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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efixed_point_is_fixed_pt (hf : contracting_with K f) {x : α} (hx : edist x (f x) ≠ ∞) :
is_fixed_pt f (efixed_point f hf x hx) | (classical.some_spec $ hf.exists_fixed_point x hx).1 | lemma | contracting_with.efixed_point_is_fixed_pt | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_iterate_efixed_point (hf : contracting_with K f) {x : α} (hx : edist x (f x) ≠ ∞) :
tendsto (λn, f^[n] x) at_top (𝓝 $ efixed_point f hf x hx) | (classical.some_spec $ hf.exists_fixed_point x hx).2.1 | lemma | contracting_with.tendsto_iterate_efixed_point | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apriori_edist_iterate_efixed_point_le (hf : contracting_with K f)
{x : α} (hx : edist x (f x) ≠ ∞) (n : ℕ) :
edist (f^[n] x) (efixed_point f hf x hx) ≤ (edist x (f x)) * K^n / (1 - K) | (classical.some_spec $ hf.exists_fixed_point x hx).2.2 n | lemma | contracting_with.apriori_edist_iterate_efixed_point_le | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_efixed_point_le (hf : contracting_with K f) {x : α} (hx : edist x (f x) ≠ ∞) :
edist x (efixed_point f hf x hx) ≤ (edist x (f x)) / (1 - K) | by { convert hf.apriori_edist_iterate_efixed_point_le hx 0, simp only [pow_zero, mul_one] } | lemma | contracting_with.edist_efixed_point_le | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with",
"mul_one",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_efixed_point_lt_top (hf : contracting_with K f) {x : α} (hx : edist x (f x) ≠ ∞) :
edist x (efixed_point f hf x hx) < ∞ | (hf.edist_efixed_point_le hx).trans_lt (ennreal.mul_lt_top hx $
ennreal.inv_ne_top.2 hf.one_sub_K_ne_zero) | lemma | contracting_with.edist_efixed_point_lt_top | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with",
"ennreal.mul_lt_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
efixed_point_eq_of_edist_lt_top (hf : contracting_with K f) {x : α} (hx : edist x (f x) ≠ ∞)
{y : α} (hy : edist y (f y) ≠ ∞) (h : edist x y ≠ ∞) :
efixed_point f hf x hx = efixed_point f hf y hy | begin
refine (hf.eq_or_edist_eq_top_of_fixed_points _ _).elim id (λ h', false.elim (ne_of_lt _ h'));
try { apply efixed_point_is_fixed_pt },
change edist_lt_top_setoid.rel _ _,
transitivity x, by { symmetry, exact hf.edist_efixed_point_lt_top hx },
transitivity y,
exacts [lt_top_iff_ne_top.2 h, hf.edist_e... | lemma | contracting_with.efixed_point_eq_of_edist_lt_top | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_fixed_point' {s : set α} (hsc : is_complete s) (hsf : maps_to f s s)
(hf : contracting_with K $ hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞) :
∃ y ∈ s, is_fixed_pt f y ∧ tendsto (λ n, f^[n] x) at_top (𝓝 y) ∧
∀ n:ℕ, edist (f^[n] x) y ≤ (edist x (f x)) * K^n / (1 - K) | begin
haveI := hsc.complete_space_coe,
rcases hf.exists_fixed_point ⟨x, hxs⟩ hx with ⟨y, hfy, h_tendsto, hle⟩,
refine ⟨y, y.2, subtype.ext_iff_val.1 hfy, _, λ n, _⟩,
{ convert (continuous_subtype_coe.tendsto _).comp h_tendsto, ext n,
simp only [(∘), maps_to.iterate_restrict, maps_to.coe_restrict_apply, subt... | theorem | contracting_with.exists_fixed_point' | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with",
"is_complete",
"subtype.coe_mk"
] | Banach fixed-point theorem for maps contracting on a complete subset. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
efixed_point' {s : set α} (hsc : is_complete s) (hsf : maps_to f s s)
(hf : contracting_with K $ hsf.restrict f s s) (x : α) (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞) :
α | classical.some $ hf.exists_fixed_point' hsc hsf hxs hx | def | contracting_with.efixed_point' | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with",
"is_complete"
] | Let `s` be a complete forward-invariant set of a self-map `f`. If `f` contracts on `s`
and `x ∈ s` satisfies `edist x (f x) ≠ ∞`, then `efixed_point'` is the unique fixed point
of the restriction of `f` to `s ∩ emetric.ball x ∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
