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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continuous_to_loop (i : N) : continuous (@to_loop N X _ x _ i) | path.continuous_uncurry_iff.1 $ continuous.subtype_mk (continuous_map.continuous_eval'.comp $
continuous.prod_map (continuous_map.continuous_curry.comp $
(continuous_map.continuous_comp_left _).comp continuous_subtype_coe) continuous_id) _ | lemma | gen_loop.continuous_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"
] | [
"continuous",
"continuous.prod_map",
"continuous.subtype_mk",
"continuous_id",
"continuous_map.continuous_comp_left",
"continuous_subtype_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_loop (i : N) (p : Ω (Ω^{j // j ≠ i} X x) const) : Ω^N X x | ⟨(continuous_map.comp ⟨coe⟩ p.to_continuous_map).uncurry.comp (cube.split_at i).to_continuous_map,
begin
rintros y ⟨j, Hj⟩,
simp only [subtype.val_eq_coe, continuous_map.comp_apply, to_continuous_map_apply,
fun_split_at_apply, continuous_map.uncurry_apply, continuous_map.coe_mk,
function.uncurry_apply_pair]... | def | gen_loop.from_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"
] | [
"continuous_map.coe_mk",
"continuous_map.comp",
"continuous_map.comp_apply",
"cube.split_at",
"eq_or_ne",
"gen_loop.boundary",
"subtype.val_eq_coe"
] | Generalized loop from a loop by uncurrying $I → (I^{N\setminus\{j\}} → X)$ into $I^N → X$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_from_loop (i : N) : continuous (@from_loop N X _ x _ i) | ((continuous_map.continuous_comp_left _).comp $ continuous_map.continuous_uncurry.comp $
(continuous_map.continuous_comp _).comp continuous_induced_dom).subtype_mk _ | lemma | gen_loop.continuous_from_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"
] | [
"continuous",
"continuous_induced_dom",
"continuous_map.continuous_comp",
"continuous_map.continuous_comp_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_from (i : N) (p : Ω (Ω^{j // j ≠ i} X x) const) : to_loop i (from_loop i p) = p | begin
simp_rw [to_loop, from_loop, continuous_map.comp_assoc, to_continuous_map_as_coe,
to_continuous_map_comp_symm, continuous_map.comp_id], ext, refl,
end | lemma | gen_loop.to_from | 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.comp_assoc",
"continuous_map.comp_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
loop_homeo (i : N) : Ω^N X x ≃ₜ Ω (Ω^{j // j ≠ i} X x) const | { to_fun := to_loop i,
inv_fun := from_loop i,
left_inv := λ p, by { ext, exact congr_arg p (equiv.apply_symm_apply _ _) },
right_inv := to_from i,
continuous_to_fun := continuous_to_loop i,
continuous_inv_fun := continuous_from_loop i } | def | gen_loop.loop_homeo | 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.apply_symm_apply",
"inv_fun"
] | The `n+1`-dimensional loops are in bijection with the loops in the space of
`n`-dimensional loops with base point `const`.
We allow an arbitrary indexing type `N` in place of `fin n` here. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_loop_apply (i : N) {p : Ω^N X x} {t} {tn} :
to_loop i p t tn = p (cube.insert_at i ⟨t, tn⟩) | rfl | lemma | gen_loop.to_loop_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"
] | [
"cube.insert_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_loop_apply (i : N) {p : Ω (Ω^{j // j ≠ i} X x) const} {t : I^N} :
from_loop i p t = p (t i) (cube.split_at i t).snd | rfl | lemma | gen_loop.from_loop_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"
] | [
"cube.split_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
c_comp_insert (i : N) : C(C(I^N, X), C(I × I^{j // j ≠ i}, X)) | ⟨λ f, f.comp (cube.insert_at i).to_continuous_map,
(cube.insert_at i).to_continuous_map.continuous_comp_left⟩ | def | gen_loop.c_comp_insert | 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"
] | Composition with `cube.insert_at` as a continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_to (i : N) {p q : Ω^N X x} (H : p.1.homotopy_rel q.1 (cube.boundary N)) :
C(I × I, C(I^{j // j ≠ i}, X)) | ((⟨_, continuous_map.continuous_curry⟩: C(_,_)).comp $
(c_comp_insert i).comp H.to_continuous_map.curry).uncurry | def | gen_loop.homotopy_to | 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"
] | A homotopy between `n+1`-dimensional loops `p` and `q` constant on the boundary
seen as a homotopy between two paths in the space of `n`-dimensional paths. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_to_apply (i : N) {p q : Ω^N X x} (H : p.1.homotopy_rel q.1 $ cube.boundary N)
(t : I × I) (tₙ : I^{j // j ≠ i}) :
homotopy_to i H t tₙ = H (t.fst, cube.insert_at i (t.snd, tₙ)) | rfl | lemma | gen_loop.homotopy_to_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"
] | [
"cube.boundary",
"cube.insert_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopic_to (i : N) {p q : Ω^N X x} :
homotopic p q → (to_loop i p).homotopic (to_loop i q) | begin
refine nonempty.map (λ H, ⟨⟨⟨λ t, ⟨homotopy_to i H t, _⟩, _⟩, _, _⟩, _⟩),
{ rintros y ⟨i, iH⟩,
rw [homotopy_to_apply, H.eq_fst, p.2],
all_goals { apply cube.insert_at_boundary, right, exact ⟨i, iH⟩} },
{ continuity },
show ∀ _ _ _, _,
{ intros t y yH,
split; ext; erw homotopy_to_apply,
a... | lemma | gen_loop.homotopic_to | 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"
] | [
"continuity",
"cube.insert_at_boundary",
"homotopic",
"nonempty.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopy_from (i : N) {p q : Ω^N X x}
