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
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
to_pou_fun_zero_of_zero {i : ι} {x : X} (h : f i x = 0) :
f.to_pou_fun i x = 0 | by rw [to_pou_fun, h, zero_mul] | lemma | bump_covering.to_pou_fun_zero_of_zero | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_to_pou_fun_subset (i : ι) :
support (f.to_pou_fun i) ⊆ support (f i) | λ x, mt $ f.to_pou_fun_zero_of_zero | lemma | bump_covering.support_to_pou_fun_subset | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_pou_fun_eq_mul_prod (i : ι) (x : X) (t : finset ι)
(ht : ∀ j, well_ordering_rel j i → f j x ≠ 0 → j ∈ t) :
f.to_pou_fun i x = f i x * ∏ j in t.filter (λ j, well_ordering_rel j i), (1 - f j x) | begin
refine congr_arg _ (finprod_cond_eq_prod_of_cond_iff _ (λ j hj, _)),
rw [ne.def, sub_eq_self] at hj,
rw [finset.mem_filter, iff.comm, and_iff_right_iff_imp],
exact flip (ht j) hj
end | lemma | bump_covering.to_pou_fun_eq_mul_prod | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"and_iff_right_iff_imp",
"finprod_cond_eq_prod_of_cond_iff",
"finset",
"finset.mem_filter",
"well_ordering_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_to_pou_fun_eq (x : X) :
∑ᶠ i, f.to_pou_fun i x = 1 - ∏ᶠ i, (1 - f i x) | begin
set s := (f.point_finite x).to_finset,
have hs : (s : set ι) = {i | f i x ≠ 0} := finite.coe_to_finset _,
have A : support (λ i, to_pou_fun f i x) ⊆ s,
{ rw hs,
exact λ i hi, f.support_to_pou_fun_subset i hi },
have B : mul_support (λ i, 1 - f i x) ⊆ s,
{ rw [hs, mul_support_one_sub], exact λ i, i... | lemma | bump_covering.sum_to_pou_fun_eq | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"finprod_eq_prod_of_mul_support_subset",
"finset.prod_one_sub_ordered",
"linear_order_of_STO",
"well_ordering_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_finset_to_pou_fun_eventually_eq (i : ι) (x : X) :
∃ t : finset ι, f.to_pou_fun i =ᶠ[𝓝 x]
f i * ∏ j in t.filter (λ j, well_ordering_rel j i), (1 - f j) | begin
rcases f.locally_finite x with ⟨U, hU, hf⟩,
use hf.to_finset,
filter_upwards [hU] with y hyU,
simp only [pi.mul_apply, finset.prod_apply],
apply to_pou_fun_eq_mul_prod,
intros j hji hj,
exact hf.mem_to_finset.2 ⟨y, ⟨hj, hyU⟩⟩
end | lemma | bump_covering.exists_finset_to_pou_fun_eventually_eq | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"finset",
"finset.prod_apply",
"pi.mul_apply",
"well_ordering_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_to_pou_fun (i : ι) : continuous (f.to_pou_fun i) | begin
refine ((f i).continuous.mul $
continuous_finprod_cond (λ j _, continuous_const.sub (f j).continuous) _),
simp only [mul_support_one_sub],
exact f.locally_finite
end | lemma | bump_covering.continuous_to_pou_fun | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"continuous",
"continuous.mul",
"continuous_finprod_cond"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_partition_of_unity : partition_of_unity ι X s | { to_fun := λ i, ⟨f.to_pou_fun i, f.continuous_to_pou_fun i⟩,
locally_finite' := f.locally_finite.subset f.support_to_pou_fun_subset,
nonneg' := λ i x, mul_nonneg (f.nonneg i x)
(finprod_cond_nonneg $ λ j hj, sub_nonneg.2 $ f.le_one j x),
sum_eq_one' := λ x hx,
begin
simp only [continuous_map.coe_mk... | def | bump_covering.to_partition_of_unity | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"continuous_map.coe_mk",
"finprod_cond_nonneg",
"finprod_eq_zero",
"finprod_nonneg",
"partition_of_unity"
] | The partition of unity defined by a `bump_covering`.
