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/Topology/Instances/ENNReal.lean
|
ENNReal.tsum_biUnion_le_tsum
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.277982\nγ : Type ?u.277985\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns✝ : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nι : Type u_1\nf : α → ℝ≥0∞\ns : Set ι\nt : ι → Set α\n⊢ (⋃ (i : ι) (_ : i ∈ s), t i) = ⋃ (i : ↑s), t ↑i",
"tactic": "simp"
}
] |
[
1008,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1005,
1
] |
Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean
|
TensorAlgebra.lift_unique
|
[
{
"state_after": "R : Type u_2\ninst✝⁴ : CommSemiring R\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_1\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\ng : TensorAlgebra R M →ₐ[R] A\n⊢ LinearMap.comp (AlgHom.toLinearMap g) (ι R) = f ↔ ↑(lift R).symm g = f",
"state_before": "R : Type u_2\ninst✝⁴ : CommSemiring R\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_1\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\ng : TensorAlgebra R M →ₐ[R] A\n⊢ LinearMap.comp (AlgHom.toLinearMap g) (ι R) = f ↔ g = ↑(lift R) f",
"tactic": "rw [← (lift R).symm_apply_eq]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁴ : CommSemiring R\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_1\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\ng : TensorAlgebra R M →ₐ[R] A\n⊢ LinearMap.comp (AlgHom.toLinearMap g) (ι R) = f ↔ ↑(lift R).symm g = f",
"tactic": "simp only [lift, Equiv.coe_fn_symm_mk]"
}
] |
[
144,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/Data/Pi/Algebra.lean
|
Pi.mulSingle_inj
|
[] |
[
331,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
AEMeasurable.mono_measure
|
[] |
[
50,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.exp_injective
|
[] |
[
1529,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1528,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.enum_zero_le
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.194526\nγ : Type ?u.194529\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nh0 : 0 < type r\na : α\n⊢ 0 ≤ typein r a",
"state_before": "α : Type u_1\nβ : Type ?u.194526\nγ : Type ?u.194529\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nh0 : 0 < type r\na : α\n⊢ ¬r a (enum r 0 h0)",
"tactic": "rw [← enum_typein r a, enum_le_enum r]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.194526\nγ : Type ?u.194529\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nh0 : 0 < type r\na : α\n⊢ 0 ≤ typein r a",
"tactic": "apply Ordinal.zero_le"
}
] |
[
1168,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1165,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean
|
Submodule.mem_mk
|
[] |
[
83,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
fderiv_pi
|
[] |
[
462,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
460,
1
] |
Mathlib/GroupTheory/Perm/List.lean
|
List.formPerm_apply_nthLe_zero
|
[
{
"state_after": "case nil\nα : Type u_1\nβ : Type ?u.668958\ninst✝ : DecidableEq α\nl : List α\nx : α\nh : Nodup []\nhl : 1 < length []\n⊢ ↑(formPerm []) (nthLe [] 0 (_ : 0 < length [])) = nthLe [] 1 hl\n\ncase cons.nil\nα : Type u_1\nβ : Type ?u.668958\ninst✝ : DecidableEq α\nl : List α\nx✝ x : α\nh : Nodup [x]\nhl : 1 < length [x]\n⊢ ↑(formPerm [x]) (nthLe [x] 0 (_ : 0 < length [x])) = nthLe [x] 1 hl\n\ncase cons.cons\nα : Type u_1\nβ : Type ?u.668958\ninst✝ : DecidableEq α\nl : List α\nx✝ x y : α\ntl : List α\nh : Nodup (x :: y :: tl)\nhl : 1 < length (x :: y :: tl)\n⊢ ↑(formPerm (x :: y :: tl)) (nthLe (x :: y :: tl) 0 (_ : 0 < length (x :: y :: tl))) = nthLe (x :: y :: tl) 1 hl",
"state_before": "α : Type u_1\nβ : Type ?u.668958\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nhl : 1 < length l\n⊢ ↑(formPerm l) (nthLe l 0 (_ : 0 < length l)) = nthLe l 1 hl",
"tactic": "rcases l with (_ | ⟨x, _ | ⟨y, tl⟩⟩)"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nβ : Type ?u.668958\ninst✝ : DecidableEq α\nl : List α\nx : α\nh : Nodup []\nhl : 1 < length []\n⊢ ↑(formPerm []) (nthLe [] 0 (_ : 0 < length [])) = nthLe [] 1 hl",
"tactic": "simp at hl"
},
{
"state_after": "no goals",
"state_before": "case cons.nil\nα : Type u_1\nβ : Type ?u.668958\ninst✝ : DecidableEq α\nl : List α\nx✝ x : α\nh : Nodup [x]\nhl : 1 < length [x]\n⊢ ↑(formPerm [x]) (nthLe [x] 0 (_ : 0 < length [x])) = nthLe [x] 1 hl",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons.cons\nα : Type u_1\nβ : Type ?u.668958\ninst✝ : DecidableEq α\nl : List α\nx✝ x y : α\ntl : List α\nh : Nodup (x :: y :: tl)\nhl : 1 < length (x :: y :: tl)\n⊢ ↑(formPerm (x :: y :: tl)) (nthLe (x :: y :: tl) 0 (_ : 0 < length (x :: y :: tl))) = nthLe (x :: y :: tl) 1 hl",
"tactic": "simpa using formPerm_apply_head _ _ _ h"
}
] |
[
158,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigOWith.congr
|
[] |
[
309,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/MeasureTheory/PiSystem.lean
|
generateFrom_piiUnionInter_measurableSet
|
[
{
"state_after": "case refine'_1\nα : Type u_1\nι : Type u_2\nm : ι → MeasurableSpace α\nS : Set ι\n⊢ generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) ≤ ⨆ (i : ι) (_ : i ∈ S), m i\n\ncase refine'_2\nα : Type u_1\nι : Type u_2\nm : ι → MeasurableSpace α\nS : Set ι\n⊢ (⨆ (i : ι) (_ : i ∈ S), m i) ≤ generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S)",
"state_before": "α : Type u_1\nι : Type u_2\nm : ι → MeasurableSpace α\nS : Set ι\n⊢ generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) = ⨆ (i : ι) (_ : i ∈ S), m i",
"tactic": "refine' le_antisymm _ _"
},
{
"state_after": "case refine'_1\nα : Type u_1\nι : Type u_2\nm : ι → MeasurableSpace α\nS : Set ι\n⊢ generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) ≤ generateFrom {s | MeasurableSet s}",
"state_before": "case refine'_1\nα : Type u_1\nι : Type u_2\nm : ι → MeasurableSpace α\nS : Set ι\n⊢ generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) ≤ ⨆ (i : ι) (_ : i ∈ S), m i",
"tactic": "rw [← @generateFrom_measurableSet α (⨆ i ∈ S, m i)]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nι : Type u_2\nm : ι → MeasurableSpace α\nS : Set ι\n⊢ generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S) ≤ generateFrom {s | MeasurableSet s}",
"tactic": "exact generateFrom_mono (measurableSet_iSup_of_mem_piiUnionInter m S)"
},
{
"state_after": "case refine'_2\nα : Type u_1\nι : Type u_2\nm : ι → MeasurableSpace α\nS : Set ι\ni : ι\nhi : i ∈ S\n⊢ m i ≤ generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S)",
"state_before": "case refine'_2\nα : Type u_1\nι : Type u_2\nm : ι → MeasurableSpace α\nS : Set ι\n⊢ (⨆ (i : ι) (_ : i ∈ S), m i) ≤ generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S)",
"tactic": "refine' iSup₂_le fun i hi => _"
},
{
"state_after": "case refine'_2\nα : Type u_1\nι : Type u_2\nm : ι → MeasurableSpace α\nS : Set ι\ni : ι\nhi : i ∈ S\n⊢ generateFrom {s | MeasurableSet s} ≤ generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S)",
"state_before": "case refine'_2\nα : Type u_1\nι : Type u_2\nm : ι → MeasurableSpace α\nS : Set ι\ni : ι\nhi : i ∈ S\n⊢ m i ≤ generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S)",
"tactic": "rw [← @generateFrom_measurableSet α (m i)]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nι : Type u_2\nm : ι → MeasurableSpace α\nS : Set ι\ni : ι\nhi : i ∈ S\n⊢ generateFrom {s | MeasurableSet s} ≤ generateFrom (piiUnionInter (fun n => {s | MeasurableSet s}) S)",
"tactic": "exact generateFrom_mono (mem_piiUnionInter_of_measurableSet m hi)"
}
] |
[
523,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
516,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toReal_inf
|
[] |
[
2064,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2063,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pushout.condition
|
[] |
[
1220,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1218,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.option_getD
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.116351\nγ : Type ?u.116354\nδ : Type ?u.116357\nσ : Type ?u.116360\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nx✝ : Option α × α\no : Option α\na : α\n⊢ (Option.casesOn ((fun a x => a) (o, a).fst (o, a).snd) ((fun x b => b) (o, a).fst (o, a).snd) fun b => b) =\n Option.getD (o, a).fst (o, a).snd",
"tactic": "cases o <;> rfl"
}
] |
[
652,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
650,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
Set.EqOn.iteratedFDerivWithin
|
[] |
[
915,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
913,
11
] |
Std/Data/Option/Lemmas.lean
|
Option.bind_eq_none
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\no : Option α\nf : α → Option β\n⊢ Option.bind o f = none ↔ ∀ (b : β) (a : α), a ∈ o → ¬b ∈ f a",
"tactic": "simp only [eq_none_iff_forall_not_mem, not_exists, not_and, mem_def, bind_eq_some]"
}
] |
[
98,
85
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
96,
9
] |
Mathlib/Logic/Function/Basic.lean
|
InvImage.equivalence
|
[] |
[
1064,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1062,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.spanPoints_subset_coe_of_subset_coe
|
[
{
"state_after": "case intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s ⊆ ↑s1\np p1 : P\nhp1 : p1 ∈ s\nv : V\nhv : v ∈ vectorSpan k s\nhp : p = v +ᵥ p1\n⊢ p ∈ ↑s1",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s ⊆ ↑s1\n⊢ spanPoints k s ⊆ ↑s1",
"tactic": "rintro p ⟨p1, hp1, v, hv, hp⟩"
},
{
"state_after": "case intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s ⊆ ↑s1\np p1 : P\nhp1 : p1 ∈ s\nv : V\nhv : v ∈ vectorSpan k s\nhp : p = v +ᵥ p1\n⊢ v +ᵥ p1 ∈ ↑s1",
"state_before": "case intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s ⊆ ↑s1\np p1 : P\nhp1 : p1 ∈ s\nv : V\nhv : v ∈ vectorSpan k s\nhp : p = v +ᵥ p1\n⊢ p ∈ ↑s1",
"tactic": "rw [hp]"
},
{
"state_after": "case intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s ⊆ ↑s1\np p1 : P\nhp1 : p1 ∈ s\nv : V\nhv : v ∈ vectorSpan k s\nhp : p = v +ᵥ p1\nhp1s1 : p1 ∈ ↑s1\n⊢ v +ᵥ p1 ∈ ↑s1",
"state_before": "case intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s ⊆ ↑s1\np p1 : P\nhp1 : p1 ∈ s\nv : V\nhv : v ∈ vectorSpan k s\nhp : p = v +ᵥ p1\n⊢ v +ᵥ p1 ∈ ↑s1",
"tactic": "have hp1s1 : p1 ∈ (s1 : Set P) := Set.mem_of_mem_of_subset hp1 h"
},
{
"state_after": "case intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s ⊆ ↑s1\np p1 : P\nhp1 : p1 ∈ s\nv : V\nhv : v ∈ vectorSpan k s\nhp : p = v +ᵥ p1\nhp1s1 : p1 ∈ ↑s1\n⊢ v ∈ direction s1",
"state_before": "case intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s ⊆ ↑s1\np p1 : P\nhp1 : p1 ∈ s\nv : V\nhv : v ∈ vectorSpan k s\nhp : p = v +ᵥ p1\nhp1s1 : p1 ∈ ↑s1\n⊢ v +ᵥ p1 ∈ ↑s1",
"tactic": "refine' vadd_mem_of_mem_direction _ hp1s1"
},
{
"state_after": "case intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s ⊆ ↑s1\np p1 : P\nhp1 : p1 ∈ s\nv : V\nhv : v ∈ vectorSpan k s\nhp : p = v +ᵥ p1\nhp1s1 : p1 ∈ ↑s1\nhs : vectorSpan k s ≤ direction s1\n⊢ v ∈ direction s1",
