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/Analysis/SpecificLimits/Basic.lean
|
ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1
|
[
{
"state_after": "case intro.intro\nα : Type ?u.42152\nβ : Type ?u.42155\nι : Type ?u.42158\nr : ℝ≥0\nhr' : ↑r < 1\nhr : ↑r < 1\n⊢ Tendsto (fun n => ↑r ^ n) atTop (𝓝 0)",
"state_before": "α : Type ?u.42152\nβ : Type ?u.42155\nι : Type ?u.42158\nr : ℝ≥0∞\nhr : r < 1\n⊢ Tendsto (fun n => r ^ n) atTop (𝓝 0)",
"tactic": "rcases ENNReal.lt_iff_exists_coe.1 hr with ⟨r, rfl, hr'⟩"
},
{
"state_after": "case intro.intro\nα : Type ?u.42152\nβ : Type ?u.42155\nι : Type ?u.42158\nr : ℝ≥0\nhr' : ↑r < 1\nhr : ↑r < 1\n⊢ Tendsto (fun n => ↑r ^ n) atTop (𝓝 ↑0)",
"state_before": "case intro.intro\nα : Type ?u.42152\nβ : Type ?u.42155\nι : Type ?u.42158\nr : ℝ≥0\nhr' : ↑r < 1\nhr : ↑r < 1\n⊢ Tendsto (fun n => ↑r ^ n) atTop (𝓝 0)",
"tactic": "rw [← ENNReal.coe_zero]"
},
{
"state_after": "case intro.intro\nα : Type ?u.42152\nβ : Type ?u.42155\nι : Type ?u.42158\nr : ℝ≥0\nhr' hr : r < 1\n⊢ Tendsto (fun a => r ^ a) atTop (𝓝 0)",
"state_before": "case intro.intro\nα : Type ?u.42152\nβ : Type ?u.42155\nι : Type ?u.42158\nr : ℝ≥0\nhr' : ↑r < 1\nhr : ↑r < 1\n⊢ Tendsto (fun n => ↑r ^ n) atTop (𝓝 ↑0)",
"tactic": "norm_cast at *"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type ?u.42152\nβ : Type ?u.42155\nι : Type ?u.42158\nr : ℝ≥0\nhr' hr : r < 1\n⊢ Tendsto (fun a => r ^ a) atTop (𝓝 0)",
"tactic": "apply NNReal.tendsto_pow_atTop_nhds_0_of_lt_1 hr"
}
] |
[
174,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
le_abs
|
[] |
[
57,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
MultilinearMap.coe_injective
|
[
{
"state_after": "R : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝¹⁰ : Semiring R\ninst✝⁹ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\ninst✝¹ : Module R M₃\ninst✝ : Module R M'\nf✝ f' f g : MultilinearMap R M₁ M₂\nh : ↑f = ↑g\n⊢ f = g",
"state_before": "R : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝¹⁰ : Semiring R\ninst✝⁹ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\ninst✝¹ : Module R M₃\ninst✝ : Module R M'\nf f' : MultilinearMap R M₁ M₂\n⊢ Injective FunLike.coe",
"tactic": "intro f g h"
},
{
"state_after": "case mk\nR : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝¹⁰ : Semiring R\ninst✝⁹ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\ninst✝¹ : Module R M₃\ninst✝ : Module R M'\nf f' g : MultilinearMap R M₁ M₂\ntoFun✝ : ((i : ι) → M₁ i) → M₂\nmap_add'✝ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i),\n toFun✝ (update m i (x + y)) = toFun✝ (update m i x) + toFun✝ (update m i y)\nmap_smul'✝ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i),\n toFun✝ (update m i (c • x)) = c • toFun✝ (update m i x)\nh : ↑{ toFun := toFun✝, map_add' := map_add'✝, map_smul' := map_smul'✝ } = ↑g\n⊢ { toFun := toFun✝, map_add' := map_add'✝, map_smul' := map_smul'✝ } = g",
"state_before": "R : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝¹⁰ : Semiring R\ninst✝⁹ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\ninst✝¹ : Module R M₃\ninst✝ : Module R M'\nf✝ f' f g : MultilinearMap R M₁ M₂\nh : ↑f = ↑g\n⊢ f = g",
"tactic": "cases f"
},
{
"state_after": "case mk.mk\nR : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝¹⁰ : Semiring R\ninst✝⁹ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\ninst✝¹ : Module R M₃\ninst✝ : Module R M'\nf f' : MultilinearMap R M₁ M₂\ntoFun✝¹ : ((i : ι) → M₁ i) → M₂\nmap_add'✝¹ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i),\n toFun✝¹ (update m i (x + y)) = toFun✝¹ (update m i x) + toFun✝¹ (update m i y)\nmap_smul'✝¹ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i),\n toFun✝¹ (update m i (c • x)) = c • toFun✝¹ (update m i x)\ntoFun✝ : ((i : ι) → M₁ i) → M₂\nmap_add'✝ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i),\n toFun✝ (update m i (x + y)) = toFun✝ (update m i x) + toFun✝ (update m i y)\nmap_smul'✝ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i),\n toFun✝ (update m i (c • x)) = c • toFun✝ (update m i x)\nh :\n ↑{ toFun := toFun✝¹, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹ } =\n ↑{ toFun := toFun✝, map_add' := map_add'✝, map_smul' := map_smul'✝ }\n⊢ { toFun := toFun✝¹, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹ } =\n { toFun := toFun✝, map_add' := map_add'✝, map_smul' := map_smul'✝ }",
"state_before": "case mk\nR : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝¹⁰ : Semiring R\ninst✝⁹ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\ninst✝¹ : Module R M₃\ninst✝ : Module R M'\nf f' g : MultilinearMap R M₁ M₂\ntoFun✝ : ((i : ι) → M₁ i) → M₂\nmap_add'✝ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i),\n toFun✝ (update m i (x + y)) = toFun✝ (update m i x) + toFun✝ (update m i y)\nmap_smul'✝ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i),\n toFun✝ (update m i (c • x)) = c • toFun✝ (update m i x)\nh : ↑{ toFun := toFun✝, map_add' := map_add'✝, map_smul' := map_smul'✝ } = ↑g\n⊢ { toFun := toFun✝, map_add' := map_add'✝, map_smul' := map_smul'✝ } = g",
"tactic": "cases g"
},
{
"state_after": "case mk.mk.refl\nR : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝¹⁰ : Semiring R\ninst✝⁹ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\ninst✝¹ : Module R M₃\ninst✝ : Module R M'\nf f' : MultilinearMap R M₁ M₂\ntoFun✝ : ((i : ι) → M₁ i) → M₂\nmap_add'✝¹ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i),\n toFun✝ (update m i (x + y)) = toFun✝ (update m i x) + toFun✝ (update m i y)\nmap_smul'✝¹ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i),\n toFun✝ (update m i (c • x)) = c • toFun✝ (update m i x)\nmap_add'✝ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i),\n toFun✝ (update m i (x + y)) = toFun✝ (update m i x) + toFun✝ (update m i y)\nmap_smul'✝ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i),\n toFun✝ (update m i (c • x)) = c • toFun✝ (update m i x)\n⊢ { toFun := toFun✝, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹ } =\n { toFun := toFun✝, map_add' := map_add'✝, map_smul' := map_smul'✝ }",
"state_before": "case mk.mk\nR : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝¹⁰ : Semiring R\ninst✝⁹ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\ninst✝¹ : Module R M₃\ninst✝ : Module R M'\nf f' : MultilinearMap R M₁ M₂\ntoFun✝¹ : ((i : ι) → M₁ i) → M₂\nmap_add'✝¹ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i),\n toFun✝¹ (update m i (x + y)) = toFun✝¹ (update m i x) + toFun✝¹ (update m i y)\nmap_smul'✝¹ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i),\n toFun✝¹ (update m i (c • x)) = c • toFun✝¹ (update m i x)\ntoFun✝ : ((i : ι) → M₁ i) → M₂\nmap_add'✝ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i),\n toFun✝ (update m i (x + y)) = toFun✝ (update m i x) + toFun✝ (update m i y)\nmap_smul'✝ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i),\n toFun✝ (update m i (c • x)) = c • toFun✝ (update m i x)\nh :\n ↑{ toFun := toFun✝¹, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹ } =\n ↑{ toFun := toFun✝, map_add' := map_add'✝, map_smul' := map_smul'✝ }\n⊢ { toFun := toFun✝¹, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹ } =\n { toFun := toFun✝, map_add' := map_add'✝, map_smul' := map_smul'✝ }",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.refl\nR : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝¹⁰ : Semiring R\ninst✝⁹ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\ninst✝¹ : Module R M₃\ninst✝ : Module R M'\nf f' : MultilinearMap R M₁ M₂\ntoFun✝ : ((i : ι) → M₁ i) → M₂\nmap_add'✝¹ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i),\n toFun✝ (update m i (x + y)) = toFun✝ (update m i x) + toFun✝ (update m i y)\nmap_smul'✝¹ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i),\n toFun✝ (update m i (c • x)) = c • toFun✝ (update m i x)\nmap_add'✝ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i),\n toFun✝ (update m i (x + y)) = toFun✝ (update m i x) + toFun✝ (update m i y)\nmap_smul'✝ :\n ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i),\n toFun✝ (update m i (c • x)) = c • toFun✝ (update m i x)\n⊢ { toFun := toFun✝, map_add' := map_add'✝¹, map_smul' := map_smul'✝¹ } =\n { toFun := toFun✝, map_add' := map_add'✝, map_smul' := map_smul'✝ }",
"tactic": "rfl"
}
] |
[
145,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Order/Max.lean
|
isMax_iff_forall_not_lt
|
[] |
[
336,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.coe_def
|
[] |
[
1681,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1680,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.coe_inj
|
[] |
[
1030,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1029,
1
] |
Mathlib/Algebra/Module/LinearMap.lean
|
LinearMap.toAddMonoidHom_coe
|
[] |
[
435,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
NormedRing.summable_geometric_of_norm_lt_1
|
[
{
"state_after": "α : Type ?u.1232352\nβ : Type ?u.1232355\nι : Type ?u.1232358\nR : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : R\nh : ‖x‖ < 1\nh1 : Summable fun n => ‖x‖ ^ n\n⊢ Summable fun n => x ^ n",
"state_before": "α : Type ?u.1232352\nβ : Type ?u.1232355\nι : Type ?u.1232358\nR : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : R\nh : ‖x‖ < 1\n⊢ Summable fun n => x ^ n",
"tactic": "have h1 : Summable fun n : ℕ ↦ ‖x‖ ^ n := summable_geometric_of_lt_1 (norm_nonneg _) h"
},
{
"state_after": "α : Type ?u.1232352\nβ : Type ?u.1232355\nι : Type ?u.1232358\nR : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : R\nh : ‖x‖ < 1\nh1 : Summable fun n => ‖x‖ ^ n\n⊢ ∀ᶠ (i : ℕ) in cofinite, ‖x ^ i‖ ≤ ‖x‖ ^ i",
"state_before": "α : Type ?u.1232352\nβ : Type ?u.1232355\nι : Type ?u.1232358\nR : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : R\nh : ‖x‖ < 1\nh1 : Summable fun n => ‖x‖ ^ n\n⊢ Summable fun n => x ^ n",
"tactic": "refine' summable_of_norm_bounded_eventually _ h1 _"
},
{
"state_after": "α : Type ?u.1232352\nβ : Type ?u.1232355\nι : Type ?u.1232358\nR : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : R\nh : ‖x‖ < 1\nh1 : Summable fun n => ‖x‖ ^ n\n⊢ ∀ᶠ (i : ℕ) in atTop, ‖x ^ i‖ ≤ ‖x‖ ^ i",