The partition of unity is given by the formula `g i x = f i x * ∏ᶠ j < i, (1 - f j x)`. In other
words, `g i x = ∏ᶠ j < i, (1 - f j x) - ∏ᶠ j ≤ i, (1 - f j x)`, so
`∑ᶠ i, g i x = 1 - ∏ᶠ j, (1 - f j x)`. If `x ∈ s`, then one of `f j x` equals one, hence the product
o... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_partition_of_unity_apply (i : ι) (x : X) :
f.to_partition_of_unity i x = f i x * ∏ᶠ j (hj : well_ordering_rel j i), (1 - f j x) | rfl | lemma | bump_covering.to_partition_of_unity_apply | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"well_ordering_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_partition_of_unity_eq_mul_prod (i : ι) (x : X) (t : finset ι)
(ht : ∀ j, well_ordering_rel j i → f j x ≠ 0 → j ∈ t) :
f.to_partition_of_unity i x = f i x * ∏ j in t.filter (λ j, well_ordering_rel j i), (1 - f j x) | f.to_pou_fun_eq_mul_prod i x t ht | lemma | bump_covering.to_partition_of_unity_eq_mul_prod | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"finset",
"well_ordering_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_finset_to_partition_of_unity_eventually_eq (i : ι) (x : X) :
∃ t : finset ι, f.to_partition_of_unity i =ᶠ[𝓝 x]
f i * ∏ j in t.filter (λ j, well_ordering_rel j i), (1 - f j) | f.exists_finset_to_pou_fun_eventually_eq i x | lemma | bump_covering.exists_finset_to_partition_of_unity_eventually_eq | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"finset",
"well_ordering_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_partition_of_unity_zero_of_zero {i : ι} {x : X} (h : f i x = 0) :
f.to_partition_of_unity i x = 0 | f.to_pou_fun_zero_of_zero h | lemma | bump_covering.to_partition_of_unity_zero_of_zero | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_to_partition_of_unity_subset (i : ι) :
support (f.to_partition_of_unity i) ⊆ support (f i) | f.support_to_pou_fun_subset i | lemma | bump_covering.support_to_partition_of_unity_subset | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_to_partition_of_unity_eq (x : X) :
∑ᶠ i, f.to_partition_of_unity i x = 1 - ∏ᶠ i, (1 - f i x) | f.sum_to_pou_fun_eq x | lemma | bump_covering.sum_to_partition_of_unity_eq | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_subordinate.to_partition_of_unity {f : bump_covering ι X s} {U : ι → set X}
(h : f.is_subordinate U) :
f.to_partition_of_unity.is_subordinate U | λ i, subset.trans (closure_mono $ f.support_to_partition_of_unity_subset i) (h i) | lemma | bump_covering.is_subordinate.to_partition_of_unity | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"bump_covering",
"closure_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_is_subordinate_of_locally_finite [normal_space X] (hs : is_closed s)
(U : ι → set X) (ho : ∀ i, is_open (U i)) (hf : locally_finite U)
(hU : s ⊆ ⋃ i, U i) :
∃ f : partition_of_unity ι X s, f.is_subordinate U | let ⟨f, hf⟩ := bump_covering.exists_is_subordinate_of_locally_finite hs U ho hf hU
in ⟨f.to_partition_of_unity, hf.to_partition_of_unity⟩ | lemma | partition_of_unity.exists_is_subordinate_of_locally_finite | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"bump_covering.exists_is_subordinate_of_locally_finite",
"is_closed",
"is_open",
"locally_finite",
"normal_space",
"partition_of_unity"
] | If `X` is a normal topological space and `U` is a locally finite open covering of a closed set
`s`, then there exists a `partition_of_unity ι X s` that is subordinate to `U`. If `X` is a
paracompact space, then the assumption `hf : locally_finite U` can be omitted, see
`bump_covering.exists_is_subordinate`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_is_subordinate [normal_space X] [paracompact_space X] (hs : is_closed s)
(U : ι → set X) (ho : ∀ i, is_open (U i)) (hU : s ⊆ ⋃ i, U i) :
∃ f : partition_of_unity ι X s, f.is_subordinate U | let ⟨f, hf⟩ := bump_covering.exists_is_subordinate hs U ho hU
in ⟨f.to_partition_of_unity, hf.to_partition_of_unity⟩ | lemma | partition_of_unity.exists_is_subordinate | topology | src/topology/partition_of_unity.lean | [
"algebra.big_operators.finprod",
"set_theory.ordinal.basic",
"topology.continuous_function.algebra",
"topology.paracompact",
"topology.shrinking_lemma",
"topology.urysohns_lemma"
] | [
"bump_covering.exists_is_subordinate",
"is_closed",
"is_open",
"normal_space",
"paracompact_space",
"partition_of_unity"
] | If `X` is a paracompact normal topological space and `U` is an open covering of a closed set
`s`, then there exists a `partition_of_unity ι X s` that is subordinate to `U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
path (x y : X) extends C(I, X) | (source' : to_fun 0 = x)