"state_before": "case intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s ⊆ ↑s1\np p1 : P\nhp1 : p1 ∈ s\nv : V\nhv : v ∈ vectorSpan k s\nhp : p = v +ᵥ p1\nhp1s1 : p1 ∈ ↑s1\n⊢ v ∈ direction s1",
"tactic": "have hs : vectorSpan k s ≤ s1.direction := vectorSpan_mono k h"
},
{
"state_after": "case intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s ⊆ ↑s1\np p1 : P\nhp1 : p1 ∈ s\nv : V\nhv : v ∈ vectorSpan k s\nhp : p = v +ᵥ p1\nhp1s1 : p1 ∈ ↑s1\nhs : ∀ ⦃x : V⦄, x ∈ vectorSpan k s → x ∈ direction s1\n⊢ v ∈ direction s1",
"state_before": "case intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s ⊆ ↑s1\np p1 : P\nhp1 : p1 ∈ s\nv : V\nhv : v ∈ vectorSpan k s\nhp : p = v +ᵥ p1\nhp1s1 : p1 ∈ ↑s1\nhs : vectorSpan k s ≤ direction s1\n⊢ v ∈ direction s1",
"tactic": "rw [SetLike.le_def] at hs"
},
{
"state_after": "case intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s ⊆ ↑s1\np p1 : P\nhp1 : p1 ∈ s\nv : V\nhv : v ∈ vectorSpan k s\nhp : p = v +ᵥ p1\nhp1s1 : p1 ∈ ↑s1\nhs : ∀ ⦃x : V⦄, x ∈ vectorSpan k s → x ∈ direction s1\n⊢ v ∈ ↑(direction s1)",
"state_before": "case intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s ⊆ ↑s1\np p1 : P\nhp1 : p1 ∈ s\nv : V\nhv : v ∈ vectorSpan k s\nhp : p = v +ᵥ p1\nhp1s1 : p1 ∈ ↑s1\nhs : ∀ ⦃x : V⦄, x ∈ vectorSpan k s → x ∈ direction s1\n⊢ v ∈ direction s1",
"tactic": "rw [← SetLike.mem_coe]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s ⊆ ↑s1\np p1 : P\nhp1 : p1 ∈ s\nv : V\nhv : v ∈ vectorSpan k s\nhp : p = v +ᵥ p1\nhp1s1 : p1 ∈ ↑s1\nhs : ∀ ⦃x : V⦄, x ∈ vectorSpan k s → x ∈ direction s1\n⊢ v ∈ ↑(direction s1)",
"tactic": "exact Set.mem_of_mem_of_subset hv hs"
}
] |
[
519,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
510,
1
] |
Mathlib/Algebra/Support.lean
|
Function.support_prod
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nA : Type u_1\nB : Type ?u.102217\nM : Type ?u.102220\nN : Type ?u.102223\nP : Type ?u.102226\nR : Type ?u.102229\nS : Type ?u.102232\nG : Type ?u.102235\nM₀ : Type ?u.102238\nG₀ : Type ?u.102241\nι : Sort ?u.102244\ninst✝² : CommMonoidWithZero A\ninst✝¹ : NoZeroDivisors A\ninst✝ : Nontrivial A\ns : Finset α\nf : α → β → A\nx : β\n⊢ (x ∈ support fun x => ∏ i in s, f i x) ↔ x ∈ ⋂ (i : α) (_ : i ∈ s), support (f i)",
"tactic": "simp [support, Ne.def, Finset.prod_eq_zero_iff, mem_setOf_eq, Set.mem_iInter, not_exists]"
}
] |
[
410,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean
|
CategoryTheory.Limits.pullback.diagonal_isKernelPair
|
[] |
[
73,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.isSeparator_sigma_of_isSeparator
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nβ : Type w\nf : β → C\ninst✝ : HasCoproduct f\nb : β\nhb : IsSeparator (f b)\n⊢ {f b} ⊆ Set.range f",
"tactic": "simp"
}
] |
[
585,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
583,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
Antitone.mul_const'
|
[] |
[
1326,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1325,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
edist_pi_const
|
[] |
[
498,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/Analysis/NormedSpace/RieszLemma.lean
|
riesz_lemma
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nhF : ∃ x, ¬x ∈ F\nr : ℝ\nhr : r < 1\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"tactic": "classical\n obtain ⟨x, hx⟩ : ∃ x : E, x ∉ F := hF\n let d := Metric.infDist x F\n have hFn : (F : Set E).Nonempty := ⟨_, F.zero_mem⟩\n have hdp : 0 < d :=\n lt_of_le_of_ne Metric.infDist_nonneg fun heq =>\n hx ((hFc.mem_iff_infDist_zero hFn).2 heq.symm)\n let r' := max r 2⁻¹\n have hr' : r' < 1 := by\n simp [hr]\n norm_num\n have hlt : 0 < r' := lt_of_lt_of_le (by norm_num) (le_max_right r 2⁻¹)\n have hdlt : d < d / r' := (lt_div_iff hlt).mpr ((mul_lt_iff_lt_one_right hdp).2 hr')\n obtain ⟨y₀, hy₀F, hxy₀⟩ : ∃ y ∈ F, dist x y < d / r' := (Metric.infDist_lt_iff hFn).mp hdlt\n have x_ne_y₀ : x - y₀ ∉ F := by\n by_contra h\n have : x - y₀ + y₀ ∈ F := F.add_mem h hy₀F\n simp only [neg_add_cancel_right, sub_eq_add_neg] at this\n exact hx this\n refine' ⟨x - y₀, x_ne_y₀, fun y hy => le_of_lt _⟩\n have hy₀y : y₀ + y ∈ F := F.add_mem hy₀F hy\n calc\n r * ‖x - y₀‖ ≤ r' * ‖x - y₀‖ := by gcongr; apply le_max_left\n _ < d := by\n rw [← dist_eq_norm]\n exact (lt_div_iff' hlt).1 hxy₀\n _ ≤ dist x (y₀ + y) := (Metric.infDist_le_dist_of_mem hy₀y)\n _ = ‖x - y₀ - y‖ := by rw [sub_sub, dist_eq_norm]"
},
{
"state_after": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nhF : ∃ x, ¬x ∈ F\nr : ℝ\nhr : r < 1\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"tactic": "obtain ⟨x, hx⟩ : ∃ x : E, x ∉ F := hF"
},
{
"state_after": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"state_before": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"tactic": "let d := Metric.infDist x F"
},
{
"state_after": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"state_before": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"tactic": "have hFn : (F : Set E).Nonempty := ⟨_, F.zero_mem⟩"
},
{
"state_after": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"state_before": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"tactic": "have hdp : 0 < d :=\n lt_of_le_of_ne Metric.infDist_nonneg fun heq =>\n hx ((hFc.mem_iff_infDist_zero hFn).2 heq.symm)"
},
{
"state_after": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"state_before": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"tactic": "let r' := max r 2⁻¹"
},
{
"state_after": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"state_before": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"tactic": "have hr' : r' < 1 := by\n simp [hr]\n norm_num"
},
{
"state_after": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"state_before": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"tactic": "have hlt : 0 < r' := lt_of_lt_of_le (by norm_num) (le_max_right r 2⁻¹)"
},
{
"state_after": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"state_before": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"tactic": "have hdlt : d < d / r' := (lt_div_iff hlt).mpr ((mul_lt_iff_lt_one_right hdp).2 hr')"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"state_before": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"tactic": "obtain ⟨y₀, hy₀F, hxy₀⟩ : ∃ y ∈ F, dist x y < d / r' := (Metric.infDist_lt_iff hFn).mp hdlt"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nx_ne_y₀ : ¬x - y₀ ∈ F\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"tactic": "have x_ne_y₀ : x - y₀ ∉ F := by\n by_contra h\n have : x - y₀ + y₀ ∈ F := F.add_mem h hy₀F\n simp only [neg_add_cancel_right, sub_eq_add_neg] at this\n exact hx this"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nx_ne_y₀ : ¬x - y₀ ∈ F\ny : E\nhy : y ∈ F\n⊢ r * ‖x - y₀‖ < ‖x - y₀ - y‖",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nx_ne_y₀ : ¬x - y₀ ∈ F\n⊢ ∃ x₀, ¬x₀ ∈ F ∧ ∀ (y : E), y ∈ F → r * ‖x₀‖ ≤ ‖x₀ - y‖",
"tactic": "refine' ⟨x - y₀, x_ne_y₀, fun y hy => le_of_lt _⟩"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nx_ne_y₀ : ¬x - y₀ ∈ F\ny : E\nhy : y ∈ F\nhy₀y : y₀ + y ∈ F\n⊢ r * ‖x - y₀‖ < ‖x - y₀ - y‖",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nx_ne_y₀ : ¬x - y₀ ∈ F\ny : E\nhy : y ∈ F\n⊢ r * ‖x - y₀‖ < ‖x - y₀ - y‖",
"tactic": "have hy₀y : y₀ + y ∈ F := F.add_mem hy₀F hy"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nx_ne_y₀ : ¬x - y₀ ∈ F\ny : E\nhy : y ∈ F\nhy₀y : y₀ + y ∈ F\n⊢ r * ‖x - y₀‖ < ‖x - y₀ - y‖",
"tactic": "calc\n r * ‖x - y₀‖ ≤ r' * ‖x - y₀‖ := by gcongr; apply le_max_left\n _ < d := by\n rw [← dist_eq_norm]\n exact (lt_div_iff' hlt).1 hxy₀\n _ ≤ dist x (y₀ + y) := (Metric.infDist_le_dist_of_mem hy₀y)\n _ = ‖x - y₀ - y‖ := by rw [sub_sub, dist_eq_norm]"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\n⊢ 2⁻¹ < 1",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\n⊢ r' < 1",
"tactic": "simp [hr]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\n⊢ 2⁻¹ < 1",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\n⊢ 0 < 2⁻¹",
"tactic": "norm_num"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nh : x - y₀ ∈ F\n⊢ False",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\n⊢ ¬x - y₀ ∈ F",
"tactic": "by_contra h"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nh : x - y₀ ∈ F\nthis : x - y₀ + y₀ ∈ F\n⊢ False",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nh : x - y₀ ∈ F\n⊢ False",
"tactic": "have : x - y₀ + y₀ ∈ F := F.add_mem h hy₀F"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nh : x - y₀ ∈ F\nthis : x ∈ F\n⊢ False",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nh : x - y₀ ∈ F\nthis : x - y₀ + y₀ ∈ F\n⊢ False",
"tactic": "simp only [neg_add_cancel_right, sub_eq_add_neg] at this"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nh : x - y₀ ∈ F\nthis : x ∈ F\n⊢ False",
"tactic": "exact hx this"
},
{
"state_after": "case h\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nx_ne_y₀ : ¬x - y₀ ∈ F\ny : E\nhy : y ∈ F\nhy₀y : y₀ + y ∈ F\n⊢ r ≤ r'",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nx_ne_y₀ : ¬x - y₀ ∈ F\ny : E\nhy : y ∈ F\nhy₀y : y₀ + y ∈ F\n⊢ r * ‖x - y₀‖ ≤ r' * ‖x - y₀‖",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nx_ne_y₀ : ¬x - y₀ ∈ F\ny : E\nhy : y ∈ F\nhy₀y : y₀ + y ∈ F\n⊢ r ≤ r'",
"tactic": "apply le_max_left"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nx_ne_y₀ : ¬x - y₀ ∈ F\ny : E\nhy : y ∈ F\nhy₀y : y₀ + y ∈ F\n⊢ r' * dist x y₀ < d",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nx_ne_y₀ : ¬x - y₀ ∈ F\ny : E\nhy : y ∈ F\nhy₀y : y₀ + y ∈ F\n⊢ r' * ‖x - y₀‖ < d",
"tactic": "rw [← dist_eq_norm]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nx_ne_y₀ : ¬x - y₀ ∈ F\ny : E\nhy : y ∈ F\nhy₀y : y₀ + y ∈ F\n⊢ r' * dist x y₀ < d",
"tactic": "exact (lt_div_iff' hlt).1 hxy₀"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF✝ : Type ?u.309\ninst✝¹ : SeminormedAddCommGroup F✝\ninst✝ : NormedSpace ℝ F✝\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : ¬x ∈ F\nd : ℝ := infDist x ↑F\nhFn : Set.Nonempty ↑F\nhdp : 0 < d\nr' : ℝ := max r 2⁻¹\nhr' : r' < 1\nhlt : 0 < r'\nhdlt : d < d / r'\ny₀ : E\nhy₀F : y₀ ∈ F\nhxy₀ : dist x y₀ < d / r'\nx_ne_y₀ : ¬x - y₀ ∈ F\ny : E\nhy : y ∈ F\nhy₀y : y₀ + y ∈ F\n⊢ dist x (y₀ + y) = ‖x - y₀ - y‖",
"tactic": "rw [sub_sub, dist_eq_norm]"
}
] |
[
74,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.map_add_le
|
[
{
"state_after": "case intro.intro\nR : Type u_1\nR₁ : Type ?u.488097\nR₂ : Type u_2\nR₃ : Type ?u.488103\nR₄ : Type ?u.488106\nS : Type ?u.488109\nK : Type ?u.488112\nK₂ : Type ?u.488115\nM : Type u_3\nM' : Type ?u.488121\nM₁ : Type ?u.488124\nM₂ : Type u_4\nM₃ : Type ?u.488130\nM₄ : Type ?u.488133\nN : Type ?u.488136\nN₂ : Type ?u.488139\nι : Type ?u.488142\nV : Type ?u.488145\nV₂ : Type ?u.488148\ninst✝¹⁴ : Semiring R\ninst✝¹³ : Semiring R₂\ninst✝¹² : Semiring R₃\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M'\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\ninst✝⁵ : Module R₂ M₂\ninst✝⁴ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₂₁ : R₂ →+* R\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\np p' : Submodule R M\nq q' : Submodule R₂ M₂\nq₁ q₁' : Submodule R M'\nr : R\nx y : M\ninst✝ : RingHomSurjective σ₁₂\nF : Type ?u.488729\nsc : SemilinearMapClass F σ₁₂ M M₂\nf g : M →ₛₗ[σ₁₂] M₂\nm : M\nhm : m ∈ ↑p\n⊢ ↑(f + g) m ∈ map f p ⊔ map g p",