(H : (to_loop i p).homotopy (to_loop i q)) : C(I × I^N, X) | (continuous_map.comp ⟨_, continuous_map.continuous_uncurry⟩
(continuous_map.comp ⟨coe⟩ H.to_continuous_map).curry).uncurry.comp $
(continuous_map.id I).prod_map (cube.split_at i).to_continuous_map | def | gen_loop.homotopy_from | 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.comp",
"continuous_map.id",
"cube.split_at",
"homotopy",
"prod_map"
] | The converse to `gen_loop.homotopy_to`: a homotopy between two loops in the space of
`n`-dimensional loops can be seen as a homotopy between two `n+1`-dimensional paths. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_from_apply (i : N) {p q : Ω^N X x} (H : (to_loop i p).homotopy (to_loop i q))
(t : I × I^N) : homotopy_from i H t = H (t.fst, t.snd i) (λ j, t.snd ↑j) | rfl | lemma | gen_loop.homotopy_from_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"
] | [
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopic_from (i : N) {p q : Ω^N X x} :
(to_loop i p).homotopic (to_loop i q) → homotopic p q | begin
refine nonempty.map (λ H, ⟨⟨homotopy_from i H, _, _⟩, _⟩),
show ∀ _ _ _, _,
{ rintros t y ⟨j, jH⟩,
erw homotopy_from_apply,
obtain rfl | h := eq_or_ne j i,
{ split,
{ rw H.eq_fst, exacts [congr_arg p (equiv.right_inv _ _), jH] },
{ rw H.eq_snd, exacts [congr_arg q (equiv.right_inv _ ... | lemma | gen_loop.homotopic_from | 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"
] | [
"eq_or_ne",
"homotopic",
"nonempty.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_at (i : N) (f g : Ω^N X x) : Ω^N X x | copy (from_loop i $ (to_loop i f).trans $ to_loop i g)
(λ t, if (t i : ℝ) ≤ 1/2
then f (t.update i $ set.proj_Icc 0 1 zero_le_one (2 * t i))
else g (t.update i $ set.proj_Icc 0 1 zero_le_one (2 * t i - 1)))
begin
ext1, symmetry,
dsimp only [path.trans, from_loop, path.coe_mk, function.comp_app,
mk_app... | def | gen_loop.trans_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"
] | [
"continuous_map.coe_mk",
"continuous_map.comp_apply",
"path.coe_mk",
"path.trans",
"set.proj_Icc",
"zero_le_one"
] | Concatenation of two `gen_loop`s along the `i`th coordinate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_at (i : N) (f : Ω^N X x) : Ω^N X x | copy (from_loop i (to_loop i f).symm)
(λ t, f $ λ j, if j = i then σ (t i) else t j) $
by { ext1, change _ = f _, congr, ext1, simp } | def | gen_loop.symm_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"
] | [] | Reversal of a `gen_loop` along the `i`th coordinate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_at_distrib {i j : N} (h : i ≠ j) (a b c d : Ω^N X x) :
trans_at i (trans_at j a b) (trans_at j c d) = trans_at j (trans_at i a c) (trans_at i b d) | begin
ext, simp_rw [trans_at, coe_copy, function.update_apply, if_neg h, if_neg h.symm],
split_ifs; { congr' 1, ext1, simp only [function.update, eq_rec_constant, dite_eq_ite],
apply ite_ite_comm, rintro rfl, exact h.symm },
end | lemma | gen_loop.trans_at_distrib | 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"
] | [
"dite_eq_ite",
"eq_rec_constant",
"ite_ite_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_loop_trans_to_loop {i : N} {p q : Ω^N X x} :
from_loop i ((to_loop i p).trans $ to_loop i q) = trans_at i p q | (copy_eq _ _).symm | lemma | gen_loop.from_loop_trans_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_loop_symm_to_loop {i : N} {p : Ω^N X x} :
from_loop i (to_loop i p).symm = symm_at i p | (copy_eq _ _).symm | lemma | gen_loop.from_loop_symm_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopy_group (N) (X : Type*) [topological_space X] (x : X) : Type* | quotient (gen_loop.homotopic.setoid N x) | def | homotopy_group | 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"
] | [
"gen_loop.homotopic.setoid",
"topological_space"
] | The `n`th homotopy group at `x` defined as the quotient of `Ω^n x` by the
`gen_loop.homotopic` relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_group_equiv_fundamental_group (i : N) :
homotopy_group N X x ≃ fundamental_group (Ω^{j // j ≠ i} X x) const | begin
refine equiv.trans _ (category_theory.groupoid.iso_equiv_hom _ _).symm,
apply quotient.congr (loop_homeo i).to_equiv,
exact λ p q, ⟨homotopic_to i, homotopic_from i⟩,
end | def | homotopy_group_equiv_fundamental_group | 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"
] | [
"category_theory.groupoid.iso_equiv_hom",
"equiv.trans",
"fundamental_group",
"homotopy_group",
"quotient.congr"
] | Equivalence between the homotopy group of X and the fundamental group of
`Ω^{j // j ≠ i} x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_group.pi (n) (X : Type*) [topological_space X] (x : X) | homotopy_group (fin n) _ x | def | homotopy_group.pi | 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"
] | [
"homotopy_group",
"topological_space"
] | Homotopy group of finite index. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gen_loop_homeo_of_is_empty (N x) [is_empty N] : Ω^N X x ≃ₜ X | { to_fun := λ f, f 0,
inv_fun := λ y, ⟨continuous_map.const _ y, λ _ ⟨i, _⟩, is_empty_elim i⟩,
left_inv := λ f, by { ext, exact congr_arg f (subsingleton.elim _ _) },
right_inv := λ _, rfl,
continuous_to_fun :=
(continuous_map.continuous_eval_const' (0 : N → I)).comp continuous_induced_dom,
continuous_inv... | def | gen_loop_homeo_of_is_empty | 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_induced_dom",