(target' : to_fun 1 = y) | structure | path | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [] | Continuous path connecting two points `x` and `y` in a topological space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
path.ext : ∀ {γ₁ γ₂ : path x y}, (γ₁ : I → X) = γ₂ → γ₁ = γ₂ | | ⟨⟨x, h11⟩, h12, h13⟩ ⟨⟨.(x), h21⟩, h22, h23⟩ rfl := rfl | lemma | path.ext | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : I → X) (h₁ h₂ h₃) : ⇑(mk ⟨f, h₁⟩ h₂ h₃ : path x y) = f | rfl | lemma | path.coe_mk | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous : continuous γ | γ.continuous_to_fun | lemma | path.continuous | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
source : γ 0 = x | γ.source' | lemma | path.source | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
target : γ 1 = y | γ.target' | lemma | path.target | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simps.apply : I → X | γ
initialize_simps_projections path (to_continuous_map_to_fun → simps.apply, -to_continuous_map) | def | path.simps.apply | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path"
] | 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 |
coe_to_continuous_map : ⇑γ.to_continuous_map = γ | rfl | lemma | path.coe_to_continuous_map | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_uncurry_path {X α : Type*} [topological_space X] {x y : α → X} :
has_uncurry (Π (a : α), path (x a) (y a)) (α × I) X | ⟨λ φ p, φ p.1 p.2⟩ | instance | path.has_uncurry_path | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"has_uncurry",
"path",
"topological_space"
] | Any function `φ : Π (a : α), path (x a) (y a)` can be seen as a function `α × I → X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl (x : X) : path x x | { to_fun := λ t, x,
continuous_to_fun := continuous_const,
source' := rfl,
target' := rfl } | def | path.refl | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous_const",
"path"
] | The constant path from a point to itself | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_range {a : X} : range (path.refl a) = {a} | by simp [path.refl, has_coe_to_fun.coe, coe_fn] | lemma | path.refl_range | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (γ : path x y) : path y x | { to_fun := γ ∘ σ,
continuous_to_fun := by continuity,
source' := by simpa [-path.target] using γ.target,
target' := by simpa [-path.source] using γ.source } | def | path.symm | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuity",
"path",
"path.source",
"path.target"
] | The reverse of a path from `x` to `y`, as a path from `y` to `x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_symm {γ : path x y} : γ.symm.symm = γ | by { ext, simp } | lemma | path.symm_symm | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_symm {a : X} : (path.refl a).symm = path.refl a | by { ext, refl } | lemma | path.refl_symm | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_range {a b : X} (γ : path a b) : range γ.symm = range γ | begin
ext x,
simp only [mem_range, path.symm, has_coe_to_fun.coe, coe_fn, unit_interval.symm, set_coe.exists,
comp_app, subtype.coe_mk, subtype.val_eq_coe],
split; rintros ⟨y, hy, hxy⟩; refine ⟨1-y, mem_iff_one_sub_mem.mp hy, _⟩; convert hxy,
simp
end | lemma | path.symm_range | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"path.symm",
"set_coe.exists",
"subtype.coe_mk",
"subtype.val_eq_coe",
"unit_interval.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_eval : continuous (λ p : path x y × I, p.1 p.2) | continuous_eval'.comp $ continuous_induced_dom.prod_map continuous_id | lemma | path.continuous_eval | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"continuous_id",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous.path_eval {Y} [topological_space Y]
{f : Y → path x y} {g : Y → I} (hf : continuous f) (hg : continuous g) :
continuous (λ y, f y (g y)) | continuous.comp continuous_eval (hf.prod_mk hg) | lemma | continuous.path_eval | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"continuous.comp",
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_uncurry_iff {Y} [topological_space Y] {g : Y → path x y} :
continuous ↿g ↔ continuous g | iff.symm $ continuous_induced_rng.trans
⟨λ h, continuous_uncurry_of_continuous ⟨_, h⟩, continuous_of_continuous_uncurry ↑g⟩ | lemma | path.continuous_uncurry_iff | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend : ℝ → X | Icc_extend zero_le_one γ | def | path.extend | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"extend",
"zero_le_one"
] | A continuous map extending a path to `ℝ`, constant before `0` and after `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.continuous.path_extend {γ : Y → path x y} {f : Y → ℝ} (hγ : continuous ↿γ)
