file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Order/Filter/Prod.lean
|
Filter.Eventually.diag_of_prod_left
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.9875\nγ : Type u_2\nδ : Type ?u.9881\nι : Sort ?u.9884\ns : Set α\nt : Set β\nf✝ : Filter α\ng✝ : Filter β\nf : Filter α\ng : Filter γ\np : (α × α) × γ → Prop\nh : ∀ᶠ (x : (α × α) × γ) in (f ×ˢ f) ×ˢ g, p x\n⊢ ∀ᶠ (x : α × γ) in f ×ˢ g, p ((x.fst, x.fst), x.snd)",
"state_before": "α : Type u_1\nβ : Type ?u.9875\nγ : Type u_2\nδ : Type ?u.9881\nι : Sort ?u.9884\ns : Set α\nt : Set β\nf✝ : Filter α\ng✝ : Filter β\nf : Filter α\ng : Filter γ\np : (α × α) × γ → Prop\n⊢ (∀ᶠ (x : (α × α) × γ) in (f ×ˢ f) ×ˢ g, p x) → ∀ᶠ (x : α × γ) in f ×ˢ g, p ((x.fst, x.fst), x.snd)",
"tactic": "intro h"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.9875\nγ : Type u_2\nδ : Type ?u.9881\nι : Sort ?u.9884\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng✝ : Filter β\nf : Filter α\ng : Filter γ\np : (α × α) × γ → Prop\nh : ∀ᶠ (x : (α × α) × γ) in (f ×ˢ f) ×ˢ g, p x\nt : α × α → Prop\nht : ∀ᶠ (x : α × α) in f ×ˢ f, t x\ns : γ → Prop\nhs : ∀ᶠ (y : γ) in g, s y\nhst : ∀ {x : α × α}, t x → ∀ {y : γ}, s y → p (x, y)\n⊢ ∀ᶠ (x : α × γ) in f ×ˢ g, p ((x.fst, x.fst), x.snd)",
"state_before": "α : Type u_1\nβ : Type ?u.9875\nγ : Type u_2\nδ : Type ?u.9881\nι : Sort ?u.9884\ns : Set α\nt : Set β\nf✝ : Filter α\ng✝ : Filter β\nf : Filter α\ng : Filter γ\np : (α × α) × γ → Prop\nh : ∀ᶠ (x : (α × α) × γ) in (f ×ˢ f) ×ˢ g, p x\n⊢ ∀ᶠ (x : α × γ) in f ×ˢ g, p ((x.fst, x.fst), x.snd)",
"tactic": "obtain ⟨t, ht, s, hs, hst⟩ := eventually_prod_iff.1 h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.9875\nγ : Type u_2\nδ : Type ?u.9881\nι : Sort ?u.9884\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng✝ : Filter β\nf : Filter α\ng : Filter γ\np : (α × α) × γ → Prop\nh : ∀ᶠ (x : (α × α) × γ) in (f ×ˢ f) ×ˢ g, p x\nt : α × α → Prop\nht : ∀ᶠ (x : α × α) in f ×ˢ f, t x\ns : γ → Prop\nhs : ∀ᶠ (y : γ) in g, s y\nhst : ∀ {x : α × α}, t x → ∀ {y : γ}, s y → p (x, y)\n⊢ ∀ᶠ (x : α × γ) in f ×ˢ g, p ((x.fst, x.fst), x.snd)",
"tactic": "refine' (ht.diag_of_prod.prod_mk hs).mono fun x hx => by simp only [hst hx.1 hx.2, Prod.mk.eta]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.9875\nγ : Type u_2\nδ : Type ?u.9881\nι : Sort ?u.9884\ns✝ : Set α\nt✝ : Set β\nf✝ : Filter α\ng✝ : Filter β\nf : Filter α\ng : Filter γ\np : (α × α) × γ → Prop\nh : ∀ᶠ (x : (α × α) × γ) in (f ×ˢ f) ×ˢ g, p x\nt : α × α → Prop\nht : ∀ᶠ (x : α × α) in f ×ˢ f, t x\ns : γ → Prop\nhs : ∀ᶠ (y : γ) in g, s y\nhst : ∀ {x : α × α}, t x → ∀ {y : γ}, s y → p (x, y)\nx : α × γ\nhx : t (x.fst, x.fst) ∧ s x.snd\n⊢ p ((x.fst, x.fst), x.snd)",
"tactic": "simp only [hst hx.1 hx.2, Prod.mk.eta]"
}
] |
[
202,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
stronglyMeasurable_bot_iff
|
[
{
"state_after": "case inl\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : IsEmpty α\n⊢ StronglyMeasurable f ↔ ∃ c, f = fun x => c\n\ncase inr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\n⊢ StronglyMeasurable f ↔ ∃ c, f = fun x => c",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\n⊢ StronglyMeasurable f ↔ ∃ c, f = fun x => c",
"tactic": "cases' isEmpty_or_nonempty α with hα hα"
},
{
"state_after": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\n⊢ ∃ c, f = fun x => c\n\ncase inr.refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf_eq : ∃ c, f = fun x => c\n⊢ StronglyMeasurable f",
"state_before": "case inr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\n⊢ StronglyMeasurable f ↔ ∃ c, f = fun x => c",
"tactic": "refine' ⟨fun hf => _, fun hf_eq => _⟩"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : IsEmpty α\n⊢ StronglyMeasurable f ↔ ∃ c, f = fun x => c",
"tactic": "simp only [@Subsingleton.stronglyMeasurable' _ _ ⊥ _ _ f,\n eq_iff_true_of_subsingleton, exists_const]"
},
{
"state_after": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\n⊢ f = fun x => f (Nonempty.some hα)",
"state_before": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\n⊢ ∃ c, f = fun x => c",
"tactic": "refine' ⟨f hα.some, _⟩"
},
{
"state_after": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\n⊢ f = fun x => f (Nonempty.some hα)",
"state_before": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\n⊢ f = fun x => f (Nonempty.some hα)",
"tactic": "let fs := hf.approx"
},
{
"state_after": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\nh_fs_tendsto : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\n⊢ f = fun x => f (Nonempty.some hα)",
"state_before": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\n⊢ f = fun x => f (Nonempty.some hα)",
"tactic": "have h_fs_tendsto : ∀ x, Tendsto (fun n => fs n x) atTop (𝓝 (f x)) := hf.tendsto_approx"
},
{
"state_after": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\nh_fs_tendsto : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nthis : ∀ (n : ℕ), ∃ c, ∀ (x : α), ↑(fs n) x = c\n⊢ f = fun x => f (Nonempty.some hα)",
"state_before": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\nh_fs_tendsto : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\n⊢ f = fun x => f (Nonempty.some hα)",
"tactic": "have : ∀ n, ∃ c, ∀ x, fs n x = c := fun n => SimpleFunc.simpleFunc_bot (fs n)"
},
{
"state_after": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\nh_fs_tendsto : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nthis : ∀ (n : ℕ), ∃ c, ∀ (x : α), ↑(fs n) x = c\ncs : ℕ → β := fun n => Exists.choose (_ : ∃ c, ∀ (x : α), ↑(fs n) x = c)\n⊢ f = fun x => f (Nonempty.some hα)",
"state_before": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\nh_fs_tendsto : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nthis : ∀ (n : ℕ), ∃ c, ∀ (x : α), ↑(fs n) x = c\n⊢ f = fun x => f (Nonempty.some hα)",
"tactic": "let cs n := (this n).choose"
},
{
"state_after": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\nh_fs_tendsto : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nthis : ∀ (n : ℕ), ∃ c, ∀ (x : α), ↑(fs n) x = c\ncs : ℕ → β := fun n => Exists.choose (_ : ∃ c, ∀ (x : α), ↑(fs n) x = c)\nh_cs_eq : ∀ (n : ℕ), ↑(fs n) = fun x => cs n\n⊢ f = fun x => f (Nonempty.some hα)",
"state_before": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\nh_fs_tendsto : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nthis : ∀ (n : ℕ), ∃ c, ∀ (x : α), ↑(fs n) x = c\ncs : ℕ → β := fun n => Exists.choose (_ : ∃ c, ∀ (x : α), ↑(fs n) x = c)\n⊢ f = fun x => f (Nonempty.some hα)",
"tactic": "have h_cs_eq : ∀ n, ⇑(fs n) = fun _ => cs n := fun n => funext (this n).choose_spec"
},
{
"state_after": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\nthis : ∀ (n : ℕ), ∃ c, ∀ (x : α), ↑(fs n) x = c\ncs : ℕ → β := fun n => Exists.choose (_ : ∃ c, ∀ (x : α), ↑(fs n) x = c)\nh_fs_tendsto : ∀ (x : α), Tendsto (fun n => (fun x => cs n) x) atTop (𝓝 (f x))\nh_cs_eq : ∀ (n : ℕ), ↑(fs n) = fun x => cs n\n⊢ f = fun x => f (Nonempty.some hα)",
"state_before": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\nh_fs_tendsto : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nthis : ∀ (n : ℕ), ∃ c, ∀ (x : α), ↑(fs n) x = c\ncs : ℕ → β := fun n => Exists.choose (_ : ∃ c, ∀ (x : α), ↑(fs n) x = c)\nh_cs_eq : ∀ (n : ℕ), ↑(fs n) = fun x => cs n\n⊢ f = fun x => f (Nonempty.some hα)",
"tactic": "conv at h_fs_tendsto => enter [x, 1, n]; rw [h_cs_eq]"
},
{
"state_after": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\nthis : ∀ (n : ℕ), ∃ c, ∀ (x : α), ↑(fs n) x = c\ncs : ℕ → β := fun n => Exists.choose (_ : ∃ c, ∀ (x : α), ↑(fs n) x = c)\nh_fs_tendsto : ∀ (x : α), Tendsto (fun n => (fun x => cs n) x) atTop (𝓝 (f x))\nh_cs_eq : ∀ (n : ℕ), ↑(fs n) = fun x => cs n\nh_tendsto : Tendsto cs atTop (𝓝 (f (Nonempty.some hα)))\n⊢ f = fun x => f (Nonempty.some hα)",
"state_before": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\nthis : ∀ (n : ℕ), ∃ c, ∀ (x : α), ↑(fs n) x = c\ncs : ℕ → β := fun n => Exists.choose (_ : ∃ c, ∀ (x : α), ↑(fs n) x = c)\nh_fs_tendsto : ∀ (x : α), Tendsto (fun n => (fun x => cs n) x) atTop (𝓝 (f x))\nh_cs_eq : ∀ (n : ℕ), ↑(fs n) = fun x => cs n\n⊢ f = fun x => f (Nonempty.some hα)",
"tactic": "have h_tendsto : Tendsto cs atTop (𝓝 (f hα.some)) := h_fs_tendsto hα.some"
},
{
"state_after": "case inr.refine'_1.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\nthis : ∀ (n : ℕ), ∃ c, ∀ (x : α), ↑(fs n) x = c\ncs : ℕ → β := fun n => Exists.choose (_ : ∃ c, ∀ (x : α), ↑(fs n) x = c)\nh_fs_tendsto : ∀ (x : α), Tendsto (fun n => (fun x => cs n) x) atTop (𝓝 (f x))\nh_cs_eq : ∀ (n : ℕ), ↑(fs n) = fun x => cs n\nh_tendsto : Tendsto cs atTop (𝓝 (f (Nonempty.some hα)))\nx : α\n⊢ f x = f (Nonempty.some hα)",
"state_before": "case inr.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\nthis : ∀ (n : ℕ), ∃ c, ∀ (x : α), ↑(fs n) x = c\ncs : ℕ → β := fun n => Exists.choose (_ : ∃ c, ∀ (x : α), ↑(fs n) x = c)\nh_fs_tendsto : ∀ (x : α), Tendsto (fun n => (fun x => cs n) x) atTop (𝓝 (f x))\nh_cs_eq : ∀ (n : ℕ), ↑(fs n) = fun x => cs n\nh_tendsto : Tendsto cs atTop (𝓝 (f (Nonempty.some hα)))\n⊢ f = fun x => f (Nonempty.some hα)",
"tactic": "ext1 x"
},
{
"state_after": "no goals",
"state_before": "case inr.refine'_1.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := StronglyMeasurable.approx hf\nthis : ∀ (n : ℕ), ∃ c, ∀ (x : α), ↑(fs n) x = c\ncs : ℕ → β := fun n => Exists.choose (_ : ∃ c, ∀ (x : α), ↑(fs n) x = c)\nh_fs_tendsto : ∀ (x : α), Tendsto (fun n => (fun x => cs n) x) atTop (𝓝 (f x))\nh_cs_eq : ∀ (n : ℕ), ↑(fs n) = fun x => cs n\nh_tendsto : Tendsto cs atTop (𝓝 (f (Nonempty.some hα)))\nx : α\n⊢ f x = f (Nonempty.some hα)",
"tactic": "exact tendsto_nhds_unique (h_fs_tendsto x) h_tendsto"
},
{
"state_after": "case inr.refine'_2.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\ng : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nc : β\n⊢ StronglyMeasurable fun x => c",
"state_before": "case inr.refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\nf g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf_eq : ∃ c, f = fun x => c\n⊢ StronglyMeasurable f",
"tactic": "obtain ⟨c, rfl⟩ := hf_eq"
},
{
"state_after": "no goals",
"state_before": "case inr.refine'_2.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48568\nι : Type ?u.48571\ninst✝³ : Countable ι\ng : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nc : β\n⊢ StronglyMeasurable fun x => c",
"tactic": "exact stronglyMeasurable_const"
}
] |
[
292,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
275,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.factorization_eq_zero_iff
|
[
{
"state_after": "n p : ℕ\n⊢ ¬p ∈ factors n ↔ ¬Prime p ∨ ¬p ∣ n ∨ n = 0",
"state_before": "n p : ℕ\n⊢ ↑(factorization n) p = 0 ↔ ¬Prime p ∨ ¬p ∣ n ∨ n = 0",
"tactic": "rw [← not_mem_support_iff, support_factorization, mem_toFinset]"
},
{
"state_after": "case inl\np : ℕ\n⊢ ¬p ∈ factors 0 ↔ ¬Prime p ∨ ¬p ∣ 0 ∨ 0 = 0\n\ncase inr\nn p : ℕ\nhn : n ≠ 0\n⊢ ¬p ∈ factors n ↔ ¬Prime p ∨ ¬p ∣ n ∨ n = 0",
"state_before": "n p : ℕ\n⊢ ¬p ∈ factors n ↔ ¬Prime p ∨ ¬p ∣ n ∨ n = 0",
"tactic": "rcases eq_or_ne n 0 with (rfl | hn)"
},
{
"state_after": "no goals",
"state_before": "case inl\np : ℕ\n⊢ ¬p ∈ factors 0 ↔ ¬Prime p ∨ ¬p ∣ 0 ∨ 0 = 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr\nn p : ℕ\nhn : n ≠ 0\n⊢ ¬p ∈ factors n ↔ ¬Prime p ∨ ¬p ∣ n ∨ n = 0",
"tactic": "simp [hn, Nat.mem_factors, not_and_or, -not_and]"
}
] |
[
160,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean
|
CauchySeq.comp_tendsto
|
[] |
[
204,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
BoundedContinuousFunction.range_toLp
|
[] |
[
1664,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1661,
1
] |
Mathlib/Data/Nat/PSub.lean
|
Nat.ppred_eq_none
|
[
{
"state_after": "no goals",
"state_before": "⊢ ppred 0 = none ↔ 0 = 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ppred (n + 1) = none ↔ n + 1 = 0",
"tactic": "constructor <;> intro <;> contradiction"
}
] |
[
77,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
FractionalIdeal.one_mem_inv_coe_ideal
|
[
{
"state_after": "R : Type ?u.275932\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI : Ideal A\nhI : I ≠ ⊥\n⊢ ∀ (y : K), y ∈ ↑I → 1 * y ∈ 1",
"state_before": "R : Type ?u.275932\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI : Ideal A\nhI : I ≠ ⊥\n⊢ 1 ∈ (↑I)⁻¹",
"tactic": "rw [mem_inv_iff (coeIdeal_ne_zero.mpr hI)]"
},
{
"state_after": "R : Type ?u.275932\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI : Ideal A\nhI : I ≠ ⊥\ny : K\nhy : y ∈ ↑I\n⊢ 1 * y ∈ 1",
"state_before": "R : Type ?u.275932\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI : Ideal A\nhI : I ≠ ⊥\n⊢ ∀ (y : K), y ∈ ↑I → 1 * y ∈ 1",
"tactic": "intro y hy"
},
{
"state_after": "R : Type ?u.275932\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI : Ideal A\nhI : I ≠ ⊥\ny : K\nhy : y ∈ ↑I\n⊢ y ∈ 1",
"state_before": "R : Type ?u.275932\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI : Ideal A\nhI : I ≠ ⊥\ny : K\nhy : y ∈ ↑I\n⊢ 1 * y ∈ 1",