efixed_point_mem' {s : set α} (hsc : is_complete s) (hsf : maps_to f s s)
(hf : contracting_with K $ hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞) :
efixed_point' f hsc hsf hf x hxs hx ∈ s | (classical.some_spec $ hf.exists_fixed_point' hsc hsf hxs hx).fst | lemma | contracting_with.efixed_point_mem' | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with",
"is_complete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
efixed_point_is_fixed_pt' {s : set α} (hsc : is_complete s) (hsf : maps_to f s s)
(hf : contracting_with K $ hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞) :
is_fixed_pt f (efixed_point' f hsc hsf hf x hxs hx) | (classical.some_spec $ hf.exists_fixed_point' hsc hsf hxs hx).snd.1 | lemma | contracting_with.efixed_point_is_fixed_pt' | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with",
"is_complete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_iterate_efixed_point' {s : set α} (hsc : is_complete s) (hsf : maps_to f s s)
(hf : contracting_with K $ hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞) :
tendsto (λn, f^[n] x) at_top (𝓝 $ efixed_point' f hsc hsf hf x hxs hx) | (classical.some_spec $ hf.exists_fixed_point' hsc hsf hxs hx).snd.2.1 | lemma | contracting_with.tendsto_iterate_efixed_point' | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with",
"is_complete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apriori_edist_iterate_efixed_point_le' {s : set α} (hsc : is_complete s)
(hsf : maps_to f s s) (hf : contracting_with K $ hsf.restrict f s s) {x : α} (hxs : x ∈ s)
(hx : edist x (f x) ≠ ∞) (n : ℕ) :
edist (f^[n] x) (efixed_point' f hsc hsf hf x hxs hx) ≤ (edist x (f x)) * K^n / (1 - K) | (classical.some_spec $ hf.exists_fixed_point' hsc hsf hxs hx).snd.2.2 n | lemma | contracting_with.apriori_edist_iterate_efixed_point_le' | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with",
"is_complete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_efixed_point_le' {s : set α} (hsc : is_complete s) (hsf : maps_to f s s)
(hf : contracting_with K $ hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞) :
edist x (efixed_point' f hsc hsf hf x hxs hx) ≤ (edist x (f x)) / (1 - K) | by { convert hf.apriori_edist_iterate_efixed_point_le' hsc hsf hxs hx 0,
rw [pow_zero, mul_one] } | lemma | contracting_with.edist_efixed_point_le' | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with",
"is_complete",
"mul_one",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_efixed_point_lt_top' {s : set α} (hsc : is_complete s) (hsf : maps_to f s s)
(hf : contracting_with K $ hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞) :
edist x (efixed_point' f hsc hsf hf x hxs hx) < ∞ | (hf.edist_efixed_point_le' hsc hsf hxs hx).trans_lt (ennreal.mul_lt_top hx $
ennreal.inv_ne_top.2 hf.one_sub_K_ne_zero) | lemma | contracting_with.edist_efixed_point_lt_top' | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with",
"ennreal.mul_lt_top",
"is_complete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
efixed_point_eq_of_edist_lt_top' (hf : contracting_with K f)
{s : set α} (hsc : is_complete s) (hsf : maps_to f s s)
(hfs : contracting_with K $ hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞)
{t : set α} (htc : is_complete t) (htf : maps_to f t t)
(hft : contracting_with K $ htf.restrict f t... | begin
refine (hf.eq_or_edist_eq_top_of_fixed_points _ _).elim id (λ h', false.elim (ne_of_lt _ h'));
try { apply efixed_point_is_fixed_pt' },
change edist_lt_top_setoid.rel _ _,
transitivity x, by { symmetry, apply edist_efixed_point_lt_top' },
transitivity y,
exact lt_top_iff_ne_top.2 hxy,
apply edist_... | lemma | contracting_with.efixed_point_eq_of_edist_lt_top' | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with",
"is_complete"
] | If a globally contracting map `f` has two complete forward-invariant sets `s`, `t`,
and `x ∈ s` is at a finite distance from `y ∈ t`, then the `efixed_point'` constructed by `x`
is the same as the `efixed_point'` constructed by `y`.