(hf : continuous f) : continuous (λ t, (γ t).extend (f t)) | continuous.Icc_extend hγ hf | lemma | continuous.path_extend | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"continuous.Icc_extend",
"extend",
"path"
] | See Note [continuity lemma statement]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_extend : continuous γ.extend | γ.continuous.Icc_extend' | lemma | path.continuous_extend | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous"
] | A useful special case of `continuous.path_extend`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.filter.tendsto.path_extend {X Y : Type*} [topological_space X] [topological_space Y]
{l r : Y → X} {y : Y} {l₁ : filter ℝ} {l₂ : filter X} {γ : ∀ y, path (l y) (r y)}
(hγ : tendsto ↿γ (𝓝 y ×ᶠ l₁.map (proj_Icc 0 1 zero_le_one)) l₂) :
tendsto ↿(λ x, (γ x).extend) (𝓝 y ×ᶠ l₁) l₂ | filter.tendsto.Icc_extend _ hγ | lemma | filter.tendsto.path_extend | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"extend",
"filter",
"filter.tendsto.Icc_extend",
"path",
"topological_space",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous_at.path_extend {g : Y → ℝ} {l r : Y → X} (γ : ∀ y, path (l y) (r y)) {y : Y}
(hγ : continuous_at ↿γ (y, proj_Icc 0 1 zero_le_one (g y)))
(hg : continuous_at g y) : continuous_at (λ i, (γ i).extend (g i)) y | hγ.Icc_extend (λ x, γ x) hg | lemma | continuous_at.path_extend | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous_at",
"extend",
"path",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_extends {X : Type*} [topological_space X] {a b : X}
(γ : path a b) {t : ℝ} (ht : t ∈ (Icc 0 1 : set ℝ)) : γ.extend t = γ ⟨t, ht⟩ | Icc_extend_of_mem _ γ ht | lemma | path.extend_extends | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_zero : γ.extend 0 = x | by simp | lemma | path.extend_zero | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_one : γ.extend 1 = y | by simp | lemma | path.extend_one | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_extends' {X : Type*} [topological_space X] {a b : X}
(γ : path a b) (t : (Icc 0 1 : set ℝ)) : γ.extend t = γ t | Icc_extend_coe _ γ t | lemma | path.extend_extends' | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_range {X : Type*} [topological_space X] {a b : X}
(γ : path a b) : range γ.extend = range γ | Icc_extend_range _ γ | lemma | path.extend_range | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_of_le_zero {X : Type*} [topological_space X] {a b : X}
(γ : path a b) {t : ℝ} (ht : t ≤ 0) : γ.extend t = a | (Icc_extend_of_le_left _ _ ht).trans γ.source | lemma | path.extend_of_le_zero | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_of_one_le {X : Type*} [topological_space X] {a b : X}
(γ : path a b) {t : ℝ} (ht : 1 ≤ t) : γ.extend t = b | (Icc_extend_of_right_le _ _ ht).trans γ.target | lemma | path.extend_of_one_le | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_extend {X : Type*} [topological_space X] {a : X} :
(path.refl a).extend = λ _, a | rfl | lemma | path.refl_extend | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"extend",
"path.refl",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_line {f : ℝ → X} (hf : continuous_on f I) (h₀ : f 0 = x) (h₁ : f 1 = y) : path x y | { to_fun := f ∘ coe,
continuous_to_fun := hf.comp_continuous continuous_subtype_coe subtype.prop,
source' := h₀,
target' := h₁ } | def | path.of_line | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous_on",
"continuous_subtype_coe",
"path",
"subtype.prop"
] | The path obtained from a map defined on `ℝ` by restriction to the unit interval. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_line_mem {f : ℝ → X} (hf : continuous_on f I) (h₀ : f 0 = x) (h₁ : f 1 = y) :
∀ t, of_line hf h₀ h₁ t ∈ f '' I | λ ⟨t, t_in⟩, ⟨t, t_in, rfl⟩ | lemma | path.of_line_mem | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (γ : path x y) (γ' : path y z) : path x z | { to_fun := (λ t : ℝ, if t ≤ 1/2 then γ.extend (2*t) else γ'.extend (2*t-1)) ∘ coe,
continuous_to_fun :=
begin
refine (continuous.if_le _ _ continuous_id continuous_const (by norm_num)).comp
continuous_subtype_coe,
-- TODO: the following are provable by `continuity` but it is too slow
exacts [γ.co... | def | path.trans | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous.if_le",
"continuous_const",
"continuous_id",
"continuous_subtype_coe",
"path"
] | Concatenation of two paths from `x` to `y` and from `y` to `z`, putting the first
path on `[0, 1/2]` and the second one on `[1/2, 1]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_apply (γ : path x y) (γ' : path y z) (t : I) : (γ.trans γ') t =
if h : (t : ℝ) ≤ 1/2 then
γ ⟨2 * t, (mul_pos_mem_iff zero_lt_two).2 ⟨t.2.1, h⟩⟩