"state_before": "R : Type u_1\nR₁ : Type ?u.488097\nR₂ : Type u_2\nR₃ : Type ?u.488103\nR₄ : Type ?u.488106\nS : Type ?u.488109\nK : Type ?u.488112\nK₂ : Type ?u.488115\nM : Type u_3\nM' : Type ?u.488121\nM₁ : Type ?u.488124\nM₂ : Type u_4\nM₃ : Type ?u.488130\nM₄ : Type ?u.488133\nN : Type ?u.488136\nN₂ : Type ?u.488139\nι : Type ?u.488142\nV : Type ?u.488145\nV₂ : Type ?u.488148\ninst✝¹⁴ : Semiring R\ninst✝¹³ : Semiring R₂\ninst✝¹² : Semiring R₃\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M'\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\ninst✝⁵ : Module R₂ M₂\ninst✝⁴ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₂₁ : R₂ →+* R\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\np p' : Submodule R M\nq q' : Submodule R₂ M₂\nq₁ q₁' : Submodule R M'\nr : R\nx y : M\ninst✝ : RingHomSurjective σ₁₂\nF : Type ?u.488729\nsc : SemilinearMapClass F σ₁₂ M M₂\nf g : M →ₛₗ[σ₁₂] M₂\n⊢ map (f + g) p ≤ map f p ⊔ map g p",
"tactic": "rintro x ⟨m, hm, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_1\nR₁ : Type ?u.488097\nR₂ : Type u_2\nR₃ : Type ?u.488103\nR₄ : Type ?u.488106\nS : Type ?u.488109\nK : Type ?u.488112\nK₂ : Type ?u.488115\nM : Type u_3\nM' : Type ?u.488121\nM₁ : Type ?u.488124\nM₂ : Type u_4\nM₃ : Type ?u.488130\nM₄ : Type ?u.488133\nN : Type ?u.488136\nN₂ : Type ?u.488139\nι : Type ?u.488142\nV : Type ?u.488145\nV₂ : Type ?u.488148\ninst✝¹⁴ : Semiring R\ninst✝¹³ : Semiring R₂\ninst✝¹² : Semiring R₃\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M'\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\ninst✝⁵ : Module R₂ M₂\ninst✝⁴ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nσ₂₁ : R₂ →+* R\ninst✝³ : RingHomInvPair σ₁₂ σ₂₁\ninst✝² : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\np p' : Submodule R M\nq q' : Submodule R₂ M₂\nq₁ q₁' : Submodule R M'\nr : R\nx y : M\ninst✝ : RingHomSurjective σ₁₂\nF : Type ?u.488729\nsc : SemilinearMapClass F σ₁₂ M M₂\nf g : M →ₛₗ[σ₁₂] M₂\nm : M\nhm : m ∈ ↑p\n⊢ ↑(f + g) m ∈ map f p ⊔ map g p",
"tactic": "exact add_mem_sup (mem_map_of_mem hm) (mem_map_of_mem hm)"
}
] |
[
746,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
744,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
tendsto_pow_const_mul_const_pow_of_abs_lt_one
|
[
{
"state_after": "case pos\nα : Type ?u.120365\nβ : Type ?u.120368\nι : Type ?u.120371\nk : ℕ\nr : ℝ\nhr : abs r < 1\nh0 : r = 0\n⊢ Tendsto (fun n => ↑n ^ k * r ^ n) atTop (𝓝 0)\n\ncase neg\nα : Type ?u.120365\nβ : Type ?u.120368\nι : Type ?u.120371\nk : ℕ\nr : ℝ\nhr : abs r < 1\nh0 : ¬r = 0\n⊢ Tendsto (fun n => ↑n ^ k * r ^ n) atTop (𝓝 0)",
"state_before": "α : Type ?u.120365\nβ : Type ?u.120368\nι : Type ?u.120371\nk : ℕ\nr : ℝ\nhr : abs r < 1\n⊢ Tendsto (fun n => ↑n ^ k * r ^ n) atTop (𝓝 0)",
"tactic": "by_cases h0 : r = 0"
},
{
"state_after": "case neg\nα : Type ?u.120365\nβ : Type ?u.120368\nι : Type ?u.120371\nk : ℕ\nr : ℝ\nhr : abs r < 1\nh0 : ¬r = 0\nhr' : 1 < (abs r)⁻¹\n⊢ Tendsto (fun n => ↑n ^ k * r ^ n) atTop (𝓝 0)",
"state_before": "case neg\nα : Type ?u.120365\nβ : Type ?u.120368\nι : Type ?u.120371\nk : ℕ\nr : ℝ\nhr : abs r < 1\nh0 : ¬r = 0\n⊢ Tendsto (fun n => ↑n ^ k * r ^ n) atTop (𝓝 0)",
"tactic": "have hr' : 1 < (|r|)⁻¹ := one_lt_inv (abs_pos.2 h0) hr"
},
{
"state_after": "case neg\nα : Type ?u.120365\nβ : Type ?u.120368\nι : Type ?u.120371\nk : ℕ\nr : ℝ\nhr : abs r < 1\nh0 : ¬r = 0\nhr' : 1 < (abs r)⁻¹\n⊢ Tendsto (fun e => ‖↑e ^ k * r ^ e‖) atTop (𝓝 0)",
"state_before": "case neg\nα : Type ?u.120365\nβ : Type ?u.120368\nι : Type ?u.120371\nk : ℕ\nr : ℝ\nhr : abs r < 1\nh0 : ¬r = 0\nhr' : 1 < (abs r)⁻¹\n⊢ Tendsto (fun n => ↑n ^ k * r ^ n) atTop (𝓝 0)",
"tactic": "rw [tendsto_zero_iff_norm_tendsto_zero]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type ?u.120365\nβ : Type ?u.120368\nι : Type ?u.120371\nk : ℕ\nr : ℝ\nhr : abs r < 1\nh0 : ¬r = 0\nhr' : 1 < (abs r)⁻¹\n⊢ Tendsto (fun e => ‖↑e ^ k * r ^ e‖) atTop (𝓝 0)",
"tactic": "simpa [div_eq_mul_inv] using tendsto_pow_const_div_const_pow_of_one_lt k hr'"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type ?u.120365\nβ : Type ?u.120368\nι : Type ?u.120371\nk : ℕ\nr : ℝ\nhr : abs r < 1\nh0 : r = 0\n⊢ Tendsto (fun n => ↑n ^ k * r ^ n) atTop (𝓝 0)",
"tactic": "exact tendsto_const_nhds.congr'\n (mem_atTop_sets.2 ⟨1, fun n hn ↦ by simp [zero_lt_one.trans_le hn, h0]⟩)"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.120365\nβ : Type ?u.120368\nι : Type ?u.120371\nk : ℕ\nr : ℝ\nhr : abs r < 1\nh0 : r = 0\nn : ℕ\nhn : n ≥ 1\n⊢ n ∈ {x | (fun x => 0 = (fun n => ↑n ^ k * r ^ n) x) x}",
"tactic": "simp [zero_lt_one.trans_le hn, h0]"
}
] |
[
243,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/FieldTheory/PerfectClosure.lean
|
PerfectClosure.induction_on
|
[] |
[
184,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/CategoryTheory/Preadditive/Biproducts.lean
|
CategoryTheory.Limits.biproduct.total
|
[] |
[
212,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Data/List/Sublists.lean
|
List.sublists'_map
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf : α → β\n⊢ sublists' (map f []) = map (map f) (sublists' [])",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf : α → β\na : α\nl : List α\n⊢ sublists' (map f (a :: l)) = map (map f) (sublists' (a :: l))",
"tactic": "simp [map_cons, sublists'_cons, sublists'_map f l, Function.comp]"
}
] |
[
413,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/Data/Set/Sups.lean
|
Set.sup_mem_sups
|
[] |
[
82,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Init/Data/Nat/Lemmas.lean
|
Nat.bit1_ne
|
[] |
[
144,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
11
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
isCountablySpanning_measurableSet
|
[] |
[
1763,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1761,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.mem_sup
|
[] |
[
559,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
558,
1
] |
Std/Data/RBMap/Lemmas.lean
|
Std.RBNode.foldr_cons
|
[
{
"state_after": "α : Type u_1\nt : RBNode α\nl : List α\n⊢ foldr (fun x x_1 => x :: x_1) t l = foldr (fun x x_1 => x :: x_1) t [] ++ l",
"state_before": "α : Type u_1\nt : RBNode α\nl : List α\n⊢ foldr (fun x x_1 => x :: x_1) t l = toList t ++ l",
"tactic": "unfold toList"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nt : RBNode α\nl : List α\n⊢ foldr (fun x x_1 => x :: x_1) t l = foldr (fun x x_1 => x :: x_1) t [] ++ l",
"tactic": "induction t generalizing l with\n| nil => rfl\n| node _ a _ b iha ihb => rw [foldr, foldr, iha, iha (_::_), ihb]; simp"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nl : List α\n⊢ foldr (fun x x_1 => x :: x_1) nil l = foldr (fun x x_1 => x :: x_1) nil [] ++ l",
"tactic": "rfl"
},
{
"state_after": "case node\nα : Type u_1\nc✝ : RBColor\na : RBNode α\nv✝ : α\nb : RBNode α\niha : ∀ (l : List α), foldr (fun x x_1 => x :: x_1) a l = foldr (fun x x_1 => x :: x_1) a [] ++ l\nihb : ∀ (l : List α), foldr (fun x x_1 => x :: x_1) b l = foldr (fun x x_1 => x :: x_1) b [] ++ l\nl : List α\n⊢ foldr (fun x x_1 => x :: x_1) a [] ++ v✝ :: (foldr (fun x x_1 => x :: x_1) b [] ++ l) =\n foldr (fun x x_1 => x :: x_1) a [] ++ v✝ :: foldr (fun x x_1 => x :: x_1) b [] ++ l",
"state_before": "case node\nα : Type u_1\nc✝ : RBColor\na : RBNode α\nv✝ : α\nb : RBNode α\niha : ∀ (l : List α), foldr (fun x x_1 => x :: x_1) a l = foldr (fun x x_1 => x :: x_1) a [] ++ l\nihb : ∀ (l : List α), foldr (fun x x_1 => x :: x_1) b l = foldr (fun x x_1 => x :: x_1) b [] ++ l\nl : List α\n⊢ foldr (fun x x_1 => x :: x_1) (node c✝ a v✝ b) l = foldr (fun x x_1 => x :: x_1) (node c✝ a v✝ b) [] ++ l",
"tactic": "rw [foldr, foldr, iha, iha (_::_), ihb]"
},
{
"state_after": "no goals",
"state_before": "case node\nα : Type u_1\nc✝ : RBColor\na : RBNode α\nv✝ : α\nb : RBNode α\niha : ∀ (l : List α), foldr (fun x x_1 => x :: x_1) a l = foldr (fun x x_1 => x :: x_1) a [] ++ l\nihb : ∀ (l : List α), foldr (fun x x_1 => x :: x_1) b l = foldr (fun x x_1 => x :: x_1) b [] ++ l\nl : List α\n⊢ foldr (fun x x_1 => x :: x_1) a [] ++ v✝ :: (foldr (fun x x_1 => x :: x_1) b [] ++ l) =\n foldr (fun x x_1 => x :: x_1) a [] ++ v✝ :: foldr (fun x x_1 => x :: x_1) b [] ++ l",
"tactic": "simp"
}
] |
[
333,
74
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
329,
1
] |
Mathlib/AlgebraicGeometry/PresheafedSpace/HasColimits.lean
|
AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.desc_fac
|
[
{
"state_after": "case w\nJ : Type u'\ninst✝⁴ : Category J\nC : Type u\ninst✝³ : Category C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nj : J\n⊢ ((colimitCocone F).ι.app j ≫ desc F s).base = (s.ι.app j).base\n\ncase h\nJ : Type u'\ninst✝⁴ : Category J\nC : Type u\ninst✝³ : Category C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nj : J\n⊢ ((colimitCocone F).ι.app j ≫ desc F s).c ≫\n whiskerRight\n (eqToHom\n (_ :\n Functor.op (Opens.map ((colimitCocone F).ι.app j ≫ desc F s).base) =\n Functor.op (Opens.map (s.ι.app j).base)))\n (F.obj j).presheaf =\n (s.ι.app j).c",
"state_before": "J : Type u'\ninst✝⁴ : Category J\nC : Type u\ninst✝³ : Category C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nj : J\n⊢ (colimitCocone F).ι.app j ≫ desc F s = s.ι.app j",
"tactic": "fapply PresheafedSpace.ext"
},
{
"state_after": "no goals",
"state_before": "case w\nJ : Type u'\ninst✝⁴ : Category J\nC : Type u\ninst✝³ : Category C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nj : J\n⊢ ((colimitCocone F).ι.app j ≫ desc F s).base = (s.ι.app j).base",
"tactic": "simp [desc]"
},
{
"state_after": "case h\nJ : Type u'\ninst✝⁴ : Category J\nC : Type u\ninst✝³ : Category C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nj : J\nU : (Opens ↑↑s.pt)ᵒᵖ\n⊢ (((colimitCocone F).ι.app j ≫ desc F s).c ≫\n whiskerRight\n (eqToHom\n (_ :\n Functor.op (Opens.map ((colimitCocone F).ι.app j ≫ desc F s).base) =\n Functor.op (Opens.map (s.ι.app j).base)))\n (F.obj j).presheaf).app\n U =\n (s.ι.app j).c.app U",
"state_before": "case h\nJ : Type u'\ninst✝⁴ : Category J\nC : Type u\ninst✝³ : Category C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nj : J\n⊢ ((colimitCocone F).ι.app j ≫ desc F s).c ≫\n whiskerRight\n (eqToHom\n (_ :\n Functor.op (Opens.map ((colimitCocone F).ι.app j ≫ desc F s).base) =\n Functor.op (Opens.map (s.ι.app j).base)))\n (F.obj j).presheaf =\n (s.ι.app j).c",
"tactic": "refine NatTrans.ext _ _ (funext fun U => ?_)"
},
{
"state_after": "case h\nJ : Type u'\ninst✝⁴ : Category J\nC : Type u\ninst✝³ : Category C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nj : J\nU : (Opens ↑↑s.pt)ᵒᵖ\n⊢ ((desc F s).c.app U ≫ ((colimitCocone F).ι.app j).c.app ((Opens.map (desc F s).base).obj U.unop).op) ≫\n (F.obj j).presheaf.map\n ((eqToHom\n (_ :\n Functor.op (Opens.map ((colimitCocone F).ι.app j ≫ desc F s).base) =\n Functor.op (Opens.map (s.ι.app j).base))).app\n U) =\n (s.ι.app j).c.app U",