This lemma takes additional arguments stating that `f` contracts on `s` and `t` becaus... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_sub_K_pos (hf : contracting_with K f) : (0:ℝ) < 1 - K | sub_pos.2 hf.1 | lemma | contracting_with.one_sub_K_pos | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_mul (x y : α) : dist (f x) (f y) ≤ K * dist x y | hf.to_lipschitz_with.dist_le_mul x y | lemma | contracting_with.dist_le_mul | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_inequality (x y) : dist x y ≤ (dist x (f x) + dist y (f y)) / (1 - K) | suffices dist x y ≤ dist x (f x) + dist y (f y) + K * dist x y,
by rwa [le_div_iff hf.one_sub_K_pos, mul_comm, sub_mul, one_mul, sub_le_iff_le_add],
calc dist x y ≤ dist x (f x) + dist y (f y) + dist (f x) (f y) : dist_triangle4_right _ _ _ _
... ≤ dist x (f x) + dist y (f y) + K * dist x y :
add_le_add_l... | lemma | contracting_with.dist_inequality | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"dist_triangle4_right",
"le_div_iff",
"mul_comm",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_le_of_fixed_point (x) {y} (hy : is_fixed_pt f y) :
dist x y ≤ (dist x (f x)) / (1 - K) | by simpa only [hy.eq, dist_self, add_zero] using hf.dist_inequality x y | lemma | contracting_with.dist_le_of_fixed_point | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"dist_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fixed_point_unique' {x y} (hx : is_fixed_pt f x) (hy : is_fixed_pt f y) : x = y | (hf.eq_or_edist_eq_top_of_fixed_points hx hy).resolve_right (edist_ne_top _ _) | theorem | contracting_with.fixed_point_unique' | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"edist_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_fixed_point_fixed_point_of_dist_le' (g : α → α)
{x y} (hx : is_fixed_pt f x) (hy : is_fixed_pt g y) {C} (hfg : ∀ z, dist (f z) (g z) ≤ C) :
dist x y ≤ C / (1 - K) | calc dist x y = dist y x : dist_comm x y
... ≤ (dist y (f y)) / (1 - K) : hf.dist_le_of_fixed_point y hx
... = (dist (f y) (g y)) / (1 - K) : by rw [hy.eq, dist_comm]
... ≤ C / (1 - K) : (div_le_div_right hf.one_sub_K_pos).2 (hfg y) | lemma | contracting_with.dist_fixed_point_fixed_point_of_dist_le' | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"dist_comm",
"div_le_div_right"
] | Let `f` be a contracting map with constant `K`; let `g` be another map uniformly
`C`-close to `f`. If `x` and `y` are their fixed points, then `dist x y ≤ C / (1 - K)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fixed_point : α | efixed_point f hf _ (edist_ne_top (classical.choice ‹nonempty α›) _) | def | contracting_with.fixed_point | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"edist_ne_top"
] | The unique fixed point of a contracting map in a nonempty complete metric space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fixed_point_is_fixed_pt : is_fixed_pt f (fixed_point f hf) | hf.efixed_point_is_fixed_pt _ | lemma | contracting_with.fixed_point_is_fixed_pt | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [] | The point provided by `contracting_with.fixed_point` is actually a fixed point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fixed_point_unique {x} (hx : is_fixed_pt f x) : x = fixed_point f hf | hf.fixed_point_unique' hx hf.fixed_point_is_fixed_pt | lemma | contracting_with.fixed_point_unique | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_fixed_point_le (x) : dist x (fixed_point f hf) ≤ (dist x (f x)) / (1 - K) | hf.dist_le_of_fixed_point x hf.fixed_point_is_fixed_pt | lemma | contracting_with.dist_fixed_point_le | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aposteriori_dist_iterate_fixed_point_le (x n) :
dist (f^[n] x) (fixed_point f hf) ≤ (dist (f^[n] x) (f^[n+1] x)) / (1 - K) | by { rw [iterate_succ'], apply hf.dist_fixed_point_le } | lemma | contracting_with.aposteriori_dist_iterate_fixed_point_le | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [] | Aposteriori estimates on the convergence of iterates to the fixed point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apriori_dist_iterate_fixed_point_le (x n) :
dist (f^[n] x) (fixed_point f hf) ≤ (dist x (f x)) * K^n / (1 - K) | le_trans (hf.aposteriori_dist_iterate_fixed_point_le x n) $
(div_le_div_right hf.one_sub_K_pos).2 $
hf.to_lipschitz_with.dist_iterate_succ_le_geometric x n | lemma | contracting_with.apriori_dist_iterate_fixed_point_le | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"div_le_div_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_iterate_fixed_point (x) :
tendsto (λn, f^[n] x) at_top (𝓝 $ fixed_point f hf) | begin
convert tendsto_iterate_efixed_point hf (edist_ne_top x _),
refine (fixed_point_unique _ _).symm,
apply efixed_point_is_fixed_pt
end | lemma | contracting_with.tendsto_iterate_fixed_point | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"edist_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fixed_point_lipschitz_in_map {g : α → α} (hg : contracting_with K g)
{C} (hfg : ∀ z, dist (f z) (g z) ≤ C) :
dist (fixed_point f hf) (fixed_point g hg) ≤ C / (1 - K) | hf.dist_fixed_point_fixed_point_of_dist_le' g hf.fixed_point_is_fixed_pt
hg.fixed_point_is_fixed_pt hfg | lemma | contracting_with.fixed_point_lipschitz_in_map | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_fixed_pt_fixed_point_iterate {n : ℕ} (hf : contracting_with K (f^[n])) :
is_fixed_pt f (hf.fixed_point (f^[n])) | begin
set x := hf.fixed_point (f^[n]),
have hx : (f^[n] x) = x := hf.fixed_point_is_fixed_pt,
have := hf.to_lipschitz_with.dist_le_mul x (f x),
rw [← iterate_succ_apply, iterate_succ_apply', hx] at this,
contrapose! this,
have := dist_pos.2 (ne.symm this),
simpa only [nnreal.coe_one, one_mul, nnreal.val_e... | lemma | contracting_with.is_fixed_pt_fixed_point_iterate | topology.metric_space | src/topology/metric_space/contracting.lean | [
"analysis.specific_limits.basic",
"data.setoid.basic",
"dynamics.fixed_points.topology"
] | [
"contracting_with",
"mul_lt_mul_right",
"nnreal.coe_one",
"nnreal.val_eq_coe",
"one_mul"
] | If a map `f` has a contracting iterate `f^[n]`, then the fixed point of `f^[n]` is also a fixed
point of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dilation | (to_fun : α → β)
(edist_eq' : ∃ r : ℝ≥0, r ≠ 0 ∧ ∀ x y : α, edist (to_fun x) (to_fun y) = r * edist x y) | structure | dilation | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [] | A dilation is a map that uniformly scales the edistance between any two points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dilation_class (F : Type*) (α β : out_param $ Type*)
[pseudo_emetric_space α] [pseudo_emetric_space β] extends fun_like F α (λ _, β) | (edist_eq' : ∀ (f : F), ∃ r : ℝ≥0, r ≠ 0 ∧ ∀ (x y : α), edist (f x) (f y) = r * edist x y) | class | dilation_class | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"fun_like",
"pseudo_emetric_space"
] | `dilation_class F α β r` states that `F` is a type of `r`-dilations.