"continuous_map.continuous_eval_const'",
"inv_fun",
"is_empty",
"is_empty_elim"
] | The 0-dimensional generalized loops based at `x` are in bijection with `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_group_equiv_zeroth_homotopy_of_is_empty (N x) [is_empty N] :
homotopy_group N X x ≃ zeroth_homotopy X | quotient.congr (gen_loop_homeo_of_is_empty N x).to_equiv
begin
-- joined iff homotopic
intros, split; rintro ⟨H⟩,
exacts
[⟨{ to_fun := λ t, H ⟨t, is_empty_elim⟩,
source' := (H.apply_zero _).trans (congr_arg a₁ $ subsingleton.elim _ _),
target' := (H.apply_one _).trans (congr_arg a₂ $ subsingleton.el... | def | homotopy_group_equiv_zeroth_homotopy_of_is_empty | 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"
] | [
"gen_loop_homeo_of_is_empty",
"homotopy_group",
"is_empty",
"is_empty_elim",
"quotient.congr",
"zeroth_homotopy"
] | The homotopy "group" indexed by an empty type is in bijection with
the path components of `X`, aka the `zeroth_homotopy`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_group.pi_0_equiv_zeroth_homotopy : π_ 0 X x ≃ zeroth_homotopy X | homotopy_group_equiv_zeroth_homotopy_of_is_empty (fin 0) x | def | homotopy_group.pi_0_equiv_zeroth_homotopy | 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"
] | [
"homotopy_group_equiv_zeroth_homotopy_of_is_empty",
"zeroth_homotopy"
] | The 0th homotopy "group" is in bijection with `zeroth_homotopy`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gen_loop_equiv_of_unique (N) [unique N] : Ω^N X x ≃ Ω X x | { to_fun := λ p, path.mk ⟨λ t, p (λ _, t), by continuity⟩
(gen_loop.boundary _ (λ _, 0) ⟨default, or.inl rfl⟩)
(gen_loop.boundary _ (λ _, 1) ⟨default, or.inr rfl⟩),
inv_fun := λ p, ⟨⟨λ c, p (c default), by continuity⟩,
begin
rintro y ⟨i, iH|iH⟩; cases unique.eq_default i; apply (congr_arg p iH).trans,
... | def | gen_loop_equiv_of_unique | 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"
] | [
"eq_const_of_unique",
"gen_loop.boundary",
"inv_fun",
"unique",
"unique.eq_default"
] | The 1-dimensional generalized loops based at `x` are in bijection with loops at `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_group_equiv_fundamental_group_of_unique (N) [unique N] :
homotopy_group N X x ≃ fundamental_group X x | begin
refine equiv.trans _ (category_theory.groupoid.iso_equiv_hom _ _).symm,
refine quotient.congr (gen_loop_equiv_of_unique N) _,
intros, split; rintros ⟨H⟩,
{ exact
⟨ { to_fun := λ tx, H (tx.fst, λ _, tx.snd),
map_zero_left' := λ _, H.apply_zero _,
map_one_left' := λ _, H.apply_one _,
... | def | homotopy_group_equiv_fundamental_group_of_unique | 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"
] | [
"category_theory.groupoid.iso_equiv_hom",
"continuous_apply",
"continuous_snd",
"eq_const_of_unique",
"equiv.trans",
"fundamental_group",
"gen_loop_equiv_of_unique",
"homotopy_group",
"quotient.congr",
"unique",
"unique.eq_default"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopy_group.pi_1_equiv_fundamental_group : π_ 1 X x ≃ fundamental_group X x | homotopy_group_equiv_fundamental_group_of_unique (fin 1) | def | homotopy_group.pi_1_equiv_fundamental_group | 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"
] | [
"fundamental_group",
"homotopy_group_equiv_fundamental_group_of_unique"
] | The first homotopy group at `x` is in bijection with the fundamental group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group (N) [decidable_eq N] [nonempty N] : group (homotopy_group N X x) | (homotopy_group_equiv_fundamental_group $ classical.arbitrary N).group | instance | homotopy_group.group | 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"
] | [
"classical.arbitrary",
"group",
"homotopy_group",
"homotopy_group_equiv_fundamental_group"
] | Group structure on `homotopy_group N X x` for nonempty `N` (in particular `π_(n+1) X x`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aux_group (i : N) : group (homotopy_group N X x) | (homotopy_group_equiv_fundamental_group i).group | def | homotopy_group.aux_group | 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"
] | [
"group",
"homotopy_group",
"homotopy_group_equiv_fundamental_group"
] | Group structure on `homotopy_group` obtained by pulling back path composition along the
`i`th direction. The group structures for two different `i j : N` distribute over each
other, and therefore are equal by the Eckmann-Hilton argument. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unital_aux_group (i : N) :
eckmann_hilton.is_unital (aux_group i).mul (⟦const⟧ : homotopy_group N X x) | ⟨⟨(aux_group i).one_mul⟩, ⟨(aux_group i).mul_one⟩⟩ | lemma | homotopy_group.is_unital_aux_group | 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"
] | [
"eckmann_hilton.is_unital",
"homotopy_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aux_group_indep (i j : N) : (aux_group i : group (homotopy_group N X x)) = aux_group j | begin
by_cases h : i = j, { rw h },
refine group.ext (eckmann_hilton.mul (is_unital_aux_group i) (is_unital_aux_group j) _),
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ⟨d⟩,
change quotient.mk _ = _,
apply congr_arg quotient.mk,
simp only [from_loop_trans_to_loop, trans_at_distrib h,
coe_to_equiv, loop_homeo_apply, coe_symm_to_... | lemma | homotopy_group.aux_group_indep | 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"
] | [