"tactic": "rw [one_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.275932\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI : Ideal A\nhI : I ≠ ⊥\ny : K\nhy : y ∈ ↑I\n⊢ y ∈ 1",
"tactic": "exact coeIdeal_le_one hy"
}
] |
[
459,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
454,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
differentiableAt_id'
|
[] |
[
1013,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1012,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
ContinuousWithinAt.tendsto_nhdsWithin
|
[] |
[
554,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
552,
1
] |
Mathlib/Algebra/Module/LinearMap.lean
|
LinearMap.comp_neg
|
[] |
[
922,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
921,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.L1.norm_sub_eq_lintegral
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1303508\nδ : Type ?u.1303511\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\n⊢ ENNReal.toReal (∫⁻ (a : α), ↑‖↑↑(f - g) a‖₊ ∂μ) = ENNReal.toReal (∫⁻ (x : α), ↑‖↑↑f x - ↑↑g x‖₊ ∂μ)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1303508\nδ : Type ?u.1303511\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\n⊢ ‖f - g‖ = ENNReal.toReal (∫⁻ (x : α), ↑‖↑↑f x - ↑↑g x‖₊ ∂μ)",
"tactic": "rw [norm_def]"
},
{
"state_after": "case e_a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.1303508\nδ : Type ?u.1303511\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\n⊢ (∫⁻ (a : α), ↑‖↑↑(f - g) a‖₊ ∂μ) = ∫⁻ (x : α), ↑‖↑↑f x - ↑↑g x‖₊ ∂μ",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1303508\nδ : Type ?u.1303511\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\n⊢ ENNReal.toReal (∫⁻ (a : α), ↑‖↑↑(f - g) a‖₊ ∂μ) = ENNReal.toReal (∫⁻ (x : α), ↑‖↑↑f x - ↑↑g x‖₊ ∂μ)",
"tactic": "congr 1"
},
{
"state_after": "case e_a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.1303508\nδ : Type ?u.1303511\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\n⊢ (fun a => ↑‖↑↑(f - g) a‖₊) =ᵐ[μ] fun a => ↑‖↑↑f a - ↑↑g a‖₊",
"state_before": "case e_a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.1303508\nδ : Type ?u.1303511\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\n⊢ (∫⁻ (a : α), ↑‖↑↑(f - g) a‖₊ ∂μ) = ∫⁻ (x : α), ↑‖↑↑f x - ↑↑g x‖₊ ∂μ",
"tactic": "rw [lintegral_congr_ae]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.1303508\nδ : Type ?u.1303511\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\na✝ : α\nha : ↑↑(f - g) a✝ = (↑↑f - ↑↑g) a✝\n⊢ ↑‖↑↑(f - g) a✝‖₊ = ↑‖↑↑f a✝ - ↑↑g a✝‖₊",
"state_before": "case e_a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.1303508\nδ : Type ?u.1303511\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\n⊢ (fun a => ↑‖↑↑(f - g) a‖₊) =ᵐ[μ] fun a => ↑‖↑↑f a - ↑↑g a‖₊",
"tactic": "filter_upwards [Lp.coeFn_sub f g] with _ ha"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.1303508\nδ : Type ?u.1303511\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\na✝ : α\nha : ↑↑(f - g) a✝ = (↑↑f - ↑↑g) a✝\n⊢ ↑‖↑↑(f - g) a✝‖₊ = ↑‖↑↑f a✝ - ↑↑g a✝‖₊",
"tactic": "simp only [ha, Pi.sub_apply]"
}
] |
[
1321,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1315,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.diam_triple
|
[
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.529537\nι : Type ?u.529540\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\n⊢ ENNReal.toReal (max (max (edist x y) (edist x z)) (edist y z)) =\n max (max (ENNReal.toReal (edist x y)) (ENNReal.toReal (edist x z))) (ENNReal.toReal (edist y z))",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.529537\nι : Type ?u.529540\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\n⊢ diam {x, y, z} = max (max (dist x y) (dist x z)) (dist y z)",
"tactic": "simp only [Metric.diam, EMetric.diam_triple, dist_edist]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.529537\nι : Type ?u.529540\ninst✝ : PseudoMetricSpace α\ns : Set α\nx y z : α\n⊢ ENNReal.toReal (max (max (edist x y) (edist x z)) (edist y z)) =\n max (max (ENNReal.toReal (edist x y)) (ENNReal.toReal (edist x z))) (ENNReal.toReal (edist y z))",
"tactic": "rw [ENNReal.toReal_max, ENNReal.toReal_max] <;> apply_rules [ne_of_lt, edist_lt_top, max_lt]"
}
] |
[
2610,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2607,
1
] |
Mathlib/Topology/PathConnected.lean
|
mem_pathComponent_of_mem
|
[] |
[
913,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
912,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearEquiv.coe_inj
|
[] |
[
1824,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1823,
1
] |
src/lean/Init/Control/Lawful.lean
|
StateT.ext
|
[] |
[
243,
11
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
242,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.SummableFamily.embDomain_succ_smul_powers
|
[
{
"state_after": "case h\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\nhx : 0 < ↑(addVal Γ R) x\n⊢ ∀ (a : ℕ),\n toFun (embDomain (x • powers x hx) { toFun := Nat.succ, inj' := Nat.succ_injective }) a =\n toFun (powers x hx - ofFinsupp (Finsupp.single 0 1)) a",
"state_before": "Γ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\nhx : 0 < ↑(addVal Γ R) x\n⊢ embDomain (x • powers x hx) { toFun := Nat.succ, inj' := Nat.succ_injective } =\n powers x hx - ofFinsupp (Finsupp.single 0 1)",
"tactic": "apply SummableFamily.ext"
},
{
"state_after": "case h.zero\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\nhx : 0 < ↑(addVal Γ R) x\n⊢ toFun (embDomain (x • powers x hx) { toFun := Nat.succ, inj' := Nat.succ_injective }) Nat.zero =\n toFun (powers x hx - ofFinsupp (Finsupp.single 0 1)) Nat.zero\n\ncase h.succ\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\nhx : 0 < ↑(addVal Γ R) x\nn : ℕ\n⊢ toFun (embDomain (x • powers x hx) { toFun := Nat.succ, inj' := Nat.succ_injective }) (Nat.succ n) =\n toFun (powers x hx - ofFinsupp (Finsupp.single 0 1)) (Nat.succ n)",
"state_before": "case h\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\nhx : 0 < ↑(addVal Γ R) x\n⊢ ∀ (a : ℕ),\n toFun (embDomain (x • powers x hx) { toFun := Nat.succ, inj' := Nat.succ_injective }) a =\n toFun (powers x hx - ofFinsupp (Finsupp.single 0 1)) a",
"tactic": "rintro (_ | n)"
},
{
"state_after": "case h.zero.h\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\nhx : 0 < ↑(addVal Γ R) x\n⊢ ¬Nat.zero ∈ Set.range ↑{ toFun := Nat.succ, inj' := Nat.succ_injective }",
"state_before": "case h.zero\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\nhx : 0 < ↑(addVal Γ R) x\n⊢ toFun (embDomain (x • powers x hx) { toFun := Nat.succ, inj' := Nat.succ_injective }) Nat.zero =\n toFun (powers x hx - ofFinsupp (Finsupp.single 0 1)) Nat.zero",
"tactic": "rw [embDomain_notin_range, sub_apply, coe_powers, pow_zero, coe_ofFinsupp,\n Finsupp.single_eq_same, sub_self]"
},
{
"state_after": "case h.zero.h\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\nhx : 0 < ↑(addVal Γ R) x\n⊢ ∀ (x : ℕ), ¬↑{ toFun := Nat.succ, inj' := Nat.succ_injective } x = Nat.zero",
"state_before": "case h.zero.h\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\nhx : 0 < ↑(addVal Γ R) x\n⊢ ¬Nat.zero ∈ Set.range ↑{ toFun := Nat.succ, inj' := Nat.succ_injective }",
"tactic": "rw [Set.mem_range, not_exists]"
},
{
"state_after": "no goals",
"state_before": "case h.zero.h\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\nhx : 0 < ↑(addVal Γ R) x\n⊢ ∀ (x : ℕ), ¬↑{ toFun := Nat.succ, inj' := Nat.succ_injective } x = Nat.zero",
"tactic": "exact Nat.succ_ne_zero"
},
{
"state_after": "case h.succ\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\nhx : 0 < ↑(addVal Γ R) x\nn : ℕ\n⊢ toFun (x • powers x hx) n = toFun (powers x hx - ofFinsupp (Finsupp.single 0 1)) (Nat.succ n)",
"state_before": "case h.succ\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\nhx : 0 < ↑(addVal Γ R) x\nn : ℕ\n⊢ toFun (embDomain (x • powers x hx) { toFun := Nat.succ, inj' := Nat.succ_injective }) (Nat.succ n) =\n toFun (powers x hx - ofFinsupp (Finsupp.single 0 1)) (Nat.succ n)",
"tactic": "refine' Eq.trans (embDomain_image _ ⟨Nat.succ, Nat.succ_injective⟩) _"
},
{
"state_after": "case h.succ\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\nhx : 0 < ↑(addVal Γ R) x\nn : ℕ\n⊢ x * x ^ n = x * x ^ n - ↑(Finsupp.single 0 1) (Nat.succ n)",
"state_before": "case h.succ\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\nhx : 0 < ↑(addVal Γ R) x\nn : ℕ\n⊢ toFun (x • powers x hx) n = toFun (powers x hx - ofFinsupp (Finsupp.single 0 1)) (Nat.succ n)",
"tactic": "simp only [pow_succ, coe_powers, coe_sub, smul_apply, coe_ofFinsupp, Pi.sub_apply]"
},
{
"state_after": "no goals",
"state_before": "case h.succ\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : HahnSeries Γ R\nhx : 0 < ↑(addVal Γ R) x\nn : ℕ\n⊢ x * x ^ n = x * x ^ n - ↑(Finsupp.single 0 1) (Nat.succ n)",
"tactic": "rw [Finsupp.single_eq_of_ne n.succ_ne_zero.symm, sub_zero]"
}
] |
[
1804,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1793,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.prod_sInter
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.241940\nι : Sort ?u.241943\nι' : Sort ?u.241946\nι₂ : Sort ?u.241949\nκ : ι → Sort ?u.241954\nκ₁ : ι → Sort ?u.241959\nκ₂ : ι → Sort ?u.241964\nκ' : ι' → Sort ?u.241969\nT : Set (Set β)\nhT : Set.Nonempty T\ns : Set α\n⊢ (⋂ (r : Set α × Set β) (_ : r ∈ {s} ×ˢ T), r.fst ×ˢ r.snd) = ⋂ (t : Set β) (_ : t ∈ T), s ×ˢ t",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.241940\nι : Sort ?u.241943\nι' : Sort ?u.241946\nι₂ : Sort ?u.241949\nκ : ι → Sort ?u.241954\nκ₁ : ι → Sort ?u.241959\nκ₂ : ι → Sort ?u.241964\nκ' : ι' → Sort ?u.241969\nT : Set (Set β)\nhT : Set.Nonempty T\ns : Set α\n⊢ s ×ˢ ⋂₀ T = ⋂ (t : Set β) (_ : t ∈ T), s ×ˢ t",
"tactic": "rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.241940\nι : Sort ?u.241943\nι' : Sort ?u.241946\nι₂ : Sort ?u.241949\nκ : ι → Sort ?u.241954\nκ₁ : ι → Sort ?u.241959\nκ₂ : ι → Sort ?u.241964\nκ' : ι' → Sort ?u.241969\nT : Set (Set β)\nhT : Set.Nonempty T\ns : Set α\n⊢ (⋂ (r : Set α × Set β) (_ : r ∈ {s} ×ˢ T), r.fst ×ˢ r.snd) = ⋂ (t : Set β) (_ : t ∈ T), s ×ˢ t",
"tactic": "simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]"
}
] |
[
1853,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1850,
1
] |
Mathlib/GroupTheory/PGroup.lean
|
IsPGroup.cyclic_center_quotient_of_card_eq_prime_sq
|
[
{
"state_after": "case intro.intro\np : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nhG : card G = p ^ 2\nk : ℕ\nhk0 : k > 0\nhk : card { x // x ∈ center G } = p ^ k\n⊢ IsCyclic (G ⧸ center G)",
"state_before": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nhG : card G = p ^ 2\n⊢ IsCyclic (G ⧸ center G)",
"tactic": "rcases card_center_eq_prime_pow hG zero_lt_two with ⟨k, hk0, hk⟩"
},
{
"state_after": "case intro.intro\np : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nk : ℕ\nhG : p ^ k * card (G ⧸ center G) = p ^ 2\nhk0 : k > 0\nhk : card { x // x ∈ center G } = p ^ k\n⊢ IsCyclic (G ⧸ center G)",
"state_before": "case intro.intro\np : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nhG : card G = p ^ 2\nk : ℕ\nhk0 : k > 0\nhk : card { x // x ∈ center G } = p ^ k\n⊢ IsCyclic (G ⧸ center G)",
"tactic": "rw [card_eq_card_quotient_mul_card_subgroup (center G), mul_comm, hk] at hG"
},
{
"state_after": "case intro.intro\np : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nk : ℕ\nhG : p ^ k * card (G ⧸ center G) = p ^ 2\nhk0 : k > 0\nhk : card { x // x ∈ center G } = p ^ k\nhk2 : k ≤ 2\n⊢ IsCyclic (G ⧸ center G)",
"state_before": "case intro.intro\np : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nk : ℕ\nhG : p ^ k * card (G ⧸ center G) = p ^ 2\nhk0 : k > 0\nhk : card { x // x ∈ center G } = p ^ k\n⊢ IsCyclic (G ⧸ center G)",
"tactic": "have hk2 := (Nat.pow_dvd_pow_iff_le_right (Fact.out (p := p.Prime)).one_lt).1 ⟨_, hG.symm⟩"
},
{
"state_after": "case intro.intro.«1»\np : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nk : ℕ\nhG : p ^ 1 * card (G ⧸ center G) = p ^ 2\nhk0 : 1 > 0\nhk : card { x // x ∈ center G } = p ^ 1\nhk2 : 1 ≤ 2\n⊢ IsCyclic (G ⧸ center G)\n\ncase intro.intro.«2»\np : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nk : ℕ\nhG : p ^ 2 * card (G ⧸ center G) = p ^ 2\nhk0 : 2 > 0\nhk : card { x // x ∈ center G } = p ^ 2\nhk2 : 2 ≤ 2\n⊢ IsCyclic (G ⧸ center G)",
"state_before": "case intro.intro\np : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nk : ℕ\nhG : p ^ k * card (G ⧸ center G) = p ^ 2\nhk0 : k > 0\nhk : card { x // x ∈ center G } = p ^ k\nhk2 : k ≤ 2\n⊢ IsCyclic (G ⧸ center G)",
"tactic": "interval_cases k"
},
{
"state_after": "case intro.intro.«1»\np : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nk : ℕ\nhG : card (G ⧸ center G) = p\nhk0 : 1 > 0\nhk : card { x // x ∈ center G } = p ^ 1\nhk2 : 1 ≤ 2\n⊢ IsCyclic (G ⧸ center G)",
"state_before": "case intro.intro.«1»\np : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nk : ℕ\nhG : p ^ 1 * card (G ⧸ center G) = p ^ 2\nhk0 : 1 > 0\nhk : card { x // x ∈ center G } = p ^ 1\nhk2 : 1 ≤ 2\n⊢ IsCyclic (G ⧸ center G)",