"state_before": "α : Type ?u.1232352\nβ : Type ?u.1232355\nι : Type ?u.1232358\nR : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : R\nh : ‖x‖ < 1\nh1 : Summable fun n => ‖x‖ ^ n\n⊢ ∀ᶠ (i : ℕ) in cofinite, ‖x ^ i‖ ≤ ‖x‖ ^ i",
"tactic": "rw [Nat.cofinite_eq_atTop]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.1232352\nβ : Type ?u.1232355\nι : Type ?u.1232358\nR : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : R\nh : ‖x‖ < 1\nh1 : Summable fun n => ‖x‖ ^ n\n⊢ ∀ᶠ (i : ℕ) in atTop, ‖x ^ i‖ ≤ ‖x‖ ^ i",
"tactic": "exact eventually_norm_pow_le x"
}
] |
[
466,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
1
] |
Mathlib/Order/CompleteLattice.lean
|
sInf_le_sInf_of_forall_exists_le
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.39374\nβ₂ : Type ?u.39377\nγ : Type ?u.39380\nι : Sort ?u.39383\nι' : Sort ?u.39386\nκ : ι → Sort ?u.39391\nκ' : ι' → Sort ?u.39396\ninst✝ : CompleteSemilatticeInf α\ns t : Set α\na b : α\nh : ∀ (x : α), x ∈ s → ∃ y, y ∈ t ∧ y ≤ x\n⊢ ∀ (c : α), (∀ (b : α), b ∈ t → c ≤ b) → ∀ (b : α), b ∈ s → c ≤ b",
"state_before": "α : Type u_1\nβ : Type ?u.39374\nβ₂ : Type ?u.39377\nγ : Type ?u.39380\nι : Sort ?u.39383\nι' : Sort ?u.39386\nκ : ι → Sort ?u.39391\nκ' : ι' → Sort ?u.39396\ninst✝ : CompleteSemilatticeInf α\ns t : Set α\na b : α\nh : ∀ (x : α), x ∈ s → ∃ y, y ∈ t ∧ y ≤ x\n⊢ ∀ (c : α), c ≤ sInf t → c ≤ sInf s",
"tactic": "simp only [le_sInf_iff]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.39374\nβ₂ : Type ?u.39377\nγ : Type ?u.39380\nι : Sort ?u.39383\nι' : Sort ?u.39386\nκ : ι → Sort ?u.39391\nκ' : ι' → Sort ?u.39396\ninst✝ : CompleteSemilatticeInf α\ns t : Set α\na b✝ : α\nh : ∀ (x : α), x ∈ s → ∃ y, y ∈ t ∧ y ≤ x\nc : α\nh₀ : ∀ (b : α), b ∈ t → c ≤ b\nb : α\nh₁ : b ∈ s\n⊢ c ≤ b",
"state_before": "α : Type u_1\nβ : Type ?u.39374\nβ₂ : Type ?u.39377\nγ : Type ?u.39380\nι : Sort ?u.39383\nι' : Sort ?u.39386\nκ : ι → Sort ?u.39391\nκ' : ι' → Sort ?u.39396\ninst✝ : CompleteSemilatticeInf α\ns t : Set α\na b : α\nh : ∀ (x : α), x ∈ s → ∃ y, y ∈ t ∧ y ≤ x\n⊢ ∀ (c : α), (∀ (b : α), b ∈ t → c ≤ b) → ∀ (b : α), b ∈ s → c ≤ b",
"tactic": "introv h₀ h₁"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.39374\nβ₂ : Type ?u.39377\nγ : Type ?u.39380\nι : Sort ?u.39383\nι' : Sort ?u.39386\nκ : ι → Sort ?u.39391\nκ' : ι' → Sort ?u.39396\ninst✝ : CompleteSemilatticeInf α\ns t : Set α\na b✝ : α\nh : ∀ (x : α), x ∈ s → ∃ y, y ∈ t ∧ y ≤ x\nc : α\nh₀ : ∀ (b : α), b ∈ t → c ≤ b\nb : α\nh₁ : b ∈ s\ny : α\nhy : y ∈ t\nhy' : y ≤ b\n⊢ c ≤ b",
"state_before": "α : Type u_1\nβ : Type ?u.39374\nβ₂ : Type ?u.39377\nγ : Type ?u.39380\nι : Sort ?u.39383\nι' : Sort ?u.39386\nκ : ι → Sort ?u.39391\nκ' : ι' → Sort ?u.39396\ninst✝ : CompleteSemilatticeInf α\ns t : Set α\na b✝ : α\nh : ∀ (x : α), x ∈ s → ∃ y, y ∈ t ∧ y ≤ x\nc : α\nh₀ : ∀ (b : α), b ∈ t → c ≤ b\nb : α\nh₁ : b ∈ s\n⊢ c ≤ b",
"tactic": "rcases h _ h₁ with ⟨y, hy, hy'⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.39374\nβ₂ : Type ?u.39377\nγ : Type ?u.39380\nι : Sort ?u.39383\nι' : Sort ?u.39386\nκ : ι → Sort ?u.39391\nκ' : ι' → Sort ?u.39396\ninst✝ : CompleteSemilatticeInf α\ns t : Set α\na b✝ : α\nh : ∀ (x : α), x ∈ s → ∃ y, y ∈ t ∧ y ≤ x\nc : α\nh₀ : ∀ (b : α), b ∈ t → c ≤ b\nb : α\nh₁ : b ∈ s\ny : α\nhy : y ∈ t\nhy' : y ≤ b\n⊢ c ≤ b",
"tactic": "solve_by_elim [le_trans _ hy']"
}
] |
[
250,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/Analysis/Calculus/Deriv/Prod.lean
|
hasDerivAt_pi
|
[] |
[
101,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
AddMonoidAlgebra.lift_apply
|
[
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.2579273\ninst✝⁵ : CommSemiring k\ninst✝⁴ : AddMonoid G\nA : Type u₃\ninst✝³ : Semiring A\ninst✝² : Algebra k A\nB : Type ?u.2579310\ninst✝¹ : Semiring B\ninst✝ : Algebra k B\nF : Multiplicative G →* A\nf : MonoidAlgebra k G\n⊢ ↑(↑(lift k G A) F) f = sum f fun a b => b • ↑F (↑Multiplicative.ofAdd a)",
"tactic": "simp only [lift_apply', Algebra.smul_def]"
}
] |
[
1974,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1972,
1
] |
Mathlib/Topology/Constructions.lean
|
isClosedMap_toAdd
|
[] |
[
127,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/LinearAlgebra/Determinant.lean
|
LinearMap.isUnit_det
|
[
{
"state_after": "case intro\nR : Type ?u.1057928\ninst✝¹¹ : CommRing R\nM : Type u_2\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\nM' : Type ?u.1058519\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nι : Type ?u.1059061\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ne : Basis ι R M\nA✝ : Type ?u.1059539\ninst✝⁴ : CommRing A✝\ninst✝³ : Module A✝ M\nκ : Type ?u.1060037\ninst✝² : Fintype κ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Module A M\nf : M →ₗ[A] M\nhf : IsUnit f\ng : M →ₗ[A] M\nhg : comp f g = 1\n⊢ IsUnit (↑LinearMap.det f)",
"state_before": "R : Type ?u.1057928\ninst✝¹¹ : CommRing R\nM : Type u_2\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\nM' : Type ?u.1058519\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nι : Type ?u.1059061\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ne : Basis ι R M\nA✝ : Type ?u.1059539\ninst✝⁴ : CommRing A✝\ninst✝³ : Module A✝ M\nκ : Type ?u.1060037\ninst✝² : Fintype κ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Module A M\nf : M →ₗ[A] M\nhf : IsUnit f\n⊢ IsUnit (↑LinearMap.det f)",
"tactic": "obtain ⟨g, hg⟩ : ∃ g, f.comp g = 1 := hf.exists_right_inv"
},
{
"state_after": "case intro\nR : Type ?u.1057928\ninst✝¹¹ : CommRing R\nM : Type u_2\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\nM' : Type ?u.1058519\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nι : Type ?u.1059061\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ne : Basis ι R M\nA✝ : Type ?u.1059539\ninst✝⁴ : CommRing A✝\ninst✝³ : Module A✝ M\nκ : Type ?u.1060037\ninst✝² : Fintype κ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Module A M\nf : M →ₗ[A] M\nhf : IsUnit f\ng : M →ₗ[A] M\nhg : comp f g = 1\nthis : ↑LinearMap.det f * ↑LinearMap.det g = 1\n⊢ IsUnit (↑LinearMap.det f)",
"state_before": "case intro\nR : Type ?u.1057928\ninst✝¹¹ : CommRing R\nM : Type u_2\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\nM' : Type ?u.1058519\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nι : Type ?u.1059061\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ne : Basis ι R M\nA✝ : Type ?u.1059539\ninst✝⁴ : CommRing A✝\ninst✝³ : Module A✝ M\nκ : Type ?u.1060037\ninst✝² : Fintype κ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Module A M\nf : M →ₗ[A] M\nhf : IsUnit f\ng : M →ₗ[A] M\nhg : comp f g = 1\n⊢ IsUnit (↑LinearMap.det f)",
"tactic": "have : LinearMap.det f * LinearMap.det g = 1 := by\n simp only [← LinearMap.det_comp, hg, MonoidHom.map_one]"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type ?u.1057928\ninst✝¹¹ : CommRing R\nM : Type u_2\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\nM' : Type ?u.1058519\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nι : Type ?u.1059061\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ne : Basis ι R M\nA✝ : Type ?u.1059539\ninst✝⁴ : CommRing A✝\ninst✝³ : Module A✝ M\nκ : Type ?u.1060037\ninst✝² : Fintype κ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Module A M\nf : M →ₗ[A] M\nhf : IsUnit f\ng : M →ₗ[A] M\nhg : comp f g = 1\nthis : ↑LinearMap.det f * ↑LinearMap.det g = 1\n⊢ IsUnit (↑LinearMap.det f)",
"tactic": "exact isUnit_of_mul_eq_one _ _ this"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1057928\ninst✝¹¹ : CommRing R\nM : Type u_2\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\nM' : Type ?u.1058519\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : Module R M'\nι : Type ?u.1059061\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : Fintype ι\ne : Basis ι R M\nA✝ : Type ?u.1059539\ninst✝⁴ : CommRing A✝\ninst✝³ : Module A✝ M\nκ : Type ?u.1060037\ninst✝² : Fintype κ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Module A M\nf : M →ₗ[A] M\nhf : IsUnit f\ng : M →ₗ[A] M\nhg : comp f g = 1\n⊢ ↑LinearMap.det f * ↑LinearMap.det g = 1",
"tactic": "simp only [← LinearMap.det_comp, hg, MonoidHom.map_one]"
}
] |
[
337,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.mem_of_mem_dropn
|
[] |
[
986,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
984,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
Complex.hasDerivAt_sin
|
[] |
[
49,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
1
] |
Mathlib/Topology/PathConnected.lean
|
Path.symm_continuous_family
|
[] |
[
490,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
487,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.op_norm_flip
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type u_6\n𝕜₃ : Type u_2\nE : Type u_3\nEₗ : Type ?u.1114653\nF : Type u_5\nFₗ : Type ?u.1114659\nG : Type u_4\nGₗ : Type ?u.1114665\n𝓕 : Type ?u.1114668\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₂₃\ninst✝ : RingHomIsometric σ₁₃\nf : E →SL[σ₁₃] F →SL[σ₂₃] G\n⊢ ‖flip f‖ ≤ ‖f‖",
"tactic": "simpa only [flip_flip] using le_norm_flip f.flip"
}
] |
[
810,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
809,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.aestronglyMeasurable_one
|
[] |
[
1158,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1156,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.to_subtype
|
[] |
[
98,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
11
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean
|
Real.hasDerivAt_tan_of_mem_Ioo
|
[] |
[
79,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
FiniteDimensional.proper
|
[
{
"state_after": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : ProperSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\ne : E ≃L[𝕜] Fin (finrank 𝕜 E) → 𝕜 :=\n ContinuousLinearEquiv.ofFinrankEq (_ : finrank 𝕜 E = finrank 𝕜 (Fin (finrank 𝕜 E) → 𝕜))\n⊢ ProperSpace E",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : ProperSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\n⊢ ProperSpace E",