"eckmann_hilton.mul",
"group",
"group.ext",
"homotopy_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_at_indep {i} (j) (f g : Ω^N X x) : ⟦trans_at i f g⟧ = ⟦trans_at j f g⟧ | begin
simp_rw ← from_loop_trans_to_loop,
have := congr_arg (@group.mul _) (aux_group_indep i j),
exact congr_fun₂ this ⟦g⟧ ⟦f⟧,
end | lemma | homotopy_group.trans_at_indep | 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"
] | [
"congr_fun₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_at_indep {i} (j) (f : Ω^N X x) : ⟦symm_at i f⟧ = ⟦symm_at j f⟧ | begin
simp_rw ← from_loop_symm_to_loop,
have := congr_arg (@group.inv _) (aux_group_indep i j),
exact congr_fun this ⟦f⟧,
end | lemma | homotopy_group.symm_at_indep | 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 | |
one_def [nonempty N] : (1 : homotopy_group N X x) = ⟦const⟧ | rfl | lemma | homotopy_group.one_def | 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"
] | [
"homotopy_group"
] | Characterization of multiplicative identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_spec [nonempty N] {i} {p q : Ω^N X x} :
(⟦p⟧ * ⟦q⟧ : homotopy_group N X x) = ⟦trans_at i q p⟧ | by { rw [trans_at_indep _ q, ← from_loop_trans_to_loop], apply quotient.sound, refl } | lemma | homotopy_group.mul_spec | 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"
] | [
"homotopy_group"
] | Characterization of multiplication | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_spec [nonempty N] {i} {p : Ω^N X x} : (⟦p⟧⁻¹ : homotopy_group N X x) = ⟦symm_at i p⟧ | by { rw [symm_at_indep _ p, ← from_loop_symm_to_loop], apply quotient.sound, refl } | lemma | homotopy_group.inv_spec | 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"
] | [
"homotopy_group"
] | Characterization of multiplicative inverse | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_group [nontrivial N] : comm_group (homotopy_group N X x) | let h := exists_ne (classical.arbitrary N) in
@eckmann_hilton.comm_group (homotopy_group N X x) _ 1 (is_unital_aux_group h.some) _
begin
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ⟨d⟩,
apply congr_arg quotient.mk,
simp only [from_loop_trans_to_loop, trans_at_distrib h.some_spec,
coe_to_equiv, loop_homeo_apply, coe_symm_to_equiv, loo... | instance | homotopy_group.comm_group | 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"
] | [
"classical.arbitrary",
"comm_group",
"eckmann_hilton.comm_group",
"exists_ne",
"homotopy_group",
"nontrivial"
] | Multiplication on `homotopy_group N X x` is commutative for nontrivial `N`.
In particular, multiplication on `π_(n+2)` is commutative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
H_space (X : Type u) [topological_space X] | (Hmul : C(X × X, X))
(e : X)
(Hmul_e_e : Hmul (e, e) = e)
(e_Hmul : (Hmul.comp $ (const X e).prod_mk $ continuous_map.id X).homotopy_rel
(continuous_map.id X) {e})
(Hmul_e : (Hmul.comp $ (continuous_map.id X).prod_mk $ const X e).homotopy_rel
(continuous_map.id X) {e}) | class | H_space | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"continuous_map.id",
"topological_space"
] | A topological space `X` is an H-space if it behaves like a (potentially non-associative)
topological group, but where the axioms for a group only hold up to homotopy. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
H_space.prod (X : Type u) (Y : Type v) [topological_space X] [topological_space Y]
[H_space X] [H_space Y] : H_space (X × Y) | { Hmul := ⟨λ p, ((p.1.1 ⋀ p.2.1), p.1.2 ⋀ p.2.2), by continuity⟩,
e := (H_space.e, H_space.e),
Hmul_e_e := by {simp only [continuous_map.coe_mk, prod.mk.inj_iff],
exact ⟨H_space.Hmul_e_e, H_space.Hmul_e_e⟩},
e_Hmul :=
begin
let G : I × (X × Y) → X × Y :=
(λ p, (H_space.e_Hmul (p.1, p.2.1), H_spac... | instance | H_space.prod | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"H_space",
"continuous",
"continuous.comp",
"continuous_map.coe_mk",
"continuous_snd",
"prod.mk.inj_iff",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_H_space (M : Type u) [mul_one_class M] [topological_space M]
[has_continuous_mul M] : H_space M | { Hmul := ⟨function.uncurry has_mul.mul, continuous_mul⟩,
e := 1,
Hmul_e_e := one_mul 1,
e_Hmul := (homotopy_rel.refl _ _).cast rfl (by {ext1, apply one_mul}),
Hmul_e := (homotopy_rel.refl _ _).cast rfl (by {ext1, apply mul_one}) } | definition | topological_group.to_H_space | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"H_space",
"has_continuous_mul",
"mul_one",
"mul_one_class",
"one_mul",
"topological_space"
] | The definition `to_H_space` is not an instance because its `@additive` version would
lead to a diamond since a topological field would inherit two `H_space` structures, one from the
`mul_one_class` and one from the `add_zero_class`. In the case of a group, we make
`topological_group.H_space` an instance." | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
H_space (G : Type u)
[topological_space G] [group G] [topological_group G] : H_space G | to_H_space G | instance | topological_group.H_space | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"H_space",
"group",
"topological_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_eq_H_space_e {G : Type u} [topological_space G] [group G] [topological_group G] :
(1 : G) = H_space.e | rfl | lemma | topological_group.one_eq_H_space_e | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"group",