else
γ' ⟨2 * t - 1, two_mul_sub_one_mem_iff.2 ⟨(not_le.1 h).le, t.2.2⟩⟩ | show ite _ _ _ = _,
by split_ifs; rw extend_extends | lemma | path.trans_apply | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_symm (γ : path x y) (γ' : path y z) :
(γ.trans γ').symm = γ'.symm.trans γ.symm | begin
ext t,
simp only [trans_apply, ← one_div, symm_apply, not_le, comp_app],
split_ifs with h h₁ h₂ h₃ h₄; rw [coe_symm_eq] at h,
{ have ht : (t : ℝ) = 1/2,
{ linarith [unit_interval.nonneg t, unit_interval.le_one t] },
norm_num [ht] },
{ refine congr_arg _ (subtype.ext _),
norm_num [sub_sub_eq_... | lemma | path.trans_symm | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"one_div",
"path",
"subtype.ext",
"unit_interval.le_one",
"unit_interval.nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_trans_refl {X : Type*} [topological_space X] {a : X} :
(path.refl a).trans (path.refl a) = path.refl a | begin
ext,
simp only [path.trans, if_t_t, one_div, path.refl_extend],
refl
end | lemma | path.refl_trans_refl | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"one_div",
"path.refl",
"path.refl_extend",
"path.trans",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_range {X : Type*} [topological_space X] {a b c : X}
(γ₁ : path a b) (γ₂ : path b c) : range (γ₁.trans γ₂) = range γ₁ ∪ range γ₂ | begin
rw path.trans,
apply eq_of_subset_of_subset,
{ rintros x ⟨⟨t, ht0, ht1⟩, hxt⟩,
by_cases h : t ≤ 1/2,
{ left,
use [2*t, ⟨by linarith, by linarith⟩],
rw ← γ₁.extend_extends,
unfold_coes at hxt,
simp only [h, comp_app, if_true] at hxt,
exact hxt },
{ right,
use [... | lemma | path.trans_range | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"mul_one_div_cancel",
"path",
"path.trans",
"topological_space",
"two_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (γ : path x y) {Y : Type*} [topological_space Y]
{f : X → Y} (h : continuous f) : path (f x) (f y) | { to_fun := f ∘ γ,
continuous_to_fun := by continuity,
source' := by simp,
target' := by simp } | def | path.map | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuity",
"continuous",
"path",
"topological_space"
] | Image of a path from `x` to `y` by a continuous map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_coe (γ : path x y) {Y : Type*} [topological_space Y]
{f : X → Y} (h : continuous f) :
(γ.map h : I → Y) = f ∘ γ | by { ext t, refl } | lemma | path.map_coe | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_symm (γ : path x y) {Y : Type*} [topological_space Y]
{f : X → Y} (h : continuous f) :
(γ.map h).symm = γ.symm.map h | rfl | lemma | path.map_symm | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_trans (γ : path x y) (γ' : path y z) {Y : Type*} [topological_space Y]
{f : X → Y} (h : continuous f) : (γ.trans γ').map h = (γ.map h).trans (γ'.map h) | by { ext t, rw [trans_apply, map_coe, comp_app, trans_apply], split_ifs; refl } | lemma | path.map_trans | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id (γ : path x y) : γ.map continuous_id = γ | by { ext, refl } | lemma | path.map_id | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous_id",
"map_id",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map (γ : path x y) {Y : Type*} [topological_space Y] {Z : Type*}
[topological_space Z] {f : X → Y} (hf : continuous f) {g : Y → Z} (hg : continuous g) :
(γ.map hf).map hg = γ.map (hg.comp hf) | by { ext, refl } | lemma | path.map_map | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast (γ : path x y) {x' y'} (hx : x' = x) (hy : y' = y) : path x' y' | { to_fun := γ,
continuous_to_fun := γ.continuous,
source' := by simp [hx],
target' := by simp [hy] } | def | path.cast | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path"
] | Casting a path from `x` to `y` to a path from `x'` to `y'` when `x' = x` and `y' = y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_cast {X : Type*} [topological_space X] {a₁ a₂ b₁ b₂ : X}
(γ : path a₂ b₂) (ha : a₁ = a₂) (hb : b₁ = b₂) :
(γ.cast ha hb).symm = (γ.symm).cast hb ha | rfl | lemma | path.symm_cast | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_cast {X : Type*} [topological_space X] {a₁ a₂ b₁ b₂ c₁ c₂ : X}
(γ : path a₂ b₂) (γ' : path b₂ c₂) (ha : a₁ = a₂) (hb : b₁ = b₂) (hc : c₁ = c₂) :
(γ.cast ha hb).trans (γ'.cast hb hc) = (γ.trans γ').cast ha hc | rfl | lemma | path.trans_cast | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_coe (γ : path x y) {x' y'} (hx : x' = x) (hy : y' = y) :
(γ.cast hx hy : I → X) = γ | rfl | lemma | path.cast_coe | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_continuous_family {X ι : Type*} [topological_space X] [topological_space ι]