"tactic": "rw [sq, pow_one, mul_right_inj' (Fact.out (p := p.Prime)).ne_zero] at hG"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.«1»\np : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nk : ℕ\nhG : card (G ⧸ center G) = p\nhk0 : 1 > 0\nhk : card { x // x ∈ center G } = p ^ 1\nhk2 : 1 ≤ 2\n⊢ IsCyclic (G ⧸ center G)",
"tactic": "exact isCyclic_of_prime_card hG"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.«2»\np : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nk : ℕ\nhG : p ^ 2 * card (G ⧸ center G) = p ^ 2\nhk0 : 2 > 0\nhk : card { x // x ∈ center G } = p ^ 2\nhk2 : 2 ≤ 2\n⊢ IsCyclic (G ⧸ center G)",
"tactic": "exact\n @isCyclic_of_subsingleton _ _\n ⟨Fintype.card_le_one_iff.1\n (mul_right_injective₀ (pow_ne_zero 2 (NeZero.ne p))\n (hG.trans (mul_one (p ^ 2)).symm)).le⟩"
}
] |
[
396,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
HasStrictDerivAt.csinh
|
[] |
[
267,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
HasDerivWithinAt.cpow
|
[
{
"state_after": "no goals",
"state_before": "f g : ℂ → ℂ\ns : Set ℂ\nf' g' x c : ℂ\nhf : HasDerivWithinAt f f' s x\nhg : HasDerivWithinAt g g' s x\nh0 : 0 < (f x).re ∨ (f x).im ≠ 0\n⊢ HasDerivWithinAt (fun x => f x ^ g x) (g x * f x ^ (g x - 1) * f' + f x ^ g x * Complex.log (f x) * g') s x",
"tactic": "simpa only [aux] using (hf.hasFDerivWithinAt.cpow hg h0).hasDerivWithinAt"
}
] |
[
196,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Monad.lean
|
Pmf.mem_support_bindOnSupport_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.107281\np : Pmf α\nf : (a : α) → a ∈ support p → Pmf β\nb : β\n⊢ b ∈ support (bindOnSupport p f) ↔ ∃ a h, b ∈ support (f a h)",
"tactic": "simp only [support_bindOnSupport, Set.mem_setOf_eq, Set.mem_iUnion]"
}
] |
[
243,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
LinearMap.toMatrix'_algebraMap
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommSemiring R\nk : Type ?u.932592\nl : Type ?u.932595\nm : Type ?u.932598\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nx : R\n⊢ ↑toMatrix' (↑(algebraMap R (Module.End R (n → R))) x) = ↑(scalar n) x",
"tactic": "simp [Module.algebraMap_end_eq_smul_id]"
}
] |
[
412,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
PosNum.bit_to_nat
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.357130\nb : Bool\nn : PosNum\n⊢ ↑(bit b n) = Nat.bit b ↑n",
"tactic": "cases b <;> rfl"
}
] |
[
659,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
659,
1
] |
Mathlib/Data/Int/Order/Basic.lean
|
Int.ediv_mul_le
|
[
{
"state_after": "a✝ b✝ : ℤ\nn : ℕ\na b : ℤ\nH : b ≠ 0\n⊢ 0 ≤ a % b",
"state_before": "a✝ b✝ : ℤ\nn : ℕ\na b : ℤ\nH : b ≠ 0\n⊢ 0 ≤ a - a / b * b",
"tactic": "rw [mul_comm, ← emod_def]"
},
{
"state_after": "no goals",
"state_before": "a✝ b✝ : ℤ\nn : ℕ\na b : ℤ\nH : b ≠ 0\n⊢ 0 ≤ a % b",
"tactic": "apply emod_nonneg _ H"
}
] |
[
422,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
11
] |
Mathlib/Data/Set/Intervals/OrdConnected.lean
|
Set.ordConnected_iff_uIcc_subset_left
|
[] |
[
282,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion_of_empty
|
[] |
[
1331,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1330,
1
] |
Mathlib/Algebra/Periodic.lean
|
Function.Periodic.exists_mem_Ioc
|
[] |
[
306,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
303,
1
] |
Mathlib/CategoryTheory/Shift/Basic.lean
|
CategoryTheory.shift_shiftFunctorCompIsoId_add_neg_self_hom_app
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\nA : Type u_1\ninst✝² : Category C\ninst✝¹ : AddGroup A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nn : A\nX : C\n⊢ (shiftFunctor C n).map ((shiftFunctorCompIsoId C n (-n) (_ : n + -n = 0)).hom.app X) =\n (shiftFunctorCompIsoId C (-n) n (_ : -n + n = 0)).hom.app ((shiftFunctor C n).obj X)",
"tactic": "apply shift_shiftFunctorCompIsoId_hom_app"
}
] |
[
526,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
523,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.coe_nsmul
|
[] |
[
555,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
554,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.arcsin_lt_iff_lt_sin'
|
[] |
[
182,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.Dart.edge_mem
|
[] |
[
736,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
735,
1
] |
Mathlib/FieldTheory/Laurent.lean
|
RatFunc.laurent_injective
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nhdomain : IsDomain R\nr s : R\np q : R[X]\nf x✝¹ x✝ : RatFunc R\nh : ↑(laurent r) x✝¹ = ↑(laurent r) x✝\n⊢ x✝¹ = x✝",
"tactic": "simpa [laurent_laurent] using congr_arg (laurent (-r)) h"
}
] |
[
121,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Algebra/Lie/TensorProduct.lean
|
LieModule.toModuleHom_apply
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁶ : CommRing R\nL : Type v\nM : Type w\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nx : L\nm : M\n⊢ ↑(toModuleHom R L M) (x ⊗ₜ[R] m) = ⁅x, m⁆",
"tactic": "simp only [toModuleHom, TensorProduct.LieModule.liftLie_apply, LieModuleHom.coe_mk,\n LinearMap.coe_mk, LinearMap.coe_toAddHom, LieHom.coe_toLinearMap, toEndomorphism_apply_apply]"
}
] |
[
199,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Algebra/Ring/BooleanRing.lean
|
BooleanRing.inf_sup_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.14539\nγ : Type ?u.14542\ninst✝² : BooleanRing α\ninst✝¹ : BooleanRing β\ninst✝ : BooleanRing γ\na b : α\n⊢ a * (a + b + a * b) = a",
"tactic": "rw [mul_add, mul_add, mul_self, ← mul_assoc, mul_self, add_assoc, add_self, add_zero]"
}
] |
[
225,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Order/SuccPred/Limit.lean
|
Order.IsSuccLimit.isMax
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Preorder α\na : α\ninst✝ : SuccOrder α\nh : IsSuccLimit (succ a)\nH : ¬IsMax a\n⊢ False",
"state_before": "α : Type u_1\ninst✝¹ : Preorder α\na : α\ninst✝ : SuccOrder α\nh : IsSuccLimit (succ a)\n⊢ IsMax a",
"tactic": "by_contra H"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Preorder α\na : α\ninst✝ : SuccOrder α\nh : IsSuccLimit (succ a)\nH : ¬IsMax a\n⊢ False",
"tactic": "exact h a (covby_succ_of_not_isMax H)"
}
] |
[
75,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
11
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.mul_eq_mul
|
[] |
[
547,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
546,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.mk_zero
|
[] |
[
406,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Icc_subset_Ico_right
|
[
{
"state_after": "ι : Type ?u.23148\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\nh : b₁ < b₂\n⊢ Set.Icc a b₁ ⊆ Set.Ico a b₂",
"state_before": "ι : Type ?u.23148\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\nh : b₁ < b₂\n⊢ Icc a b₁ ⊆ Ico a b₂",
"tactic": "rw [← coe_subset, coe_Icc, coe_Ico]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.23148\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\nh : b₁ < b₂\n⊢ Set.Icc a b₁ ⊆ Set.Ico a b₂",
"tactic": "exact Set.Icc_subset_Ico_right h"
}
] |
[
222,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/Data/Nat/Factors.lean
|
Nat.coprime_factors_disjoint
|
[
{
"state_after": "a b : ℕ\nhab : coprime a b\nq : ℕ\nhqa : q ∈ factors a\nhqb : q ∈ factors b\n⊢ False",
"state_before": "a b : ℕ\nhab : coprime a b\n⊢ List.Disjoint (factors a) (factors b)",
"tactic": "intro q hqa hqb"
},
{
"state_after": "a b : ℕ\nhab : coprime a b\nq : ℕ\nhqa : q ∈ factors a\nhqb : q ∈ factors b\n⊢ Prime 1",
"state_before": "a b : ℕ\nhab : coprime a b\nq : ℕ\nhqa : q ∈ factors a\nhqb : q ∈ factors b\n⊢ False",
"tactic": "apply not_prime_one"
},
{
"state_after": "a b : ℕ\nhab : coprime a b\nq : ℕ\nhqa : q ∈ factors a\nhqb : q ∈ factors b\n⊢ Prime q",
"state_before": "a b : ℕ\nhab : coprime a b\nq : ℕ\nhqa : q ∈ factors a\nhqb : q ∈ factors b\n⊢ Prime 1",
"tactic": "rw [← eq_one_of_dvd_coprimes hab (dvd_of_mem_factors hqa) (dvd_of_mem_factors hqb)]"
},
{
"state_after": "no goals",
"state_before": "a b : ℕ\nhab : coprime a b\nq : ℕ\nhqa : q ∈ factors a\nhqb : q ∈ factors b\n⊢ Prime q",
"tactic": "exact prime_of_mem_factors hqa"
}
] |
[
270,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Order/Height.lean
|
Set.chainHeight_eq_iSup_subtype
|
[] |
[
93,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieModuleHom.map_neg
|
[] |
[
746,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
745,
1
] |
Mathlib/Algebra/Group/Opposite.lean
|
MulOpposite.semiconjBy_op
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Mul α\na x y : α\n⊢ SemiconjBy (op a) (op y) (op x) ↔ SemiconjBy a x y",
"tactic": "simp only [SemiconjBy, ← op_mul, op_inj, eq_comm]"
}
] |
[
218,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Data/Fintype/Card.lean
|
Finset.card_eq_iff_eq_univ
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.8142\nγ : Type ?u.8145\ninst✝ : Fintype α\n⊢ card univ = Fintype.card α",
"state_before": "α : Type u_1\nβ : Type ?u.8142\nγ : Type ?u.8145\ninst✝ : Fintype α\ns : Finset α\n⊢ s = univ → card s = Fintype.card α",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.8142\nγ : Type ?u.8145\ninst✝ : Fintype α\n⊢ card univ = Fintype.card α",
"tactic": "exact Finset.card_univ"
}
] |
[
263,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.apply_lt_nfpBFamily_iff
|
[
{
"state_after": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nho : o ≠ 0\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\nb : Ordinal\nh : ∀ (i : Ordinal) (hi : i < o), f i hi b < nfpBFamily o f a\nthis : Nonempty (Quotient.out o).α\n⊢ b < nfpBFamily o f a",
"state_before": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nho : o ≠ 0\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\nb : Ordinal\nh : ∀ (i : Ordinal) (hi : i < o), f i hi b < nfpBFamily o f a\n⊢ b < nfpBFamily o f a",
"tactic": "haveI := out_nonempty_iff_ne_zero.2 ho"
},
{
"state_after": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nho : o ≠ 0\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\nb : Ordinal\nh : ∀ (i : Ordinal) (hi : i < o), f i hi b < nfpBFamily o f a\nthis : Nonempty (Quotient.out o).α\n⊢ ∀ (i : (Quotient.out o).α), IsNormal (familyOfBFamily o f i)",
"state_before": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nho : o ≠ 0\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\nb : Ordinal\nh : ∀ (i : Ordinal) (hi : i < o), f i hi b < nfpBFamily o f a\nthis : Nonempty (Quotient.out o).α\n⊢ b < nfpBFamily o f a",
"tactic": "refine' (apply_lt_nfpFamily_iff.{u, v} _).1 fun _ => h _ _"
},
{
"state_after": "no goals",
"state_before": "o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal → Ordinal\nho : o ≠ 0\nH : ∀ (i : Ordinal) (hi : i < o), IsNormal (f i hi)\na : Ordinal\nb : Ordinal\nh : ∀ (i : Ordinal) (hi : i < o), f i hi b < nfpBFamily o f a\nthis : Nonempty (Quotient.out o).α\n⊢ ∀ (i : (Quotient.out o).α), IsNormal (familyOfBFamily o f i)",
"tactic": "exact fun _ => H _ _"
}
] |
[
307,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.dvd_mul_right
|
[] |
[
657,
68
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
657,
11
] |
Mathlib/Order/Monotone/Extension.lean
|
MonotoneOn.exists_monotone_extension
|
[
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\n⊢ ∃ g, Monotone g ∧ EqOn f g s",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na b : α\nh : MonotoneOn f s\nhl : BddBelow (f '' s)\nhu : BddAbove (f '' s)\n⊢ ∃ g, Monotone g ∧ EqOn f g s",
"tactic": "rcases hl with ⟨a, ha⟩"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\n⊢ ∃ g, Monotone g ∧ EqOn f g s",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\n⊢ ∃ g, Monotone g ∧ EqOn f g s",
"tactic": "have hu' : ∀ x, BddAbove (f '' (Iic x ∩ s)) := fun x =>\n hu.mono (image_subset _ (inter_subset_right _ _))"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\n⊢ ∃ g, Monotone g ∧ EqOn f g s",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\n⊢ ∃ g, Monotone g ∧ EqOn f g s",
"tactic": "let g : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\n⊢ ∃ g, Monotone g ∧ EqOn f g s",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\n⊢ ∃ g, Monotone g ∧ EqOn f g s",
"tactic": "have hgs : EqOn f g s := by\n intro x hx\n simp only []\n have : IsGreatest (Iic x ∩ s) x := ⟨⟨right_mem_Iic, hx⟩, fun y hy => hy.1⟩\n rw [if_neg this.nonempty.not_disjoint,\n ((h.mono <| inter_subset_right _ _).map_isGreatest this).csSup_eq]"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\nx y : α\nhxy : x ≤ y\n⊢ g x ≤ g y",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\n⊢ ∃ g, Monotone g ∧ EqOn f g s",
"tactic": "refine' ⟨g, fun x y hxy => _, hgs⟩"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\nx y : α\nhxy : x ≤ y\nhx : Disjoint (Iic x) s\nhy : ¬Disjoint (Iic y) s\n⊢ a ≤ sSup (f '' (Iic y ∩ s))\n\ncase pos\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\nx y : α\nhxy : x ≤ y\nhx : ¬Disjoint (Iic x) s\nhy : Disjoint (Iic y) s\n⊢ sSup (f '' (Iic x ∩ s)) ≤ a\n\ncase neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\nx y : α\nhxy : x ≤ y\nhx : ¬Disjoint (Iic x) s\nhy : ¬Disjoint (Iic y) s\n⊢ sSup (f '' (Iic x ∩ s)) ≤ sSup (f '' (Iic y ∩ s))",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\nx y : α\nhxy : x ≤ y\n⊢ g x ≤ g y",