"topological_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Q_right (p : I × I) : I | set.proj_Icc 0 1 zero_le_one (2 * p.1 / (1 + p.2)) | def | unit_interval.Q_right | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"set.proj_Icc",
"zero_le_one"
] | `Q_right` is analogous to the function `Q` defined on p. 475 of [serre1951] that helps proving
continuity of `delay_refl_right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_Q_right : continuous Q_right | continuous_proj_Icc.comp $ continuous.div (by continuity) (by continuity)
(λ x, (add_pos zero_lt_one).ne') | lemma | unit_interval.continuous_Q_right | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"continuity",
"continuous",
"continuous.div",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Q_right_zero_left (θ : I) : Q_right (0, θ) = 0 | set.proj_Icc_of_le_left _ $ by simp only [coe_zero, mul_zero, zero_div] | lemma | unit_interval.Q_right_zero_left | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"mul_zero",
"set.proj_Icc_of_le_left",
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Q_right_one_left (θ : I) : Q_right (1, θ) = 1 | set.proj_Icc_of_right_le _ $ (le_div_iff $ add_pos zero_lt_one).2 $
by { dsimp only, rw [coe_one, one_mul, mul_one], apply add_le_add_left (le_one _) } | lemma | unit_interval.Q_right_one_left | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"le_div_iff",
"mul_one",
"one_mul",
"set.proj_Icc_of_right_le",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Q_right_zero_right (t : I) : (Q_right (t, 0) : ℝ) = if (t : ℝ) ≤ 1 / 2 then 2 * t else 1 | begin
simp only [Q_right, coe_zero, add_zero, div_one],
split_ifs,
{ rw set.proj_Icc_of_mem _ ((mul_pos_mem_iff zero_lt_two).2 _), exacts [rfl, ⟨t.2.1, h⟩] },
{ rw (set.proj_Icc_eq_right _).2, { refl }, { linarith }, { exact zero_lt_one } },
end | lemma | unit_interval.Q_right_zero_right | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"div_one",
"set.proj_Icc_eq_right",
"set.proj_Icc_of_mem",
"zero_lt_one",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Q_right_one_right (t : I) : Q_right (t, 1) = t | eq.trans (by {rw Q_right, congr, apply mul_div_cancel_left, exact two_ne_zero}) $
set.proj_Icc_coe zero_le_one _ | lemma | unit_interval.Q_right_one_right | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"mul_div_cancel_left",
"set.proj_Icc_coe",
"two_ne_zero",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
delay_refl_right (θ : I) (γ : path x y) : path x y | { to_fun := λ t, γ (Q_right (t, θ)),
continuous_to_fun := γ.continuous.comp (continuous_Q_right.comp $ continuous.prod.mk_left θ),
source' := by { dsimp only, rw [Q_right_zero_left, γ.source] },
target' := by { dsimp only, rw [Q_right_one_left, γ.target] } } | def | path.delay_refl_right | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"continuous.prod.mk_left",
"path"
] | This is the function analogous to the one on p. 475 of [serre1951], defining a homotopy from
the product path `γ ∧ e` to `γ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_delay_refl_right : continuous (λ p : I × path x y, delay_refl_right p.1 p.2) | continuous_uncurry_iff.mp $ (continuous_snd.comp continuous_fst).path_eval $
continuous_Q_right.comp $ continuous_snd.prod_mk $ continuous_fst.comp continuous_fst | lemma | path.continuous_delay_refl_right | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"continuous",
"continuous_fst",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
delay_refl_right_zero (γ : path x y) : delay_refl_right 0 γ = γ.trans (path.refl y) | begin
ext t,
simp only [delay_refl_right,
trans_apply, refl_extend, path.coe_mk, function.comp_app, refl_apply],
split_ifs, swap, conv_rhs { rw ← γ.target },
all_goals { apply congr_arg γ, ext1, rw Q_right_zero_right },
exacts [if_neg h, if_pos h],
end | lemma | path.delay_refl_right_zero | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"path",
"path.coe_mk",
"path.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
delay_refl_right_one (γ : path x y) : delay_refl_right 1 γ = γ | by { ext t, exact congr_arg γ (Q_right_one_right t) } | lemma | path.delay_refl_right_one | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
delay_refl_left (θ : I) (γ : path x y) : path x y | (delay_refl_right θ γ.symm).symm | def | path.delay_refl_left | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"path"
] | This is the function on p. 475 of [serre1951], defining a homotopy from a path `γ` to the
product path `e ∧ γ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_delay_refl_left : continuous (λ p : I × path x y, delay_refl_left p.1 p.2) | path.continuous_symm.comp $ continuous_delay_refl_right.comp $ continuous_fst.prod_mk $
path.continuous_symm.comp continuous_snd | lemma | path.continuous_delay_refl_left | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"continuous",
"continuous_snd",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
delay_refl_left_zero (γ : path x y) : delay_refl_left 0 γ = (path.refl x).trans γ | by simp only [delay_refl_left, delay_refl_right_zero, trans_symm, refl_symm, path.symm_symm] | lemma | path.delay_refl_left_zero | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"path",
"path.refl",
"path.symm_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
delay_refl_left_one (γ : path x y) : delay_refl_left 1 γ = γ | by simp only [delay_refl_left, delay_refl_right_one, path.symm_symm] | lemma | path.delay_refl_left_one | topology.homotopy | src/topology/homotopy/H_spaces.lean | [