You should extend this typeclass when you extend `dilation`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_dilation_class :
dilation_class (dilation α β) α β | { coe := to_fun,
coe_injective' := λ f g h, by { cases f; cases g; congr', },
edist_eq' := λ f, edist_eq' f } | instance | dilation.to_dilation_class | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"dilation_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe {f : dilation α β} : f.to_fun = (f : α → β) | rfl | lemma | dilation.to_fun_eq_coe | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : α → β) (h) : ⇑(⟨f, h⟩ : dilation α β) = f | rfl | lemma | dilation.coe_mk | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun {f g : dilation α β} (h : f = g) (x : α) : f x = g x | fun_like.congr_fun h x | lemma | dilation.congr_fun | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"fun_like.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_arg (f : dilation α β) {x y : α} (h : x = y) : f x = f y | fun_like.congr_arg f h | lemma | dilation.congr_arg | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"fun_like.congr_arg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : dilation α β} (h : ∀ x, f x = g x) : f = g | fun_like.ext f g h | theorem | dilation.ext | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : dilation α β} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | lemma | dilation.ext_iff | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (f : dilation α β) (h) : dilation.mk f h = f | ext $ λ _, rfl | lemma | dilation.mk_coe | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : dilation α β) (f' : α → β) (h : f' = ⇑f) : dilation α β | { to_fun := f',
edist_eq' := h.symm ▸ f.edist_eq' } | def | dilation.copy | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation"
] | Copy of a `dilation` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
copy_eq_self (f : dilation α β) {f' : α → β} (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | dilation.copy_eq_self | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ratio [dilation_class F α β] (f : F) : ℝ≥0 | if ∀ x y : α, edist x y = 0 ∨ edist x y = ⊤ then 1 else (dilation_class.edist_eq' f).some | def | dilation.ratio | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation_class"
] | The ratio of a dilation `f`. If the ratio is undefined (i.e., the distance between any two
points in `α` is either zero or infinity), then we choose one as the ratio. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ratio_ne_zero [dilation_class F α β] (f : F) : ratio f ≠ 0 | begin
rw ratio, split_ifs,
{ exact one_ne_zero, },
exact (dilation_class.edist_eq' f).some_spec.1,
end | lemma | dilation.ratio_ne_zero | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation_class",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ratio_pos [dilation_class F α β] (f : F) : 0 < ratio f | (ratio_ne_zero f).bot_lt | lemma | dilation.ratio_pos | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_eq [dilation_class F α β] (f : F) (x y : α) :
edist (f x) (f y) = ratio f * edist x y | begin
rw ratio, split_ifs with key,
{ rcases dilation_class.edist_eq' f with ⟨r, hne, hr⟩,
replace hr := hr x y,
cases key x y,
{ simp only [hr, h, mul_zero] },
{ simp [hr, h, hne] } },
exact (dilation_class.edist_eq' f).some_spec.2 x y,
end | lemma | dilation.edist_eq | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation_class",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_eq {α β F : Type*} [pseudo_metric_space α] [pseudo_metric_space β]
[dilation_class F α β] (f : F) (x y : α) : nndist (f x) (f y) = ratio f * nndist x y | by simp only [← ennreal.coe_eq_coe, ← edist_nndist, ennreal.coe_mul, edist_eq] | lemma | dilation.nndist_eq | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation_class",
"edist_nndist",
"ennreal.coe_eq_coe",
"ennreal.coe_mul",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_eq {α β F : Type*} [pseudo_metric_space α] [pseudo_metric_space β]
[dilation_class F α β] (f : F) (x y : α) : dist (f x) (f y) = ratio f * dist x y | by simp only [dist_nndist, nndist_eq, nnreal.coe_mul] | lemma | dilation.dist_eq | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation_class",
"dist_nndist",
"nnreal.coe_mul",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ratio_unique [dilation_class F α β] {f : F} {x y : α} {r : ℝ≥0}
(h₀ : edist x y ≠ 0) (htop : edist x y ≠ ⊤) (hr : edist (f x) (f y) = r * edist x y) :
r = ratio f | by simpa only [hr, ennreal.mul_eq_mul_right h₀ htop, ennreal.coe_eq_coe] using edist_eq f x y | lemma | dilation.ratio_unique | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation_class",
"ennreal.coe_eq_coe",
"ennreal.mul_eq_mul_right"
] | The `ratio` is equal to the distance ratio for any two points with nonzero finite distance.