"tactic": "set e := ContinuousLinearEquiv.ofFinrankEq (@finrank_fin_fun 𝕜 _ _ (finrank 𝕜 E)).symm"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : ProperSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\ne : E ≃L[𝕜] Fin (finrank 𝕜 E) → 𝕜 :=\n ContinuousLinearEquiv.ofFinrankEq (_ : finrank 𝕜 E = finrank 𝕜 (Fin (finrank 𝕜 E) → 𝕜))\n⊢ ProperSpace E",
"tactic": "exact e.symm.antilipschitz.properSpace e.symm.continuous e.symm.surjective"
}
] |
[
606,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
604,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.castPred_castSucc
|
[
{
"state_after": "no goals",
"state_before": "n m : ℕ\ni : Fin (n + 1)\n⊢ castPred (↑castSucc i) = i",
"tactic": "simp [castPred, predAbove, not_lt.mpr (le_last i)]"
}
] |
[
2453,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2452,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval_mul
|
[] |
[
1053,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1052,
1
] |
Std/Data/List/Lemmas.lean
|
List.eq_of_mem_singleton
|
[] |
[
73,
20
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
72,
1
] |
Mathlib/Analysis/Calculus/Deriv/Inverse.lean
|
HasDerivAt.hasFDerivAt_equiv
|
[] |
[
61,
5
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/MeasureTheory/Group/Measure.lean
|
MeasureTheory.Measure.inv_inv
|
[] |
[
408,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
11
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isLittleO_norm_norm
|
[] |
[
842,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
841,
1
] |
Mathlib/Data/Real/Sqrt.lean
|
Real.lt_sqrt
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx : 0 ≤ x\n⊢ x < sqrt y ↔ x ^ 2 < y",
"tactic": "rw [← sqrt_lt_sqrt_iff (sq_nonneg _), sqrt_sq hx]"
}
] |
[
433,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.smul_pure
|
[] |
[
1025,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1024,
1
] |
Mathlib/MeasureTheory/Function/Jacobian.lean
|
MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image
|
[
{
"state_after": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\n⊢ (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) ≤ ↑↑μ (f '' s)",
"state_before": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n⊢ (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) ≤ ↑↑μ (f '' s)",
"tactic": "let u n := disjointed (spanningSets μ) n"
},
{
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"tactic": "have u_meas : ∀ n, MeasurableSet (u n) := by\n intro n\n apply MeasurableSet.disjointed fun i => ?_\n exact measurable_spanningSets μ i"
},
{
"state_after": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) ≤ ↑↑μ (f '' s)",
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"tactic": "have A : s = ⋃ n, s ∩ u n := by\n rw [← inter_iUnion, iUnion_disjointed, iUnion_spanningSets, inter_univ]"
},
{
"state_after": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nn : ℕ\n⊢ MeasurableSet (u n)",
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"tactic": "intro n"
},
{
"state_after": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nn i : ℕ\n⊢ MeasurableSet (spanningSets μ i)",
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"tactic": "apply MeasurableSet.disjointed fun i => ?_"
},
{
"state_after": "no goals",
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"tactic": "exact measurable_spanningSets μ i"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\n⊢ s = ⋃ (n : ℕ), s ∩ u n",
"tactic": "rw [← inter_iUnion, iUnion_disjointed, iUnion_spanningSets, inter_univ]"
},
{
"state_after": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ (∫⁻ (x : E) in ⋃ (n : ℕ), s ∩ u n, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) =\n ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ u n, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ",
"state_before": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) =\n ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ u n, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ",
"tactic": "conv_lhs => rw [A]"
},
{
"state_after": "case hm\nE : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ ∀ (i : ℕ), MeasurableSet (s ∩ u i)\n\ncase hd\nE : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ Pairwise (Disjoint on fun n => s ∩ u n)",
"state_before": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ (∫⁻ (x : E) in ⋃ (n : ℕ), s ∩ u n, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) =\n ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ u n, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ",
"tactic": "rw [lintegral_iUnion]"
},
{
"state_after": "case hm\nE : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\nn : ℕ\n⊢ MeasurableSet (s ∩ u n)",
"state_before": "case hm\nE : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ ∀ (i : ℕ), MeasurableSet (s ∩ u i)",
"tactic": "intro n"
},
{
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"state_before": "case hm\nE : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\nn : ℕ\n⊢ MeasurableSet (s ∩ u n)",
"tactic": "exact hs.inter (u_meas n)"
},
{
"state_after": "no goals",
"state_before": "case hd\nE : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ Pairwise (Disjoint on fun n => s ∩ u n)",
"tactic": "exact pairwise_disjoint_mono (disjoint_disjointed _) fun n => inter_subset_right _ _"
},
{
"state_after": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\nn : ℕ\n⊢ (∫⁻ (x : E) in s ∩ u n, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) ≤ ↑↑μ (f '' (s ∩ u n))",
"state_before": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ (∑' (n : ℕ), ∫⁻ (x : E) in s ∩ u n, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) ≤\n ∑' (n : ℕ), ↑↑μ (f '' (s ∩ u n))",
"tactic": "apply ENNReal.tsum_le_tsum fun n => ?_"
},
{
"state_after": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\nn : ℕ\n⊢ ↑↑μ (s ∩ u n) ≠ ⊤",
"state_before": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\nn : ℕ\n⊢ (∫⁻ (x : E) in s ∩ u n, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) ≤ ↑↑μ (f '' (s ∩ u n))",
"tactic": "apply\n lintegral_abs_det_fderiv_le_addHaar_image_aux2 μ (hs.inter (u_meas n)) _\n (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _)) (hf.mono (inter_subset_left _ _))"
},
{
"state_after": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\nn : ℕ\nthis : ↑↑μ (u n) < ⊤\n⊢ ↑↑μ (s ∩ u n) ≠ ⊤",
"state_before": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\nn : ℕ\n⊢ ↑↑μ (s ∩ u n) ≠ ⊤",
"tactic": "have : μ (u n) < ∞ :=\n lt_of_le_of_lt (measure_mono (disjointed_subset _ _)) (measure_spanningSets_lt_top μ n)"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\nn : ℕ\nthis : ↑↑μ (u n) < ⊤\n⊢ ↑↑μ (s ∩ u n) ≠ ⊤",
"tactic": "exact ne_of_lt (lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) this)"
},
{
"state_after": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ (∑' (n : ℕ), ↑↑μ (f '' (s ∩ u n))) = ↑↑μ (⋃ (i : ℕ), f '' (s ∩ u i))",
"state_before": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ (∑' (n : ℕ), ↑↑μ (f '' (s ∩ u n))) = ↑↑μ (f '' s)",
"tactic": "conv_rhs => rw [A, image_iUnion]"
},
{
"state_after": "case hn\nE : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ Pairwise (Disjoint on fun i => f '' (s ∩ u i))\n\ncase h\nE : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ ∀ (i : ℕ), MeasurableSet (f '' (s ∩ u i))",
"state_before": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ (∑' (n : ℕ), ↑↑μ (f '' (s ∩ u n))) = ↑↑μ (⋃ (i : ℕ), f '' (s ∩ u i))",
"tactic": "rw [measure_iUnion]"
},
{
"state_after": "case hn\nE : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\ni j : ℕ\nhij : i ≠ j\n⊢ (Disjoint on fun i => f '' (s ∩ u i)) i j",
"state_before": "case hn\nE : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ Pairwise (Disjoint on fun i => f '' (s ∩ u i))",
"tactic": "intro i j hij"
},
{
"state_after": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\ni j : ℕ\nhij : i ≠ j\n⊢ Disjoint (s ∩ u i) (s ∩ u j)",
"state_before": "case hn\nE : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\ni j : ℕ\nhij : i ≠ j\n⊢ (Disjoint on fun i => f '' (s ∩ u i)) i j",
"tactic": "apply Disjoint.image _ hf (inter_subset_left _ _) (inter_subset_left _ _)"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\ni j : ℕ\nhij : i ≠ j\n⊢ Disjoint (s ∩ u i) (s ∩ u j)",
"tactic": "exact\n Disjoint.mono (inter_subset_right _ _) (inter_subset_right _ _)\n (disjoint_disjointed _ hij)"
},
{
"state_after": "case h\nE : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\ni : ℕ\n⊢ MeasurableSet (f '' (s ∩ u i))",
"state_before": "case h\nE : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\n⊢ ∀ (i : ℕ), MeasurableSet (f '' (s ∩ u i))",
"tactic": "intro i"
},
{
"state_after": "no goals",
"state_before": "case h\nE : Type u_1\nF : Type ?u.837957\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : ℕ → Set E := fun n => disjointed (spanningSets μ) n\nu_meas : ∀ (n : ℕ), MeasurableSet (u n)\nA : s = ⋃ (n : ℕ), s ∩ u n\ni : ℕ\n⊢ MeasurableSet (f '' (s ∩ u i))",
"tactic": "exact\n measurable_image_of_fderivWithin (hs.inter (u_meas i))\n (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _))\n (hf.mono (inter_subset_left _ _))"
}
] |
[
1099,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1060,
1
] |
Mathlib/Algebra/Order/UpperLower.lean
|
lowerClosure_smul
|
[] |
[
283,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
uniformContinuous_ofMul
|
[] |
[
1448,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1447,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
ne_zero_and_ne_zero_of_mul
|
[] |
[
64,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.ceil_int
|
[] |
[
619,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
618,
1
] |
Mathlib/FieldTheory/IntermediateField.lean
|
IntermediateField.multiset_prod_mem
|
[] |
[
217,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