"state_before": "case h\nJ : Type u'\ninst✝⁴ : Category J\nC : Type u\ninst✝³ : Category C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nj : J\nU : (Opens ↑↑s.pt)ᵒᵖ\n⊢ (((colimitCocone F).ι.app j ≫ desc F s).c ≫\n whiskerRight\n (eqToHom\n (_ :\n Functor.op (Opens.map ((colimitCocone F).ι.app j ≫ desc F s).base) =\n Functor.op (Opens.map (s.ι.app j).base)))\n (F.obj j).presheaf).app\n U =\n (s.ι.app j).c.app U",
"tactic": "rw [NatTrans.comp_app, PresheafedSpace.comp_c_app, whiskerRight_app]"
},
{
"state_after": "case h\nJ : Type u'\ninst✝⁴ : Category J\nC : Type u\ninst✝³ : Category C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nj : J\nU : (Opens ↑↑s.pt)ᵒᵖ\n⊢ ((limit.lift\n ((pushforwardDiagramToColimit F).leftOp ⋙\n (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op)\n { pt := s.pt.presheaf.obj U,\n π :=\n NatTrans.mk fun j =>\n (s.ι.app j.unop).c.app U ≫\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Functor.op (Opens.map (s.ι.app j.unop).base)).obj U =\n (Functor.op (Opens.map (colimit.ι (F ⋙ forget C) j.unop))).obj\n ((Functor.op (Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s)))).obj\n U))) } ≫\n (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op).inv) ≫\n (limit.π (pushforwardDiagramToColimit F).leftOp j.op).app\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op) ≫\n (F.obj j).presheaf.map\n ((eqToHom\n (_ :\n Functor.op\n (Opens.map (colimit.ι (F ⋙ forget C) j ≫ colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))) =\n Functor.op (Opens.map (s.ι.app j).base))).app\n U) =\n (s.ι.app j).c.app U",
"state_before": "case h\nJ : Type u'\ninst✝⁴ : Category J\nC : Type u\ninst✝³ : Category C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nj : J\nU : (Opens ↑↑s.pt)ᵒᵖ\n⊢ ((desc F s).c.app U ≫ ((colimitCocone F).ι.app j).c.app ((Opens.map (desc F s).base).obj U.unop).op) ≫\n (F.obj j).presheaf.map\n ((eqToHom\n (_ :\n Functor.op (Opens.map ((colimitCocone F).ι.app j ≫ desc F s).base) =\n Functor.op (Opens.map (s.ι.app j).base))).app\n U) =\n (s.ι.app j).c.app U",
"tactic": "dsimp [desc, descCApp]"
},
{
"state_after": "case h\nJ : Type u'\ninst✝⁴ : Category J\nC : Type u\ninst✝³ : Category C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nj : J\nU : (Opens ↑↑s.pt)ᵒᵖ\n⊢ limit.lift\n (CategoryTheory.Functor.mk\n { obj := fun X => ((pushforwardDiagramToColimit F).obj X.unop).unop,\n map := fun X Y f => ((pushforwardDiagramToColimit F).map f.unop).unop } ⋙\n (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op)\n { pt := s.pt.presheaf.obj U,\n π :=\n NatTrans.mk fun j =>\n (s.ι.app j.unop).c.app U ≫\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Functor.op (Opens.map (s.ι.app j.unop).base)).obj U =\n (Functor.op (Opens.map (colimit.ι (F ⋙ forget C) j.unop))).obj\n ((Functor.op (Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s)))).obj U))) } ≫\n (limitObjIsoLimitCompEvaluation\n (CategoryTheory.Functor.mk\n { obj := fun X => ((pushforwardDiagramToColimit F).obj X.unop).unop,\n map := fun X Y f => ((pushforwardDiagramToColimit F).map f.unop).unop })\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op).inv ≫\n (limit.π\n (CategoryTheory.Functor.mk\n { obj := fun X => ((pushforwardDiagramToColimit F).obj X.unop).unop,\n map := fun X Y f => ((pushforwardDiagramToColimit F).map f.unop).unop })\n j.op).app\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op ≫\n eqToHom\n (_ :\n (F.obj j).presheaf.obj\n ((Opens.map (colimit.ι (F ⋙ forget C) j)).obj\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop)).op =\n (F.obj j).presheaf.obj ((Opens.map (s.ι.app j).base).obj U.unop).op) =\n (s.ι.app j).c.app U",
"state_before": "case h\nJ : Type u'\ninst✝⁴ : Category J\nC : Type u\ninst✝³ : Category C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nj : J\nU : (Opens ↑↑s.pt)ᵒᵖ\n⊢ ((limit.lift\n ((pushforwardDiagramToColimit F).leftOp ⋙\n (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op)\n { pt := s.pt.presheaf.obj U,\n π :=\n NatTrans.mk fun j =>\n (s.ι.app j.unop).c.app U ≫\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Functor.op (Opens.map (s.ι.app j.unop).base)).obj U =\n (Functor.op (Opens.map (colimit.ι (F ⋙ forget C) j.unop))).obj\n ((Functor.op (Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s)))).obj\n U))) } ≫\n (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op).inv) ≫\n (limit.π (pushforwardDiagramToColimit F).leftOp j.op).app\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op) ≫\n (F.obj j).presheaf.map\n ((eqToHom\n (_ :\n Functor.op\n (Opens.map (colimit.ι (F ⋙ forget C) j ≫ colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))) =\n Functor.op (Opens.map (s.ι.app j).base))).app\n U) =\n (s.ι.app j).c.app U",
"tactic": "simp only [eqToHom_app, op_obj, Opens.map_comp_obj, eqToHom_map, Functor.leftOp, assoc]"
},
{
"state_after": "case h\nJ : Type u'\ninst✝⁴ : Category J\nC : Type u\ninst✝³ : Category C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nj : J\nU : (Opens ↑↑s.pt)ᵒᵖ\n⊢ limit.lift\n (CategoryTheory.Functor.mk\n { obj := fun X => ((pushforwardDiagramToColimit F).obj X.unop).unop,\n map := fun X Y f => ((pushforwardDiagramToColimit F).map f.unop).unop } ⋙\n (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op)\n { pt := s.pt.presheaf.obj U,\n π :=\n NatTrans.mk fun j =>\n (s.ι.app j.unop).c.app U ≫\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Functor.op (Opens.map (s.ι.app j.unop).base)).obj U =\n (Functor.op (Opens.map (colimit.ι (F ⋙ forget C) j.unop))).obj\n ((Functor.op (Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s)))).obj U))) } ≫\n limit.π\n (CategoryTheory.Functor.mk\n { obj := fun X => ((pushforwardDiagramToColimit F).obj X.unop).unop,\n map := fun X Y f => ((pushforwardDiagramToColimit F).map f.unop).unop } ⋙\n (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op)\n j.op ≫\n eqToHom\n (_ :\n (F.obj j).presheaf.obj\n ((Opens.map (colimit.ι (F ⋙ forget C) j)).obj\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop)).op =\n (F.obj j).presheaf.obj ((Opens.map (s.ι.app j).base).obj U.unop).op) =\n (s.ι.app j).c.app U",
"state_before": "case h\nJ : Type u'\ninst✝⁴ : Category J\nC : Type u\ninst✝³ : Category C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nj : J\nU : (Opens ↑↑s.pt)ᵒᵖ\n⊢ limit.lift\n (CategoryTheory.Functor.mk\n { obj := fun X => ((pushforwardDiagramToColimit F).obj X.unop).unop,\n map := fun X Y f => ((pushforwardDiagramToColimit F).map f.unop).unop } ⋙\n (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op)\n { pt := s.pt.presheaf.obj U,\n π :=\n NatTrans.mk fun j =>\n (s.ι.app j.unop).c.app U ≫\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Functor.op (Opens.map (s.ι.app j.unop).base)).obj U =\n (Functor.op (Opens.map (colimit.ι (F ⋙ forget C) j.unop))).obj\n ((Functor.op (Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s)))).obj U))) } ≫\n (limitObjIsoLimitCompEvaluation\n (CategoryTheory.Functor.mk\n { obj := fun X => ((pushforwardDiagramToColimit F).obj X.unop).unop,\n map := fun X Y f => ((pushforwardDiagramToColimit F).map f.unop).unop })\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op).inv ≫\n (limit.π\n (CategoryTheory.Functor.mk\n { obj := fun X => ((pushforwardDiagramToColimit F).obj X.unop).unop,\n map := fun X Y f => ((pushforwardDiagramToColimit F).map f.unop).unop })\n j.op).app\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op ≫\n eqToHom\n (_ :\n (F.obj j).presheaf.obj\n ((Opens.map (colimit.ι (F ⋙ forget C) j)).obj\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop)).op =\n (F.obj j).presheaf.obj ((Opens.map (s.ι.app j).base).obj U.unop).op) =\n (s.ι.app j).c.app U",
"tactic": "rw [limitObjIsoLimitCompEvaluation_inv_π_app_assoc]"
},
{
"state_after": "no goals",
"state_before": "case h\nJ : Type u'\ninst✝⁴ : Category J\nC : Type u\ninst✝³ : Category C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nj : J\nU : (Opens ↑↑s.pt)ᵒᵖ\n⊢ limit.lift\n (CategoryTheory.Functor.mk\n { obj := fun X => ((pushforwardDiagramToColimit F).obj X.unop).unop,\n map := fun X Y f => ((pushforwardDiagramToColimit F).map f.unop).unop } ⋙\n (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op)\n { pt := s.pt.presheaf.obj U,\n π :=\n NatTrans.mk fun j =>\n (s.ι.app j.unop).c.app U ≫\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Functor.op (Opens.map (s.ι.app j.unop).base)).obj U =\n (Functor.op (Opens.map (colimit.ι (F ⋙ forget C) j.unop))).obj\n ((Functor.op (Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s)))).obj U))) } ≫\n limit.π\n (CategoryTheory.Functor.mk\n { obj := fun X => ((pushforwardDiagramToColimit F).obj X.unop).unop,\n map := fun X Y f => ((pushforwardDiagramToColimit F).map f.unop).unop } ⋙\n (evaluation (Opens ↑(Limits.colimit (F ⋙ forget C)))ᵒᵖ C).obj\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop).op)\n j.op ≫\n eqToHom\n (_ :\n (F.obj j).presheaf.obj\n ((Opens.map (colimit.ι (F ⋙ forget C) j)).obj\n ((Opens.map (colimit.desc (F ⋙ forget C) ((forget C).mapCocone s))).obj U.unop)).op =\n (F.obj j).presheaf.obj ((Opens.map (s.ι.app j).base).obj U.unop).op) =\n (s.ι.app j).c.app U",
"tactic": "simp"
}
] |
[
317,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.subtype_comp_inclusion
|
[] |
[
190,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.total_eq_fintype_total
|
[] |
[
1083,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1080,
1
] |
Mathlib/ModelTheory/ElementaryMaps.lean
|
FirstOrder.Language.ElementaryEmbedding.map_formula
|
[
{
"state_after": "no goals",
"state_before": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.238691\nQ : Type ?u.238694\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nf : M ↪ₑ[L] N\nα : Type u_5\nφ : Formula L α\nx : α → M\n⊢ Formula.Realize φ (↑f ∘ x) ↔ Formula.Realize φ x",
"tactic": "rw [Formula.Realize, Formula.Realize, ← f.map_boundedFormula, Unique.eq_default (f ∘ default)]"
}
] |
[
110,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Data/Matrix/DMatrix.lean
|
AddMonoidHom.mapDMatrix_apply
|
[] |
[
190,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Topology/Maps.lean
|
Function.LeftInverse.embedding
|
[] |
[
226,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
11
] |
Mathlib/Analysis/Fourier/FourierTransform.lean
|
Real.fourierIntegral_def
|
[] |
[
270,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.integral_comp_div_add
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.14892009\n𝕜 : Type ?u.14892012\nE : Type u_1\nF : Type ?u.14892018\nA : Type ?u.14892021\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na b c d✝ : ℝ\nf : ℝ → E\nhc : c ≠ 0\nd : ℝ\n⊢ (∫ (x : ℝ) in a..b, f (x / c + d)) = c • ∫ (x : ℝ) in a / c + d..b / c + d, f x",
"tactic": "simpa only [div_eq_inv_mul, inv_inv] using integral_comp_mul_add f (inv_ne_zero hc) d"
