X _ x) t = x | rfl | lemma | gen_loop.const_apply | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inhabited : inhabited (Ω^N X x) | ⟨const⟩ | instance | gen_loop.inhabited | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopic (f g : Ω^N X x) : Prop | f.1.homotopic_rel g.1 (cube.boundary N) | def | gen_loop.homotopic | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [
"cube.boundary",
"homotopic"
] | The "homotopic relative to boundary" relation between `gen_loop`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl (f : Ω^N X x) : homotopic f f | continuous_map.homotopic_rel.refl _ | lemma | gen_loop.homotopic.refl | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [
"continuous_map.homotopic_rel.refl",
"homotopic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (H : homotopic f g) : homotopic g f | H.symm | lemma | gen_loop.homotopic.symm | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [
"homotopic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (H0 : homotopic f g) (H1 : homotopic g h) : homotopic f h | H0.trans H1 | lemma | gen_loop.homotopic.trans | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [
"homotopic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv : equivalence (@homotopic N X _ x) | ⟨homotopic.refl, λ _ _, homotopic.symm, λ _ _ _, homotopic.trans⟩ | lemma | gen_loop.homotopic.equiv | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [
"equiv",
"homotopic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
setoid (N) (x : X) : setoid (Ω^N X x) | ⟨homotopic, equiv⟩ | instance | gen_loop.homotopic.setoid | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_loop (i : N) (p : Ω^N X x) : Ω (Ω^{j // j ≠ i} X x) const | { to_fun := λ t, ⟨(p.val.comp (cube.insert_at i).to_continuous_map).curry t,
λ y yH, p.property (cube.insert_at i (t, y)) (cube.insert_at_boundary i $ or.inr yH)⟩,
source' := by { ext t, refine p.property (cube.insert_at i (0, t)) ⟨i, or.inl _⟩, simp },
target' := by { ext t, refine p.property (cube.insert_at i... | def | gen_loop.to_loop | topology.homotopy | src/topology/homotopy/homotopy_group.lean | [
"algebraic_topology.fundamental_groupoid.fundamental_group",
"group_theory.eckmann_hilton",
"logic.equiv.transfer_instance",
"algebra.group.ext"
] | [
"cube.insert_at",
"cube.insert_at_boundary"
] | Loop from a generalized loop by currying $I^N → X$ into $I → (I^{N\setminus\{j\}} → X)$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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