{a b : ι → X} (γ : Π (t : ι), path (a t) (b t)) (h : continuous ↿γ) :
continuous ↿(λ t, (γ t).symm) | h.comp (continuous_id.prod_map continuous_symm) | lemma | path.symm_continuous_family | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_symm : continuous (symm : path x y → path y x) | continuous_uncurry_iff.mp $ symm_continuous_family _ (continuous_fst.path_eval continuous_snd) | lemma | path.continuous_symm | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"continuous_snd",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_uncurry_extend_of_continuous_family {X ι : Type*} [topological_space X]
[topological_space ι] {a b : ι → X} (γ : Π (t : ι), path (a t) (b t)) (h : continuous ↿γ) :
continuous ↿(λ t, (γ t).extend) | h.comp (continuous_id.prod_map continuous_proj_Icc) | lemma | path.continuous_uncurry_extend_of_continuous_family | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"continuous_proj_Icc",
"extend",
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_continuous_family {X ι : Type*} [topological_space X] [topological_space ι]
{a b c : ι → X}
(γ₁ : Π (t : ι), path (a t) (b t)) (h₁ : continuous ↿γ₁)
(γ₂ : Π (t : ι), path (b t) (c t)) (h₂ : continuous ↿γ₂) :
continuous ↿(λ t, (γ₁ t).trans (γ₂ t)) | begin
have h₁' := path.continuous_uncurry_extend_of_continuous_family γ₁ h₁,
have h₂' := path.continuous_uncurry_extend_of_continuous_family γ₂ h₂,
simp only [has_uncurry.uncurry, has_coe_to_fun.coe, coe_fn, path.trans, (∘)],
refine continuous.if_le _ _ (continuous_subtype_coe.comp continuous_snd) continuous_co... | lemma | path.trans_continuous_family | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"continuous.if_le",
"continuous_const",
"continuous_snd",
"continuous_subtype_coe",
"extend",
"mul_inv_cancel",
"path",
"path.continuous_uncurry_extend_of_continuous_family",
"path.trans",
"topological_space",
"two_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous.path_trans {f : Y → path x y} {g : Y → path y z} : continuous f →
continuous g → continuous (λ t, (f t).trans (g t)) | begin
intros hf hg,
apply continuous_uncurry_iff.mp,
exact trans_continuous_family _ (continuous_uncurry_iff.mpr hf)
_ (continuous_uncurry_iff.mpr hg),
end | lemma | continuous.path_trans | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_trans {x y z : X} : continuous (λ ρ : path x y × path y z, ρ.1.trans ρ.2) | continuous_fst.path_trans continuous_snd | lemma | path.continuous_trans | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"continuous_snd",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod (γ₁ : path a₁ a₂) (γ₂ : path b₁ b₂) :
path (a₁, b₁) (a₂, b₂) | { to_continuous_map := continuous_map.prod_mk γ₁.to_continuous_map γ₂.to_continuous_map,
source' := by simp,
target' := by simp, } | def | path.prod | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous_map.prod_mk",
"path"
] | Given a path in `X` and a path in `Y`, we can take their pointwise product to get a path in
`X × Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_coe_fn (γ₁ : path a₁ a₂) (γ₂ : path b₁ b₂) :
(coe_fn (γ₁.prod γ₂)) = λ t, (γ₁ t, γ₂ t) | rfl | lemma | path.prod_coe_fn | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_prod_eq_prod_trans
(γ₁ : path a₁ a₂) (δ₁ : path a₂ a₃) (γ₂ : path b₁ b₂) (δ₂ : path b₂ b₃) :
(γ₁.prod γ₂).trans (δ₁.prod δ₂) = (γ₁.trans δ₁).prod (γ₂.trans δ₂) | begin
ext t;
unfold path.trans;
simp only [path.coe_mk, path.prod_coe_fn, function.comp_app];
split_ifs; refl,
end | lemma | path.trans_prod_eq_prod_trans | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"path.coe_mk",
"path.prod_coe_fn",
"path.trans"
] | Path composition commutes with products | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi (γ : Π i, path (as i) (bs i)) : path as bs | { to_continuous_map := continuous_map.pi (λ i, (γ i).to_continuous_map),
source' := by simp,
target' := by simp, } | def | path.pi | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous_map.pi",
"path"
] | Given a family of paths, one in each Xᵢ, we take their pointwise product to get a path in
Π i, Xᵢ. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_coe_fn (γ : Π i, path (as i) (bs i)) : (coe_fn (path.pi γ)) = λ t i, γ i t | rfl | lemma | path.pi_coe_fn | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"path.pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_pi_eq_pi_trans (γ₀ : Π i, path (as i) (bs i)) (γ₁ : Π i, path (bs i) (cs i)) :
(path.pi γ₀).trans (path.pi γ₁) = path.pi (λ i, (γ₀ i).trans (γ₁ i)) | begin
ext t i,
unfold path.trans,
simp only [path.coe_mk, function.comp_app, pi_coe_fn],
split_ifs; refl,
end | lemma | path.trans_pi_eq_pi_trans | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"path.coe_mk",