}
] |
[
779,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
777,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
one_lt_of_lt_mul_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.23830\ninst✝² : MulOneClass α\ninst✝¹ : LT α\ninst✝ : ContravariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : α\nh : b < a * b\n⊢ 1 * ?m.24201 h < a * ?m.24201 h",
"tactic": "simpa only [one_mul]"
}
] |
[
492,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.pure_div_pure
|
[] |
[
507,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
LatticeHom.coe_comp_inf_hom
|
[] |
[
1135,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1133,
1
] |
Mathlib/Algebra/GroupPower/Order.lean
|
sq_pos_of_ne_zero
|
[] |
[
662,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
661,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.measure_add_diff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.21168\nγ : Type ?u.21171\nδ : Type ?u.21174\nι : Type ?u.21177\nR : Type ?u.21180\nR' : Type ?u.21183\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t✝ : Set α\nhs : MeasurableSet s\nt : Set α\n⊢ ↑↑μ s + ↑↑μ (t \\ s) = ↑↑μ (s ∪ t)",
"tactic": "rw [← measure_union' (@disjoint_sdiff_right _ s t) hs, union_diff_self]"
}
] |
[
232,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Order/Lattice.lean
|
eq_of_inf_eq_sup_eq
|
[] |
[
799,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
795,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
StrictAntiOn.mul_antitone'
|
[] |
[
1548,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1546,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieModuleHom.coe_zsmul
|
[] |
[
923,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
922,
1
] |
Mathlib/Probability/Independence/ZeroOne.lean
|
ProbabilityTheory.measure_eq_zero_or_one_of_indepSetCat_self
|
[
{
"state_after": "Ω : Type u_1\nι : Type ?u.108607\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsFiniteMeasure μ\nt : Set Ω\nh_indep : IndepSet t t\nh_0_1_top : ↑↑μ t = 0 ∨ ↑↑μ t = 1 ∨ ↑↑μ t = ⊤\n⊢ ↑↑μ t = 0 ∨ ↑↑μ t = 1",
"state_before": "Ω : Type u_1\nι : Type ?u.108607\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsFiniteMeasure μ\nt : Set Ω\nh_indep : IndepSet t t\n⊢ ↑↑μ t = 0 ∨ ↑↑μ t = 1",
"tactic": "have h_0_1_top := measure_eq_zero_or_one_or_top_of_indepSet_self h_indep"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\nι : Type ?u.108607\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsFiniteMeasure μ\nt : Set Ω\nh_indep : IndepSet t t\nh_0_1_top : ↑↑μ t = 0 ∨ ↑↑μ t = 1 ∨ ↑↑μ t = ⊤\n⊢ ↑↑μ t = 0 ∨ ↑↑μ t = 1",
"tactic": "simpa [measure_ne_top μ] using h_0_1_top"
}
] |
[
50,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/RingTheory/WittVector/WittPolynomial.lean
|
xInTermsOfW_vars_subset
|
[] |
[
288,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/Data/Nat/Choose/Cast.lean
|
Nat.cast_choose
|
[
{
"state_after": "K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : CharZero K\na b : ℕ\nh : a ≤ b\nthis : ∀ {n : ℕ}, ↑n ! ≠ 0\n⊢ ↑(choose b a) = ↑b ! / (↑a ! * ↑(b - a)!)",
"state_before": "K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : CharZero K\na b : ℕ\nh : a ≤ b\n⊢ ↑(choose b a) = ↑b ! / (↑a ! * ↑(b - a)!)",
"tactic": "have : ∀ {n : ℕ}, (n ! : K) ≠ 0 := Nat.cast_ne_zero.2 (factorial_ne_zero _)"
},
{
"state_after": "K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : CharZero K\na b : ℕ\nh : a ≤ b\nthis : ∀ {n : ℕ}, ↑n ! ≠ 0\n⊢ ↑(choose b a) * (↑a ! * ↑(b - a)!) = ↑b !",
"state_before": "K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : CharZero K\na b : ℕ\nh : a ≤ b\nthis : ∀ {n : ℕ}, ↑n ! ≠ 0\n⊢ ↑(choose b a) = ↑b ! / (↑a ! * ↑(b - a)!)",
"tactic": "rw [eq_div_iff_mul_eq (mul_ne_zero this this)]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : CharZero K\na b : ℕ\nh : a ≤ b\nthis : ∀ {n : ℕ}, ↑n ! ≠ 0\n⊢ ↑(choose b a) * (↑a ! * ↑(b - a)!) = ↑b !",
"tactic": "rw_mod_cast [← mul_assoc, choose_mul_factorial_mul_factorial h]"
}
] |
[
31,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
28,
1
] |
Mathlib/Control/Traversable/Instances.lean
|
Sum.map_traverse
|
[
{
"state_after": "no goals",
"state_before": "σ : Type u\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα : Type u_1\nβ γ : Type u\ng : α → G β\nf : β → γ\nx : σ ⊕ α\n⊢ (fun x x_1 => x <$> x_1) f <$> Sum.traverse g x = Sum.traverse ((fun x x_1 => x <$> x_1) f ∘ g) x",
"tactic": "cases x <;> simp [Sum.traverse, id_map, functor_norm] <;> congr"
}
] |
[
184,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
11
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.destruct_flatten
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\n⊢ Computation.IsBisimulation fun c1 c2 => c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\n⊢ destruct (flatten c) = c >>= destruct",
"tactic": "refine'\n Computation.eq_of_bisim\n (fun c1 c2 => c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct) _\n (Or.inr ⟨c, rfl, rfl⟩)"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\nc1 c2 : Computation (Option (α × WSeq α))\nh : c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct\n⊢ Computation.BisimO (fun c1 c2 => c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct)\n (Computation.destruct c1) (Computation.destruct c2)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\n⊢ Computation.IsBisimulation fun c1 c2 => c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct",
"tactic": "intro c1 c2 h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nc✝ : Computation (WSeq α)\nc1 c2 : Computation (Option (α × WSeq α))\nh : c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct\nc : Computation (Option (α × WSeq α))\n⊢ Computation.BisimO (fun c1 c2 => c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct)\n (Computation.destruct c) (Computation.destruct c)",
"tactic": "cases c.destruct <;> simp"
},
{
"state_after": "case h1\nα : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\nc1 c2 : Computation (Option (α × WSeq α))\nh : c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct\na : WSeq α\n⊢ match Computation.destruct (destruct a), Computation.destruct (destruct a) with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' => s = s' ∨ ∃ c, s = destruct (flatten c) ∧ s' = Computation.bind c destruct\n | x, x_1 => False\n\ncase h2\nα : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\nc1 c2 : Computation (Option (α × WSeq α))\nh : c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct\nc' : Computation (WSeq α)\n⊢ destruct (flatten c') = Computation.bind c' destruct ∨\n ∃ c, destruct (flatten c') = destruct (flatten c) ∧ Computation.bind c' destruct = Computation.bind c destruct",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nc✝ : Computation (WSeq α)\nc1 c2 : Computation (Option (α × WSeq α))\nh : c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct\nc : Computation (WSeq α)\n⊢ Computation.BisimO (fun c1 c2 => c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct)\n (Computation.destruct (destruct (flatten c))) (Computation.destruct (Computation.bind c destruct))",
"tactic": "induction' c using Computation.recOn with a c' <;> simp"
},
{
"state_after": "no goals",
"state_before": "case h1\nα : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\nc1 c2 : Computation (Option (α × WSeq α))\nh : c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct\na : WSeq α\n⊢ match Computation.destruct (destruct a), Computation.destruct (destruct a) with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' => s = s' ∨ ∃ c, s = destruct (flatten c) ∧ s' = Computation.bind c destruct\n | x, x_1 => False",
"tactic": "cases (destruct a).destruct <;> simp"
},
{
"state_after": "no goals",
"state_before": "case h2\nα : Type u\nβ : Type v\nγ : Type w\nc : Computation (WSeq α)\nc1 c2 : Computation (Option (α × WSeq α))\nh : c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct\nc' : Computation (WSeq α)\n⊢ destruct (flatten c') = Computation.bind c' destruct ∨\n ∃ c, destruct (flatten c') = destruct (flatten c) ∧ Computation.bind c' destruct = Computation.bind c destruct",
"tactic": "exact Or.inr ⟨c', rfl, rfl⟩"
}
] |
[
698,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
686,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
wbtw_lineMap_iff
|
[
{
"state_after": "case pos\nR : Type u_1\nV : Type u_2\nV' : Type ?u.236526\nP : Type u_3\nP' : Type ?u.236532\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nx y : P\nr : R\nhxy : x = y\n⊢ Wbtw R x (↑(lineMap x y) r) y ↔ x = y ∨ r ∈ Set.Icc 0 1\n\ncase neg\nR : Type u_1\nV : Type u_2\nV' : Type ?u.236526\nP : Type u_3\nP' : Type ?u.236532\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nx y : P\nr : R\nhxy : ¬x = y\n⊢ Wbtw R x (↑(lineMap x y) r) y ↔ x = y ∨ r ∈ Set.Icc 0 1",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.236526\nP : Type u_3\nP' : Type ?u.236532\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nx y : P\nr : R\n⊢ Wbtw R x (↑(lineMap x y) r) y ↔ x = y ∨ r ∈ Set.Icc 0 1",
"tactic": "by_cases hxy : x = y"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_1\nV : Type u_2\nV' : Type ?u.236526\nP : Type u_3\nP' : Type ?u.236532\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nx y : P\nr : R\nhxy : ¬x = y\n⊢ Wbtw R x (↑(lineMap x y) r) y ↔ x = y ∨ r ∈ Set.Icc 0 1",
"tactic": "rw [or_iff_right hxy, Wbtw, affineSegment, (lineMap_injective R hxy).mem_set_image]"
},
{
"state_after": "case pos\nR : Type u_1\nV : Type u_2\nV' : Type ?u.236526\nP : Type u_3\nP' : Type ?u.236532\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nx y : P\nr : R\nhxy : x = y\n⊢ Wbtw R y y y ↔ y = y ∨ r ∈ Set.Icc 0 1",
"state_before": "case pos\nR : Type u_1\nV : Type u_2\nV' : Type ?u.236526\nP : Type u_3\nP' : Type ?u.236532\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nx y : P\nr : R\nhxy : x = y\n⊢ Wbtw R x (↑(lineMap x y) r) y ↔ x = y ∨ r ∈ Set.Icc 0 1",
"tactic": "rw [hxy, lineMap_same_apply]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\nV : Type u_2\nV' : Type ?u.236526\nP : Type u_3\nP' : Type ?u.236532\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nx y : P\nr : R\nhxy : x = y\n⊢ Wbtw R y y y ↔ y = y ∨ r ∈ Set.Icc 0 1",
"tactic": "simp"
}
] |
[
421,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
416,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.factors_decreasing
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\na b : α\ninst✝² : CommSemiring α\nI : Ideal α\ninst✝¹ : CommRing β\ninst✝ : IsDomain β\nb₁ b₂ : β\nh₁ : b₁ ≠ 0\nh₂ : ¬IsUnit b₂\nh : span {b₁} ≤ span {b₁ * b₂}\n⊢ b₁ * b₂ ∣ b₁ * 1",
"tactic": "rwa [mul_one, ← Ideal.span_singleton_le_span_singleton]"
}
] |
[
560,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
555,
1
] |
Mathlib/Topology/MetricSpace/Equicontinuity.lean
|
Metric.equicontinuousAt_iff_pair
|
[
{