11
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
gcd_mul_right'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\n⊢ Associated (gcd (b * a) (c * a)) (gcd b c * a)",
"tactic": "simp only [mul_comm, gcd_mul_left']"
}
] |
[
470,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
469,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.exists_finset_of_mem_supr''
|
[
{
"state_after": "F : Type u_2\ninst✝² : Field F\nE : Type u_3\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nα : E\nι : Type u_1\nf : ι → IntermediateField F E\nh : ∀ (i : ι), Algebra.IsAlgebraic F { x // x ∈ f i }\nx : E\nhx : x ∈ ⨆ (i : ι), f i\ni : ι\n⊢ f i ≤ ⨆ (i : (i : ι) × { x // x ∈ f i }), adjoin F (rootSet (minpoly F i.snd) E)",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_3\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nα : E\nι : Type u_1\nf : ι → IntermediateField F E\nh : ∀ (i : ι), Algebra.IsAlgebraic F { x // x ∈ f i }\nx : E\nhx : x ∈ ⨆ (i : ι), f i\n⊢ ∃ s, x ∈ ⨆ (i : (i : ι) × { x // x ∈ f i }) (_ : i ∈ s), adjoin F (rootSet (minpoly F i.snd) E)",
"tactic": "refine' exists_finset_of_mem_iSup (SetLike.le_def.mp (iSup_le (fun i => _)) hx)"
},
{
"state_after": "F : Type u_2\ninst✝² : Field F\nE : Type u_3\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nα : E\nι : Type u_1\nf : ι → IntermediateField F E\nh : ∀ (i : ι), Algebra.IsAlgebraic F { x // x ∈ f i }\nx : E\nhx : x ∈ ⨆ (i : ι), f i\ni : ι\nx1 : E\nhx1 : x1 ∈ f i\n⊢ x1 ∈ ⨆ (i : (i : ι) × { x // x ∈ f i }), adjoin F (rootSet (minpoly F i.snd) E)",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_3\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nα : E\nι : Type u_1\nf : ι → IntermediateField F E\nh : ∀ (i : ι), Algebra.IsAlgebraic F { x // x ∈ f i }\nx : E\nhx : x ∈ ⨆ (i : ι), f i\ni : ι\n⊢ f i ≤ ⨆ (i : (i : ι) × { x // x ∈ f i }), adjoin F (rootSet (minpoly F i.snd) E)",
"tactic": "intro x1 hx1"
},
{
"state_after": "case refine'_1\nF : Type u_2\ninst✝² : Field F\nE : Type u_3\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nα : E\nι : Type u_1\nf : ι → IntermediateField F E\nh : ∀ (i : ι), Algebra.IsAlgebraic F { x // x ∈ f i }\nx : E\nhx : x ∈ ⨆ (i : ι), f i\ni : ι\nx1 : E\nhx1 : x1 ∈ f i\n⊢ adjoin F (rootSet (minpoly F x1) E) ≤\n adjoin F (rootSet (minpoly F { fst := i, snd := { val := x1, property := hx1 } }.snd) E)\n\ncase refine'_2\nF : Type u_2\ninst✝² : Field F\nE : Type u_3\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nα : E\nι : Type u_1\nf : ι → IntermediateField F E\nh : ∀ (i : ι), Algebra.IsAlgebraic F { x // x ∈ f i }\nx : E\nhx : x ∈ ⨆ (i : ι), f i\ni : ι\nx1 : E\nhx1 : x1 ∈ f i\n⊢ x1 ∈ rootSet (minpoly F x1) E",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_3\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nα : E\nι : Type u_1\nf : ι → IntermediateField F E\nh : ∀ (i : ι), Algebra.IsAlgebraic F { x // x ∈ f i }\nx : E\nhx : x ∈ ⨆ (i : ι), f i\ni : ι\nx1 : E\nhx1 : x1 ∈ f i\n⊢ x1 ∈ ⨆ (i : (i : ι) × { x // x ∈ f i }), adjoin F (rootSet (minpoly F i.snd) E)",
"tactic": "refine' SetLike.le_def.mp (le_iSup_of_le ⟨i, x1, hx1⟩ _)\n (subset_adjoin F (rootSet (minpoly F x1) E) _)"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nF : Type u_2\ninst✝² : Field F\nE : Type u_3\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nα : E\nι : Type u_1\nf : ι → IntermediateField F E\nh : ∀ (i : ι), Algebra.IsAlgebraic F { x // x ∈ f i }\nx : E\nhx : x ∈ ⨆ (i : ι), f i\ni : ι\nx1 : E\nhx1 : x1 ∈ f i\n⊢ adjoin F (rootSet (minpoly F x1) E) ≤\n adjoin F (rootSet (minpoly F { fst := i, snd := { val := x1, property := hx1 } }.snd) E)",
"tactic": "rw [IntermediateField.minpoly_eq, Subtype.coe_mk]"
},
{
"state_after": "case refine'_2.hp\nF : Type u_2\ninst✝² : Field F\nE : Type u_3\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nα : E\nι : Type u_1\nf : ι → IntermediateField F E\nh : ∀ (i : ι), Algebra.IsAlgebraic F { x // x ∈ f i }\nx : E\nhx : x ∈ ⨆ (i : ι), f i\ni : ι\nx1 : E\nhx1 : x1 ∈ f i\n⊢ minpoly F x1 ≠ 0",
"state_before": "case refine'_2\nF : Type u_2\ninst✝² : Field F\nE : Type u_3\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nα : E\nι : Type u_1\nf : ι → IntermediateField F E\nh : ∀ (i : ι), Algebra.IsAlgebraic F { x // x ∈ f i }\nx : E\nhx : x ∈ ⨆ (i : ι), f i\ni : ι\nx1 : E\nhx1 : x1 ∈ f i\n⊢ x1 ∈ rootSet (minpoly F x1) E",
"tactic": "rw [mem_rootSet_of_ne, minpoly.aeval]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.hp\nF : Type u_2\ninst✝² : Field F\nE : Type u_3\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nα : E\nι : Type u_1\nf : ι → IntermediateField F E\nh : ∀ (i : ι), Algebra.IsAlgebraic F { x // x ∈ f i }\nx : E\nhx : x ∈ ⨆ (i : ι), f i\ni : ι\nx1 : E\nhx1 : x1 ∈ f i\n⊢ minpoly F x1 ≠ 0",
"tactic": "exact minpoly.ne_zero (isIntegral_iff.mp (isAlgebraic_iff_isIntegral.mp (h i ⟨x1, hx1⟩)))"
}
] |
[
664,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
654,
1
] |
Mathlib/Data/Set/Intervals/WithBotTop.lean
|
WithBot.preimage_coe_Iio
|
[] |
[
159,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Basic.lean
|
Pmf.toMeasure_apply_eq_one_iff
|
[] |
[
274,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/Order/Filter/Extr.lean
|
IsMinOn.comp_mapsTo
|
[] |
[
431,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
429,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.iSup_add_iSup_of_monotone
|
[] |
[
627,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
625,
1
] |
Std/Logic.lean
|
true_iff_false
|
[] |
[
94,
85
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
94,
1
] |
Mathlib/CategoryTheory/Sites/Sieves.lean
|
CategoryTheory.Sieve.functor_galoisConnection
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : Y ⟶ X✝\nS✝ R✝ : Sieve X✝\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n⊢ functorPushforward F R ≤ S ↔ R ≤ functorPullback F S",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : Y ⟶ X✝\nS R : Sieve X✝\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX : C\n⊢ GaloisConnection (functorPushforward F) (functorPullback F)",
"tactic": "intro R S"
},
{
"state_after": "case mp\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : Y ⟶ X✝\nS✝ R✝ : Sieve X✝\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n⊢ functorPushforward F R ≤ S → R ≤ functorPullback F S\n\ncase mpr\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : Y ⟶ X✝\nS✝ R✝ : Sieve X✝\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n⊢ R ≤ functorPullback F S → functorPushforward F R ≤ S",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : Y ⟶ X✝\nS✝ R✝ : Sieve X✝\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n⊢ functorPushforward F R ≤ S ↔ R ≤ functorPullback F S",
"tactic": "constructor"
},
{
"state_after": "case mp\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝¹ Y Z : C\nf✝ : Y ⟶ X✝¹\nS✝ R✝ : Sieve X✝¹\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX✝ : C\nR : Sieve X✝\nS : Sieve (F.obj X✝)\nhle : functorPushforward F R ≤ S\nX : C\nf : X ⟶ X✝\nhf : R.arrows f\n⊢ (functorPullback F S).arrows f",
"state_before": "case mp\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : Y ⟶ X✝\nS✝ R✝ : Sieve X✝\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n⊢ functorPushforward F R ≤ S → R ≤ functorPullback F S",
"tactic": "intro hle X f hf"
},
{
"state_after": "case mp.a\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝¹ Y Z : C\nf✝ : Y ⟶ X✝¹\nS✝ R✝ : Sieve X✝¹\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX✝ : C\nR : Sieve X✝\nS : Sieve (F.obj X✝)\nhle : functorPushforward F R ≤ S\nX : C\nf : X ⟶ X✝\nhf : R.arrows f\n⊢ (functorPushforward F R).arrows (F.map f)",
"state_before": "case mp\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝¹ Y Z : C\nf✝ : Y ⟶ X✝¹\nS✝ R✝ : Sieve X✝¹\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX✝ : C\nR : Sieve X✝\nS : Sieve (F.obj X✝)\nhle : functorPushforward F R ≤ S\nX : C\nf : X ⟶ X✝\nhf : R.arrows f\n⊢ (functorPullback F S).arrows f",
"tactic": "apply hle"
},
{
"state_after": "case mp.a\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝¹ Y Z : C\nf✝ : Y ⟶ X✝¹\nS✝ R✝ : Sieve X✝¹\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX✝ : C\nR : Sieve X✝\nS : Sieve (F.obj X✝)\nhle : functorPushforward F R ≤ S\nX : C\nf : X ⟶ X✝\nhf : R.arrows f\n⊢ F.map f = 𝟙 (F.obj X) ≫ F.map f",
"state_before": "case mp.a\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝¹ Y Z : C\nf✝ : Y ⟶ X✝¹\nS✝ R✝ : Sieve X✝¹\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX✝ : C\nR : Sieve X✝\nS : Sieve (F.obj X✝)\nhle : functorPushforward F R ≤ S\nX : C\nf : X ⟶ X✝\nhf : R.arrows f\n⊢ (functorPushforward F R).arrows (F.map f)",
"tactic": "refine' ⟨X, f, 𝟙 _, hf, _⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.a\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝¹ Y Z : C\nf✝ : Y ⟶ X✝¹\nS✝ R✝ : Sieve X✝¹\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX✝ : C\nR : Sieve X✝\nS : Sieve (F.obj X✝)\nhle : functorPushforward F R ≤ S\nX : C\nf : X ⟶ X✝\nhf : R.arrows f\n⊢ F.map f = 𝟙 (F.obj X) ≫ F.map f",
"tactic": "rw [id_comp]"
},
{
"state_after": "case mpr.intro.intro.intro.intro\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝¹ Y✝ Z : C\nf : Y✝ ⟶ X✝¹\nS✝ R✝ : Sieve X✝¹\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX✝ : C\nR : Sieve X✝\nS : Sieve (F.obj X✝)\nhle : R ≤ functorPullback F S\nY : D\nX : C\ng : X ⟶ X✝\nh : Y ⟶ F.obj X\nhg : R.arrows g\n⊢ S.arrows (h ≫ F.map g)",
"state_before": "case mpr\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : Y ⟶ X✝\nS✝ R✝ : Sieve X✝\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n⊢ R ≤ functorPullback F S → functorPushforward F R ≤ S",
"tactic": "rintro hle Y f ⟨X, g, h, hg, rfl⟩"
},
{
"state_after": "case mpr.intro.intro.intro.intro.x\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝¹ Y✝ Z : C\nf : Y✝ ⟶ X✝¹\nS✝ R✝ : Sieve X✝¹\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX✝ : C\nR : Sieve X✝\nS : Sieve (F.obj X✝)\nhle : R ≤ functorPullback F S\nY : D\nX : C\ng : X ⟶ X✝\nh : Y ⟶ F.obj X\nhg : R.arrows g\n⊢ S.arrows (F.map g)",