`dist` and `nndist` versions below | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ratio_unique_of_nndist_ne_zero {α β F : Type*} [pseudo_metric_space α] [pseudo_metric_space β]
[dilation_class F α β] {f : F} {x y : α} {r : ℝ≥0} (hxy : nndist x y ≠ 0)
(hr : nndist (f x) (f y) = r * nndist x y) : r = ratio f | ratio_unique (by rwa [edist_nndist, ennreal.coe_ne_zero]) (edist_ne_top x y)
(by rw [edist_nndist, edist_nndist, hr, ennreal.coe_mul]) | lemma | dilation.ratio_unique_of_nndist_ne_zero | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation_class",
"edist_ne_top",
"edist_nndist",
"ennreal.coe_mul",
"ennreal.coe_ne_zero",
"pseudo_metric_space"
] | The `ratio` is equal to the distance ratio for any two points
with nonzero finite distance; `nndist` version | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ratio_unique_of_dist_ne_zero {α β} {F : Type*} [pseudo_metric_space α] [pseudo_metric_space β]
[dilation_class F α β] {f : F} {x y : α} {r : ℝ≥0}
(hxy : dist x y ≠ 0) (hr : dist (f x) (f y) = r * dist x y) :
r = ratio f | ratio_unique_of_nndist_ne_zero (nnreal.coe_ne_zero.1 hxy) $ nnreal.eq $
by rw [coe_nndist, hr, nnreal.coe_mul, coe_nndist] | lemma | dilation.ratio_unique_of_dist_ne_zero | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"coe_nndist",
"dilation_class",
"nnreal.coe_mul",
"nnreal.eq",
"pseudo_metric_space"
] | The `ratio` is equal to the distance ratio for any two points
with nonzero finite distance; `dist` version | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_of_nndist_eq {α β}
[pseudo_metric_space α] [pseudo_metric_space β]
(f : α → β) (h : ∃ (r : ℝ≥0), r ≠ 0 ∧ ∀ (x y : α), nndist (f x) (f y) = r * nndist x y) :
dilation α β | { to_fun := f,
edist_eq' :=
begin
rcases h with ⟨r, hne, h⟩,
refine ⟨r, hne, λ x y, _⟩,
rw [edist_nndist, edist_nndist, ← ennreal.coe_mul, h x y],
end } | def | dilation.mk_of_nndist_eq | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"edist_nndist",
"ennreal.coe_mul",
"pseudo_metric_space"
] | Alternative `dilation` constructor when the distance hypothesis is over `nndist` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mk_of_nndist_eq {α β}
[pseudo_metric_space α] [pseudo_metric_space β]
(f : α → β) (h) : ⇑(mk_of_nndist_eq f h : dilation α β) = f | rfl | lemma | dilation.coe_mk_of_nndist_eq | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe_of_nndist_eq {α β}
[pseudo_metric_space α] [pseudo_metric_space β]
(f : dilation α β) (h) : dilation.mk_of_nndist_eq f h = f | ext $ λ _, rfl | lemma | dilation.mk_coe_of_nndist_eq | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"dilation.mk_of_nndist_eq",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_of_dist_eq {α β}
[pseudo_metric_space α] [pseudo_metric_space β]
(f : α → β) (h : ∃ (r : ℝ≥0), r ≠ 0 ∧ ∀ (x y : α), dist (f x) (f y) = r * dist x y) :
dilation α β | mk_of_nndist_eq f $ h.imp $ λ r hr,
⟨hr.1, λ x y, nnreal.eq $ by rw [coe_nndist, hr.2, nnreal.coe_mul, coe_nndist]⟩ | def | dilation.mk_of_dist_eq | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"coe_nndist",
"dilation",
"nnreal.coe_mul",
"nnreal.eq",
"pseudo_metric_space"
] | Alternative `dilation` constructor when the distance hypothesis is over `dist` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mk_of_dist_eq {α β}
[pseudo_metric_space α] [pseudo_metric_space β]
(f : α → β) (h) : ⇑(mk_of_dist_eq f h : dilation α β) = f | rfl | lemma | dilation.coe_mk_of_dist_eq | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe_of_dist_eq {α β}
[pseudo_metric_space α] [pseudo_metric_space β]
(f : dilation α β) (h) : dilation.mk_of_dist_eq f h = f | ext $ λ _, rfl | lemma | dilation.mk_coe_of_dist_eq | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"dilation.mk_of_dist_eq",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz : lipschitz_with (ratio f) (f : α → β) | λ x y, (edist_eq f x y).le | lemma | dilation.lipschitz | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz : antilipschitz_with (ratio f)⁻¹ (f : α → β) | λ x y, have hr : ratio f ≠ 0 := ratio_ne_zero f, by exact_mod_cast
(ennreal.mul_le_iff_le_inv (ennreal.coe_ne_zero.2 hr) ennreal.coe_ne_top).1 (edist_eq f x y).ge | lemma | dilation.antilipschitz | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"antilipschitz_with",
"ennreal.coe_ne_top",
"ennreal.mul_le_iff_le_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective {α : Type*} [emetric_space α] [dilation_class F α β] (f : F) :
injective f | (antilipschitz f).injective | lemma | dilation.injective | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation_class",
"emetric_space"
] | A dilation from an emetric space is injective | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id (α) [pseudo_emetric_space α] : dilation α α | { to_fun := _root_.id,
edist_eq' := ⟨1, one_ne_zero, λ x y, by simp only [id.def, ennreal.coe_one, one_mul]⟩ } | def | dilation.id | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"ennreal.coe_one",