"tactic": "by_cases hx : Disjoint (Iic x) s <;> by_cases hy : Disjoint (Iic y) s <;>\n simp only [if_pos, if_neg, not_false_iff, *, refl]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nx : α\nhx : x ∈ s\n⊢ f x = g x",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\n⊢ EqOn f g s",
"tactic": "intro x hx"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nx : α\nhx : x ∈ s\n⊢ f x = if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nx : α\nhx : x ∈ s\n⊢ f x = g x",
"tactic": "simp only []"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nx : α\nhx : x ∈ s\nthis : IsGreatest (Iic x ∩ s) x\n⊢ f x = if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nx : α\nhx : x ∈ s\n⊢ f x = if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))",
"tactic": "have : IsGreatest (Iic x ∩ s) x := ⟨⟨right_mem_Iic, hx⟩, fun y hy => hy.1⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nx : α\nhx : x ∈ s\nthis : IsGreatest (Iic x ∩ s) x\n⊢ f x = if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))",
"tactic": "rw [if_neg this.nonempty.not_disjoint,\n ((h.mono <| inter_subset_right _ _).map_isGreatest this).csSup_eq]"
},
{
"state_after": "case neg.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\nx y : α\nhxy : x ≤ y\nhx : Disjoint (Iic x) s\nhy : ¬Disjoint (Iic y) s\nz : α\nhz : z ∈ Iic y ∩ s\n⊢ a ≤ sSup (f '' (Iic y ∩ s))",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\nx y : α\nhxy : x ≤ y\nhx : Disjoint (Iic x) s\nhy : ¬Disjoint (Iic y) s\n⊢ a ≤ sSup (f '' (Iic y ∩ s))",
"tactic": "rcases not_disjoint_iff_nonempty_inter.1 hy with ⟨z, hz⟩"
},
{
"state_after": "no goals",
"state_before": "case neg.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\nx y : α\nhxy : x ≤ y\nhx : Disjoint (Iic x) s\nhy : ¬Disjoint (Iic y) s\nz : α\nhz : z ∈ Iic y ∩ s\n⊢ a ≤ sSup (f '' (Iic y ∩ s))",
"tactic": "exact le_csSup_of_le (hu' _) (mem_image_of_mem _ hz) (ha <| mem_image_of_mem _ hz.2)"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\nx y : α\nhxy : x ≤ y\nhx : ¬Disjoint (Iic x) s\nhy : Disjoint (Iic y) s\n⊢ sSup (f '' (Iic x ∩ s)) ≤ a",
"tactic": "exact (hx <| hy.mono_left <| Iic_subset_Iic.2 hxy).elim"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\nx y : α\nhxy : x ≤ y\nhx : Set.Nonempty (Iic x ∩ s)\nhy : Set.Nonempty (Iic y ∩ s)\n⊢ sSup (f '' (Iic x ∩ s)) ≤ sSup (f '' (Iic y ∩ s))",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\nx y : α\nhxy : x ≤ y\nhx : ¬Disjoint (Iic x) s\nhy : ¬Disjoint (Iic y) s\n⊢ sSup (f '' (Iic x ∩ s)) ≤ sSup (f '' (Iic y ∩ s))",
"tactic": "rw [not_disjoint_iff_nonempty_inter] at hx hy"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\nx y : α\nhxy : x ≤ y\nhx : Set.Nonempty (Iic x ∩ s)\nhy : Set.Nonempty (Iic y ∩ s)\n⊢ Iic x ∩ s ⊆ Iic y ∩ s",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\nx y : α\nhxy : x ≤ y\nhx : Set.Nonempty (Iic x ∩ s)\nhy : Set.Nonempty (Iic y ∩ s)\n⊢ sSup (f '' (Iic x ∩ s)) ≤ sSup (f '' (Iic y ∩ s))",
"tactic": "refine' csSup_le_csSup (hu' _) (hx.image _) (image_subset _ _)"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\na✝ b : α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x ∩ s))\nhgs : EqOn f g s\nx y : α\nhxy : x ≤ y\nhx : Set.Nonempty (Iic x ∩ s)\nhy : Set.Nonempty (Iic y ∩ s)\n⊢ Iic x ∩ s ⊆ Iic y ∩ s",
"tactic": "exact inter_subset_inter_left _ (Iic_subset_Iic.2 hxy)"
}
] |
[
51,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
28,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.card_one
|
[] |
[
639,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
638,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.lt_mul_self_iff
|
[] |
[
333,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Mathlib/Algebra/Group/Commute.lean
|
Commute.units_inv_left_iff
|
[] |
[
241,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
RelIso.ordinal_type_eq
|
[] |
[
215,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean
|
Ultrafilter.eventually_imp
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.12644\nf g : Ultrafilter α\ns t : Set α\np q : α → Prop\n⊢ (∀ᶠ (x : α) in ↑f, p x → q x) ↔ (∀ᶠ (x : α) in ↑f, p x) → ∀ᶠ (x : α) in ↑f, q x",
"tactic": "simp only [imp_iff_not_or, eventually_or, eventually_not]"
}
] |
[
199,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
LinearEquiv.coord_self
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₂ : Type ?u.393929\nK : Type ?u.393932\nM : Type u_2\nM₂ : Type ?u.393938\nV : Type ?u.393941\nS : Type ?u.393944\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : NoZeroSMulDivisors R M\nx : M\nh : x ≠ 0\n⊢ ↑(coord R M x h) { val := x, property := (_ : x ∈ Submodule.span R {x}) } = 1",
"tactic": "rw [← toSpanNonzeroSingleton_one R M x h, LinearEquiv.symm_apply_apply]"
}
] |
[
1042,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1041,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.toSubgraph_map
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\nf : G →g G'\np : Walk G u v\n⊢ Walk.toSubgraph (Walk.map f p) = Subgraph.map f (Walk.toSubgraph p)",
"tactic": "induction p <;> simp [*, Subgraph.map_sup]"
}
] |
[
2276,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2275,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.add_le_add_three
|
[] |
[
1148,
43
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1146,
11
] |
Mathlib/Algebra/GeomSum.lean
|
geom_sum_inv
|
[
{
"state_after": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\nh₁ : x⁻¹ ≠ 1\n⊢ ∑ i in range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)",
"state_before": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\n⊢ ∑ i in range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)",
"tactic": "have h₁ : x⁻¹ ≠ 1 := by rwa [inv_eq_one_div, Ne.def, div_eq_iff_mul_eq hx0, one_mul]"
},
{
"state_after": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\nh₁ : x⁻¹ ≠ 1\nh₂ : x⁻¹ - 1 ≠ 0\n⊢ ∑ i in range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)",
"state_before": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\nh₁ : x⁻¹ ≠ 1\n⊢ ∑ i in range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)",
"tactic": "have h₂ : x⁻¹ - 1 ≠ 0 := mt sub_eq_zero.1 h₁"
},
{
"state_after": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\nh₁ : x⁻¹ ≠ 1\nh₂ : x⁻¹ - 1 ≠ 0\nh₃ : x - 1 ≠ 0\n⊢ ∑ i in range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)",
"state_before": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\nh₁ : x⁻¹ ≠ 1\nh₂ : x⁻¹ - 1 ≠ 0\n⊢ ∑ i in range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)",
"tactic": "have h₃ : x - 1 ≠ 0 := mt sub_eq_zero.1 hx1"
},
{
"state_after": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\nh₁ : x⁻¹ ≠ 1\nh₂ : x⁻¹ - 1 ≠ 0\nh₃ : x - 1 ≠ 0\nh₄ : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x\n⊢ ∑ i in range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)",
"state_before": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\nh₁ : x⁻¹ ≠ 1\nh₂ : x⁻¹ - 1 ≠ 0\nh₃ : x - 1 ≠ 0\n⊢ ∑ i in range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)",
"tactic": "have h₄ : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x :=\n Nat.recOn n (by simp) fun n h => by\n rw [pow_succ, mul_inv_rev, ← mul_assoc, h, mul_assoc, mul_inv_cancel hx0, mul_assoc,\n inv_mul_cancel hx0]"
},
{
"state_after": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\nh₁ : x⁻¹ ≠ 1\nh₂ : x⁻¹ - 1 ≠ 0\nh₃ : x - 1 ≠ 0\nh₄ : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x\n⊢ 1 * (x - x⁻¹ ^ n * x) * (x⁻¹ - 1) = (x - 1) * (x⁻¹ ^ n - 1)",
"state_before": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\nh₁ : x⁻¹ ≠ 1\nh₂ : x⁻¹ - 1 ≠ 0\nh₃ : x - 1 ≠ 0\nh₄ : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x\n⊢ ∑ i in range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)",
"tactic": "rw [geom_sum_eq h₁, div_eq_iff_mul_eq h₂, ← mul_right_inj' h₃, ← mul_assoc, ← mul_assoc,\n mul_inv_cancel h₃]"
},
{
"state_after": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\nh₁ : x⁻¹ ≠ 1\nh₂ : x⁻¹ - 1 ≠ 0\nh₃ : x - 1 ≠ 0\nh₄ : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x\n⊢ (x ^ n)⁻¹ * x + -x + (-(x ^ n)⁻¹ + 1) = -x + ((x ^ n)⁻¹ * x + -(x ^ n)⁻¹ + 1)",
"state_before": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\nh₁ : x⁻¹ ≠ 1\nh₂ : x⁻¹ - 1 ≠ 0\nh₃ : x - 1 ≠ 0\nh₄ : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x\n⊢ 1 * (x - x⁻¹ ^ n * x) * (x⁻¹ - 1) = (x - 1) * (x⁻¹ ^ n - 1)",
"tactic": "simp [mul_add, add_mul, mul_inv_cancel hx0, mul_assoc, h₄, sub_eq_add_neg, add_comm,\n add_left_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\nh₁ : x⁻¹ ≠ 1\nh₂ : x⁻¹ - 1 ≠ 0\nh₃ : x - 1 ≠ 0\nh₄ : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x\n⊢ (x ^ n)⁻¹ * x + -x + (-(x ^ n)⁻¹ + 1) = -x + ((x ^ n)⁻¹ * x + -(x ^ n)⁻¹ + 1)",
"tactic": "rw [add_comm _ (-x), add_assoc, add_assoc _ _ 1]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\n⊢ x⁻¹ ≠ 1",
"tactic": "rwa [inv_eq_one_div, Ne.def, div_eq_iff_mul_eq hx0, one_mul]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn : ℕ\nh₁ : x⁻¹ ≠ 1\nh₂ : x⁻¹ - 1 ≠ 0\nh₃ : x - 1 ≠ 0\n⊢ x * (x ^ Nat.zero)⁻¹ = (x ^ Nat.zero)⁻¹ * x",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : DivisionRing α\nx : α\nhx1 : x ≠ 1\nhx0 : x ≠ 0\nn✝ : ℕ\nh₁ : x⁻¹ ≠ 1\nh₂ : x⁻¹ - 1 ≠ 0\nh₃ : x - 1 ≠ 0\nn : ℕ\nh : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x\n⊢ x * (x ^ Nat.succ n)⁻¹ = (x ^ Nat.succ n)⁻¹ * x",
"tactic": "rw [pow_succ, mul_inv_rev, ← mul_assoc, h, mul_assoc, mul_inv_cancel hx0, mul_assoc,\n inv_mul_cancel hx0]"
}
] |
[
391,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
1
] |
Mathlib/LinearAlgebra/SesquilinearForm.lean
|
LinearMap.orthogonal_span_singleton_eq_to_lin_ker
|
[
{
"state_after": "case h\nR : Type ?u.247696\nR₁ : Type ?u.247699\nR₂ : Type ?u.247702\nR₃ : Type ?u.247705\nM : Type ?u.247708\nM₁ : Type ?u.247711\nM₂ : Type ?u.247714\nMₗ₁ : Type ?u.247717\nMₗ₁' : Type ?u.247720\nMₗ₂ : Type ?u.247723\nMₗ₂' : Type ?u.247726\nK : Type u_1\nK₁ : Type ?u.247732\nK₂ : Type ?u.247735\nV : Type u_2\nV₁ : Type ?u.247741\nV₂ : Type ?u.247744\nn : Type ?u.247747\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₛₗ[J] K\nx y : V\n⊢ y ∈ Submodule.orthogonalBilin (Submodule.span K {x}) B ↔ y ∈ ker (↑B x)",
"state_before": "R : Type ?u.247696\nR₁ : Type ?u.247699\nR₂ : Type ?u.247702\nR₃ : Type ?u.247705\nM : Type ?u.247708\nM₁ : Type ?u.247711\nM₂ : Type ?u.247714\nMₗ₁ : Type ?u.247717\nMₗ₁' : Type ?u.247720\nMₗ₂ : Type ?u.247723\nMₗ₂' : Type ?u.247726\nK : Type u_1\nK₁ : Type ?u.247732\nK₂ : Type ?u.247735\nV : Type u_2\nV₁ : Type ?u.247741\nV₂ : Type ?u.247744\nn : Type ?u.247747\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₛₗ[J] K\nx : V\n⊢ Submodule.orthogonalBilin (Submodule.span K {x}) B = ker (↑B x)",
"tactic": "ext y"
},
{
"state_after": "case h\nR : Type ?u.247696\nR₁ : Type ?u.247699\nR₂ : Type ?u.247702\nR₃ : Type ?u.247705\nM : Type ?u.247708\nM₁ : Type ?u.247711\nM₂ : Type ?u.247714\nMₗ₁ : Type ?u.247717\nMₗ₁' : Type ?u.247720\nMₗ₂ : Type ?u.247723\nMₗ₂' : Type ?u.247726\nK : Type u_1\nK₁ : Type ?u.247732\nK₂ : Type ?u.247735\nV : Type u_2\nV₁ : Type ?u.247741\nV₂ : Type ?u.247744\nn : Type ?u.247747\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₛₗ[J] K\nx y : V\n⊢ (∀ (n : V), (∃ a, a • x = n) → IsOrtho B n y) ↔ ↑(↑B x) y = 0",
"state_before": "case h\nR : Type ?u.247696\nR₁ : Type ?u.247699\nR₂ : Type ?u.247702\nR₃ : Type ?u.247705\nM : Type ?u.247708\nM₁ : Type ?u.247711\nM₂ : Type ?u.247714\nMₗ₁ : Type ?u.247717\nMₗ₁' : Type ?u.247720\nMₗ₂ : Type ?u.247723\nMₗ₂' : Type ?u.247726\nK : Type u_1\nK₁ : Type ?u.247732\nK₂ : Type ?u.247735\nV : Type u_2\nV₁ : Type ?u.247741\nV₂ : Type ?u.247744\nn : Type ?u.247747\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₛₗ[J] K\nx y : V\n⊢ y ∈ Submodule.orthogonalBilin (Submodule.span K {x}) B ↔ y ∈ ker (↑B x)",
"tactic": "simp_rw [Submodule.mem_orthogonalBilin_iff, LinearMap.mem_ker, Submodule.mem_span_singleton]"
},
{
"state_after": "case h.mp\nR : Type ?u.247696\nR₁ : Type ?u.247699\nR₂ : Type ?u.247702\nR₃ : Type ?u.247705\nM : Type ?u.247708\nM₁ : Type ?u.247711\nM₂ : Type ?u.247714\nMₗ₁ : Type ?u.247717\nMₗ₁' : Type ?u.247720\nMₗ₂ : Type ?u.247723\nMₗ₂' : Type ?u.247726\nK : Type u_1\nK₁ : Type ?u.247732\nK₂ : Type ?u.247735\nV : Type u_2\nV₁ : Type ?u.247741\nV₂ : Type ?u.247744\nn : Type ?u.247747\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₛₗ[J] K\nx y : V\n⊢ (∀ (n : V), (∃ a, a • x = n) → IsOrtho B n y) → ↑(↑B x) y = 0\n\ncase h.mpr\nR : Type ?u.247696\nR₁ : Type ?u.247699\nR₂ : Type ?u.247702\nR₃ : Type ?u.247705\nM : Type ?u.247708\nM₁ : Type ?u.247711\nM₂ : Type ?u.247714\nMₗ₁ : Type ?u.247717\nMₗ₁' : Type ?u.247720\nMₗ₂ : Type ?u.247723\nMₗ₂' : Type ?u.247726\nK : Type u_1\nK₁ : Type ?u.247732\nK₂ : Type ?u.247735\nV : Type u_2\nV₁ : Type ?u.247741\nV₂ : Type ?u.247744\nn : Type ?u.247747\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₛₗ[J] K\nx y : V\n⊢ ↑(↑B x) y = 0 → ∀ (n : V), (∃ a, a • x = n) → IsOrtho B n y",