"state_before": "case mpr.intro.intro.intro.intro\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝¹ Y✝ Z : C\nf : Y✝ ⟶ X✝¹\nS✝ R✝ : Sieve X✝¹\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX✝ : C\nR : Sieve X✝\nS : Sieve (F.obj X✝)\nhle : R ≤ functorPullback F S\nY : D\nX : C\ng : X ⟶ X✝\nh : Y ⟶ F.obj X\nhg : R.arrows g\n⊢ S.arrows (h ≫ F.map g)",
"tactic": "apply Sieve.downward_closed S"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro.intro.x\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX✝¹ Y✝ Z : C\nf : Y✝ ⟶ X✝¹\nS✝ R✝ : Sieve X✝¹\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nX✝ : C\nR : Sieve X✝\nS : Sieve (F.obj X✝)\nhle : R ≤ functorPullback F S\nY : D\nX : C\ng : X ⟶ X✝\nh : Y ⟶ F.obj X\nhg : R.arrows g\n⊢ S.arrows (F.map g)",
"tactic": "exact hle g hg"
}
] |
[
666,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
655,
1
] |
Mathlib/Order/InitialSeg.lean
|
PrincipalSeg.ofElement_apply
|
[] |
[
387,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
compl_unique
|
[] |
[
868,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
867,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.singleton_sups_singleton
|
[] |
[
160,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/Order/BooleanAlgebra.lean
|
sdiff_sdiff_right
|
[
{
"state_after": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x \\ (y \\ z) = z ⊓ x ⊔ x \\ y",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x \\ (y \\ z) = x \\ y ⊔ x ⊓ y ⊓ z",
"tactic": "rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]"
},
{
"state_after": "case s\nα : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y \\ z ⊔ (z ⊓ x ⊔ x \\ y) = x\n\ncase i\nα : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y \\ z ⊓ (z ⊓ x ⊔ x \\ y) = ⊥",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x \\ (y \\ z) = z ⊓ x ⊔ x \\ y",
"tactic": "apply sdiff_unique"
},
{
"state_after": "no goals",
"state_before": "case s\nα : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y \\ z ⊔ (z ⊓ x ⊔ x \\ y) = x",
"tactic": "calc\n x ⊓ y \\ z ⊔ (z ⊓ x ⊔ x \\ y) = (x ⊔ (z ⊓ x ⊔ x \\ y)) ⊓ (y \\ z ⊔ (z ⊓ x ⊔ x \\ y)) :=\n by rw [sup_inf_right]\n _ = (x ⊔ x ⊓ z ⊔ x \\ y) ⊓ (y \\ z ⊔ (x ⊓ z ⊔ x \\ y)) := by ac_rfl\n _ = x ⊓ (y \\ z ⊔ x ⊓ z ⊔ x \\ y) := by rw [sup_inf_self, sup_sdiff_left, ← sup_assoc]\n _ = x ⊓ (y \\ z ⊓ (z ⊔ y) ⊔ x ⊓ (z ⊔ y) ⊔ x \\ y) :=\n by rw [sup_inf_left, sdiff_sup_self', inf_sup_right, @sup_comm _ _ y]\n _ = x ⊓ (y \\ z ⊔ (x ⊓ z ⊔ x ⊓ y) ⊔ x \\ y) :=\n by rw [inf_sdiff_sup_right, @inf_sup_left _ _ x z y]\n _ = x ⊓ (y \\ z ⊔ (x ⊓ z ⊔ (x ⊓ y ⊔ x \\ y))) := by ac_rfl\n _ = x ⊓ (y \\ z ⊔ (x ⊔ x ⊓ z)) := by rw [sup_inf_sdiff, @sup_comm _ _ (x ⊓ z)]\n _ = x := by rw [sup_inf_self, sup_comm, inf_sup_self]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y \\ z ⊔ (z ⊓ x ⊔ x \\ y) = (x ⊔ (z ⊓ x ⊔ x \\ y)) ⊓ (y \\ z ⊔ (z ⊓ x ⊔ x \\ y))",
"tactic": "rw [sup_inf_right]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ (x ⊔ (z ⊓ x ⊔ x \\ y)) ⊓ (y \\ z ⊔ (z ⊓ x ⊔ x \\ y)) = (x ⊔ x ⊓ z ⊔ x \\ y) ⊓ (y \\ z ⊔ (x ⊓ z ⊔ x \\ y))",
"tactic": "ac_rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ (x ⊔ x ⊓ z ⊔ x \\ y) ⊓ (y \\ z ⊔ (x ⊓ z ⊔ x \\ y)) = x ⊓ (y \\ z ⊔ x ⊓ z ⊔ x \\ y)",
"tactic": "rw [sup_inf_self, sup_sdiff_left, ← sup_assoc]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ (y \\ z ⊔ x ⊓ z ⊔ x \\ y) = x ⊓ (y \\ z ⊓ (z ⊔ y) ⊔ x ⊓ (z ⊔ y) ⊔ x \\ y)",
"tactic": "rw [sup_inf_left, sdiff_sup_self', inf_sup_right, @sup_comm _ _ y]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ (y \\ z ⊓ (z ⊔ y) ⊔ x ⊓ (z ⊔ y) ⊔ x \\ y) = x ⊓ (y \\ z ⊔ (x ⊓ z ⊔ x ⊓ y) ⊔ x \\ y)",
"tactic": "rw [inf_sdiff_sup_right, @inf_sup_left _ _ x z y]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ (y \\ z ⊔ (x ⊓ z ⊔ x ⊓ y) ⊔ x \\ y) = x ⊓ (y \\ z ⊔ (x ⊓ z ⊔ (x ⊓ y ⊔ x \\ y)))",
"tactic": "ac_rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ (y \\ z ⊔ (x ⊓ z ⊔ (x ⊓ y ⊔ x \\ y))) = x ⊓ (y \\ z ⊔ (x ⊔ x ⊓ z))",
"tactic": "rw [sup_inf_sdiff, @sup_comm _ _ (x ⊓ z)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ (y \\ z ⊔ (x ⊔ x ⊓ z)) = x",
"tactic": "rw [sup_inf_self, sup_comm, inf_sup_self]"
},
{
"state_after": "no goals",
"state_before": "case i\nα : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y \\ z ⊓ (z ⊓ x ⊔ x \\ y) = ⊥",
"tactic": "calc\n x ⊓ y \\ z ⊓ (z ⊓ x ⊔ x \\ y) = x ⊓ y \\ z ⊓ (z ⊓ x) ⊔ x ⊓ y \\ z ⊓ x \\ y := by rw [inf_sup_left]\n _ = x ⊓ (y \\ z ⊓ z ⊓ x) ⊔ x ⊓ y \\ z ⊓ x \\ y := by ac_rfl\n _ = x ⊓ y \\ z ⊓ x \\ y := by rw [inf_sdiff_self_left, bot_inf_eq, inf_bot_eq, bot_sup_eq]\n _ = x ⊓ (y \\ z ⊓ y) ⊓ x \\ y := by conv_lhs => rw [← inf_sdiff_left]\n _ = x ⊓ (y \\ z ⊓ (y ⊓ x \\ y)) := by ac_rfl\n _ = ⊥ := by rw [inf_sdiff_self_right, inf_bot_eq, inf_bot_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y \\ z ⊓ (z ⊓ x ⊔ x \\ y) = x ⊓ y \\ z ⊓ (z ⊓ x) ⊔ x ⊓ y \\ z ⊓ x \\ y",
"tactic": "rw [inf_sup_left]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y \\ z ⊓ (z ⊓ x) ⊔ x ⊓ y \\ z ⊓ x \\ y = x ⊓ (y \\ z ⊓ z ⊓ x) ⊔ x ⊓ y \\ z ⊓ x \\ y",
"tactic": "ac_rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ (y \\ z ⊓ z ⊓ x) ⊔ x ⊓ y \\ z ⊓ x \\ y = x ⊓ y \\ z ⊓ x \\ y",
"tactic": "rw [inf_sdiff_self_left, bot_inf_eq, inf_bot_eq, bot_sup_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y \\ z ⊓ x \\ y = x ⊓ (y \\ z ⊓ y) ⊓ x \\ y",
"tactic": "conv_lhs => rw [← inf_sdiff_left]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ (y \\ z ⊓ y) ⊓ x \\ y = x ⊓ (y \\ z ⊓ (y ⊓ x \\ y))",
"tactic": "ac_rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.23496\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ (y \\ z ⊓ (y ⊓ x \\ y)) = ⊥",
"tactic": "rw [inf_sdiff_self_right, inf_bot_eq, inf_bot_eq]"
}
] |
[
365,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.fromBlocks_inj
|
[] |
[
144,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean
|
TopCat.Sheaf.objSupIsoProdEqLocus_hom_fst
|
[] |
[
475,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
470,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.pred_le_pred_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : PredOrder α\na b : α\ninst✝ : NoMinOrder α\n⊢ pred a ≤ pred b ↔ a ≤ b",
"tactic": "simp"
}
] |
[
709,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
709,
1
] |
Mathlib/NumberTheory/Padics/RingHoms.lean
|
PadicInt.appr_lt
|
[
{
"state_after": "case zero\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ x : ℤ_[p]\n⊢ appr x zero < p ^ zero\n\ncase succ\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\n⊢ appr x (succ n) < p ^ succ n",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝ x : ℤ_[p]\nn : ℕ\n⊢ appr x n < p ^ n",
"tactic": "induction' n with n ih generalizing x"
},
{
"state_after": "case succ\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\n⊢ (if h : x - ↑(appr x n) = 0 then appr x n\n else\n appr x n +\n p ^ n *\n ZMod.val\n (↑toZMod ↑(unitCoeff (_ : ¬x - ↑(appr x n) = 0)) *\n 0 ^\n match valuation (x - ↑(appr x n)) - ↑n with\n | Int.ofNat m => m\n | Int.negSucc m => succ m)) <\n p ^ succ n",
"state_before": "case succ\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\n⊢ appr x (succ n) < p ^ succ n",
"tactic": "simp only [appr, map_natCast, ZMod.nat_cast_self, RingHom.map_pow, Int.natAbs, RingHom.map_mul]"
},
{
"state_after": "case succ\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\n⊢ (if h : x - ↑(appr x n) = 0 then appr x n\n else\n appr x n +\n p ^ n *\n ZMod.val\n (↑toZMod ↑(unitCoeff (_ : ¬x - ↑(appr x n) = 0)) *\n 0 ^\n match valuation (x - ↑(appr x n)) - ↑n with\n | Int.ofNat m => m\n | Int.negSucc m => succ m)) <\n p ^ succ n",
"state_before": "case succ\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\n⊢ (if h : x - ↑(appr x n) = 0 then appr x n\n else\n appr x n +\n p ^ n *\n ZMod.val\n (↑toZMod ↑(unitCoeff (_ : ¬x - ↑(appr x n) = 0)) *\n 0 ^\n match valuation (x - ↑(appr x n)) - ↑n with\n | Int.ofNat m => m\n | Int.negSucc m => succ m)) <\n p ^ succ n",
"tactic": "have hp : p ^ n < p ^ (n + 1) := by apply pow_lt_pow hp_prime.1.one_lt (lt_add_one n)"
},
{
"state_after": "case succ.inl\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : x - ↑(appr x n) = 0\n⊢ appr x n < p ^ succ n\n\ncase succ.inr\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : ¬x - ↑(appr x n) = 0\n⊢ appr x n +\n p ^ n *\n ZMod.val\n (↑toZMod ↑(unitCoeff (_ : ¬x - ↑(appr x n) = 0)) *\n 0 ^\n match valuation (x - ↑(appr x n)) - ↑n with\n | Int.ofNat m => m\n | Int.negSucc m => succ m) <\n p ^ succ n",
"state_before": "case succ\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\n⊢ (if h : x - ↑(appr x n) = 0 then appr x n\n else\n appr x n +\n p ^ n *\n ZMod.val\n (↑toZMod ↑(unitCoeff (_ : ¬x - ↑(appr x n) = 0)) *\n 0 ^\n match valuation (x - ↑(appr x n)) - ↑n with\n | Int.ofNat m => m\n | Int.negSucc m => succ m)) <\n p ^ succ n",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case zero\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ x : ℤ_[p]\n⊢ appr x zero < p ^ zero",
"tactic": "simp only [appr, zero_eq, _root_.pow_zero, zero_lt_one]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\n⊢ p ^ n < p ^ (n + 1)",
"tactic": "apply pow_lt_pow hp_prime.1.one_lt (lt_add_one n)"
},
{
"state_after": "no goals",
"state_before": "case succ.inl\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : x - ↑(appr x n) = 0\n⊢ appr x n < p ^ succ n",
"tactic": "apply lt_trans (ih _) hp"
},
{
"state_after": "case succ.inr.calc_1\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : ¬x - ↑(appr x n) = 0\n⊢ appr x n +\n p ^ n *\n ZMod.val\n (↑toZMod ↑(unitCoeff (_ : ¬x - ↑(appr x n) = 0)) *\n 0 ^\n match valuation (x - ↑(appr x n)) - ↑n with\n | Int.ofNat m => m\n | Int.negSucc m => succ m) <\n p ^ n + p ^ n * (p - 1)\n\ncase succ.inr.calc_2\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : ¬x - ↑(appr x n) = 0\n⊢ p ^ n + p ^ n * (p - 1) = p ^ (n + 1)",