"state_after": "α : Type u_3\nβ : Type u_2\nι✝ : Type ?u.1103\ninst✝¹ : PseudoMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nx₀ : β\n⊢ (∀ (U : Set (α × α)), U ∈ 𝓤 α → ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ V → ∀ (y : β), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U) ↔\n ∀ (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ U → ∀ (x' : β), x' ∈ U → ∀ (i : ι), dist (F i x) (F i x') < ε",
"state_before": "α : Type u_3\nβ : Type u_2\nι✝ : Type ?u.1103\ninst✝¹ : PseudoMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nx₀ : β\n⊢ EquicontinuousAt F x₀ ↔\n ∀ (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ U → ∀ (x' : β), x' ∈ U → ∀ (i : ι), dist (F i x) (F i x') < ε",
"tactic": "rw [equicontinuousAt_iff_pair]"
},
{
"state_after": "case mp\nα : Type u_3\nβ : Type u_2\nι✝ : Type ?u.1103\ninst✝¹ : PseudoMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nx₀ : β\nH : ∀ (U : Set (α × α)), U ∈ 𝓤 α → ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ V → ∀ (y : β), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U\n⊢ ∀ (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ U → ∀ (x' : β), x' ∈ U → ∀ (i : ι), dist (F i x) (F i x') < ε\n\ncase mpr\nα : Type u_3\nβ : Type u_2\nι✝ : Type ?u.1103\ninst✝¹ : PseudoMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nx₀ : β\nH : ∀ (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ U → ∀ (x' : β), x' ∈ U → ∀ (i : ι), dist (F i x) (F i x') < ε\n⊢ ∀ (U : Set (α × α)), U ∈ 𝓤 α → ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ V → ∀ (y : β), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U",
"state_before": "α : Type u_3\nβ : Type u_2\nι✝ : Type ?u.1103\ninst✝¹ : PseudoMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nx₀ : β\n⊢ (∀ (U : Set (α × α)), U ∈ 𝓤 α → ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ V → ∀ (y : β), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U) ↔\n ∀ (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ U → ∀ (x' : β), x' ∈ U → ∀ (i : ι), dist (F i x) (F i x') < ε",
"tactic": "constructor <;> intro H"
},
{
"state_after": "case mp\nα : Type u_3\nβ : Type u_2\nι✝ : Type ?u.1103\ninst✝¹ : PseudoMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nx₀ : β\nH : ∀ (U : Set (α × α)), U ∈ 𝓤 α → ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ V → ∀ (y : β), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U\nε : ℝ\nhε : ε > 0\n⊢ ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ U → ∀ (x' : β), x' ∈ U → ∀ (i : ι), dist (F i x) (F i x') < ε",
"state_before": "case mp\nα : Type u_3\nβ : Type u_2\nι✝ : Type ?u.1103\ninst✝¹ : PseudoMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nx₀ : β\nH : ∀ (U : Set (α × α)), U ∈ 𝓤 α → ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ V → ∀ (y : β), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U\n⊢ ∀ (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ U → ∀ (x' : β), x' ∈ U → ∀ (i : ι), dist (F i x) (F i x') < ε",
"tactic": "intro ε hε"
},
{
"state_after": "no goals",
"state_before": "case mp\nα : Type u_3\nβ : Type u_2\nι✝ : Type ?u.1103\ninst✝¹ : PseudoMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nx₀ : β\nH : ∀ (U : Set (α × α)), U ∈ 𝓤 α → ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ V → ∀ (y : β), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U\nε : ℝ\nhε : ε > 0\n⊢ ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ U → ∀ (x' : β), x' ∈ U → ∀ (i : ι), dist (F i x) (F i x') < ε",
"tactic": "exact H _ (dist_mem_uniformity hε)"
},
{
"state_after": "case mpr\nα : Type u_3\nβ : Type u_2\nι✝ : Type ?u.1103\ninst✝¹ : PseudoMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nx₀ : β\nH : ∀ (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ U → ∀ (x' : β), x' ∈ U → ∀ (i : ι), dist (F i x) (F i x') < ε\nU : Set (α × α)\nhU : U ∈ 𝓤 α\n⊢ ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ V → ∀ (y : β), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U",
"state_before": "case mpr\nα : Type u_3\nβ : Type u_2\nι✝ : Type ?u.1103\ninst✝¹ : PseudoMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nx₀ : β\nH : ∀ (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ U → ∀ (x' : β), x' ∈ U → ∀ (i : ι), dist (F i x) (F i x') < ε\n⊢ ∀ (U : Set (α × α)), U ∈ 𝓤 α → ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ V → ∀ (y : β), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U",
"tactic": "intro U hU"
},
{
"state_after": "case mpr.intro.intro\nα : Type u_3\nβ : Type u_2\nι✝ : Type ?u.1103\ninst✝¹ : PseudoMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nx₀ : β\nH : ∀ (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ U → ∀ (x' : β), x' ∈ U → ∀ (i : ι), dist (F i x) (F i x') < ε\nU : Set (α × α)\nhU : U ∈ 𝓤 α\nε : ℝ\nhε : ε > 0\nhεU : ∀ {a b : α}, dist a b < ε → (a, b) ∈ U\n⊢ ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ V → ∀ (y : β), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U",
"state_before": "case mpr\nα : Type u_3\nβ : Type u_2\nι✝ : Type ?u.1103\ninst✝¹ : PseudoMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nx₀ : β\nH : ∀ (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ U → ∀ (x' : β), x' ∈ U → ∀ (i : ι), dist (F i x) (F i x') < ε\nU : Set (α × α)\nhU : U ∈ 𝓤 α\n⊢ ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ V → ∀ (y : β), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U",
"tactic": "rcases mem_uniformity_dist.mp hU with ⟨ε, hε, hεU⟩"
},
{
"state_after": "case mpr.intro.intro\nα : Type u_3\nβ : Type u_2\nι✝ : Type ?u.1103\ninst✝¹ : PseudoMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nx₀ : β\nH : ∀ (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ U → ∀ (x' : β), x' ∈ U → ∀ (i : ι), dist (F i x) (F i x') < ε\nU : Set (α × α)\nhU : U ∈ 𝓤 α\nε : ℝ\nhε : ε > 0\nhεU : ∀ {a b : α}, dist a b < ε → (a, b) ∈ U\nV : Set β\nh : ∀ (x : β), x ∈ V → ∀ (x' : β), x' ∈ V → ∀ (i : ι), dist (F i x) (F i x') < ε\n⊢ ∀ (x : β), x ∈ V → ∀ (y : β), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U",
"state_before": "case mpr.intro.intro\nα : Type u_3\nβ : Type u_2\nι✝ : Type ?u.1103\ninst✝¹ : PseudoMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nx₀ : β\nH : ∀ (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ U → ∀ (x' : β), x' ∈ U → ∀ (i : ι), dist (F i x) (F i x') < ε\nU : Set (α × α)\nhU : U ∈ 𝓤 α\nε : ℝ\nhε : ε > 0\nhεU : ∀ {a b : α}, dist a b < ε → (a, b) ∈ U\n⊢ ∃ V, V ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ V → ∀ (y : β), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U",
"tactic": "refine' Exists.imp (fun V => And.imp_right fun h => _) (H _ hε)"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\nα : Type u_3\nβ : Type u_2\nι✝ : Type ?u.1103\ninst✝¹ : PseudoMetricSpace α\nι : Type u_1\ninst✝ : TopologicalSpace β\nF : ι → β → α\nx₀ : β\nH : ∀ (ε : ℝ), ε > 0 → ∃ U, U ∈ 𝓝 x₀ ∧ ∀ (x : β), x ∈ U → ∀ (x' : β), x' ∈ U → ∀ (i : ι), dist (F i x) (F i x') < ε\nU : Set (α × α)\nhU : U ∈ 𝓤 α\nε : ℝ\nhε : ε > 0\nhεU : ∀ {a b : α}, dist a b < ε → (a, b) ∈ U\nV : Set β\nh : ∀ (x : β), x ∈ V → ∀ (x' : β), x' ∈ V → ∀ (i : ι), dist (F i x) (F i x') < ε\n⊢ ∀ (x : β), x ∈ V → ∀ (y : β), y ∈ V → ∀ (i : ι), (F i x, F i y) ∈ U",
"tactic": "exact fun x hx x' hx' i => hεU (h _ hx _ hx' i)"
}
] |
[
72,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
11
] |
Mathlib/Data/Real/Sqrt.lean
|
Real.sqrt_one
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ sqrt 1 = 1",
"tactic": "simp [sqrt]"
}
] |
[
258,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/Order/Category/NonemptyFinLinOrdCat.lean
|
NonemptyFinLinOrdCat.epi_iff_surjective
|
[
{
"state_after": "case mp\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\n⊢ Epi f → Function.Surjective ↑f\n\ncase mpr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\n⊢ Function.Surjective ↑f → Epi f",
"state_before": "A B : NonemptyFinLinOrdCat\nf : A ⟶ B\n⊢ Epi f ↔ Function.Surjective ↑f",
"tactic": "constructor"
},
{
"state_after": "case mp\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\n⊢ Function.Surjective ↑f",
"state_before": "case mp\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\n⊢ Epi f → Function.Surjective ↑f",
"tactic": "intro"
},
{
"state_after": "case mp\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\n⊢ ∀ (b : ↑B), ∃ a, ↑f a = b",
"state_before": "case mp\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\n⊢ Function.Surjective ↑f",
"tactic": "dsimp only [Function.Surjective]"
},
{
"state_after": "case mp\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nhf' : ∃ b, ∀ (a : ↑A), ↑f a ≠ b\n⊢ False",
"state_before": "case mp\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\n⊢ ∀ (b : ↑B), ∃ a, ↑f a = b",
"tactic": "by_contra' hf'"
},
{
"state_after": "case mp.intro\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\n⊢ False",
"state_before": "case mp\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nhf' : ∃ b, ∀ (a : ↑A), ↑f a ≠ b\n⊢ False",
"tactic": "rcases hf' with ⟨m, hm⟩"
},
{
"state_after": "case mp.intro\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\n⊢ False",
"state_before": "case mp.intro\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\n⊢ False",
"tactic": "let Y := NonemptyFinLinOrdCat.of (ULift (Fin 2))"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\nh : ↑p₁ m = ↑p₂ m\n⊢ False",
"tactic": "simp [FunLike.coe] at h"
},
{
"state_after": "A B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\n⊢ (if x₁ < m then { down := 0 } else { down := 1 }) ≤ if x₂ < m then { down := 0 } else { down := 1 }",
"state_before": "A B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\n⊢ (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂",
"tactic": "simp only"
},
{
"state_after": "case inl.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : x₁ < m\nh₂ : x₂ < m\n⊢ { down := 0 } ≤ { down := 0 }\n\ncase inl.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : x₁ < m\nh₂ : ¬x₂ < m\n⊢ { down := 0 } ≤ { down := 1 }\n\ncase inr.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ < m\nh₂ : x₂ < m\n⊢ { down := 1 } ≤ { down := 0 }\n\ncase inr.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ < m\nh₂ : ¬x₂ < m\n⊢ { down := 1 } ≤ { down := 1 }",
"state_before": "A B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\n⊢ (if x₁ < m then { down := 0 } else { down := 1 }) ≤ if x₂ < m then { down := 0 } else { down := 1 }",
"tactic": "split_ifs with h₁ h₂ h₂"
},
{
"state_after": "case inr.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ < m\nh₂ : x₂ < m\n⊢ { down := 1 } ≤ { down := 0 }\n\ncase inr.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ < m\nh₂ : ¬x₂ < m\n⊢ { down := 1 } ≤ { down := 1 }",
"state_before": "case inl.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : x₁ < m\nh₂ : x₂ < m\n⊢ { down := 0 } ≤ { down := 0 }\n\ncase inl.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : x₁ < m\nh₂ : ¬x₂ < m\n⊢ { down := 0 } ≤ { down := 1 }\n\ncase inr.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ < m\nh₂ : x₂ < m\n⊢ { down := 1 } ≤ { down := 0 }\n\ncase inr.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ < m\nh₂ : ¬x₂ < m\n⊢ { down := 1 } ≤ { down := 1 }",