"state_before": "case h\nR : Type ?u.247696\nR₁ : Type ?u.247699\nR₂ : Type ?u.247702\nR₃ : Type ?u.247705\nM : Type ?u.247708\nM₁ : Type ?u.247711\nM₂ : Type ?u.247714\nMₗ₁ : Type ?u.247717\nMₗ₁' : Type ?u.247720\nMₗ₂ : Type ?u.247723\nMₗ₂' : Type ?u.247726\nK : Type u_1\nK₁ : Type ?u.247732\nK₂ : Type ?u.247735\nV : Type u_2\nV₁ : Type ?u.247741\nV₂ : Type ?u.247744\nn : Type ?u.247747\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₛₗ[J] K\nx y : V\n⊢ (∀ (n : V), (∃ a, a • x = n) → IsOrtho B n y) ↔ ↑(↑B x) y = 0",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case h.mp\nR : Type ?u.247696\nR₁ : Type ?u.247699\nR₂ : Type ?u.247702\nR₃ : Type ?u.247705\nM : Type ?u.247708\nM₁ : Type ?u.247711\nM₂ : Type ?u.247714\nMₗ₁ : Type ?u.247717\nMₗ₁' : Type ?u.247720\nMₗ₂ : Type ?u.247723\nMₗ₂' : Type ?u.247726\nK : Type u_1\nK₁ : Type ?u.247732\nK₂ : Type ?u.247735\nV : Type u_2\nV₁ : Type ?u.247741\nV₂ : Type ?u.247744\nn : Type ?u.247747\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₛₗ[J] K\nx y : V\n⊢ (∀ (n : V), (∃ a, a • x = n) → IsOrtho B n y) → ↑(↑B x) y = 0",
"tactic": "exact fun h ↦ h x ⟨1, one_smul _ _⟩"
},
{
"state_after": "case h.mpr.intro\nR : Type ?u.247696\nR₁ : Type ?u.247699\nR₂ : Type ?u.247702\nR₃ : Type ?u.247705\nM : Type ?u.247708\nM₁ : Type ?u.247711\nM₂ : Type ?u.247714\nMₗ₁ : Type ?u.247717\nMₗ₁' : Type ?u.247720\nMₗ₂ : Type ?u.247723\nMₗ₂' : Type ?u.247726\nK : Type u_1\nK₁ : Type ?u.247732\nK₂ : Type ?u.247735\nV : Type u_2\nV₁ : Type ?u.247741\nV₂ : Type ?u.247744\nn : Type ?u.247747\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₛₗ[J] K\nx y : V\nh : ↑(↑B x) y = 0\nz : K\n⊢ IsOrtho B (z • x) y",
"state_before": "case h.mpr\nR : Type ?u.247696\nR₁ : Type ?u.247699\nR₂ : Type ?u.247702\nR₃ : Type ?u.247705\nM : Type ?u.247708\nM₁ : Type ?u.247711\nM₂ : Type ?u.247714\nMₗ₁ : Type ?u.247717\nMₗ₁' : Type ?u.247720\nMₗ₂ : Type ?u.247723\nMₗ₂' : Type ?u.247726\nK : Type u_1\nK₁ : Type ?u.247732\nK₂ : Type ?u.247735\nV : Type u_2\nV₁ : Type ?u.247741\nV₂ : Type ?u.247744\nn : Type ?u.247747\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₛₗ[J] K\nx y : V\n⊢ ↑(↑B x) y = 0 → ∀ (n : V), (∃ a, a • x = n) → IsOrtho B n y",
"tactic": "rintro h _ ⟨z, rfl⟩"
},
{
"state_after": "case h.mpr.intro\nR : Type ?u.247696\nR₁ : Type ?u.247699\nR₂ : Type ?u.247702\nR₃ : Type ?u.247705\nM : Type ?u.247708\nM₁ : Type ?u.247711\nM₂ : Type ?u.247714\nMₗ₁ : Type ?u.247717\nMₗ₁' : Type ?u.247720\nMₗ₂ : Type ?u.247723\nMₗ₂' : Type ?u.247726\nK : Type u_1\nK₁ : Type ?u.247732\nK₂ : Type ?u.247735\nV : Type u_2\nV₁ : Type ?u.247741\nV₂ : Type ?u.247744\nn : Type ?u.247747\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₛₗ[J] K\nx y : V\nh : ↑(↑B x) y = 0\nz : K\n⊢ ↑(RingHom.id K) z = 0 ∨ ↑(↑B x) y = 0",
"state_before": "case h.mpr.intro\nR : Type ?u.247696\nR₁ : Type ?u.247699\nR₂ : Type ?u.247702\nR₃ : Type ?u.247705\nM : Type ?u.247708\nM₁ : Type ?u.247711\nM₂ : Type ?u.247714\nMₗ₁ : Type ?u.247717\nMₗ₁' : Type ?u.247720\nMₗ₂ : Type ?u.247723\nMₗ₂' : Type ?u.247726\nK : Type u_1\nK₁ : Type ?u.247732\nK₂ : Type ?u.247735\nV : Type u_2\nV₁ : Type ?u.247741\nV₂ : Type ?u.247744\nn : Type ?u.247747\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₛₗ[J] K\nx y : V\nh : ↑(↑B x) y = 0\nz : K\n⊢ IsOrtho B (z • x) y",
"tactic": "rw [isOrtho_def, map_smulₛₗ₂, smul_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro\nR : Type ?u.247696\nR₁ : Type ?u.247699\nR₂ : Type ?u.247702\nR₃ : Type ?u.247705\nM : Type ?u.247708\nM₁ : Type ?u.247711\nM₂ : Type ?u.247714\nMₗ₁ : Type ?u.247717\nMₗ₁' : Type ?u.247720\nMₗ₂ : Type ?u.247723\nMₗ₂' : Type ?u.247726\nK : Type u_1\nK₁ : Type ?u.247732\nK₂ : Type ?u.247735\nV : Type u_2\nV₁ : Type ?u.247741\nV₂ : Type ?u.247744\nn : Type ?u.247747\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : Field K₁\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K₁ V₁\nJ : K →+* K\nJ₁ J₁' : K₁ →+* K\nB : V →ₗ[K] V →ₛₗ[J] K\nx y : V\nh : ↑(↑B x) y = 0\nz : K\n⊢ ↑(RingHom.id K) z = 0 ∨ ↑(↑B x) y = 0",
"tactic": "exact Or.intro_right _ h"
}
] |
[
386,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.join_congr
|
[] |
[
1084,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1081,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.map_eq_of_inverse
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.313560\nι : Sort x\nf : Filter α\ng : Filter β\nφ : α → β\nψ : β → α\neq : φ ∘ ψ = id\nhφ : Tendsto φ f g\nhψ : Tendsto ψ g f\n⊢ g ≤ map φ (map ψ g)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.313560\nι : Sort x\nf : Filter α\ng : Filter β\nφ : α → β\nψ : β → α\neq : φ ∘ ψ = id\nhφ : Tendsto φ f g\nhψ : Tendsto ψ g f\n⊢ map φ f = g",
"tactic": "refine' le_antisymm hφ (le_trans _ (map_mono hψ))"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.313560\nι : Sort x\nf : Filter α\ng : Filter β\nφ : α → β\nψ : β → α\neq : φ ∘ ψ = id\nhφ : Tendsto φ f g\nhψ : Tendsto ψ g f\n⊢ g ≤ map φ (map ψ g)",
"tactic": "rw [map_map, eq, map_id]"
}
] |
[
2981,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2978,
1
] |
Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
|
Finpartition.isUniformOne
|
[
{
"state_after": "α : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\n⊢ card (nonUniforms P G 1) ≤ card P.parts * (card P.parts - 1)",
"state_before": "α : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\n⊢ IsUniform P G 1",
"tactic": "rw [IsUniform, mul_one, Nat.cast_le]"
},
{
"state_after": "α : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\n⊢ card (offDiag P.parts) ≤ card P.parts * (card P.parts - 1)",
"state_before": "α : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\n⊢ card (nonUniforms P G 1) ≤ card P.parts * (card P.parts - 1)",
"tactic": "refine' (card_filter_le _\n (fun uv => ¬SimpleGraph.IsUniform G 1 (Prod.fst uv) (Prod.snd uv))).trans _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\n𝕜 : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : DecidableEq α\nA : Finset α\nP : Finpartition A\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\n⊢ card (offDiag P.parts) ≤ card P.parts * (card P.parts - 1)",
"tactic": "rw [offDiag_card, Nat.mul_sub_left_distrib, mul_one]"
}
] |
[
246,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Icc_subset_Iic_iff
|
[] |
[
582,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
581,
1
] |
Mathlib/GroupTheory/Coset.lean
|
Subgroup.quotientSubgroupOfMapOfLe_apply_mk
|
[] |
[
720,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
717,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
smul_pow
|
[
{
"state_after": "case zero\nα : Type ?u.64314\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝⁶ : Monoid M\ninst✝⁵ : Monoid N\ninst✝⁴ : AddMonoid A\ninst✝³ : AddMonoid B\ninst✝² : MulAction M N\ninst✝¹ : IsScalarTower M N N\ninst✝ : SMulCommClass M N N\nk : M\nx : N\n⊢ (k • x) ^ Nat.zero = k ^ Nat.zero • x ^ Nat.zero\n\ncase succ\nα : Type ?u.64314\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝⁶ : Monoid M\ninst✝⁵ : Monoid N\ninst✝⁴ : AddMonoid A\ninst✝³ : AddMonoid B\ninst✝² : MulAction M N\ninst✝¹ : IsScalarTower M N N\ninst✝ : SMulCommClass M N N\nk : M\nx : N\np : ℕ\nIH : (k • x) ^ p = k ^ p • x ^ p\n⊢ (k • x) ^ Nat.succ p = k ^ Nat.succ p • x ^ Nat.succ p",
"state_before": "α : Type ?u.64314\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝⁶ : Monoid M\ninst✝⁵ : Monoid N\ninst✝⁴ : AddMonoid A\ninst✝³ : AddMonoid B\ninst✝² : MulAction M N\ninst✝¹ : IsScalarTower M N N\ninst✝ : SMulCommClass M N N\nk : M\nx : N\np : ℕ\n⊢ (k • x) ^ p = k ^ p • x ^ p",
"tactic": "induction' p with p IH"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type ?u.64314\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝⁶ : Monoid M\ninst✝⁵ : Monoid N\ninst✝⁴ : AddMonoid A\ninst✝³ : AddMonoid B\ninst✝² : MulAction M N\ninst✝¹ : IsScalarTower M N N\ninst✝ : SMulCommClass M N N\nk : M\nx : N\n⊢ (k • x) ^ Nat.zero = k ^ Nat.zero • x ^ Nat.zero",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type ?u.64314\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝⁶ : Monoid M\ninst✝⁵ : Monoid N\ninst✝⁴ : AddMonoid A\ninst✝³ : AddMonoid B\ninst✝² : MulAction M N\ninst✝¹ : IsScalarTower M N N\ninst✝ : SMulCommClass M N N\nk : M\nx : N\np : ℕ\nIH : (k • x) ^ p = k ^ p • x ^ p\n⊢ (k • x) ^ Nat.succ p = k ^ Nat.succ p • x ^ Nat.succ p",
"tactic": "rw [pow_succ', IH, smul_mul_smul, ← pow_succ', ← pow_succ']"
}
] |
[
117,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Order/Atoms.lean
|
Set.isCoatom_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.81807\ns : Set α\n⊢ (∃ x, sᶜ = {x}) ↔ ∃ x, s = {x}ᶜ",
"state_before": "α : Type u_1\nβ : Type ?u.81807\ns : Set α\n⊢ IsCoatom s ↔ ∃ x, s = {x}ᶜ",
"tactic": "rw [isCompl_compl.isCoatom_iff_isAtom, isAtom_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.81807\ns : Set α\n⊢ (∃ x, sᶜ = {x}) ↔ ∃ x, s = {x}ᶜ",
"tactic": "simp_rw [@eq_comm _ s, compl_eq_comm]"
}
] |
[
906,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
904,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto
|
[
{
"state_after": "α : Type u_2\nι : Type u_1\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NeBot l\ninst✝ : IsCountablyGenerated l\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nI : ℝ\nhfm : AEStronglyMeasurable f μ\nhtendsto : Tendsto (fun i => ∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ) l (𝓝 (ENNReal.ofReal I))\n⊢ ∀ᶠ (i : ι) in l, (∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ) ≤ ENNReal.ofReal (max 1 (I + 1))",
"state_before": "α : Type u_2\nι : Type u_1\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NeBot l\ninst✝ : IsCountablyGenerated l\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nI : ℝ\nhfm : AEStronglyMeasurable f μ\nhtendsto : Tendsto (fun i => ∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ) l (𝓝 (ENNReal.ofReal I))\n⊢ Integrable f",
"tactic": "refine' hφ.integrable_of_lintegral_nnnorm_bounded (max 1 (I + 1)) hfm _"
},
{
"state_after": "α : Type u_2\nι : Type u_1\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NeBot l\ninst✝ : IsCountablyGenerated l\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nI : ℝ\nhfm : AEStronglyMeasurable f μ\nhtendsto : Tendsto (fun i => ∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ) l (𝓝 (ENNReal.ofReal I))\n⊢ ENNReal.ofReal I < ENNReal.ofReal (max 1 (I + 1))",
"state_before": "α : Type u_2\nι : Type u_1\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NeBot l\ninst✝ : IsCountablyGenerated l\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nI : ℝ\nhfm : AEStronglyMeasurable f μ\nhtendsto : Tendsto (fun i => ∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ) l (𝓝 (ENNReal.ofReal I))\n⊢ ∀ᶠ (i : ι) in l, (∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ) ≤ ENNReal.ofReal (max 1 (I + 1))",
"tactic": "refine' htendsto.eventually (ge_mem_nhds _)"
},
{
"state_after": "α : Type u_2\nι : Type u_1\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NeBot l\ninst✝ : IsCountablyGenerated l\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nI : ℝ\nhfm : AEStronglyMeasurable f μ\nhtendsto : Tendsto (fun i => ∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ) l (𝓝 (ENNReal.ofReal I))\n⊢ I < max 1 (I + 1)",
"state_before": "α : Type u_2\nι : Type u_1\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NeBot l\ninst✝ : IsCountablyGenerated l\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nI : ℝ\nhfm : AEStronglyMeasurable f μ\nhtendsto : Tendsto (fun i => ∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ) l (𝓝 (ENNReal.ofReal I))\n⊢ ENNReal.ofReal I < ENNReal.ofReal (max 1 (I + 1))",
"tactic": "refine' (ENNReal.ofReal_lt_ofReal_iff (lt_max_of_lt_left zero_lt_one)).2 _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nι : Type u_1\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NeBot l\ninst✝ : IsCountablyGenerated l\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nI : ℝ\nhfm : AEStronglyMeasurable f μ\nhtendsto : Tendsto (fun i => ∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ) l (𝓝 (ENNReal.ofReal I))\n⊢ I < max 1 (I + 1)",
"tactic": "exact lt_max_of_lt_right (lt_add_one I)"
}
] |
[
422,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
415,
1
] |
Mathlib/Order/Lattice.lean
|
inf_eq_inf_iff_right
|
[] |
[
556,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
555,
1
] |
Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean
|
MeasureTheory.Lp.ae_eq_of_forall_set_integral_eq
|
[] |
[
489,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
482,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.mul_rpow
|
[
{
"state_after": "case inl.inr.hx\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : x = 0 ∨ y = 0\nh✝ : ¬z = 0\n⊢ 0 ≤ x\n\ncase inr\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬(x = 0 ∨ y = 0)\n⊢ exp (log (x * y) * z) = x ^ z * y ^ z\n\ncase hx\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\n⊢ 0 ≤ x * y",