"state_before": "case succ.inr\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : ¬x - ↑(appr x n) = 0\n⊢ appr x n +\n p ^ n *\n ZMod.val\n (↑toZMod ↑(unitCoeff (_ : ¬x - ↑(appr x n) = 0)) *\n 0 ^\n match valuation (x - ↑(appr x n)) - ↑n with\n | Int.ofNat m => m\n | Int.negSucc m => succ m) <\n p ^ succ n",
"tactic": "calc\n _ < p ^ n + p ^ n * (p - 1) := ?_\n _ = p ^ (n + 1) := ?_"
},
{
"state_after": "case succ.inr.calc_1\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : ¬x - ↑(appr x n) = 0\n⊢ p ^ n *\n ZMod.val\n (↑toZMod ↑(unitCoeff (_ : ¬x - ↑(appr x n) = 0)) *\n 0 ^\n match valuation (x - ↑(appr x n)) - ↑n with\n | Int.ofNat m => m\n | Int.negSucc m => succ m) ≤\n p ^ n * (p - 1)",
"state_before": "case succ.inr.calc_1\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : ¬x - ↑(appr x n) = 0\n⊢ appr x n +\n p ^ n *\n ZMod.val\n (↑toZMod ↑(unitCoeff (_ : ¬x - ↑(appr x n) = 0)) *\n 0 ^\n match valuation (x - ↑(appr x n)) - ↑n with\n | Int.ofNat m => m\n | Int.negSucc m => succ m) <\n p ^ n + p ^ n * (p - 1)",
"tactic": "apply add_lt_add_of_lt_of_le (ih _)"
},
{
"state_after": "case succ.inr.calc_1.h\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : ¬x - ↑(appr x n) = 0\n⊢ ZMod.val\n (↑toZMod ↑(unitCoeff (_ : ¬x - ↑(appr x n) = 0)) *\n 0 ^\n match valuation (x - ↑(appr x n)) - ↑n with\n | Int.ofNat m => m\n | Int.negSucc m => succ m) ≤\n p - 1",
"state_before": "case succ.inr.calc_1\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : ¬x - ↑(appr x n) = 0\n⊢ p ^ n *\n ZMod.val\n (↑toZMod ↑(unitCoeff (_ : ¬x - ↑(appr x n) = 0)) *\n 0 ^\n match valuation (x - ↑(appr x n)) - ↑n with\n | Int.ofNat m => m\n | Int.negSucc m => succ m) ≤\n p ^ n * (p - 1)",
"tactic": "apply Nat.mul_le_mul_left"
},
{
"state_after": "case succ.inr.calc_1.h.h\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : ¬x - ↑(appr x n) = 0\n⊢ ZMod.val\n (↑toZMod ↑(unitCoeff (_ : ¬x - ↑(appr x n) = 0)) *\n 0 ^\n match valuation (x - ↑(appr x n)) - ↑n with\n | Int.ofNat m => m\n | Int.negSucc m => succ m) <\n p",
"state_before": "case succ.inr.calc_1.h\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : ¬x - ↑(appr x n) = 0\n⊢ ZMod.val\n (↑toZMod ↑(unitCoeff (_ : ¬x - ↑(appr x n) = 0)) *\n 0 ^\n match valuation (x - ↑(appr x n)) - ↑n with\n | Int.ofNat m => m\n | Int.negSucc m => succ m) ≤\n p - 1",
"tactic": "apply le_pred_of_lt"
},
{
"state_after": "no goals",
"state_before": "case succ.inr.calc_1.h.h\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : ¬x - ↑(appr x n) = 0\n⊢ ZMod.val\n (↑toZMod ↑(unitCoeff (_ : ¬x - ↑(appr x n) = 0)) *\n 0 ^\n match valuation (x - ↑(appr x n)) - ↑n with\n | Int.ofNat m => m\n | Int.negSucc m => succ m) <\n p",
"tactic": "apply ZMod.val_lt"
},
{
"state_after": "case succ.inr.calc_2\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : ¬x - ↑(appr x n) = 0\n⊢ p ^ n + (p ^ (n + 1) - p ^ n) = p ^ (n + 1)",
"state_before": "case succ.inr.calc_2\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : ¬x - ↑(appr x n) = 0\n⊢ p ^ n + p ^ n * (p - 1) = p ^ (n + 1)",
"tactic": "rw [mul_tsub, mul_one, ← _root_.pow_succ']"
},
{
"state_after": "no goals",
"state_before": "case succ.inr.calc_2\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝¹ x✝ : ℤ_[p]\nn : ℕ\nih : ∀ (x : ℤ_[p]), appr x n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : ¬x - ↑(appr x n) = 0\n⊢ p ^ n + (p ^ (n + 1) - p ^ n) = p ^ (n + 1)",
"tactic": "apply add_tsub_cancel_of_le (le_of_lt hp)"
}
] |
[
328,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
312,
1
] |
Mathlib/CategoryTheory/Sites/Subsheaf.lean
|
CategoryTheory.GrothendieckTopology.Subpresheaf.le_sheafify
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj G U\n⊢ s ∈ obj (sheafify J G) U",
"state_before": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\n⊢ G ≤ sheafify J G",
"tactic": "intro U s hs"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj G U\n⊢ sieveOfSection G s ∈ sieves J U.unop",
"state_before": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj G U\n⊢ s ∈ obj (sheafify J G) U",
"tactic": "change _ ∈ J _"
},
{
"state_after": "case h.e'_4\nC : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj G U\n⊢ sieveOfSection G s = ⊤",
"state_before": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj G U\n⊢ sieveOfSection G s ∈ sieves J U.unop",
"tactic": "convert J.top_mem U.unop"
},
{
"state_after": "case h.e'_4\nC : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj G U\n⊢ ⊤ ≤ sieveOfSection G s",
"state_before": "case h.e'_4\nC : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj G U\n⊢ sieveOfSection G s = ⊤",
"tactic": "rw [eq_top_iff]"
},
{
"state_after": "case h.e'_4\nC : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj G U\nV : C\ni : V ⟶ U.unop\n⊢ (sieveOfSection G s).arrows i",
"state_before": "case h.e'_4\nC : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj G U\n⊢ ⊤ ≤ sieveOfSection G s",
"tactic": "rintro V i -"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nC : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj G U\nV : C\ni : V ⟶ U.unop\n⊢ (sieveOfSection G s).arrows i",
"tactic": "exact G.map i.op hs"
}
] |
[
201,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Order/Filter/Pi.lean
|
Filter.mem_pi
|
[
{
"state_after": "case mp\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\ns : Set ((i : ι) → α i)\n⊢ s ∈ pi f → ∃ I, Set.Finite I ∧ ∃ t, (∀ (i : ι), t i ∈ f i) ∧ Set.pi I t ⊆ s\n\ncase mpr\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\ns : Set ((i : ι) → α i)\n⊢ (∃ I, Set.Finite I ∧ ∃ t, (∀ (i : ι), t i ∈ f i) ∧ Set.pi I t ⊆ s) → s ∈ pi f",
"state_before": "ι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\ns : Set ((i : ι) → α i)\n⊢ s ∈ pi f ↔ ∃ I, Set.Finite I ∧ ∃ t, (∀ (i : ι), t i ∈ f i) ∧ Set.pi I t ⊆ s",
"tactic": "constructor"
},
{
"state_after": "case mp\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\ns : Set ((i : ι) → α i)\n⊢ (∃ I,\n Set.Finite I ∧\n ∃ V,\n (∀ (i : ι), ∃ t, t ∈ f i ∧ eval i ⁻¹' t ⊆ V i) ∧\n (∀ (i : ι), ¬i ∈ I → V i = univ) ∧ (s = ⋂ (i : ι) (_ : i ∈ I), V i) ∧ s = ⋂ (i : ι), V i) →\n ∃ I, Set.Finite I ∧ ∃ t, (∀ (i : ι), t i ∈ f i) ∧ (⋂ (a : ι) (_ : a ∈ I), eval a ⁻¹' t a) ⊆ s",
"state_before": "case mp\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\ns : Set ((i : ι) → α i)\n⊢ s ∈ pi f → ∃ I, Set.Finite I ∧ ∃ t, (∀ (i : ι), t i ∈ f i) ∧ Set.pi I t ⊆ s",
"tactic": "simp only [pi, mem_iInf', mem_comap, pi_def]"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\nI : Set ι\nIf : Set.Finite I\nV : ι → Set ((i : ι) → α i)\nhVf : ∀ (i : ι), ∃ t, t ∈ f i ∧ eval i ⁻¹' t ⊆ V i\n⊢ ∃ I_1,\n Set.Finite I_1 ∧\n ∃ t, (∀ (i : ι), t i ∈ f i) ∧ (⋂ (a : ι) (_ : a ∈ I_1), eval a ⁻¹' t a) ⊆ ⋂ (i : ι) (_ : i ∈ I), V i",
"state_before": "case mp\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\ns : Set ((i : ι) → α i)\n⊢ (∃ I,\n Set.Finite I ∧\n ∃ V,\n (∀ (i : ι), ∃ t, t ∈ f i ∧ eval i ⁻¹' t ⊆ V i) ∧\n (∀ (i : ι), ¬i ∈ I → V i = univ) ∧ (s = ⋂ (i : ι) (_ : i ∈ I), V i) ∧ s = ⋂ (i : ι), V i) →\n ∃ I, Set.Finite I ∧ ∃ t, (∀ (i : ι), t i ∈ f i) ∧ (⋂ (a : ι) (_ : a ∈ I), eval a ⁻¹' t a) ⊆ s",
"tactic": "rintro ⟨I, If, V, hVf, -, rfl, -⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\nI : Set ι\nIf : Set.Finite I\nV : ι → Set ((i : ι) → α i)\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhtV : ∀ (i : ι), eval i ⁻¹' t i ⊆ V i\n⊢ ∃ I_1,\n Set.Finite I_1 ∧\n ∃ t, (∀ (i : ι), t i ∈ f i) ∧ (⋂ (a : ι) (_ : a ∈ I_1), eval a ⁻¹' t a) ⊆ ⋂ (i : ι) (_ : i ∈ I), V i",
"state_before": "case mp.intro.intro.intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\nI : Set ι\nIf : Set.Finite I\nV : ι → Set ((i : ι) → α i)\nhVf : ∀ (i : ι), ∃ t, t ∈ f i ∧ eval i ⁻¹' t ⊆ V i\n⊢ ∃ I_1,\n Set.Finite I_1 ∧\n ∃ t, (∀ (i : ι), t i ∈ f i) ∧ (⋂ (a : ι) (_ : a ∈ I_1), eval a ⁻¹' t a) ⊆ ⋂ (i : ι) (_ : i ∈ I), V i",
"tactic": "choose t htf htV using hVf"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\nI : Set ι\nIf : Set.Finite I\nV : ι → Set ((i : ι) → α i)\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhtV : ∀ (i : ι), eval i ⁻¹' t i ⊆ V i\n⊢ ∃ I_1,\n Set.Finite I_1 ∧\n ∃ t, (∀ (i : ι), t i ∈ f i) ∧ (⋂ (a : ι) (_ : a ∈ I_1), eval a ⁻¹' t a) ⊆ ⋂ (i : ι) (_ : i ∈ I), V i",
"tactic": "exact ⟨I, If, t, htf, iInter₂_mono fun i _ => htV i⟩"
},
{
"state_after": "case mpr.intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\ns : Set ((i : ι) → α i)\nI : Set ι\nIf : Set.Finite I\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : Set.pi I t ⊆ s\n⊢ s ∈ pi f",
"state_before": "case mpr\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\ns : Set ((i : ι) → α i)\n⊢ (∃ I, Set.Finite I ∧ ∃ t, (∀ (i : ι), t i ∈ f i) ∧ Set.pi I t ⊆ s) → s ∈ pi f",
"tactic": "rintro ⟨I, If, t, htf, hts⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\ns : Set ((i : ι) → α i)\nI : Set ι\nIf : Set.Finite I\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : Set.pi I t ⊆ s\n⊢ s ∈ pi f",
"tactic": "exact mem_of_superset (pi_mem_pi If fun i _ => htf i) hts"
}
] |
[
84,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Topology/Instances/Int.lean
|
Int.closedEmbedding_coe_real
|
[] |
[
51,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/LogDeriv.lean
|
HasDerivWithinAt.clog
|
[
{
"state_after": "α : Type ?u.48747\ninst✝² : TopologicalSpace α\nE : Type ?u.48753\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → ℂ\nf' x : ℂ\ns : Set ℂ\nh₁ : HasDerivWithinAt f f' s x\nh₂ : 0 < (f x).re ∨ (f x).im ≠ 0\n⊢ HasDerivWithinAt (fun t => log (f t)) ((f x)⁻¹ * f') s x",
"state_before": "α : Type ?u.48747\ninst✝² : TopologicalSpace α\nE : Type ?u.48753\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → ℂ\nf' x : ℂ\ns : Set ℂ\nh₁ : HasDerivWithinAt f f' s x\nh₂ : 0 < (f x).re ∨ (f x).im ≠ 0\n⊢ HasDerivWithinAt (fun t => log (f t)) (f' / f x) s x",