"topology.compact_open",
"topology.homotopy.path"
] | [
"path",
"path.symm_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopy (p₀ p₁ : path x₀ x₁) | continuous_map.homotopy_rel p₀.to_continuous_map p₁.to_continuous_map {0, 1} | abbreviation | path.homotopy | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"continuous_map.homotopy_rel",
"homotopy",
"path"
] | The type of homotopies between two paths. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_injective : @function.injective (homotopy p₀ p₁) (I × I → X) coe_fn | continuous_map.homotopy_with.coe_fn_injective | lemma | path.homotopy.coe_fn_injective | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"continuous_map.homotopy_with.coe_fn_injective",
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
source (F : homotopy p₀ p₁) (t : I) : F (t, 0) = x₀ | begin
simp_rw [←p₀.source],
apply continuous_map.homotopy_rel.eq_fst,
simp,
end | lemma | path.homotopy.source | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"continuous_map.homotopy_rel.eq_fst",
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
target (F : homotopy p₀ p₁) (t : I) : F (t, 1) = x₁ | begin
simp_rw [←p₁.target],
apply continuous_map.homotopy_rel.eq_snd,
simp,
end | lemma | path.homotopy.target | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"continuous_map.homotopy_rel.eq_snd",
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval (F : homotopy p₀ p₁) (t : I) : path x₀ x₁ | { to_fun := F.to_homotopy.curry t,
source' := by simp,
target' := by simp } | def | path.homotopy.eval | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"homotopy",
"path"
] | Evaluating a path homotopy at an intermediate point, giving us a `path`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_zero (F : homotopy p₀ p₁) : F.eval 0 = p₀ | begin
ext t,
simp [eval],
end | lemma | path.homotopy.eval_zero | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_one (F : homotopy p₀ p₁) : F.eval 1 = p₁ | begin
ext t,
simp [eval],
end | lemma | path.homotopy.eval_one | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl (p : path x₀ x₁) : homotopy p p | continuous_map.homotopy_rel.refl p.to_continuous_map {0, 1} | def | path.homotopy.refl | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"continuous_map.homotopy_rel.refl",
"homotopy",
"path"
] | Given a path `p`, we can define a `homotopy p p` by `F (t, x) = p x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm (F : homotopy p₀ p₁) : homotopy p₁ p₀ | continuous_map.homotopy_rel.symm F | def | path.homotopy.symm | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"continuous_map.homotopy_rel.symm",
"homotopy"
] | Given a `homotopy p₀ p₁`, we can define a `homotopy p₁ p₀` by reversing the homotopy. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_symm (F : homotopy p₀ p₁) : F.symm.symm = F | continuous_map.homotopy_rel.symm_symm F | lemma | path.homotopy.symm_symm | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"continuous_map.homotopy_rel.symm_symm",
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (F : homotopy p₀ p₁) (G : homotopy p₁ p₂) : homotopy p₀ p₂ | continuous_map.homotopy_rel.trans F G | def | path.homotopy.trans | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"continuous_map.homotopy_rel.trans",
"homotopy"
] | Given `homotopy p₀ p₁` and `homotopy p₁ p₂`, we can define a `homotopy p₀ p₂` 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 p₀ p₁) (G : homotopy p₁ p₂) (x : I × I) :
(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) | continuous_map.homotopy_rel.trans_apply _ _ _ | lemma | path.homotopy.trans_apply | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"continuous_map.homotopy_rel.trans_apply",
"homotopy",
"unit_interval.mul_pos_mem_iff",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans (F : homotopy p₀ p₁) (G : homotopy p₁ p₂) :
(F.trans G).symm = G.symm.trans F.symm | continuous_map.homotopy_rel.symm_trans _ _ | lemma | path.homotopy.symm_trans | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"continuous_map.homotopy_rel.symm_trans",
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast {p₀ p₁ q₀ q₁ : path x₀ x₁} (F : homotopy p₀ p₁) (h₀ : p₀ = q₀) (h₁ : p₁ = q₁) :
homotopy q₀ q₁ | continuous_map.homotopy_rel.cast F (congr_arg _ h₀) (congr_arg _ h₁) | def | path.homotopy.cast | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"continuous_map.homotopy_rel.cast",
"homotopy",
"path"
] | Casting a `homotopy p₀ p₁` to a `homotopy q₀ q₁` where `p₀ = q₀` and `p₁ = q₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hcomp (F : homotopy p₀ q₀) (G : homotopy p₁ q₁) :
homotopy (p₀.trans p₁) (q₀.trans q₁) | { to_fun := λ x,
if (x.2 : ℝ) ≤ 1/2 then
(F.eval x.1).extend (2 * x.2)
else
(G.eval x.1).extend (2 * x.2 - 1),
continuous_to_fun := begin
refine continuous_if_le (continuous_induced_dom.comp continuous_snd) continuous_const
(F.to_homotopy.continuous.comp (by continuity)).continuous_on
(G.t... | def | path.homotopy.hcomp | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"continuity",
"continuous_const",
"continuous_if_le",
"continuous_on",
"continuous_snd",
"extend",
"homotopy",
"path.trans",
"set.mem_singleton_iff"
] | Suppose `p₀` and `q₀` are paths from `x₀` to `x₁`, `p₁` and `q₁` are paths from `x₁` to `x₂`.