"one_mul",
"one_ne_zero",
"pseudo_emetric_space"
] | The identity is a dilation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(dilation.id α) = id | rfl | lemma | dilation.coe_id | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_ratio : ratio (dilation.id α) = 1 | begin
by_cases h : ∀ x y : α, edist x y = 0 ∨ edist x y = ∞,
{ rw [ratio, if_pos h] },
{ push_neg at h,
rcases h with ⟨x, y, hne⟩,
refine (ratio_unique hne.1 hne.2 _).symm,
simp }
end | lemma | dilation.id_ratio | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (g : dilation β γ) (f : dilation α β) : dilation α γ | { to_fun := g ∘ f,
edist_eq' := ⟨ratio g * ratio f,
mul_ne_zero (ratio_ne_zero g) (ratio_ne_zero f),
λ x y, by { simp only [edist_eq, ennreal.coe_mul], ring, }⟩ } | def | dilation.comp | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"ennreal.coe_mul",
"mul_ne_zero",
"ring"
] | The composition of dilations is a dilation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_assoc {δ : Type*} [pseudo_emetric_space δ]
(f : dilation α β) (g : dilation β γ) (h : dilation γ δ) :
(h.comp g).comp f = h.comp (g.comp f) | rfl | lemma | dilation.comp_assoc | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp (g : dilation β γ) (f : dilation α β) :
(g.comp f : α → γ) = g ∘ f | rfl | lemma | dilation.coe_comp | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (g : dilation β γ) (f : dilation α β) (x : α) :
(g.comp f : α → γ) x = (g (f x)) | rfl | lemma | dilation.comp_apply | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_ratio
{g : dilation β γ} {f : dilation α β} (hne : ∃ x y : α, edist x y ≠ 0 ∧ edist x y ≠ ⊤) :
ratio (g.comp f) = ratio g * ratio f | begin
rcases hne with ⟨x, y, hα⟩,
have hgf := (edist_eq (g.comp f) x y).symm,
simp only [dist_eq, coe_comp, ← mul_assoc, mul_eq_mul_right_iff] at hgf,
rw [edist_eq, edist_eq, ← mul_assoc, ennreal.mul_eq_mul_right hα.1 hα.2] at hgf,
rwa [← ennreal.coe_eq_coe, ennreal.coe_mul],
end | lemma | dilation.comp_ratio | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"ennreal.coe_eq_coe",
"ennreal.coe_mul",
"ennreal.mul_eq_mul_right",
"mul_assoc",
"mul_eq_mul_right_iff"
] | Ratio of the composition `g.comp f` of two dilations is the product of their ratios. We assume
that the domain `α` of `f` is nontrivial, otherwise `ratio f = ratio (g.comp f) = 1` but `ratio g`
may have any value. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_id (f : dilation α β) : f.comp (dilation.id α) = f | ext $ λ x, rfl | lemma | dilation.comp_id | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"dilation.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : dilation α β) : (dilation.id β).comp f = f | ext $ λ x, rfl | lemma | dilation.id_comp | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"dilation.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_def : (1 : dilation α α) = dilation.id α | rfl | lemma | dilation.one_def | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"dilation.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_def (f g : dilation α α) : f * g = f.comp g | rfl | lemma | dilation.mul_def | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ⇑(1 : dilation α α) = _root_.id | rfl | lemma | dilation.coe_one | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (f g : dilation α α) : ⇑(f * g) = f ∘ g | rfl | lemma | dilation.coe_mul | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : dilation β γ} {f : dilation α β} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, dilation.ext $ hf.forall.2 (ext_iff.1 h), λ h, h ▸ rfl⟩ | lemma | dilation.cancel_right | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"dilation.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : dilation β γ} {f₁ f₂ : dilation α β} (hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, dilation.ext $ λ x, hg $ by rw [← comp_apply, h, comp_apply], λ h, h ▸ rfl⟩ | lemma | dilation.cancel_left | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation",
"dilation.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing : uniform_inducing (f : α → β) | (antilipschitz f).uniform_inducing (lipschitz f).uniform_continuous | theorem | dilation.uniform_inducing | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"uniform_continuous",
"uniform_inducing"
] | A dilation from a metric space is a uniform inducing map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_nhds_iff {ι : Type*} {g : ι → α} {a : filter ι} {b : α} :
filter.tendsto g a (𝓝 b) ↔ filter.tendsto ((f : α → β) ∘ g) a (𝓝 (f b)) | (dilation.uniform_inducing f).inducing.tendsto_nhds_iff | lemma | dilation.tendsto_nhds_iff | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation.uniform_inducing",
"filter",
"filter.tendsto",