"tactic": "rw [div_eq_inv_mul]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.48747\ninst✝² : TopologicalSpace α\nE : Type ?u.48753\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → ℂ\nf' x : ℂ\ns : Set ℂ\nh₁ : HasDerivWithinAt f f' s x\nh₂ : 0 < (f x).re ∨ (f x).im ≠ 0\n⊢ HasDerivWithinAt (fun t => log (f t)) ((f x)⁻¹ * f') s x",
"tactic": "exact (hasStrictDerivAt_log h₂).hasDerivAt.comp_hasDerivWithinAt x h₁"
}
] |
[
127,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.lastCases_castSucc
|
[] |
[
1835,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1832,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
tendsto_pow_atTop_atTop_of_one_lt
|
[] |
[
94,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
Differentiable.csinh
|
[] |
[
540,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
539,
1
] |
Mathlib/CategoryTheory/Filtered.lean
|
CategoryTheory.IsFiltered.coeq_condition
|
[] |
[
194,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Order/Lattice.lean
|
SemilatticeSup.ext
|
[
{
"state_after": "α✝ : Type u\nβ : Type v\ninst✝ : SemilatticeSup α✝\na b c d : α✝\nα : Type u_1\nA B : SemilatticeSup α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toSup = toSup\n⊢ A = B",
"state_before": "α✝ : Type u\nβ : Type v\ninst✝ : SemilatticeSup α✝\na b c d : α✝\nα : Type u_1\nA B : SemilatticeSup α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ A = B",
"tactic": "have ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H"
},
{
"state_after": "case mk\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeSup α✝\na b c d : α✝\nα : Type u_1\nB : SemilatticeSup α\ntoSup✝ : Sup α\ntoPartialOrder✝ : PartialOrder α\nle_sup_left✝ : ∀ (a b : α), a ≤ a ⊔ b\nle_sup_right✝ : ∀ (a b : α), b ≤ a ⊔ b\nsup_le✝ : ∀ (a b c : α), a ≤ c → b ≤ c → a ⊔ b ≤ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toSup = toSup\n⊢ mk le_sup_left✝ le_sup_right✝ sup_le✝ = B",
"state_before": "α✝ : Type u\nβ : Type v\ninst✝ : SemilatticeSup α✝\na b c d : α✝\nα : Type u_1\nA B : SemilatticeSup α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toSup = toSup\n⊢ A = B",
"tactic": "cases A"
},
{
"state_after": "case mk.mk\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeSup α✝\na b c d : α✝\nα : Type u_1\ntoSup✝¹ : Sup α\ntoPartialOrder✝¹ : PartialOrder α\nle_sup_left✝¹ : ∀ (a b : α), a ≤ a ⊔ b\nle_sup_right✝¹ : ∀ (a b : α), b ≤ a ⊔ b\nsup_le✝¹ : ∀ (a b c : α), a ≤ c → b ≤ c → a ⊔ b ≤ c\ntoSup✝ : Sup α\ntoPartialOrder✝ : PartialOrder α\nle_sup_left✝ : ∀ (a b : α), a ≤ a ⊔ b\nle_sup_right✝ : ∀ (a b : α), b ≤ a ⊔ b\nsup_le✝ : ∀ (a b c : α), a ≤ c → b ≤ c → a ⊔ b ≤ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toSup = toSup\n⊢ mk le_sup_left✝¹ le_sup_right✝¹ sup_le✝¹ = mk le_sup_left✝ le_sup_right✝ sup_le✝",
"state_before": "case mk\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeSup α✝\na b c d : α✝\nα : Type u_1\nB : SemilatticeSup α\ntoSup✝ : Sup α\ntoPartialOrder✝ : PartialOrder α\nle_sup_left✝ : ∀ (a b : α), a ≤ a ⊔ b\nle_sup_right✝ : ∀ (a b : α), b ≤ a ⊔ b\nsup_le✝ : ∀ (a b c : α), a ≤ c → b ≤ c → a ⊔ b ≤ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toSup = toSup\n⊢ mk le_sup_left✝ le_sup_right✝ sup_le✝ = B",
"tactic": "cases B"
},
{
"state_after": "case mk.mk.refl\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeSup α✝\na b c d : α✝\nα : Type u_1\ntoSup✝¹ : Sup α\ntoPartialOrder✝ : PartialOrder α\nle_sup_left✝¹ : ∀ (a b : α), a ≤ a ⊔ b\nle_sup_right✝¹ : ∀ (a b : α), b ≤ a ⊔ b\nsup_le✝¹ : ∀ (a b c : α), a ≤ c → b ≤ c → a ⊔ b ≤ c\ntoSup✝ : Sup α\nle_sup_left✝ : ∀ (a b : α), a ≤ a ⊔ b\nle_sup_right✝ : ∀ (a b : α), b ≤ a ⊔ b\nsup_le✝ : ∀ (a b c : α), a ≤ c → b ≤ c → a ⊔ b ≤ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toSup = toSup\n⊢ mk le_sup_left✝¹ le_sup_right✝¹ sup_le✝¹ = mk le_sup_left✝ le_sup_right✝ sup_le✝",
"state_before": "case mk.mk\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeSup α✝\na b c d : α✝\nα : Type u_1\ntoSup✝¹ : Sup α\ntoPartialOrder✝¹ : PartialOrder α\nle_sup_left✝¹ : ∀ (a b : α), a ≤ a ⊔ b\nle_sup_right✝¹ : ∀ (a b : α), b ≤ a ⊔ b\nsup_le✝¹ : ∀ (a b c : α), a ≤ c → b ≤ c → a ⊔ b ≤ c\ntoSup✝ : Sup α\ntoPartialOrder✝ : PartialOrder α\nle_sup_left✝ : ∀ (a b : α), a ≤ a ⊔ b\nle_sup_right✝ : ∀ (a b : α), b ≤ a ⊔ b\nsup_le✝ : ∀ (a b c : α), a ≤ c → b ≤ c → a ⊔ b ≤ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toSup = toSup\n⊢ mk le_sup_left✝¹ le_sup_right✝¹ sup_le✝¹ = mk le_sup_left✝ le_sup_right✝ sup_le✝",
"tactic": "cases PartialOrder.ext H"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.refl\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeSup α✝\na b c d : α✝\nα : Type u_1\ntoSup✝¹ : Sup α\ntoPartialOrder✝ : PartialOrder α\nle_sup_left✝¹ : ∀ (a b : α), a ≤ a ⊔ b\nle_sup_right✝¹ : ∀ (a b : α), b ≤ a ⊔ b\nsup_le✝¹ : ∀ (a b c : α), a ≤ c → b ≤ c → a ⊔ b ≤ c\ntoSup✝ : Sup α\nle_sup_left✝ : ∀ (a b : α), a ≤ a ⊔ b\nle_sup_right✝ : ∀ (a b : α), b ≤ a ⊔ b\nsup_le✝ : ∀ (a b c : α), a ≤ c → b ≤ c → a ⊔ b ≤ c\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nss : toSup = toSup\n⊢ mk le_sup_left✝¹ le_sup_right✝¹ sup_le✝¹ = mk le_sup_left✝ le_sup_right✝ sup_le✝",
"tactic": "congr"
},
{
"state_after": "case sup.h.h\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeSup α✝\na b c d : α✝\nα : Type u_1\nA B : SemilatticeSup α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nx✝¹ x✝ : α\n⊢ x✝¹ ⊔ x✝ = x✝¹ ⊔ x✝",
"state_before": "α✝ : Type u\nβ : Type v\ninst✝ : SemilatticeSup α✝\na b c d : α✝\nα : Type u_1\nA B : SemilatticeSup α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ toSup = toSup",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case sup.h.h\nα✝ : Type u\nβ : Type v\ninst✝ : SemilatticeSup α✝\na b c d : α✝\nα : Type u_1\nA B : SemilatticeSup α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nx✝¹ x✝ : α\n⊢ x✝¹ ⊔ x✝ = x✝¹ ⊔ x✝",
"tactic": "apply SemilatticeSup.ext_sup H"
}
] |
[
336,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
329,
1
] |
Mathlib/Algebra/IsPrimePow.lean
|
IsPrimePow.two_le
|
[] |
[
120,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.isTopologicalBasis_basic_opens
|
[
{
"state_after": "case h_open\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ ∀ (u : Set (PrimeSpectrum R)), (u ∈ Set.range fun r => ↑(basicOpen r)) → IsOpen u\n\ncase h_nhds\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ ∀ (a : PrimeSpectrum R) (u : Set (PrimeSpectrum R)),\n a ∈ u → IsOpen u → ∃ v, (v ∈ Set.range fun r => ↑(basicOpen r)) ∧ a ∈ v ∧ v ⊆ u",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ TopologicalSpace.IsTopologicalBasis (Set.range fun r => ↑(basicOpen r))",
"tactic": "apply TopologicalSpace.isTopologicalBasis_of_open_of_nhds"
},
{
"state_after": "case h_open.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\n⊢ IsOpen ((fun r => ↑(basicOpen r)) r)",
"state_before": "case h_open\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ ∀ (u : Set (PrimeSpectrum R)), (u ∈ Set.range fun r => ↑(basicOpen r)) → IsOpen u",
"tactic": "rintro _ ⟨r, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h_open.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nr : R\n⊢ IsOpen ((fun r => ↑(basicOpen r)) r)",
"tactic": "exact isOpen_basicOpen"
},
{
"state_after": "case h_nhds.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\np : PrimeSpectrum R\nU : Set (PrimeSpectrum R)\nhp : p ∈ U\ns : Set R\nhs : zeroLocus s = Uᶜ\n⊢ ∃ v, (v ∈ Set.range fun r => ↑(basicOpen r)) ∧ p ∈ v ∧ v ⊆ U",
"state_before": "case h_nhds\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ ∀ (a : PrimeSpectrum R) (u : Set (PrimeSpectrum R)),\n a ∈ u → IsOpen u → ∃ v, (v ∈ Set.range fun r => ↑(basicOpen r)) ∧ a ∈ v ∧ v ⊆ u",
"tactic": "rintro p U hp ⟨s, hs⟩"
},
{
"state_after": "case h_nhds.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\np : PrimeSpectrum R\nU : Set (PrimeSpectrum R)\ns : Set R\nhp : ∃ a, a ∈ s ∧ ¬a ∈ ↑p.asIdeal\nhs : zeroLocus s = Uᶜ\n⊢ ∃ v, (v ∈ Set.range fun r => ↑(basicOpen r)) ∧ p ∈ v ∧ v ⊆ U",
"state_before": "case h_nhds.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\np : PrimeSpectrum R\nU : Set (PrimeSpectrum R)\nhp : p ∈ U\ns : Set R\nhs : zeroLocus s = Uᶜ\n⊢ ∃ v, (v ∈ Set.range fun r => ↑(basicOpen r)) ∧ p ∈ v ∧ v ⊆ U",
"tactic": "rw [← compl_compl U, Set.mem_compl_iff, ← hs, mem_zeroLocus, Set.not_subset] at hp"
},
{
"state_after": "case h_nhds.intro.intro.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\np : PrimeSpectrum R\nU : Set (PrimeSpectrum R)\ns : Set R\nhs : zeroLocus s = Uᶜ\nf : R\nhfs : f ∈ s\nhfp : ¬f ∈ ↑p.asIdeal\n⊢ ∃ v, (v ∈ Set.range fun r => ↑(basicOpen r)) ∧ p ∈ v ∧ v ⊆ U",
"state_before": "case h_nhds.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\np : PrimeSpectrum R\nU : Set (PrimeSpectrum R)\ns : Set R\nhp : ∃ a, a ∈ s ∧ ¬a ∈ ↑p.asIdeal\nhs : zeroLocus s = Uᶜ\n⊢ ∃ v, (v ∈ Set.range fun r => ↑(basicOpen r)) ∧ p ∈ v ∧ v ⊆ U",
"tactic": "obtain ⟨f, hfs, hfp⟩ := hp"
},
{
"state_after": "case h_nhds.intro.intro.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\np : PrimeSpectrum R\nU : Set (PrimeSpectrum R)\ns : Set R\nhs : zeroLocus s = Uᶜ\nf : R\nhfs : f ∈ s\nhfp : ¬f ∈ ↑p.asIdeal\n⊢ ↑(basicOpen f) ⊆ U",
"state_before": "case h_nhds.intro.intro.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\np : PrimeSpectrum R\nU : Set (PrimeSpectrum R)\ns : Set R\nhs : zeroLocus s = Uᶜ\nf : R\nhfs : f ∈ s\nhfp : ¬f ∈ ↑p.asIdeal\n⊢ ∃ v, (v ∈ Set.range fun r => ↑(basicOpen r)) ∧ p ∈ v ∧ v ⊆ U",
"tactic": "refine' ⟨basicOpen f, ⟨f, rfl⟩, hfp, _⟩"
},
{
"state_after": "case h_nhds.intro.intro.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\np : PrimeSpectrum R\nU : Set (PrimeSpectrum R)\ns : Set R\nhs : zeroLocus s = Uᶜ\nf : R\nhfs : f ∈ s\nhfp : ¬f ∈ ↑p.asIdeal\n⊢ zeroLocus s ⊆ zeroLocus {f}",
"state_before": "case h_nhds.intro.intro.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\np : PrimeSpectrum R\nU : Set (PrimeSpectrum R)\ns : Set R\nhs : zeroLocus s = Uᶜ\nf : R\nhfs : f ∈ s\nhfp : ¬f ∈ ↑p.asIdeal\n⊢ ↑(basicOpen f) ⊆ U",