Furthermore, suppose `F : homotopy p₀ q₀` and `G : homotopy p₁ q₁`. Then we can define a homotopy
from `p₀.trans p₁` to `q₀.trans q₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hcomp_apply (F : homotopy p₀ q₀) (G : homotopy p₁ q₁) (x : I × I) :
F.hcomp G x =
if h : (x.2 : ℝ) ≤ 1/2 then
F.eval x.1 ⟨2 * x.2, (unit_interval.mul_pos_mem_iff zero_lt_two).2 ⟨x.2.2.1, h⟩⟩
else
G.eval x.1 ⟨2 * x.2 - 1, unit_interval.two_mul_sub_one_mem_iff.2 ⟨(not_le.1 h).le, x.2.2.2⟩⟩ | show ite _ _ _ = _, by split_ifs; exact path.extend_extends _ _ | lemma | path.homotopy.hcomp_apply | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"homotopy",
"path.extend_extends",
"unit_interval.mul_pos_mem_iff",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hcomp_half (F : homotopy p₀ q₀) (G : homotopy p₁ q₁) (t : I) :
F.hcomp G (t, ⟨1/2, by norm_num, by norm_num⟩) = x₁ | show ite _ _ _ = _, by norm_num | lemma | path.homotopy.hcomp_half | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reparam (p : path x₀ x₁) (f : I → I) (hf : continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
homotopy p (p.reparam f hf hf₀ hf₁) | { to_fun := λ x, p ⟨σ x.1 * x.2 + x.1 * f x.2,
show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I, from convex_Icc _ _ x.2.2 (f x.2).2
(by unit_interval) (by unit_interval) (by simp)⟩,
map_zero_left' := λ x, by norm_num,
map_one_left' := λ x, by norm_num,
prop' := λ t x hx,
begin
cases hx,
... | def | path.homotopy.reparam | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"continuous",
"convex_Icc",
"homotopy",
"path",
"set.mem_singleton_iff",
"unit_interval"
] | Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm₂ {p q : path x₀ x₁} (F : p.homotopy q) : p.symm.homotopy q.symm | { to_fun := λ x, F ⟨x.1, σ x.2⟩,
map_zero_left' := by simp [path.symm],
map_one_left' := by simp [path.symm],
prop' := λ t x hx, begin
cases hx,
{ rw hx, simp },
{ rw set.mem_singleton_iff at hx,
rw hx,
simp }
end } | def | path.homotopy.symm₂ | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"path",
"path.symm",
"set.mem_singleton_iff"
] | Suppose `F : homotopy p q`. Then we have a `homotopy p.symm q.symm` by reversing the second
argument. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map {p q : path x₀ x₁} (F : p.homotopy q) (f : C(X, Y)) :
homotopy (p.map f.continuous) (q.map f.continuous) | { to_fun := f ∘ F,
map_zero_left' := by simp,
map_one_left' := by simp,
prop' := λ t x hx, begin
cases hx,
{ simp [hx] },
{ rw set.mem_singleton_iff at hx,
simp [hx] }
end } | def | path.homotopy.map | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"homotopy",
"path",
"set.mem_singleton_iff"
] | Given `F : homotopy p q`, and `f : C(X, Y)`, we can define a homotopy from `p.map f.continuous` to
`q.map f.continuous`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopic (p₀ p₁ : path x₀ x₁) : Prop | nonempty (p₀.homotopy p₁) | def | path.homotopic | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"homotopic",
"path"
] | Two paths `p₀` and `p₁` are `path.homotopic` if there exists a `homotopy` between them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl (p : path x₀ x₁) : p.homotopic p | ⟨homotopy.refl p⟩ | lemma | path.homotopic.refl | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm ⦃p₀ p₁ : path x₀ x₁⦄ (h : p₀.homotopic p₁) : p₁.homotopic p₀ | h.map homotopy.symm | lemma | path.homotopic.symm | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"homotopy.symm",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans ⦃p₀ p₁ p₂ : path x₀ x₁⦄ (h₀ : p₀.homotopic p₁) (h₁ : p₁.homotopic p₂) :
p₀.homotopic p₂ | h₀.map2 homotopy.trans h₁ | lemma | path.homotopic.trans | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"homotopy.trans",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence : equivalence (@homotopic X _ x₀ x₁) | ⟨refl, symm, trans⟩ | lemma | path.homotopic.equivalence | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"homotopic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map {p q : path x₀ x₁} (h : p.homotopic q) (f : C(X, Y)) :
homotopic (p.map f.continuous) (q.map f.continuous) | h.map (λ F, F.map f) | lemma | path.homotopic.map | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"homotopic",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hcomp {p₀ p₁ : path x₀ x₁} {q₀ q₁ : path x₁ x₂} (hp : p₀.homotopic p₁)
(hq : q₀.homotopic q₁) : (p₀.trans q₀).homotopic (p₁.trans q₁) | hp.map2 homotopy.hcomp hq | lemma | path.homotopic.hcomp | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"homotopic",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
setoid (x₀ x₁ : X) : setoid (path x₀ x₁) | ⟨homotopic, equivalence⟩ | def | path.homotopic.setoid | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"path"
] | The setoid on `path`s defined by the equivalence relation `path.homotopic`. That is, two paths are
equivalent if there is a `homotopy` between them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient (x₀ x₁ : X) | quotient (homotopic.setoid x₀ x₁) | def | path.homotopic.quotient | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [] | The quotient on `path x₀ x₁` by the equivalence relation `path.homotopic`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient.comp (P₀ : path.homotopic.quotient x₀ x₁) (P₁ : path.homotopic.quotient x₁ x₂) :
path.homotopic.quotient x₀ x₂ | quotient.map₂ path.trans (λ (p₀ : path x₀ x₁) p₁ hp (q₀ : path x₁ x₂) q₁ hq, (hcomp hp hq)) P₀ P₁ | def | path.homotopic.quotient.comp | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"path",
"path.homotopic.quotient",
"path.trans",
"quotient.map₂"
] | The composition of path homotopy classes. This is `path.trans` descended to the quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_lift (P₀ : path x₀ x₁) (P₁ : path x₁ x₂) : ⟦P₀.trans P₁⟧ = quotient.comp ⟦P₀⟧ ⟦P₁⟧ | rfl | lemma | path.homotopic.comp_lift | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.map_fn (P₀ : path.homotopic.quotient x₀ x₁) (f : C(X, Y)) :
path.homotopic.quotient (f x₀) (f x₁) | quotient.map (λ (q : path x₀ x₁), q.map f.continuous) (λ p₀ p₁ h, path.homotopic.map h f) P₀ | def | path.homotopic.quotient.map_fn | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"path",
"path.homotopic.map",
"path.homotopic.quotient",
"quotient.map"
] | The image of a path homotopy class `P₀` under a map `f`.