"inducing.tendsto_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous : continuous (f : α → β) | (lipschitz f).continuous | lemma | dilation.to_continuous | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"continuous"
] | A dilation is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ediam_image (s : set α) :
emetric.diam ((f : α → β) '' s) = ratio f * emetric.diam s | begin
refine ((lipschitz f).ediam_image_le s).antisymm _,
apply ennreal.mul_le_of_le_div',
rw [div_eq_mul_inv, mul_comm, ← ennreal.coe_inv],
exacts [(antilipschitz f).le_mul_ediam_image s, ratio_ne_zero f],
end | lemma | dilation.ediam_image | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"div_eq_mul_inv",
"emetric.diam",
"ennreal.coe_inv",
"ennreal.mul_le_of_le_div'",
"mul_comm"
] | Dilations scale the diameter by `ratio f` in pseudoemetric spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ediam_range :
emetric.diam (range (f : α → β)) = ratio f * emetric.diam (univ : set α) | by { rw ← image_univ, exact ediam_image f univ } | lemma | dilation.ediam_range | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"emetric.diam"
] | A dilation scales the diameter of the range by `ratio f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
maps_to_emetric_ball (x : α) (r : ℝ≥0∞) :
maps_to (f : α → β) (emetric.ball x r) (emetric.ball (f x) (ratio f * r)) | λ y hy, (edist_eq f y x).trans_lt $
(ennreal.mul_lt_mul_left (ennreal.coe_ne_zero.2 $ ratio_ne_zero f) ennreal.coe_ne_top).2 hy | lemma | dilation.maps_to_emetric_ball | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"emetric.ball",
"ennreal.coe_ne_top",
"ennreal.mul_lt_mul_left"
] | A dilation maps balls to balls and scales the radius by `ratio f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
maps_to_emetric_closed_ball (x : α) (r' : ℝ≥0∞) :
maps_to (f : α → β) (emetric.closed_ball x r') (emetric.closed_ball (f x) (ratio f * r')) | λ y hy, (edist_eq f y x).trans_le $ mul_le_mul_left' hy _ | lemma | dilation.maps_to_emetric_closed_ball | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"emetric.closed_ball",
"mul_le_mul_left'"
] | A dilation maps closed balls to closed balls and scales the radius by `ratio f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_continuous_on_iff {γ} [topological_space γ] {g : γ → α} {s : set γ} :
continuous_on ((f : α → β) ∘ g) s ↔ continuous_on g s | (dilation.uniform_inducing f).inducing.continuous_on_iff.symm | lemma | dilation.comp_continuous_on_iff | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"continuous_on",
"dilation.uniform_inducing",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_iff {γ} [topological_space γ] {g : γ → α} :
continuous ((f : α → β) ∘ g) ↔ continuous g | (dilation.uniform_inducing f).inducing.continuous_iff.symm | lemma | dilation.comp_continuous_iff | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"continuous",
"dilation.uniform_inducing",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding [pseudo_emetric_space β] [dilation_class F α β]
(f : F) : uniform_embedding f | (antilipschitz f).uniform_embedding (lipschitz f).uniform_continuous | theorem | dilation.uniform_embedding | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation_class",
"pseudo_emetric_space",
"uniform_continuous",
"uniform_embedding"
] | A dilation from a metric space is a uniform embedding | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding [pseudo_emetric_space β] [dilation_class F α β]
(f : F) : embedding (f : α → β) | (dilation.uniform_embedding f).embedding | theorem | dilation.embedding | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"dilation.uniform_embedding",
"dilation_class",
"embedding",
"pseudo_emetric_space"
] | A dilation from a metric space is an embedding | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_embedding [complete_space α] [emetric_space β] [dilation_class F α β]
(f : F) : closed_embedding f | (antilipschitz f).closed_embedding (lipschitz f).uniform_continuous | theorem | dilation.closed_embedding | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"closed_embedding",
"complete_space",
"dilation_class",
"emetric_space",
"uniform_continuous"
] | A dilation from a complete emetric space is a closed embedding | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_image (s : set α) : metric.diam ((f : α → β) '' s) = ratio f * metric.diam s | by { simp [metric.diam, ediam_image, ennreal.to_real_mul], } | lemma | dilation.diam_image | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"ennreal.to_real_mul",
"metric.diam"
] | A dilation scales the diameter by `ratio f` in pseudometric spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_range : metric.diam (range (f : α → β)) = ratio f * metric.diam (univ : set α) | by rw [← image_univ, diam_image] | lemma | dilation.diam_range | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"metric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_ball (x : α) (r' : ℝ) :