"tactic": "any_goals apply Fin.zero_le"
},
{
"state_after": "case inr.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ < m\nh₂ : ¬x₂ < m\n⊢ { down := 1 } ≤ { down := 1 }",
"state_before": "case inr.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ < m\nh₂ : ¬x₂ < m\n⊢ { down := 1 } ≤ { down := 1 }",
"tactic": "apply Fin.zero_le"
},
{
"state_after": "case inr.inl.h\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ < m\nh₂ : x₂ < m\n⊢ False",
"state_before": "case inr.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ < m\nh₂ : x₂ < m\n⊢ { down := 1 } ≤ { down := 0 }",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.h\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ < m\nh₂ : x₂ < m\n⊢ False",
"tactic": "exact h₁ (lt_of_le_of_lt h h₂)"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ < m\nh₂ : ¬x₂ < m\n⊢ { down := 1 } ≤ { down := 1 }",
"tactic": "rfl"
},
{
"state_after": "A B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\n⊢ (if x₁ ≤ m then { down := 0 } else { down := 1 }) ≤ if x₂ ≤ m then { down := 0 } else { down := 1 }",
"state_before": "A B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\n⊢ (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂",
"tactic": "simp only"
},
{
"state_after": "case inl.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : x₁ ≤ m\nh₂ : x₂ ≤ m\n⊢ { down := 0 } ≤ { down := 0 }\n\ncase inl.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : x₁ ≤ m\nh₂ : ¬x₂ ≤ m\n⊢ { down := 0 } ≤ { down := 1 }\n\ncase inr.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ ≤ m\nh₂ : x₂ ≤ m\n⊢ { down := 1 } ≤ { down := 0 }\n\ncase inr.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ ≤ m\nh₂ : ¬x₂ ≤ m\n⊢ { down := 1 } ≤ { down := 1 }",
"state_before": "A B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\n⊢ (if x₁ ≤ m then { down := 0 } else { down := 1 }) ≤ if x₂ ≤ m then { down := 0 } else { down := 1 }",
"tactic": "split_ifs with h₁ h₂ h₂"
},
{
"state_after": "case inr.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ ≤ m\nh₂ : x₂ ≤ m\n⊢ { down := 1 } ≤ { down := 0 }\n\ncase inr.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ ≤ m\nh₂ : ¬x₂ ≤ m\n⊢ { down := 1 } ≤ { down := 1 }",
"state_before": "case inl.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : x₁ ≤ m\nh₂ : x₂ ≤ m\n⊢ { down := 0 } ≤ { down := 0 }\n\ncase inl.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : x₁ ≤ m\nh₂ : ¬x₂ ≤ m\n⊢ { down := 0 } ≤ { down := 1 }\n\ncase inr.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ ≤ m\nh₂ : x₂ ≤ m\n⊢ { down := 1 } ≤ { down := 0 }\n\ncase inr.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ ≤ m\nh₂ : ¬x₂ ≤ m\n⊢ { down := 1 } ≤ { down := 1 }",
"tactic": "any_goals apply Fin.zero_le"
},
{
"state_after": "case inr.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ ≤ m\nh₂ : ¬x₂ ≤ m\n⊢ { down := 1 } ≤ { down := 1 }",
"state_before": "case inr.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ ≤ m\nh₂ : ¬x₂ ≤ m\n⊢ { down := 1 } ≤ { down := 1 }",
"tactic": "apply Fin.zero_le"
},
{
"state_after": "case inr.inl.h\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ ≤ m\nh₂ : x₂ ≤ m\n⊢ False",
"state_before": "case inr.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ ≤ m\nh₂ : x₂ ≤ m\n⊢ { down := 1 } ≤ { down := 0 }",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.h\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ ≤ m\nh₂ : x₂ ≤ m\n⊢ False",
"tactic": "exact h₁ (h.trans h₂)"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\nx₁ x₂ : ↑B\nh : x₁ ≤ x₂\nh₁ : ¬x₁ ≤ m\nh₂ : ¬x₂ ≤ m\n⊢ { down := 1 } ≤ { down := 1 }",
"tactic": "rfl"
},
{
"state_after": "case e_a\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\n⊢ p₁ = p₂",
"state_before": "A B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\n⊢ ↑p₁ m = ↑p₂ m",
"tactic": "congr"
},
{
"state_after": "case e_a\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\n⊢ f ≫ p₁ = f ≫ p₂",
"state_before": "case e_a\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\n⊢ p₁ = p₂",
"tactic": "rw [← cancel_epi f]"
},
{
"state_after": "case e_a.w\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\n⊢ (forget NonemptyFinLinOrdCat).map (f ≫ p₁) a = (forget NonemptyFinLinOrdCat).map (f ≫ p₂) a",
"state_before": "case e_a\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\n⊢ f ≫ p₁ = f ≫ p₂",
"tactic": "ext a"
},
{
"state_after": "case e_a.w\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\n⊢ (forget NonemptyFinLinOrdCat).map\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\n ((forget NonemptyFinLinOrdCat).map f a) =\n (forget NonemptyFinLinOrdCat).map\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\n ((forget NonemptyFinLinOrdCat).map f a)",
"state_before": "case e_a.w\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\n⊢ (forget NonemptyFinLinOrdCat).map (f ≫ p₁) a = (forget NonemptyFinLinOrdCat).map (f ≫ p₂) a",
"tactic": "simp only [coe_of, FunctorToTypes.map_comp_apply]"
},
{
"state_after": "case e_a.w\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\n⊢ (if ↑f a < m then { down := 0 } else { down := 1 }) = if ↑f a ≤ m then { down := 0 } else { down := 1 }",
"state_before": "case e_a.w\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\n⊢ (forget NonemptyFinLinOrdCat).map\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\n ((forget NonemptyFinLinOrdCat).map f a) =\n (forget NonemptyFinLinOrdCat).map\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\n ((forget NonemptyFinLinOrdCat).map f a)",
"tactic": "simp only [forget_map_apply]"
},
{
"state_after": "case e_a.w.inl.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ↑f a < m\nh₂ : ↑f a ≤ m\n⊢ { down := 0 } = { down := 0 }\n\ncase e_a.w.inl.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ↑f a < m\nh₂ : ¬↑f a ≤ m\n⊢ { down := 0 } = { down := 1 }\n\ncase e_a.w.inr.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ¬↑f a < m\nh₂ : ↑f a ≤ m\n⊢ { down := 1 } = { down := 0 }\n\ncase e_a.w.inr.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ¬↑f a < m\nh₂ : ¬↑f a ≤ m\n⊢ { down := 1 } = { down := 1 }",
"state_before": "case e_a.w\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\n⊢ (if ↑f a < m then { down := 0 } else { down := 1 }) = if ↑f a ≤ m then { down := 0 } else { down := 1 }",
"tactic": "split_ifs with h₁ h₂ h₂"
},
{
"state_after": "case e_a.w.inl.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ↑f a < m\nh₂ : ¬↑f a ≤ m\n⊢ { down := 0 } = { down := 1 }\n\ncase e_a.w.inr.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ¬↑f a < m\nh₂ : ↑f a ≤ m\n⊢ { down := 1 } = { down := 0 }",
"state_before": "case e_a.w.inl.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ↑f a < m\nh₂ : ↑f a ≤ m\n⊢ { down := 0 } = { down := 0 }\n\ncase e_a.w.inl.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ↑f a < m\nh₂ : ¬↑f a ≤ m\n⊢ { down := 0 } = { down := 1 }\n\ncase e_a.w.inr.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ¬↑f a < m\nh₂ : ↑f a ≤ m\n⊢ { down := 1 } = { down := 0 }\n\ncase e_a.w.inr.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ¬↑f a < m\nh₂ : ¬↑f a ≤ m\n⊢ { down := 1 } = { down := 1 }",
"tactic": "any_goals rfl"
},
{
"state_after": "no goals",
"state_before": "case e_a.w.inr.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ¬↑f a < m\nh₂ : ¬↑f a ≤ m\n⊢ { down := 1 } = { down := 1 }",
"tactic": "rfl"
},
{
"state_after": "case e_a.w.inl.inr.h\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ↑f a < m\nh₂ : ¬↑f a ≤ m\n⊢ False",
"state_before": "case e_a.w.inl.inr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ↑f a < m\nh₂ : ¬↑f a ≤ m\n⊢ { down := 0 } = { down := 1 }",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case e_a.w.inl.inr.h\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ↑f a < m\nh₂ : ¬↑f a ≤ m\n⊢ False",
"tactic": "exact h₂ (le_of_lt h₁)"
},
{
"state_after": "case e_a.w.inr.inl.h\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ¬↑f a < m\nh₂ : ↑f a ≤ m\n⊢ False",
"state_before": "case e_a.w.inr.inl\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ¬↑f a < m\nh₂ : ↑f a ≤ m\n⊢ { down := 1 } = { down := 0 }",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case e_a.w.inr.inl.h\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\na✝ : Epi f\nm : ↑B\nhm : ∀ (a : ↑A), ↑f a ≠ m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np₁ : B ⟶ Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b < m then { down := 0 } else { down := 1 }) x₂) }\np₂ : B ⟶ Y :=\n { toFun := fun b => if b ≤ m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n ∀ (x₁ x₂ : ↑B),\n x₁ ≤ x₂ →\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₁ ≤\n (fun b => if b ≤ m then { down := 0 } else { down := 1 }) x₂) }\na : (forget NonemptyFinLinOrdCat).obj A\nh₁ : ¬↑f a < m\nh₂ : ↑f a ≤ m\n⊢ False",
"tactic": "exact hm a (eq_of_le_of_not_lt h₂ h₁)"
},
{
"state_after": "case mpr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\nh : Function.Surjective ↑f\n⊢ Epi f",
"state_before": "case mpr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\n⊢ Function.Surjective ↑f → Epi f",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mpr\nA B : NonemptyFinLinOrdCat\nf : A ⟶ B\nh : Function.Surjective ↑f\n⊢ Epi f",
"tactic": "exact ConcreteCategory.epi_of_surjective f h"
}
] |
[
208,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Topology/MetricSpace/PiNat.lean
|
PiNat.dist_eq_of_ne
|
[
{
"state_after": "no goals",
"state_before": "E : ℕ → Type u_1\nx y : (n : ℕ) → E n\nh : x ≠ y\n⊢ dist x y = (1 / 2) ^ firstDiff x y",
"tactic": "simp [dist, h]"
}
] |
[
277,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Mathlib/Analysis/Quaternion.lean
|
Quaternion.summable_coe
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf : α → ℝ\n⊢ (Summable fun a => ↑(f a)) ↔ Summable f",
"tactic": "simpa only using\n Summable.map_iff_of_leftInverse (algebraMap ℝ ℍ) (show ℍ →ₗ[ℝ] ℝ from QuaternionAlgebra.reₗ _ _)\n (continuous_algebraMap _ _) continuous_re coe_re"
}
] |
[
251,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/CategoryTheory/Limits/Creates.lean
|
CategoryTheory.hasColimit_of_created
|
[] |
[
204,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
measurableSet_Ici
|
[] |
[
438,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
|
UniformOnFun.hasBasis_uniformity_of_basis_aux₂
|
[] |
[
653,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
644,
11
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.add_lt_add_of_lt_of_le
|
[] |
[
794,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
793,
11
] |
Mathlib/Data/Nat/Parity.lean
|
Nat.two_mul_div_two_of_even
|
[] |
[
223,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/Topology/Algebra/Module/Multilinear.lean
|
ContinuousMultilinearMap.pi_apply
|
[] |
[
270,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.reduce.min
|
[
{