"state_before": "x✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\n⊢ (x * y) ^ z = x ^ z * y ^ z",
"tactic": "iterate 2 rw [Real.rpow_def_of_nonneg]; split_ifs with h_ifs <;> simp_all"
},
{
"state_after": "case inl.inr\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : x = 0 ∨ y = 0\nh✝ : ¬z = 0\n⊢ (if x = 0 then if z = 0 then 1 else 0 else exp (log x * z)) = 0 ∨ y ^ z = 0\n\ncase inl.inr.hx\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : x = 0 ∨ y = 0\nh✝ : ¬z = 0\n⊢ 0 ≤ x\n\ncase inr\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬(x = 0 ∨ y = 0)\n⊢ exp (log (x * y) * z) = x ^ z * y ^ z\n\ncase hx\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\n⊢ 0 ≤ x * y",
"state_before": "case inl.inr\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : x = 0 ∨ y = 0\nh✝ : ¬z = 0\n⊢ x ^ z = 0 ∨ y ^ z = 0\n\ncase inr\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬(x = 0 ∨ y = 0)\n⊢ exp (log (x * y) * z) = x ^ z * y ^ z\n\ncase hx\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\n⊢ 0 ≤ x * y",
"tactic": "rw [Real.rpow_def_of_nonneg]"
},
{
"state_after": "case inl.inr.hx\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : x = 0 ∨ y = 0\nh✝ : ¬z = 0\n⊢ 0 ≤ x\n\ncase inr\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬(x = 0 ∨ y = 0)\n⊢ exp (log (x * y) * z) = x ^ z * y ^ z\n\ncase hx\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\n⊢ 0 ≤ x * y",
"state_before": "case inl.inr\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : x = 0 ∨ y = 0\nh✝ : ¬z = 0\n⊢ (if x = 0 then if z = 0 then 1 else 0 else exp (log x * z)) = 0 ∨ y ^ z = 0\n\ncase inl.inr.hx\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : x = 0 ∨ y = 0\nh✝ : ¬z = 0\n⊢ 0 ≤ x\n\ncase inr\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬(x = 0 ∨ y = 0)\n⊢ exp (log (x * y) * z) = x ^ z * y ^ z\n\ncase hx\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\n⊢ 0 ≤ x * y",
"tactic": "split_ifs with h_ifs <;> simp_all"
},
{
"state_after": "no goals",
"state_before": "case inl.inr.hx\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : x = 0 ∨ y = 0\nh✝ : ¬z = 0\n⊢ 0 ≤ x",
"tactic": "exact h"
},
{
"state_after": "case inr\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\n⊢ exp (log (x * y) * z) = x ^ z * y ^ z",
"state_before": "case inr\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬(x = 0 ∨ y = 0)\n⊢ exp (log (x * y) * z) = x ^ z * y ^ z",
"tactic": "rw [not_or] at h_ifs"
},
{
"state_after": "no goals",
"state_before": "case inr\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nhx : 0 < x\nhy : 0 < y\n⊢ exp (log (x * y) * z) = x ^ z * y ^ z",
"tactic": "rw [log_mul (ne_of_gt hx) (ne_of_gt hy), add_mul, exp_add, rpow_def_of_pos hx,\n rpow_def_of_pos hy]"
},
{
"state_after": "case inl\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nh₂ : 0 < x\n⊢ 0 < x\n\ncase inr\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nh₂ : 0 = x\n⊢ 0 < x",
"state_before": "x✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\n⊢ 0 < x",
"tactic": "cases' lt_or_eq_of_le h with h₂ h₂"
},
{
"state_after": "case inr.h\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nh₂ : 0 = x\n⊢ False",
"state_before": "case inr\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nh₂ : 0 = x\n⊢ 0 < x",
"tactic": "exfalso"
},
{
"state_after": "case inr.h\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nh₂ : 0 = x\n⊢ x = 0",
"state_before": "case inr.h\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nh₂ : 0 = x\n⊢ False",
"tactic": "apply h_ifs.1"
},
{
"state_after": "no goals",
"state_before": "case inr.h\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nh₂ : 0 = x\n⊢ x = 0",
"tactic": "exact Eq.symm h₂"
},
{
"state_after": "no goals",
"state_before": "case inl\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nh₂ : 0 < x\n⊢ 0 < x",
"tactic": "exact h₂"
},
{
"state_after": "case inl\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nhx : 0 < x\nh₂ : 0 < y\n⊢ 0 < y\n\ncase inr\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nhx : 0 < x\nh₂ : 0 = y\n⊢ 0 < y",
"state_before": "x✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nhx : 0 < x\n⊢ 0 < y",
"tactic": "cases' lt_or_eq_of_le h₁ with h₂ h₂"
},
{
"state_after": "case inr.h\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nhx : 0 < x\nh₂ : 0 = y\n⊢ False",
"state_before": "case inr\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nhx : 0 < x\nh₂ : 0 = y\n⊢ 0 < y",
"tactic": "exfalso"
},
{
"state_after": "case inr.h\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nhx : 0 < x\nh₂ : 0 = y\n⊢ y = 0",
"state_before": "case inr.h\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nhx : 0 < x\nh₂ : 0 = y\n⊢ False",
"tactic": "apply h_ifs.2"
},
{
"state_after": "no goals",
"state_before": "case inr.h\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nhx : 0 < x\nh₂ : 0 = y\n⊢ y = 0",
"tactic": "exact Eq.symm h₂"
},
{
"state_after": "no goals",
"state_before": "case inl\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\nhx : 0 < x\nh₂ : 0 < y\n⊢ 0 < y",
"tactic": "exact h₂"
},
{
"state_after": "no goals",
"state_before": "case hx\nx✝ y✝ z✝ x y z : ℝ\nh : 0 ≤ x\nh₁ : 0 ≤ y\n⊢ 0 ≤ x * y",
"tactic": "exact mul_nonneg h h₁"
}
] |
[
393,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Data/MvPolynomial/CommRing.lean
|
MvPolynomial.totalDegree_neg
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommRing R\np q a : MvPolynomial σ R\n⊢ totalDegree (-a) = totalDegree a",
"tactic": "simp only [totalDegree, support_neg]"
}
] |
[
203,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Data/List/Basic.lean
|
List.splitOnP_first
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.302857\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nh : ∀ (x : α), x ∈ xs → ¬p x = true\nsep : α\nhsep : p sep = true\nas : List α\n⊢ splitOnP p (xs ++ sep :: as) = xs :: splitOnP p as",
"tactic": "induction xs with\n| nil => simp [hsep]\n| cons hd tl ih => simp [h hd _, ih <| forall_mem_of_forall_mem_cons h]"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.302857\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nsep : α\nhsep : p sep = true\nas : List α\nh : ∀ (x : α), x ∈ [] → ¬p x = true\n⊢ splitOnP p ([] ++ sep :: as) = [] :: splitOnP p as",
"tactic": "simp [hsep]"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type ?u.302857\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nsep : α\nhsep : p sep = true\nas : List α\nhd : α\ntl : List α\nih : (∀ (x : α), x ∈ tl → ¬p x = true) → splitOnP p (tl ++ sep :: as) = tl :: splitOnP p as\nh : ∀ (x : α), x ∈ hd :: tl → ¬p x = true\n⊢ splitOnP p (hd :: tl ++ sep :: as) = (hd :: tl) :: splitOnP p as",
"tactic": "simp [h hd _, ih <| forall_mem_of_forall_mem_cons h]"
}
] |
[
2981,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2977,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.sup_def
|
[] |
[
46,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.coeff_C_ne_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\nh : n ≠ 0\n⊢ coeff (↑C a) n = 0",
"tactic": "rw [coeff_C, if_neg h]"
}
] |
[
730,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
730,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.coe_zero
|
[] |
[
292,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
291,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_apply_dite
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.387864\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ns : Finset α\np : α → Prop\nhp : DecidablePred p\ninst✝ : DecidablePred fun x => ¬p x\nf : (x : α) → p x → γ\ng : (x : α) → ¬p x → γ\nh : γ → β\nx : { x // x ∈ filter p s }\n⊢ p ↑x",
"tactic": "simpa using (mem_filter.mp x.2).2"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.387864\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ns : Finset α\np : α → Prop\nhp : DecidablePred p\ninst✝ : DecidablePred fun x => ¬p x\nf : (x : α) → p x → γ\ng : (x : α) → ¬p x → γ\nh : γ → β\nx : { x // x ∈ filter (fun x => ¬p x) s }\n⊢ ¬p ↑x",
"tactic": "simpa using (mem_filter.mp x.2).2"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.387864\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ns : Finset α\np : α → Prop\nhp : DecidablePred p\ninst✝ : DecidablePred fun x => ¬p x\nf : (x : α) → p x → γ\ng : (x : α) → ¬p x → γ\nh : γ → β\nx : { x // x ∈ filter p s }\n⊢ p ↑x",
"tactic": "simpa using (mem_filter.mp x.2).2"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.387864\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ns : Finset α\np : α → Prop\nhp : DecidablePred p\ninst✝ : DecidablePred fun x => ¬p x\nf : (x : α) → p x → γ\ng : (x : α) → ¬p x → γ\nh : γ → β\nx : { x // x ∈ filter (fun x => ¬p x) s }\n⊢ ¬p ↑x",
"tactic": "simpa using (mem_filter.mp x.2).2"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.387864\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ns : Finset α\np : α → Prop\nhp : DecidablePred p\ninst✝ : DecidablePred fun x => ¬p x\nf : (x : α) → p x → γ\ng : (x : α) → ¬p x → γ\nh : γ → β\nx : { x // x ∈ filter p s }\n_hx : x ∈ attach (filter p s)\n⊢ p ↑x",
"tactic": "simpa using (mem_filter.mp x.2).2"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.387864\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ns : Finset α\np : α → Prop\nhp : DecidablePred p\ninst✝ : DecidablePred fun x => ¬p x\nf : (x : α) → p x → γ\ng : (x : α) → ¬p x → γ\nh : γ → β\nx : { x // x ∈ filter (fun x => ¬p x) s }\n_hx : x ∈ attach (filter (fun x => ¬p x) s)\n⊢ ¬p ↑x",
"tactic": "simpa using (mem_filter.mp x.2).2"
}
] |
[
968,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
951,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.measure_le_sInf
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.152195\nγ : Type ?u.152198\nδ : Type ?u.152201\nι : Type ?u.152204\nR : Type ?u.152207\nR' : Type ?u.152210\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nm : Set (Measure α)\nh : ∀ (μ' : Measure α), μ' ∈ m → μ ≤ μ'\nthis : ↑μ ≤ sInf (toOuterMeasure '' m)\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑μ s ≤ ↑(sInf (toOuterMeasure '' m)) s",
"state_before": "α : Type u_1\nβ : Type ?u.152195\nγ : Type ?u.152198\nδ : Type ?u.152201\nι : Type ?u.152204\nR : Type ?u.152207\nR' : Type ?u.152210\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nm : Set (Measure α)\nh : ∀ (μ' : Measure α), μ' ∈ m → μ ≤ μ'\nthis : ↑μ ≤ sInf (toOuterMeasure '' m)\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑μ s ≤ ↑↑(sInf m) s",
"tactic": "rw [sInf_apply hs]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.152195\nγ : Type ?u.152198\nδ : Type ?u.152201\nι : Type ?u.152204\nR : Type ?u.152207\nR' : Type ?u.152210\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nm : Set (Measure α)\nh : ∀ (μ' : Measure α), μ' ∈ m → μ ≤ μ'\nthis : ↑μ ≤ sInf (toOuterMeasure '' m)\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑μ s ≤ ↑(sInf (toOuterMeasure '' m)) s",
"tactic": "exact this s"
}
] |
[
1033,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1030,
9
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
lp.coeFn_add
|
[] |
[
374,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.map_iSup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ✝ : Type ?u.80158\nR : Type ?u.80161\nR' : Type ?u.80164\nms : Set (OuterMeasure α)\nm✝ : OuterMeasure α\nβ : Type u_1\nι : Sort u_2\nf : α → β\nm : ι → OuterMeasure α\ns : Set β\n⊢ ↑(↑(map f) (⨆ (i : ι), m i)) s = ↑(⨆ (i : ι), ↑(map f) (m i)) s",
"tactic": "simp only [map_apply, iSup_apply]"
}
] |
[
483,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
482,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsLittleO.add_isBigO
|
[] |
[
1091,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1090,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snorm_const_smul
|
[
{
"state_after": "case inl\nα : Type u_1\nE : Type ?u.5983830\nF : Type u_2\nG : Type ?u.5983836\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\n𝕜 : Type u_3\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : MulActionWithZero 𝕜 E\ninst✝² : Module 𝕜 F\ninst✝¹ : BoundedSMul 𝕜 E\ninst✝ : BoundedSMul 𝕜 F\nf : α → F\n⊢ snorm (0 • f) p μ = ↑‖0‖₊ * snorm f p μ\n\ncase inr\nα : Type u_1\nE : Type ?u.5983830\nF : Type u_2\nG : Type ?u.5983836\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\n𝕜 : Type u_3\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : MulActionWithZero 𝕜 E\ninst✝² : Module 𝕜 F\ninst✝¹ : BoundedSMul 𝕜 E\ninst✝ : BoundedSMul 𝕜 F\nc : 𝕜\nf : α → F\nhc : c ≠ 0\n⊢ snorm (c • f) p μ = ↑‖c‖₊ * snorm f p μ",
"state_before": "α : Type u_1\nE : Type ?u.5983830\nF : Type u_2\nG : Type ?u.5983836\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\n𝕜 : Type u_3\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : MulActionWithZero 𝕜 E\ninst✝² : Module 𝕜 F\ninst✝¹ : BoundedSMul 𝕜 E\ninst✝ : BoundedSMul 𝕜 F\nc : 𝕜\nf : α → F\n⊢ snorm (c • f) p μ = ↑‖c‖₊ * snorm f p μ",
"tactic": "obtain rfl | hc := eq_or_ne c 0"
},
{