maps_to (f : α → β) (metric.ball x r') (metric.ball (f x) (ratio f * r')) | λ y hy, (dist_eq f y x).trans_lt $ (mul_lt_mul_left $ nnreal.coe_pos.2 $ ratio_pos f).2 hy | lemma | dilation.maps_to_ball | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"metric.ball",
"mul_lt_mul_left"
] | A dilation maps balls to balls and scales the radius by `ratio f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
maps_to_sphere (x : α) (r' : ℝ) :
maps_to (f : α → β) (metric.sphere x r') (metric.sphere (f x) (ratio f * r')) | λ y hy, metric.mem_sphere.mp hy ▸ dist_eq f y x | lemma | dilation.maps_to_sphere | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"metric.sphere"
] | A dilation maps spheres to spheres and scales the radius by `ratio f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
maps_to_closed_ball (x : α) (r' : ℝ) :
maps_to (f : α → β) (metric.closed_ball x r') (metric.closed_ball (f x) (ratio f * r')) | λ y hy, (dist_eq f y x).trans_le $ mul_le_mul_of_nonneg_left hy (nnreal.coe_nonneg _) | lemma | dilation.maps_to_closed_ball | topology.metric_space | src/topology/metric_space/dilation.lean | [
"topology.metric_space.antilipschitz",
"data.fun_like.basic"
] | [
"metric.closed_ball",
"mul_le_mul_of_nonneg_left",
"nnreal.coe_nonneg"
] | A dilation maps closed balls to closed balls and scales the radius by `ratio f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_of_emetric [emetric_space α] : normal_space α | normal_of_paracompact_t2 | instance | emetric.normal_of_emetric | topology.metric_space | src/topology/metric_space/emetric_paracompact.lean | [
"set_theory.ordinal.basic",
"topology.metric_space.emetric_space",
"topology.paracompact"
] | [
"emetric_space",
"normal_of_paracompact_t2",
"normal_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_dist_of_mem_uniformity [linear_order β] {U : filter (α × α)} (z : β)
(D : α → α → β) (H : ∀ s, s ∈ U ↔ ∃ε>z, ∀{a b:α}, D a b < ε → (a, b) ∈ s) :
U = ⨅ ε > z, 𝓟 {p:α×α | D p.1 p.2 < ε} | has_basis.eq_binfi ⟨λ s, by simp only [H, subset_def, prod.forall, mem_set_of]⟩ | theorem | uniformity_dist_of_mem_uniformity | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"filter"
] | Characterizing uniformities associated to a (generalized) distance function `D`
in terms of the elements of the uniformity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_edist (α : Type*) | (edist : α → α → ℝ≥0∞) | class | has_edist | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | `has_edist α` means that `α` is equipped with an extended distance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space_of_edist (edist : α → α → ℝ≥0∞)
(edist_self : ∀ x : α, edist x x = 0) (edist_comm : ∀ x y : α, edist x y = edist y x)
(edist_triangle : ∀ x y z : α, edist x z ≤ edist x y + edist y z) : uniform_space α | uniform_space.of_fun edist edist_self edist_comm edist_triangle $
λ ε ε0, ⟨ε / 2, ennreal.half_pos ε0.lt.ne', λ _ h₁ _ h₂,
(ennreal.add_lt_add h₁ h₂).trans_eq (ennreal.add_halves _)⟩ | def | uniform_space_of_edist | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"ennreal.add_halves",
"ennreal.add_lt_add",
"ennreal.half_pos",
"uniform_space",
"uniform_space.of_fun"
] | Creating a uniform space from an extended distance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_emetric_space (α : Type u) extends has_edist α : Type u | (edist_self : ∀ x : α, edist x x = 0)
(edist_comm : ∀ x y : α, edist x y = edist y x)
(edist_triangle : ∀ x y z : α, edist x z ≤ edist x y + edist y z)
(to_uniform_space : uniform_space α :=
uniform_space_of_edist edist edist_self edist_comm edist_triangle)
(uniformity_edist : 𝓤 α = ⨅ ε>0, 𝓟 {p:α×α | edist p.1 p.2 ... | class | pseudo_emetric_space | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [
"has_edist",
"uniform_space",
"uniform_space_of_edist",
"uniformity_edist"
] | Extended (pseudo) metric spaces, with an extended distance `edist` possibly taking the
value ∞
Each pseudo_emetric space induces a canonical `uniform_space` and hence a canonical
`topological_space`.
This is enforced in the type class definition, by extending the `uniform_space` structure. When
instantiating a `pseudo... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_triangle_left (x y z : α) : edist x y ≤ edist z x + edist z y | by rw edist_comm z; apply edist_triangle | theorem | edist_triangle_left | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | Triangle inequality for the extended distance | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_triangle_right (x y z : α) : edist x y ≤ edist x z + edist y z | by rw edist_comm y; apply edist_triangle | theorem | edist_triangle_right | topology.metric_space | src/topology/metric_space/emetric_space.lean | [
"data.nat.interval",
"data.real.ennreal",
"topology.uniform_space.pi",
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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