This is `path.map` descended to the quotient | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_lift (P₀ : path x₀ x₁) (f : C(X, Y)) :
⟦P₀.map f.continuous⟧ = quotient.map_fn ⟦P₀⟧ f | rfl | lemma | path.homotopic.map_lift | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hpath_hext {p₁ : path x₀ x₁} {p₂ : path x₂ x₃} (hp : ∀ t, p₁ t = p₂ t) : ⟦p₁⟧ == ⟦p₂⟧ | begin
obtain rfl : x₀ = x₂ := by { convert hp 0; simp, },
obtain rfl : x₁ = x₃ := by { convert hp 1; simp, },
rw heq_iff_eq, congr, ext t, exact hp t,
end | lemma | path.homotopic.hpath_hext | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"heq_iff_eq",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_at {X : Type*} {Y : Type*} [topological_space X] [topological_space Y] {f g : C(X, Y)}
(H : continuous_map.homotopy f g) (x : X) : path (f x) (g x) | { to_fun := λ t, H (t, x),
source' := H.apply_zero x,
target' := H.apply_one x, } | def | continuous_map.homotopy.eval_at | topology.homotopy | src/topology/homotopy/path.lean | [
"topology.homotopy.basic",
"topology.path_connected",
"analysis.convex.basic"
] | [
"continuous_map.homotopy",
"path",
"topological_space"
] | Given a homotopy H: f ∼ g, get the path traced by the point `x` as it moves from
`f x` to `g x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy.pi (homotopies : Π i, homotopy (f i) (g i)) : homotopy (pi f) (pi g) | { to_fun := λ t i, homotopies i t,
map_zero_left' := λ t, by { ext i, simp only [pi_eval, homotopy.apply_zero] },
map_one_left' := λ t, by { ext i, simp only [pi_eval, homotopy.apply_one] } } | def | continuous_map.homotopy.pi | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"homotopy"
] | The product homotopy of `homotopies` between functions `f` and `g` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_rel.pi (homotopies : Π i : I, homotopy_rel (f i) (g i) S) :
homotopy_rel (pi f) (pi g) S | { prop' :=
begin
intros t x hx,
dsimp only [coe_mk, pi_eval, to_fun_eq_coe, homotopy_with.coe_to_continuous_map],
simp only [function.funext_iff, ← forall_and_distrib],
intro i,
exact (homotopies i).prop' t x hx,
end,
..(homotopy.pi (λ i, (homotopies i).to_homotopy)), } | def | continuous_map.homotopy_rel.pi | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"forall_and_distrib",
"function.funext_iff"
] | The relative product homotopy of `homotopies` between functions `f` and `g` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy.prod (F : homotopy f₀ f₁) (G : homotopy g₀ g₁) :
homotopy (prod_mk f₀ g₀) (prod_mk f₁ g₁) | { to_fun := λ t, (F t, G t),
map_zero_left' := λ x, by simp only [prod_eval, homotopy.apply_zero],
map_one_left' := λ x, by simp only [prod_eval, homotopy.apply_one] } | def | continuous_map.homotopy.prod | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"homotopy"
] | The product of homotopies `F` and `G`,
where `F` takes `f₀` to `f₁` and `G` takes `g₀` to `g₁` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_rel.prod (F : homotopy_rel f₀ f₁ S) (G : homotopy_rel g₀ g₁ S) :
homotopy_rel (prod_mk f₀ g₀) (prod_mk f₁ g₁) S | { prop' :=
begin
intros t x hx,
have hF := F.prop' t x hx,
have hG := G.prop' t x hx,
simp only [coe_mk, prod_eval, prod.mk.inj_iff, homotopy.prod] at hF hG ⊢,
exact ⟨⟨hF.1, hG.1⟩, ⟨hF.2, hG.2⟩⟩,
end,
..(homotopy.prod F.to_homotopy G.to_homotopy) } | def | continuous_map.homotopy_rel.prod | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"prod.mk.inj_iff"
] | The relative product of homotopies `F` and `G`,
where `F` takes `f₀` to `f₁` and `G` takes `g₀` to `g₁` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_homotopy (γ₀ γ₁ : Π i, path (as i) (bs i)) (H : ∀ i, path.homotopy (γ₀ i) (γ₁ i)) :
path.homotopy (path.pi γ₀) (path.pi γ₁) | continuous_map.homotopy_rel.pi H | def | path.homotopic.pi_homotopy | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"continuous_map.homotopy_rel.pi",
"path",
"path.homotopy",
"path.pi"
] | The product of a family of path homotopies. This is just a specialization of `homotopy_rel` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi (γ : Π i, path.homotopic.quotient (as i) (bs i)) : path.homotopic.quotient as bs | (quotient.map path.pi
(λ x y hxy, nonempty.map (pi_homotopy x y) (classical.nonempty_pi.mpr hxy)))
(quotient.choice γ) | def | path.homotopic.pi | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"nonempty.map",
"path.homotopic.quotient",
"path.pi",
"quotient.choice",
"quotient.map"
] | The product of a family of path homotopy classes | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_lift (γ : Π i, path (as i) (bs i)) : path.homotopic.pi (λ i, ⟦γ i⟧) = ⟦path.pi γ⟧ | by { unfold pi, simp, } | lemma | path.homotopic.pi_lift | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"path",
"path.homotopic.pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_pi_eq_pi_comp
(γ₀ : Π i, path.homotopic.quotient (as i) (bs i))
(γ₁ : Π i, path.homotopic.quotient (bs i) (cs i)) :
pi γ₀ ⬝ pi γ₁ = pi (λ i, γ₀ i ⬝ γ₁ i) | begin
apply quotient.induction_on_pi γ₁,
apply quotient.induction_on_pi γ₀,
intros,
simp only [pi_lift],
rw [← path.homotopic.comp_lift,
path.trans_pi_eq_pi_trans,
← pi_lift],
refl,
end | lemma | path.homotopic.comp_pi_eq_pi_comp | topology.homotopy | src/topology/homotopy/product.lean | [
"topology.constructions",
"topology.homotopy.path"
] | [
"path.homotopic.comp_lift",
"path.homotopic.quotient",
"path.trans_pi_eq_pi_trans",
"quotient.induction_on_pi"
] | Composition and products commute.
This is `path.trans_pi_eq_pi_trans` descended to path homotopy classes | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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