"path.pi",
"path.trans"
] | Path composition commutes with products | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul [has_mul X] [has_continuous_mul X] {a₁ b₁ a₂ b₂ : X}
(γ₁ : path a₁ b₁) (γ₂ : path a₂ b₂) : path (a₁ * a₂) (b₁ * b₂) | (γ₁.prod γ₂).map continuous_mul | def | path.mul | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous_mul",
"has_continuous_mul",
"path"
] | Pointwise multiplication of paths in a topological group. The additive version is probably more
useful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_apply [has_mul X] [has_continuous_mul X] {a₁ b₁ a₂ b₂ : X}
(γ₁ : path a₁ b₁) (γ₂ : path a₂ b₂) (t : unit_interval) : (γ₁.mul γ₂) t = γ₁ t * γ₂ t | rfl | lemma | path.mul_apply | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"has_continuous_mul",
"path",
"unit_interval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate {X : Type*} [topological_space X] {a b : X}
(γ : path a b) (t₀ t₁ : ℝ) : path (γ.extend $ min t₀ t₁) (γ.extend t₁) | { to_fun := λ s, γ.extend (min (max s t₀) t₁),
continuous_to_fun := γ.continuous_extend.comp
((continuous_subtype_coe.max continuous_const).min continuous_const),
source' :=
begin
simp only [min_def, max_def'],
norm_cast,
split_ifs with h₁ h₂ h₃ h₄,
{ simp [γ.extend_of_le_zero h₁] },
{ con... | def | path.truncate | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous_const",
"max_def'",
"path",
"topological_space"
] | `γ.truncate t₀ t₁` is the path which follows the path `γ` on the
time interval `[t₀, t₁]` and stays still otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncate_of_le {X : Type*} [topological_space X] {a b : X}
(γ : path a b) {t₀ t₁ : ℝ} (h : t₀ ≤ t₁) : path (γ.extend t₀) (γ.extend t₁) | (γ.truncate t₀ t₁).cast (by rw min_eq_left h) rfl | def | path.truncate_of_le | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"topological_space"
] | `γ.truncate_of_le t₀ t₁ h`, where `h : t₀ ≤ t₁` is `γ.truncate t₀ t₁`
casted as a path from `γ.extend t₀` to `γ.extend t₁`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncate_range {X : Type*} [topological_space X] {a b : X}
(γ : path a b) {t₀ t₁ : ℝ} : range (γ.truncate t₀ t₁) ⊆ range γ | begin
rw ← γ.extend_range,
simp only [range_subset_iff, set_coe.exists, set_coe.forall],
intros x hx,
simp only [has_coe_to_fun.coe, coe_fn, path.truncate, mem_range_self]
end | lemma | path.truncate_range | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"path.truncate",
"set_coe.exists",
"set_coe.forall",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_continuous_family {X : Type*} [topological_space X] {a b : X}
(γ : path a b) : continuous (λ x, γ.truncate x.1 x.2.1 x.2.2 : ℝ × ℝ × I → X) | γ.continuous_extend.comp
(((continuous_subtype_coe.comp (continuous_snd.comp continuous_snd)).max continuous_fst).min
(continuous_fst.comp continuous_snd)) | lemma | path.truncate_continuous_family | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"continuous_fst",
"continuous_snd",
"path",
"topological_space"
] | For a path `γ`, `γ.truncate` gives a "continuous family of paths", by which we
mean the uncurried function which maps `(t₀, t₁, s)` to `γ.truncate t₀ t₁ s` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
truncate_const_continuous_family {X : Type*} [topological_space X] {a b : X}
(γ : path a b) (t : ℝ) : continuous ↿(γ.truncate t) | have key : continuous (λ x, (t, x) : ℝ × I → ℝ × ℝ × I) := continuous_const.prod_mk continuous_id,
by convert γ.truncate_continuous_family.comp key | lemma | path.truncate_const_continuous_family | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"continuous_id",
"path",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_self {X : Type*} [topological_space X] {a b : X}
(γ : path a b) (t : ℝ) : γ.truncate t t = (path.refl $ γ.extend t).cast (by rw min_self) rfl | begin
ext x,
rw cast_coe,
simp only [truncate, has_coe_to_fun.coe, coe_fn, refl, min_def, max_def],
split_ifs with h₁ h₂; congr,
end | lemma | path.truncate_self | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"path.refl",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_zero_zero {X : Type*} [topological_space X] {a b : X} (γ : path a b) :
γ.truncate 0 0 = (path.refl a).cast (by rw [min_self, γ.extend_zero]) γ.extend_zero | by convert γ.truncate_self 0; exact γ.extend_zero.symm | lemma | path.truncate_zero_zero | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"path.refl",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_one_one {X : Type*} [topological_space X] {a b : X} (γ : path a b) :