"state_after": "case inr\nα : Type u_1\nE : Type ?u.5983830\nF : Type u_2\nG : Type ?u.5983836\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\n𝕜 : Type u_3\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : MulActionWithZero 𝕜 E\ninst✝² : Module 𝕜 F\ninst✝¹ : BoundedSMul 𝕜 E\ninst✝ : BoundedSMul 𝕜 F\nc : 𝕜\nf : α → F\nhc : c ≠ 0\n⊢ ↑‖c‖₊ * snorm f p μ ≤ snorm (c • f) p μ",
"state_before": "case inr\nα : Type u_1\nE : Type ?u.5983830\nF : Type u_2\nG : Type ?u.5983836\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\n𝕜 : Type u_3\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : MulActionWithZero 𝕜 E\ninst✝² : Module 𝕜 F\ninst✝¹ : BoundedSMul 𝕜 E\ninst✝ : BoundedSMul 𝕜 F\nc : 𝕜\nf : α → F\nhc : c ≠ 0\n⊢ snorm (c • f) p μ = ↑‖c‖₊ * snorm f p μ",
"tactic": "refine' le_antisymm (snorm_const_smul_le _ _) _"
},
{
"state_after": "case inr\nα : Type u_1\nE : Type ?u.5983830\nF : Type u_2\nG : Type ?u.5983836\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\n𝕜 : Type u_3\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : MulActionWithZero 𝕜 E\ninst✝² : Module 𝕜 F\ninst✝¹ : BoundedSMul 𝕜 E\ninst✝ : BoundedSMul 𝕜 F\nc : 𝕜\nf : α → F\nhc : c ≠ 0\nthis : snorm (c⁻¹ • c • f) p μ ≤ ‖c⁻¹‖₊ • snorm (c • f) p μ\n⊢ ↑‖c‖₊ * snorm f p μ ≤ snorm (c • f) p μ",
"state_before": "case inr\nα : Type u_1\nE : Type ?u.5983830\nF : Type u_2\nG : Type ?u.5983836\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\n𝕜 : Type u_3\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : MulActionWithZero 𝕜 E\ninst✝² : Module 𝕜 F\ninst✝¹ : BoundedSMul 𝕜 E\ninst✝ : BoundedSMul 𝕜 F\nc : 𝕜\nf : α → F\nhc : c ≠ 0\n⊢ ↑‖c‖₊ * snorm f p μ ≤ snorm (c • f) p μ",
"tactic": "have : snorm _ p μ ≤ _ := snorm_const_smul_le c⁻¹ (c • f)"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nE : Type ?u.5983830\nF : Type u_2\nG : Type ?u.5983836\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\n𝕜 : Type u_3\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : MulActionWithZero 𝕜 E\ninst✝² : Module 𝕜 F\ninst✝¹ : BoundedSMul 𝕜 E\ninst✝ : BoundedSMul 𝕜 F\nc : 𝕜\nf : α → F\nhc : c ≠ 0\nthis : snorm (c⁻¹ • c • f) p μ ≤ ‖c⁻¹‖₊ • snorm (c • f) p μ\n⊢ ↑‖c‖₊ * snorm f p μ ≤ snorm (c • f) p μ",
"tactic": "rwa [inv_smul_smul₀ hc, nnnorm_inv, ENNReal.le_inv_smul_iff (nnnorm_ne_zero_iff.mpr hc)] at this"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nE : Type ?u.5983830\nF : Type u_2\nG : Type ?u.5983836\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\n𝕜 : Type u_3\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : MulActionWithZero 𝕜 E\ninst✝² : Module 𝕜 F\ninst✝¹ : BoundedSMul 𝕜 E\ninst✝ : BoundedSMul 𝕜 F\nf : α → F\n⊢ snorm (0 • f) p μ = ↑‖0‖₊ * snorm f p μ",
"tactic": "simp"
}
] |
[
1574,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1568,
1
] |
Mathlib/Data/Polynomial/Splits.lean
|
Polynomial.splits_prod_iff
|
[
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nι : Type u\ns : ι → K[X]\nt✝ : Finset ι\na : ι\nt : Finset ι\nhat : ¬a ∈ t\nih : (∀ (j : ι), j ∈ t → s j ≠ 0) → (Splits i (∏ x in t, s x) ↔ ∀ (j : ι), j ∈ t → Splits i (s j))\nht : ∀ (j : ι), j ∈ insert a t → s j ≠ 0\n⊢ Splits i (∏ x in insert a t, s x) ↔ ∀ (j : ι), j ∈ insert a t → Splits i (s j)",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nι : Type u\ns : ι → K[X]\nt : Finset ι\n⊢ (∀ (j : ι), j ∈ t → s j ≠ 0) → (Splits i (∏ x in t, s x) ↔ ∀ (j : ι), j ∈ t → Splits i (s j))",
"tactic": "refine'\n Finset.induction_on t (fun _ =>\n ⟨fun _ _ h => by simp only [Finset.not_mem_empty] at h, fun _ => splits_one i⟩)\n fun a t hat ih ht => _"
},
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nι : Type u\ns : ι → K[X]\nt✝ : Finset ι\na : ι\nt : Finset ι\nhat : ¬a ∈ t\nih : (∀ (j : ι), j ∈ t → s j ≠ 0) → (Splits i (∏ x in t, s x) ↔ ∀ (j : ι), j ∈ t → Splits i (s j))\nht : s a ≠ 0 ∧ ∀ (x : ι), x ∈ t → s x ≠ 0\n⊢ Splits i (∏ x in insert a t, s x) ↔ Splits i (s a) ∧ ∀ (x : ι), x ∈ t → Splits i (s x)",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nι : Type u\ns : ι → K[X]\nt✝ : Finset ι\na : ι\nt : Finset ι\nhat : ¬a ∈ t\nih : (∀ (j : ι), j ∈ t → s j ≠ 0) → (Splits i (∏ x in t, s x) ↔ ∀ (j : ι), j ∈ t → Splits i (s j))\nht : ∀ (j : ι), j ∈ insert a t → s j ≠ 0\n⊢ Splits i (∏ x in insert a t, s x) ↔ ∀ (j : ι), j ∈ insert a t → Splits i (s j)",
"tactic": "rw [Finset.forall_mem_insert] at ht⊢"
},
{
"state_after": "no goals",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nι : Type u\ns : ι → K[X]\nt✝ : Finset ι\na : ι\nt : Finset ι\nhat : ¬a ∈ t\nih : (∀ (j : ι), j ∈ t → s j ≠ 0) → (Splits i (∏ x in t, s x) ↔ ∀ (j : ι), j ∈ t → Splits i (s j))\nht : s a ≠ 0 ∧ ∀ (x : ι), x ∈ t → s x ≠ 0\n⊢ Splits i (∏ x in insert a t, s x) ↔ Splits i (s a) ∧ ∀ (x : ι), x ∈ t → Splits i (s x)",
"tactic": "rw [Finset.prod_insert hat, splits_mul_iff i ht.1 (Finset.prod_ne_zero_iff.2 ht.2), ih ht.2]"
},
{
"state_after": "no goals",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nι : Type u\ns : ι → K[X]\nt : Finset ι\nx✝² : ∀ (j : ι), j ∈ ∅ → s j ≠ 0\nx✝¹ : Splits i (∏ x in ∅, s x)\nx✝ : ι\nh : x✝ ∈ ∅\n⊢ Splits i (s x✝)",
"tactic": "simp only [Finset.not_mem_empty] at h"
}
] |
[
281,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.card_eq_of_bijective
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.20200\ns t : Finset α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\n⊢ card s = n",
"tactic": "classical\n have : ∀ a : α, a ∈ s ↔ ∃ (i : _)(hi : i ∈ range n), f i (mem_range.1 hi) = a := fun a =>\n ⟨fun ha =>\n let ⟨i, hi, eq⟩ := hf a ha\n ⟨i, mem_range.2 hi, eq⟩,\n fun ⟨i, hi, eq⟩ => eq ▸ hf' i (mem_range.1 hi)⟩\n have : s = (range n).attach.image fun i => f i.1 (mem_range.1 i.2) := by\n simpa only [ext_iff, mem_image, exists_prop, Subtype.exists, mem_attach, true_and_iff]\n calc\n s.card = card ((range n).attach.image fun i => f i.1 (mem_range.1 i.2)) := by rw [this]\n _ = card (range n).attach :=\n (card_image_of_injective _) fun ⟨i, hi⟩ ⟨j, hj⟩ eq =>\n Subtype.eq <| f_inj i j (mem_range.1 hi) (mem_range.1 hj) eq\n _ = card (range n) := card_attach\n _ = n := card_range n"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.20200\ns t : Finset α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nthis : ∀ (a : α), a ∈ s ↔ ∃ i hi, f i (_ : i < n) = a\n⊢ card s = n",
"state_before": "α : Type u_1\nβ : Type ?u.20200\ns t : Finset α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\n⊢ card s = n",
"tactic": "have : ∀ a : α, a ∈ s ↔ ∃ (i : _)(hi : i ∈ range n), f i (mem_range.1 hi) = a := fun a =>\n ⟨fun ha =>\n let ⟨i, hi, eq⟩ := hf a ha\n ⟨i, mem_range.2 hi, eq⟩,\n fun ⟨i, hi, eq⟩ => eq ▸ hf' i (mem_range.1 hi)⟩"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.20200\ns t : Finset α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nthis✝ : ∀ (a : α), a ∈ s ↔ ∃ i hi, f i (_ : i < n) = a\nthis : s = image (fun i => f ↑i (_ : ↑i < n)) (attach (range n))\n⊢ card s = n",
"state_before": "α : Type u_1\nβ : Type ?u.20200\ns t : Finset α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nthis : ∀ (a : α), a ∈ s ↔ ∃ i hi, f i (_ : i < n) = a\n⊢ card s = n",
"tactic": "have : s = (range n).attach.image fun i => f i.1 (mem_range.1 i.2) := by\n simpa only [ext_iff, mem_image, exists_prop, Subtype.exists, mem_attach, true_and_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.20200\ns t : Finset α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nthis✝ : ∀ (a : α), a ∈ s ↔ ∃ i hi, f i (_ : i < n) = a\nthis : s = image (fun i => f ↑i (_ : ↑i < n)) (attach (range n))\n⊢ card s = n",
"tactic": "calc\n s.card = card ((range n).attach.image fun i => f i.1 (mem_range.1 i.2)) := by rw [this]\n _ = card (range n).attach :=\n (card_image_of_injective _) fun ⟨i, hi⟩ ⟨j, hj⟩ eq =>\n Subtype.eq <| f_inj i j (mem_range.1 hi) (mem_range.1 hj) eq\n _ = card (range n) := card_attach\n _ = n := card_range n"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.20200\ns t : Finset α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nthis : ∀ (a : α), a ∈ s ↔ ∃ i hi, f i (_ : i < n) = a\n⊢ s = image (fun i => f ↑i (_ : ↑i < n)) (attach (range n))",
"tactic": "simpa only [ext_iff, mem_image, exists_prop, Subtype.exists, mem_attach, true_and_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.20200\ns t : Finset α\nf✝ : α → β\nn : ℕ\nf : (i : ℕ) → i < n → α\nhf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a\nhf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s\nf_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j\nthis✝ : ∀ (a : α), a ∈ s ↔ ∃ i hi, f i (_ : i < n) = a\nthis : s = image (fun i => f ↑i (_ : ↑i < n)) (attach (range n))\n⊢ card s = card (image (fun i => f ↑i (_ : ↑i < n)) (attach (range n)))",
"tactic": "rw [this]"
}
] |
[
309,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/Data/Finsupp/NeLocus.lean
|
Finsupp.neLocus_add_right
|
[] |
[
132,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/Logic/Encodable/Basic.lean
|
Encodable.decode₂_eq_some
|
[] |
[
205,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.eq_of_same_basis
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.11236\nγ : Type ?u.11239\nι : Sort u_2\nι' : Sort ?u.11245\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p s\nt : Set α\n⊢ t ∈ l ↔ t ∈ l'",
"state_before": "α : Type u_1\nβ : Type ?u.11236\nγ : Type ?u.11239\nι : Sort u_2\nι' : Sort ?u.11245\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p s\n⊢ l = l'",
"tactic": "ext t"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.11236\nγ : Type ?u.11239\nι : Sort u_2\nι' : Sort ?u.11245\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p s\nt : Set α\n⊢ t ∈ l ↔ t ∈ l'",
"tactic": "rw [hl.mem_iff, hl'.mem_iff]"
}
] |
[
270,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Std/Data/String/Lemmas.lean
|
String.Iterator.ValidFor.mk'
|
[
{
"state_after": "no goals",
"state_before": "l r : List Char\n⊢ ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }",
"tactic": "simpa [List.reverseAux_eq] using mk"
}
] |
[
522,
38
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
521,
1
] |
Mathlib/Computability/RegularExpressions.lean
|
RegularExpression.plus_def
|
[] |
[
101,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Data/Option/Basic.lean
|
Option.coe_get
|
[] |
[
54,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
|
MeasureTheory.Measure.map_linearMap_add_haar_eq_smul_add_haar
|
[
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det f)⁻¹) • μ",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det f)⁻¹) • μ",
"tactic": "let ι := Fin (finrank ℝ E)"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis : FiniteDimensional ℝ (ι → ℝ)\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det f)⁻¹) • μ",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det f)⁻¹) • μ",
"tactic": "haveI : FiniteDimensional ℝ (ι → ℝ) := by infer_instance"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det f)⁻¹) • μ",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis : FiniteDimensional ℝ (ι → ℝ)\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det f)⁻¹) • μ",
"tactic": "have : finrank ℝ E = finrank ℝ (ι → ℝ) := by simp"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det f)⁻¹) • μ",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det f)⁻¹) • μ",
"tactic": "have e : E ≃ₗ[ℝ] ι → ℝ := LinearEquiv.ofFinrankEq E (ι → ℝ) this"
},
{
"state_after": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det f)⁻¹) • μ",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det f)⁻¹) • μ",
"tactic": "obtain ⟨g, hg⟩ : ∃ g, g = (e : E →ₗ[ℝ] ι → ℝ).comp (f.comp (e.symm : (ι → ℝ) →ₗ[ℝ] E)) := ⟨_, rfl⟩"
},
{
"state_after": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det f)⁻¹) • μ",
"state_before": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det f)⁻¹) • μ",
"tactic": "have gdet : LinearMap.det g = LinearMap.det f := by rw [hg]; exact LinearMap.det_conj f e"
},
{
"state_after": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"state_before": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det f)⁻¹) • μ",
"tactic": "rw [← gdet] at hf ⊢"
},
{
"state_after": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"state_before": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"tactic": "have fg : f = (e.symm : (ι → ℝ) →ₗ[ℝ] E).comp (g.comp (e : E →ₗ[ℝ] ι → ℝ)) := by\n ext x\n simp only [LinearEquiv.coe_coe, Function.comp_apply, LinearMap.coe_comp,\n LinearEquiv.symm_apply_apply, hg]"
},
{
"state_after": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\n⊢ map (↑(LinearEquiv.symm e) ∘ ↑g ∘ ↑e) μ = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"state_before": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\n⊢ map (↑f) μ = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"tactic": "simp only [fg, LinearEquiv.coe_coe, LinearMap.coe_comp]"
},
{
"state_after": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\nCe : Continuous ↑e\n⊢ map (↑(LinearEquiv.symm e) ∘ ↑g ∘ ↑e) μ = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"state_before": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\n⊢ map (↑(LinearEquiv.symm e) ∘ ↑g ∘ ↑e) μ = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"tactic": "have Ce : Continuous e := (e : E →ₗ[ℝ] ι → ℝ).continuous_of_finiteDimensional"