"state_after": "case refl\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : DecidableEq α\n⊢ reduce L₁ = reduce L₁\n\ncase tail\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : DecidableEq α\nL1 L' : List (α × Bool)\nL2 : ReflTransGen Red.Step (reduce L₁) L1\nH1 : Red.Step L1 L'\nH2 : reduce L₁ = L1\n⊢ reduce L₁ = L'",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : DecidableEq α\nH : Red (reduce L₁) L₂\n⊢ reduce L₁ = L₂",
"tactic": "induction' H with L1 L' L2 H1 H2 ih"
},
{
"state_after": "no goals",
"state_before": "case refl\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : DecidableEq α\n⊢ reduce L₁ = reduce L₁",
"tactic": "rfl"
},
{
"state_after": "case tail.not\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : DecidableEq α\nL4 L5 : List (α × Bool)\nx : α\nb : Bool\nL2 : ReflTransGen Red.Step (reduce L₁) (L4 ++ (x, b) :: (x, !b) :: L5)\nH2 : reduce L₁ = L4 ++ (x, b) :: (x, !b) :: L5\n⊢ reduce L₁ = L4 ++ L5",
"state_before": "case tail\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : DecidableEq α\nL1 L' : List (α × Bool)\nL2 : ReflTransGen Red.Step (reduce L₁) L1\nH1 : Red.Step L1 L'\nH2 : reduce L₁ = L1\n⊢ reduce L₁ = L'",
"tactic": "cases' H1 with L4 L5 x b"
},
{
"state_after": "no goals",
"state_before": "case tail.not\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : DecidableEq α\nL4 L5 : List (α × Bool)\nx : α\nb : Bool\nL2 : ReflTransGen Red.Step (reduce L₁) (L4 ++ (x, b) :: (x, !b) :: L5)\nH2 : reduce L₁ = L4 ++ (x, b) :: (x, !b) :: L5\n⊢ reduce L₁ = L4 ++ L5",
"tactic": "exact reduce.not H2"
}
] |
[
1197,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1193,
1
] |
Mathlib/Logic/Basic.lean
|
Ne.ite_ne_right_iff
|
[] |
[
1205,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1204,
11
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.mk'_mem_incidenceSet_left_iff
|
[] |
[
839,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
838,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.isSuccLimit_aleph0
|
[
{
"state_after": "case intro\nα β : Type u\nn : ℕ\nha : ↑n < ℵ₀\n⊢ succ ↑n < ℵ₀",
"state_before": "α β : Type u\na : Cardinal\nha : a < ℵ₀\n⊢ succ a < ℵ₀",
"tactic": "rcases lt_aleph0.1 ha with ⟨n, rfl⟩"
},
{
"state_after": "case intro\nα β : Type u\nn : ℕ\nha : ↑n < ℵ₀\n⊢ ↑(Nat.succ n) < ℵ₀",
"state_before": "case intro\nα β : Type u\nn : ℕ\nha : ↑n < ℵ₀\n⊢ succ ↑n < ℵ₀",
"tactic": "rw [← nat_succ]"
},
{
"state_after": "no goals",
"state_before": "case intro\nα β : Type u\nn : ℕ\nha : ↑n < ℵ₀\n⊢ ↑(Nat.succ n) < ℵ₀",
"tactic": "apply nat_lt_aleph0"
}
] |
[
1444,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1440,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.sum_apply
|
[] |
[
524,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
523,
1
] |
Mathlib/Algebra/Order/LatticeGroup.lean
|
LatticeOrderedCommGroup.m_neg_part_def
|
[] |
[
162,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Analysis/NormedSpace/AddTorsorBases.lean
|
IsOpen.exists_subset_affineIndependent_span_eq_top
|
[
{
"state_after": "case intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nu : Set P\nhu : IsOpen u\nx : P\nhx : x ∈ u\n⊢ ∃ s x, AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ s = ⊤",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nu : Set P\nhu : IsOpen u\nhne : Set.Nonempty u\n⊢ ∃ s x, AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ s = ⊤",
"tactic": "rcases hne with ⟨x, hx⟩"
},
{
"state_after": "case intro.intro.intro.intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nu : Set P\nhu : IsOpen u\nx : P\nhx : x ∈ u\ns : Set P\nhsu : s ⊆ u\nhs : AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ s = ⊤\n⊢ ∃ s x, AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ s = ⊤",
"state_before": "case intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nu : Set P\nhu : IsOpen u\nx : P\nhx : x ∈ u\n⊢ ∃ s x, AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ s = ⊤",
"tactic": "rcases hu.exists_between_affineIndependent_span_eq_top (singleton_subset_iff.mpr hx)\n (singleton_nonempty _) (affineIndependent_of_subsingleton _ _) with ⟨s, -, hsu, hs⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nu : Set P\nhu : IsOpen u\nx : P\nhx : x ∈ u\ns : Set P\nhsu : s ⊆ u\nhs : AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ s = ⊤\n⊢ ∃ s x, AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ s = ⊤",
"tactic": "exact ⟨s, hsu, hs⟩"
}
] |
[
127,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Algebra/Module/Basic.lean
|
inv_int_cast_smul_eq
|
[] |
[
522,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
520,
1
] |
Mathlib/Topology/Instances/Rat.lean
|
Rat.continuous_coe_real
|
[] |
[
56,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.totallyBounded_iff'
|
[] |
[
772,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
767,
1
] |
Mathlib/Data/Matrix/Hadamard.lean
|
Matrix.add_hadamard
|
[] |
[
82,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Topology/Order/Basic.lean
|
comap_coe_nhdsWithin_Ioi_of_Ioo_subset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nha : s ⊆ Ioi a\nhs : Set.Nonempty s → ∃ b, b > a ∧ Ioo a b ⊆ s\nh : Set.Nonempty (↑ofDual ⁻¹' s)\n⊢ ∃ a_1, a_1 < ↑toDual a ∧ Ioo a_1 (↑toDual a) ⊆ ↑ofDual ⁻¹' s",
"tactic": "simpa only [OrderDual.exists, dual_Ioo] using hs h"
}
] |
[
2466,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2463,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Lp.norm_eq_zero_iff
|
[] |
[
333,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Std/Logic.lean
|
or_and_left
|
[] |
[
331,
60
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
329,
1
] |
Mathlib/Topology/IsLocallyHomeomorph.lean
|
IsLocallyHomeomorphOn.continuousAt
|
[] |
[
68,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
11
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.normalClosure_mono
|
[] |
[
2498,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2497,
1
] |
Mathlib/LinearAlgebra/Dimension.lean
|
rank_zero_iff
|
[] |
[
517,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
516,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
IsMax.Ici_eq
|
[] |
[
953,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
952,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
div_eq_inv_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.30968\nG : Type ?u.30971\ninst✝ : DivisionCommMonoid α\na b c d : α\n⊢ a / b = b⁻¹ * a",
"tactic": "simp"
}
] |
[
505,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
505,
1
] |
Mathlib/CategoryTheory/Localization/Predicate.lean
|
CategoryTheory.Localization.liftNatTrans_app
|
[] |
[
349,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean
|
CauchySeq.mul_const
|
[] |
[
449,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
447,
1
] |
Mathlib/Data/Polynomial/Expand.lean
|
Polynomial.coeff_contract
|
[
{
"state_after": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q p : ℕ\nhp : p ≠ 0\nf : R[X]\nn : ℕ\n⊢ natDegree f + 1 ≤ n → 0 = coeff f (n * p)",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q p : ℕ\nhp : p ≠ 0\nf : R[X]\nn : ℕ\n⊢ coeff (contract p f) n = coeff f (n * p)",
"tactic": "simp only [contract, coeff_monomial, sum_ite_eq', finset_sum_coeff, mem_range, not_lt,\n ite_eq_left_iff]"
},
{
"state_after": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q p : ℕ\nhp : p ≠ 0\nf : R[X]\nn : ℕ\nhn : natDegree f + 1 ≤ n\n⊢ 0 = coeff f (n * p)",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q p : ℕ\nhp : p ≠ 0\nf : R[X]\nn : ℕ\n⊢ natDegree f + 1 ≤ n → 0 = coeff f (n * p)",
"tactic": "intro hn"
},
{
"state_after": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q p : ℕ\nhp : p ≠ 0\nf : R[X]\nn : ℕ\nhn : natDegree f + 1 ≤ n\n⊢ natDegree f < n * p",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q p : ℕ\nhp : p ≠ 0\nf : R[X]\nn : ℕ\nhn : natDegree f + 1 ≤ n\n⊢ 0 = coeff f (n * p)",
"tactic": "apply (coeff_eq_zero_of_natDegree_lt _).symm"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q p : ℕ\nhp : p ≠ 0\nf : R[X]\nn : ℕ\nhn : natDegree f + 1 ≤ n\n⊢ natDegree f < n * p",
"tactic": "calc\n f.natDegree < f.natDegree + 1 := Nat.lt_succ_self _\n _ ≤ n * 1 := by simpa only [mul_one] using hn\n _ ≤ n * p := mul_le_mul_of_nonneg_left (show 1 ≤ p from hp.bot_lt) (zero_le n)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q p : ℕ\nhp : p ≠ 0\nf : R[X]\nn : ℕ\nhn : natDegree f + 1 ≤ n\n⊢ natDegree f + 1 ≤ n * 1",
"tactic": "simpa only [mul_one] using hn"
}
] |
[
217,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/GroupTheory/Finiteness.lean
|
Group.fg_def
|
[] |
[
297,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Algebra/Parity.lean
|
Dvd.dvd.even
|
[] |
[
261,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/CategoryTheory/Limits/ConeCategory.lean
|
CategoryTheory.Limits.hasLimitsOfShape_iff_isLeftAdjoint_const
|
[] |
[
104,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.subset_iUnion_of_subset
|
[] |
[
346,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
tendsto_dist_right_cocompact_atTop
|
[] |
[
2153,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2150,
1
] |
Mathlib/Analysis/NormedSpace/Pointwise.lean
|
affinity_unitClosedBall
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.1782138\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : 0 ≤ r\nx : E\n⊢ x +ᵥ r • closedBall 0 1 = closedBall x r",
"tactic": "rw [smul_closedUnitBall, Real.norm_of_nonneg hr, vadd_closedBall_zero]"
}
] |
[
444,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.mulVec_add
|
[
{
"state_after": "case h\nl : Type ?u.856313\nm : Type u_2\nn : Type u_1\no : Type ?u.856322\nm' : o → Type ?u.856327\nn' : o → Type ?u.856332\nR : Type ?u.856335\nS : Type ?u.856338\nα : Type v\nβ : Type w\nγ : Type ?u.856345\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype n\nA : Matrix m n α\nx y : n → α\nx✝ : m\n⊢ mulVec A (x + y) x✝ = (mulVec A x + mulVec A y) x✝",
"state_before": "l : Type ?u.856313\nm : Type u_2\nn : Type u_1\no : Type ?u.856322\nm' : o → Type ?u.856327\nn' : o → Type ?u.856332\nR : Type ?u.856335\nS : Type ?u.856338\nα : Type v\nβ : Type w\nγ : Type ?u.856345\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype n\nA : Matrix m n α\nx y : n → α\n⊢ mulVec A (x + y) = mulVec A x + mulVec A y",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nl : Type ?u.856313\nm : Type u_2\nn : Type u_1\no : Type ?u.856322\nm' : o → Type ?u.856327\nn' : o → Type ?u.856332\nR : Type ?u.856335\nS : Type ?u.856338\nα : Type v\nβ : Type w\nγ : Type ?u.856345\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype n\nA : Matrix m n α\nx y : n → α\nx✝ : m\n⊢ mulVec A (x + y) x✝ = (mulVec A x + mulVec A y) x✝",
"tactic": "apply dotProduct_add"
}
] |
[
1751,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1748,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
tsum_univ
|
[] |
[
612,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
611,
1
] |
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.coe_finset_sum
|
[] |
[
103,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
ofAdd_nsmul
|
[] |
[
449,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
447,
1
] |
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