γ.truncate 1 1 = (path.refl b).cast (by rw [min_self, γ.extend_one]) γ.extend_one | by convert γ.truncate_self 1; exact γ.extend_one.symm | lemma | path.truncate_one_one | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"path.refl",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
truncate_zero_one {X : Type*} [topological_space X] {a b : X}
(γ : path a b) : γ.truncate 0 1 = γ.cast (by simp [zero_le_one, extend_zero]) (by simp) | begin
ext x,
rw cast_coe,
have : ↑x ∈ (Icc 0 1 : set ℝ) := x.2,
rw [truncate, coe_mk, max_eq_left this.1, min_eq_left this.2, extend_extends']
end | lemma | path.truncate_zero_one | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path",
"topological_space",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reparam (γ : path x y) (f : I → I) (hfcont : continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
path x y | { to_fun := γ ∘ f,
continuous_to_fun := by continuity,
source' := by simp [hf₀],
target' := by simp [hf₁] } | def | path.reparam | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuity",
"continuous",
"path"
] | Given a path `γ` and a function `f : I → I` where `f 0 = 0` and `f 1 = 1`, `γ.reparam f` is the
path defined by `γ ∘ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_fun (γ : path x y) {f : I → I} (hfcont : continuous f) (hf₀ : f 0 = 0)
(hf₁ : f 1 = 1) : ⇑(γ.reparam f hfcont hf₀ hf₁) = γ ∘ f | rfl | lemma | path.coe_to_fun | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reparam_id (γ : path x y) : γ.reparam id continuous_id rfl rfl = γ | by { ext, refl } | lemma | path.reparam_id | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous_id",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_reparam (γ : path x y) {f : I → I} (hfcont : continuous f) (hf₀ : f 0 = 0)
(hf₁ : f 1 = 1) : range ⇑(γ.reparam f hfcont hf₀ hf₁) = range γ | begin
change range (γ ∘ f) = range γ,
have : range f = univ,
{ rw range_iff_surjective,
intro t,
have h₁ : continuous (Icc_extend (zero_le_one' ℝ) f),
{ continuity },
have := intermediate_value_Icc (zero_le_one' ℝ) h₁.continuous_on,
{ rw [Icc_extend_left, Icc_extend_right] at this,
chang... | lemma | path.range_reparam | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuity",
"continuous",
"intermediate_value_Icc",
"path",
"zero_le_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_reparam {f : I → I} (hfcont : continuous f) (hf₀ : f 0 = 0)
(hf₁ : f 1 = 1) : (refl x).reparam f hfcont hf₀ hf₁ = refl x | begin
ext,
simp,
end | lemma | path.refl_reparam | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
joined (x y : X) : Prop | nonempty (path x y) | def | joined | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path"
] | The relation "being joined by a path". This is an equivalence relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
joined.refl (x : X) : joined x x | ⟨path.refl x⟩ | lemma | joined.refl | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
joined.some_path (h : joined x y) : path x y | nonempty.some h | def | joined.some_path | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined",
"nonempty.some",
"path"
] | When two points are joined, choose some path from `x` to `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
joined.symm {x y : X} (h : joined x y) : joined y x | ⟨h.some_path.symm⟩ | lemma | joined.symm | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
joined.trans {x y z : X} (hxy : joined x y) (hyz : joined y z) :
joined x z | ⟨hxy.some_path.trans hyz.some_path⟩ | lemma | joined.trans | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
path_setoid : setoid X | { r := joined,
iseqv := mk_equivalence _ joined.refl (λ x y, joined.symm) (λ x y z, joined.trans) } | def | path_setoid | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"joined",
"joined.refl",
"joined.symm",
"joined.trans"
] | The setoid corresponding the equivalence relation of being joined by a continuous path. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zeroth_homotopy | quotient (path_setoid X) | def | zeroth_homotopy | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path_setoid"
] | The quotient type of points of a topological space modulo being joined by a continuous path. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
joined_in (F : set X) (x y : X) : Prop | ∃ γ : path x y, ∀ t, γ t ∈ F | def | joined_in | topology | src/topology/path_connected.lean | [
"topology.algebra.order.proj_Icc",
"topology.compact_open",
"topology.continuous_function.basic",
"topology.unit_interval"
] | [
"path"
] | The relation "being joined by a path in `F`". Not quite an equivalence relation since it's not
reflexive for points that do not belong to `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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