},
{
"state_after": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\nCe : Continuous ↑e\nCg : Continuous ↑g\n⊢ map (↑(LinearEquiv.symm e) ∘ ↑g ∘ ↑e) μ = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"state_before": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\nCe : Continuous ↑e\n⊢ map (↑(LinearEquiv.symm e) ∘ ↑g ∘ ↑e) μ = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"tactic": "have Cg : Continuous g := LinearMap.continuous_of_finiteDimensional g"
},
{
"state_after": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\nCe : Continuous ↑e\nCg : Continuous ↑g\nCesymm : Continuous ↑(LinearEquiv.symm e)\n⊢ map (↑(LinearEquiv.symm e) ∘ ↑g ∘ ↑e) μ = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"state_before": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\nCe : Continuous ↑e\nCg : Continuous ↑g\n⊢ map (↑(LinearEquiv.symm e) ∘ ↑g ∘ ↑e) μ = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"tactic": "have Cesymm : Continuous e.symm := (e.symm : (ι → ℝ) →ₗ[ℝ] E).continuous_of_finiteDimensional"
},
{
"state_after": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\nCe : Continuous ↑e\nCg : Continuous ↑g\nCesymm : Continuous ↑(LinearEquiv.symm e)\n⊢ map (↑(LinearEquiv.symm e)) (map (↑g) (map (↑e) μ)) = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"state_before": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\nCe : Continuous ↑e\nCg : Continuous ↑g\nCesymm : Continuous ↑(LinearEquiv.symm e)\n⊢ map (↑(LinearEquiv.symm e) ∘ ↑g ∘ ↑e) μ = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"tactic": "rw [← map_map Cesymm.measurable (Cg.comp Ce).measurable, ← map_map Cg.measurable Ce.measurable]"
},
{
"state_after": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝¹ : FiniteDimensional ℝ (ι → ℝ)\nthis✝ : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\nCe : Continuous ↑e\nCg : Continuous ↑g\nCesymm : Continuous ↑(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (↑e) μ)\n⊢ map (↑(LinearEquiv.symm e)) (map (↑g) (map (↑e) μ)) = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"state_before": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\nCe : Continuous ↑e\nCg : Continuous ↑g\nCesymm : Continuous ↑(LinearEquiv.symm e)\n⊢ map (↑(LinearEquiv.symm e)) (map (↑g) (map (↑e) μ)) = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"tactic": "haveI : IsAddHaarMeasure (map e μ) := (e : E ≃+ (ι → ℝ)).isAddHaarMeasure_map μ Ce Cesymm"
},
{
"state_after": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝¹ : FiniteDimensional ℝ (ι → ℝ)\nthis✝ : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\nCe : Continuous ↑e\nCg : Continuous ↑g\nCesymm : Continuous ↑(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (↑e) μ)\necomp : ↑(LinearEquiv.symm e) ∘ ↑e = id\n⊢ map (↑(LinearEquiv.symm e)) (map (↑g) (map (↑e) μ)) = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"state_before": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝¹ : FiniteDimensional ℝ (ι → ℝ)\nthis✝ : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\nCe : Continuous ↑e\nCg : Continuous ↑g\nCesymm : Continuous ↑(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (↑e) μ)\n⊢ map (↑(LinearEquiv.symm e)) (map (↑g) (map (↑e) μ)) = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"tactic": "have ecomp : e.symm ∘ e = id := by\n ext x; simp only [id.def, Function.comp_apply, LinearEquiv.symm_apply_apply]"
},
{
"state_after": "no goals",
"state_before": "case intro\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝¹ : FiniteDimensional ℝ (ι → ℝ)\nthis✝ : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\nCe : Continuous ↑e\nCg : Continuous ↑g\nCesymm : Continuous ↑(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (↑e) μ)\necomp : ↑(LinearEquiv.symm e) ∘ ↑e = id\n⊢ map (↑(LinearEquiv.symm e)) (map (↑g) (map (↑e) μ)) = ENNReal.ofReal (abs (↑LinearMap.det g)⁻¹) • μ",
"tactic": "rw [map_linearMap_add_haar_pi_eq_smul_add_haar hf (map e μ), Measure.map_smul,\n map_map Cesymm.measurable Ce.measurable, ecomp, Measure.map_id]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\n⊢ FiniteDimensional ℝ (ι → ℝ)",
"tactic": "infer_instance"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis : FiniteDimensional ℝ (ι → ℝ)\n⊢ finrank ℝ E = finrank ℝ (ι → ℝ)",
"tactic": "simp"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\n⊢ ↑LinearMap.det (LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))) = ↑LinearMap.det f",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\n⊢ ↑LinearMap.det g = ↑LinearMap.det f",
"tactic": "rw [hg]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nhf : ↑LinearMap.det f ≠ 0\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\n⊢ ↑LinearMap.det (LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))) = ↑LinearMap.det f",
"tactic": "exact LinearMap.det_conj f e"
},
{
"state_after": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nx : E\n⊢ ↑f x = ↑(LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)) x",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\n⊢ f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nx : E\n⊢ ↑f x = ↑(LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)) x",
"tactic": "simp only [LinearEquiv.coe_coe, Function.comp_apply, LinearMap.coe_comp,\n LinearEquiv.symm_apply_apply, hg]"
},
{
"state_after": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝¹ : FiniteDimensional ℝ (ι → ℝ)\nthis✝ : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\nCe : Continuous ↑e\nCg : Continuous ↑g\nCesymm : Continuous ↑(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (↑e) μ)\nx : E\n⊢ (↑(LinearEquiv.symm e) ∘ ↑e) x = id x",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝¹ : FiniteDimensional ℝ (ι → ℝ)\nthis✝ : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\nCe : Continuous ↑e\nCg : Continuous ↑g\nCesymm : Continuous ↑(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (↑e) μ)\n⊢ ↑(LinearEquiv.symm e) ∘ ↑e = id",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.986679\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝¹ : FiniteDimensional ℝ (ι → ℝ)\nthis✝ : finrank ℝ E = finrank ℝ (ι → ℝ)\ne : E ≃ₗ[ℝ] ι → ℝ\ng : (ι → ℝ) →ₗ[ℝ] ι → ℝ\nhf : ↑LinearMap.det g ≠ 0\nhg : g = LinearMap.comp (↑e) (LinearMap.comp f ↑(LinearEquiv.symm e))\ngdet : ↑LinearMap.det g = ↑LinearMap.det f\nfg : f = LinearMap.comp (↑(LinearEquiv.symm e)) (LinearMap.comp g ↑e)\nCe : Continuous ↑e\nCg : Continuous ↑g\nCesymm : Continuous ↑(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (↑e) μ)\nx : E\n⊢ (↑(LinearEquiv.symm e) ∘ ↑e) x = id x",
"tactic": "simp only [id.def, Function.comp_apply, LinearEquiv.symm_apply_apply]"
}
] |
[
246,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.restrict_apply_superset
|
[] |
[
1610,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1608,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.ofReal_pos
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.807239\nβ : Type ?u.807242\na b c d : ℝ≥0∞\nr p✝ q : ℝ≥0\np : ℝ\n⊢ 0 < ENNReal.ofReal p ↔ 0 < p",
"tactic": "simp [ENNReal.ofReal]"
}
] |
[
2114,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2114,
1
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
SubmodulesRingBasis.toRing_subgroups_basis
|
[
{
"state_after": "ι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nB : ι → Submodule R A\nhB✝ hB : SubmodulesRingBasis B\n⊢ ∀ (x : A) (i : ι), ∃ j, ↑(Submodule.toAddSubgroup (B j)) ⊆ (fun y => x * y) ⁻¹' ↑(Submodule.toAddSubgroup (B i))",
"state_before": "ι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nB : ι → Submodule R A\nhB✝ hB : SubmodulesRingBasis B\n⊢ RingSubgroupsBasis fun i => Submodule.toAddSubgroup (B i)",
"tactic": "apply RingSubgroupsBasis.of_comm (fun i => (B i).toAddSubgroup) hB.inter hB.mul"
},
{
"state_after": "ι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nB : ι → Submodule R A\nhB✝ hB : SubmodulesRingBasis B\na : A\ni : ι\n⊢ ∃ j, ↑(Submodule.toAddSubgroup (B j)) ⊆ (fun y => a * y) ⁻¹' ↑(Submodule.toAddSubgroup (B i))",
"state_before": "ι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nB : ι → Submodule R A\nhB✝ hB : SubmodulesRingBasis B\n⊢ ∀ (x : A) (i : ι), ∃ j, ↑(Submodule.toAddSubgroup (B j)) ⊆ (fun y => x * y) ⁻¹' ↑(Submodule.toAddSubgroup (B i))",
"tactic": "intro a i"
},
{
"state_after": "case intro\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nB : ι → Submodule R A\nhB✝ hB : SubmodulesRingBasis B\na : A\ni j : ι\nhj : a • B j ≤ B i\n⊢ ∃ j, ↑(Submodule.toAddSubgroup (B j)) ⊆ (fun y => a * y) ⁻¹' ↑(Submodule.toAddSubgroup (B i))",
"state_before": "ι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nB : ι → Submodule R A\nhB✝ hB : SubmodulesRingBasis B\na : A\ni : ι\n⊢ ∃ j, ↑(Submodule.toAddSubgroup (B j)) ⊆ (fun y => a * y) ⁻¹' ↑(Submodule.toAddSubgroup (B i))",
"tactic": "rcases hB.leftMul a i with ⟨j, hj⟩"
},
{
"state_after": "case intro\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nB : ι → Submodule R A\nhB✝ hB : SubmodulesRingBasis B\na : A\ni j : ι\nhj : a • B j ≤ B i\n⊢ ↑(Submodule.toAddSubgroup (B j)) ⊆ (fun y => a * y) ⁻¹' ↑(Submodule.toAddSubgroup (B i))",
"state_before": "case intro\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nB : ι → Submodule R A\nhB✝ hB : SubmodulesRingBasis B\na : A\ni j : ι\nhj : a • B j ≤ B i\n⊢ ∃ j, ↑(Submodule.toAddSubgroup (B j)) ⊆ (fun y => a * y) ⁻¹' ↑(Submodule.toAddSubgroup (B i))",
"tactic": "use j"
},
{
"state_after": "case intro\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nB : ι → Submodule R A\nhB✝ hB : SubmodulesRingBasis B\na : A\ni j : ι\nhj : a • B j ≤ B i\nb : A\nb_in : b ∈ B j\n⊢ b ∈ (fun y => a * y) ⁻¹' ↑(Submodule.toAddSubgroup (B i))",
"state_before": "case intro\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nB : ι → Submodule R A\nhB✝ hB : SubmodulesRingBasis B\na : A\ni j : ι\nhj : a • B j ≤ B i\n⊢ ↑(Submodule.toAddSubgroup (B j)) ⊆ (fun y => a * y) ⁻¹' ↑(Submodule.toAddSubgroup (B i))",
"tactic": "rintro b (b_in : b ∈ B j)"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nB : ι → Submodule R A\nhB✝ hB : SubmodulesRingBasis B\na : A\ni j : ι\nhj : a • B j ≤ B i\nb : A\nb_in : b ∈ B j\n⊢ b ∈ (fun y => a * y) ⁻¹' ↑(Submodule.toAddSubgroup (B i))",
"tactic": "exact hj ⟨b, b_in, rfl⟩"
}
] |
[
238,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.symm_trans_self
|
[] |
[
833,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
832,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
Wbtw.right_mem_image_Ici_of_left_ne
|
[] |
[
697,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
695,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.ext_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q✝ p q : MvPolynomial σ R\nh : p = q\nm : σ →₀ ℕ\n⊢ coeff m p = coeff m q",
"tactic": "rw [h]"
}
] |
[
606,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
605,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean
|
UniformGroup.mk'
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.661\nβ : Type ?u.664\nα : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : Group α\nh₁ : UniformContinuous fun p => p.fst * p.snd\nh₂ : UniformContinuous fun p => p⁻¹\n⊢ UniformContinuous fun p => p.fst / p.snd",
"tactic": "simpa only [div_eq_mul_inv] using\nh₁.comp (uniformContinuous_fst.prod_mk (h₂.comp uniformContinuous_snd))"
}
] |
[
70,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
mem_extChartAt_source
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\nE : Type u_2\nM : Type u_1\nH : Type u_4\nE' : Type ?u.184306\nM' : Type ?u.184309\nH' : Type ?u.184312\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁵ : NormedAddCommGroup E'\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\ninst✝¹ : ChartedSpace H M\ninst✝ : ChartedSpace H' M'\n⊢ x ∈ (extChartAt I x).source",
"tactic": "simp only [extChartAt_source, mem_chart_source]"
}
] |
[
1050,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1049,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Fintype.univ_empty
|
[] |
[
872,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
871,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biprod.hom_ext
|
[] |
[
1425,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1423,
1
] |
Mathlib/LinearAlgebra/Ray.lean
|
sameRay_neg_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁴ : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.141134\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nx y : M\n⊢ SameRay R (-x) (-y) ↔ SameRay R x y",
"tactic": "simp only [SameRay, neg_eq_zero, smul_neg, neg_inj]"
}
] |
[
399,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
398,
1
] |
Mathlib/Logic/Basic.lean
|
eq_equivalence
|
[] |
[
514,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
513,
1
] |
Mathlib/Topology/Connected.lean
|
isPreconnected_iUnion
|
[] |
[
139,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
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