"tactic": "rw [← Set.compl_subset_compl, ← hs, basicOpen_eq_zeroLocus_compl, compl_compl]"
},
{
"state_after": "no goals",
"state_before": "case h_nhds.intro.intro.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\np : PrimeSpectrum R\nU : Set (PrimeSpectrum R)\ns : Set R\nhs : zeroLocus s = Uᶜ\nf : R\nhfs : f ∈ s\nhfp : ¬f ∈ ↑p.asIdeal\n⊢ zeroLocus s ⊆ zeroLocus {f}",
"tactic": "exact zeroLocus_anti_mono (Set.singleton_subset_iff.mpr hfs)"
}
] |
[
832,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
821,
1
] |
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean
|
Polynomial.natTrailingDegree_monomial
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nha : a ≠ 0\n⊢ Option.getD (↑n) 0 = n",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nha : a ≠ 0\n⊢ natTrailingDegree (↑(monomial n) a) = n",
"tactic": "rw [natTrailingDegree, trailingDegree_monomial ha]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nha : a ≠ 0\n⊢ Option.getD (↑n) 0 = n",
"tactic": "rfl"
}
] |
[
209,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
HasFPowerSeriesOnBall.uniform_geometric_approx
|
[
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.690610\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r'✝ : ℝ≥0∞\nr' : ℝ≥0\nhf : HasFPowerSeriesOnBall f p x r\nh : ↑r' < r\na : ℝ\nha : a ∈ Ioo 0 1\nC : ℝ\nhC : C > 0\nhp :\n ∀ (y : E),\n y ∈ Metric.ball 0 ↑r' →\n ∀ (n : ℕ), ‖f (x + y) - FormalMultilinearSeries.partialSum p n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n\n⊢ ∃ a,\n a ∈ Ioo 0 1 ∧\n ∃ C,\n C > 0 ∧\n ∀ (y : E),\n y ∈ Metric.ball 0 ↑r' → ∀ (n : ℕ), ‖f (x + y) - FormalMultilinearSeries.partialSum p n y‖ ≤ C * a ^ n",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.690610\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r'✝ : ℝ≥0∞\nr' : ℝ≥0\nhf : HasFPowerSeriesOnBall f p x r\nh : ↑r' < r\n⊢ ∃ a,\n a ∈ Ioo 0 1 ∧\n ∃ C,\n C > 0 ∧\n ∀ (y : E),\n y ∈ Metric.ball 0 ↑r' → ∀ (n : ℕ), ‖f (x + y) - FormalMultilinearSeries.partialSum p n y‖ ≤ C * a ^ n",
"tactic": "obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,\n ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=\n hf.uniform_geometric_approx' h"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.690610\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r'✝ : ℝ≥0∞\nr' : ℝ≥0\nhf : HasFPowerSeriesOnBall f p x r\nh : ↑r' < r\na : ℝ\nha : a ∈ Ioo 0 1\nC : ℝ\nhC : C > 0\nhp :\n ∀ (y : E),\n y ∈ Metric.ball 0 ↑r' →\n ∀ (n : ℕ), ‖f (x + y) - FormalMultilinearSeries.partialSum p n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n\ny : E\nhy : y ∈ Metric.ball 0 ↑r'\nn : ℕ\n⊢ C * (a * (‖y‖ / ↑r')) ^ n ≤ C * a ^ n",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.690610\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r'✝ : ℝ≥0∞\nr' : ℝ≥0\nhf : HasFPowerSeriesOnBall f p x r\nh : ↑r' < r\na : ℝ\nha : a ∈ Ioo 0 1\nC : ℝ\nhC : C > 0\nhp :\n ∀ (y : E),\n y ∈ Metric.ball 0 ↑r' →\n ∀ (n : ℕ), ‖f (x + y) - FormalMultilinearSeries.partialSum p n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n\n⊢ ∃ a,\n a ∈ Ioo 0 1 ∧\n ∃ C,\n C > 0 ∧\n ∀ (y : E),\n y ∈ Metric.ball 0 ↑r' → ∀ (n : ℕ), ‖f (x + y) - FormalMultilinearSeries.partialSum p n y‖ ≤ C * a ^ n",
"tactic": "refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.690610\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r'✝ : ℝ≥0∞\nr' : ℝ≥0\nhf : HasFPowerSeriesOnBall f p x r\nh : ↑r' < r\na : ℝ\nha : a ∈ Ioo 0 1\nC : ℝ\nhC : C > 0\nhp :\n ∀ (y : E),\n y ∈ Metric.ball 0 ↑r' →\n ∀ (n : ℕ), ‖f (x + y) - FormalMultilinearSeries.partialSum p n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n\ny : E\nhy : y ∈ Metric.ball 0 ↑r'\nn : ℕ\nyr' : ‖y‖ < ↑r'\n⊢ C * (a * (‖y‖ / ↑r')) ^ n ≤ C * a ^ n",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.690610\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r'✝ : ℝ≥0∞\nr' : ℝ≥0\nhf : HasFPowerSeriesOnBall f p x r\nh : ↑r' < r\na : ℝ\nha : a ∈ Ioo 0 1\nC : ℝ\nhC : C > 0\nhp :\n ∀ (y : E),\n y ∈ Metric.ball 0 ↑r' →\n ∀ (n : ℕ), ‖f (x + y) - FormalMultilinearSeries.partialSum p n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n\ny : E\nhy : y ∈ Metric.ball 0 ↑r'\nn : ℕ\n⊢ C * (a * (‖y‖ / ↑r')) ^ n ≤ C * a ^ n",
"tactic": "have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy"
},
{
"state_after": "case intro.intro.intro.intro.h.ha\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.690610\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r'✝ : ℝ≥0∞\nr' : ℝ≥0\nhf : HasFPowerSeriesOnBall f p x r\nh : ↑r' < r\na : ℝ\nha : a ∈ Ioo 0 1\nC : ℝ\nhC : C > 0\nhp :\n ∀ (y : E),\n y ∈ Metric.ball 0 ↑r' →\n ∀ (n : ℕ), ‖f (x + y) - FormalMultilinearSeries.partialSum p n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n\ny : E\nhy : y ∈ Metric.ball 0 ↑r'\nn : ℕ\nyr' : ‖y‖ < ↑r'\n⊢ 0 ≤ a * (‖y‖ / ↑r')\n\ncase intro.intro.intro.intro.h.hab\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.690610\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r'✝ : ℝ≥0∞\nr' : ℝ≥0\nhf : HasFPowerSeriesOnBall f p x r\nh : ↑r' < r\na : ℝ\nha : a ∈ Ioo 0 1\nC : ℝ\nhC : C > 0\nhp :\n ∀ (y : E),\n y ∈ Metric.ball 0 ↑r' →\n ∀ (n : ℕ), ‖f (x + y) - FormalMultilinearSeries.partialSum p n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n\ny : E\nhy : y ∈ Metric.ball 0 ↑r'\nn : ℕ\nyr' : ‖y‖ < ↑r'\n⊢ a * (‖y‖ / ↑r') ≤ a",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.690610\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r'✝ : ℝ≥0∞\nr' : ℝ≥0\nhf : HasFPowerSeriesOnBall f p x r\nh : ↑r' < r\na : ℝ\nha : a ∈ Ioo 0 1\nC : ℝ\nhC : C > 0\nhp :\n ∀ (y : E),\n y ∈ Metric.ball 0 ↑r' →\n ∀ (n : ℕ), ‖f (x + y) - FormalMultilinearSeries.partialSum p n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n\ny : E\nhy : y ∈ Metric.ball 0 ↑r'\nn : ℕ\nyr' : ‖y‖ < ↑r'\n⊢ C * (a * (‖y‖ / ↑r')) ^ n ≤ C * a ^ n",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.h.ha\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.690610\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r'✝ : ℝ≥0∞\nr' : ℝ≥0\nhf : HasFPowerSeriesOnBall f p x r\nh : ↑r' < r\na : ℝ\nha : a ∈ Ioo 0 1\nC : ℝ\nhC : C > 0\nhp :\n ∀ (y : E),\n y ∈ Metric.ball 0 ↑r' →\n ∀ (n : ℕ), ‖f (x + y) - FormalMultilinearSeries.partialSum p n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n\ny : E\nhy : y ∈ Metric.ball 0 ↑r'\nn : ℕ\nyr' : ‖y‖ < ↑r'\n⊢ 0 ≤ a * (‖y‖ / ↑r')\n\ncase intro.intro.intro.intro.h.hab\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.690610\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r'✝ : ℝ≥0∞\nr' : ℝ≥0\nhf : HasFPowerSeriesOnBall f p x r\nh : ↑r' < r\na : ℝ\nha : a ∈ Ioo 0 1\nC : ℝ\nhC : C > 0\nhp :\n ∀ (y : E),\n y ∈ Metric.ball 0 ↑r' →\n ∀ (n : ℕ), ‖f (x + y) - FormalMultilinearSeries.partialSum p n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n\ny : E\nhy : y ∈ Metric.ball 0 ↑r'\nn : ℕ\nyr' : ‖y‖ < ↑r'\n⊢ a * (‖y‖ / ↑r') ≤ a",
"tactic": "exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),\n mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.690610\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r'✝ : ℝ≥0∞\nr' : ℝ≥0\nhf : HasFPowerSeriesOnBall f p x r\nh : ↑r' < r\na : ℝ\nha : a ∈ Ioo 0 1\nC : ℝ\nhC : C > 0\nhp :\n ∀ (y : E),\n y ∈ Metric.ball 0 ↑r' →\n ∀ (n : ℕ), ‖f (x + y) - FormalMultilinearSeries.partialSum p n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n\ny : E\nhy : y ∈ Metric.ball 0 ↑r'\nn : ℕ\n⊢ ‖y‖ < ↑r'",
"tactic": "rwa [ball_zero_eq] at hy"
}
] |
[
694,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
683,
1
] |
Mathlib/CategoryTheory/Limits/Opposites.lean
|
CategoryTheory.Limits.hasFiniteProducts_of_opposite
|
[] |
[
391,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
MultilinearMap.cons_add
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝¹⁰ : Semiring R\ninst✝⁹ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁸ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\ninst✝¹ : Module R M₃\ninst✝ : Module R M'\nf✝ f' : MultilinearMap R M₁ M₂\nf : MultilinearMap R M M₂\nm : (i : Fin n) → M (succ i)\nx y : M 0\n⊢ ↑f (cons (x + y) m) = ↑f (cons x m) + ↑f (cons y m)",
"tactic": "simp_rw [← update_cons_zero x m (x + y), f.map_add, update_cons_zero]"
}
] |
[
351,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
349,
1
] |
Mathlib/Topology/Instances/Nat.lean
|
Nat.preimage_ball
|
[] |
[
52,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.contractNth_apply_of_eq
|
[
{
"state_after": "m n : ℕ\nα : Type u_1\nj : Fin (n + 1)\nop : α → α → α\ng : Fin (n + 1) → α\nk : Fin n\nh : ↑k = ↑j\nthis : ¬↑k < ↑j\n⊢ contractNth j op g k = op (g (↑castSucc k)) (g (succ k))",
"state_before": "m n : ℕ\nα : Type u_1\nj : Fin (n + 1)\nop : α → α → α\ng : Fin (n + 1) → α\nk : Fin n\nh : ↑k = ↑j\n⊢ contractNth j op g k = op (g (↑castSucc k)) (g (succ k))",
"tactic": "have : ¬(k : ℕ) < j := not_lt.2 (le_of_eq h.symm)"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ\nα : Type u_1\nj : Fin (n + 1)\nop : α → α → α\ng : Fin (n + 1) → α\nk : Fin n\nh : ↑k = ↑j\nthis : ¬↑k < ↑j\n⊢ contractNth j op g k = op (g (↑castSucc k)) (g (succ k))",
"tactic": "rw [contractNth, if_neg this, if_pos h]"
}
] |
[
970,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
967,
1
] |
Mathlib/Order/Basic.lean
|
Decidable.le_iff_eq_or_lt
|
[] |
[
370,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
368,
11
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_eq_single
|
[
{
"state_after": "ι : Type ?u.372604\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\nthis✝ : DecidableEq α\nb : α\nhb : b ∈ s\nh₀ : ∀ (b_1 : α), b_1 ∈ s → b_1 ≠ b → f b_1 = 1\nh₁ : ¬b ∈ s → f b = 1\nthis : ¬b ∈ s\n⊢ False",
"state_before": "ι : Type ?u.372604\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\na : α\nh₀ : ∀ (b : α), b ∈ s → b ≠ a → f b = 1\nh₁ : ¬a ∈ s → f a = 1\nthis✝ : DecidableEq α\nthis : ¬a ∈ s\nb : α\nhb : b ∈ s\n⊢ b ≠ a",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.372604\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\nthis✝ : DecidableEq α\nb : α\nhb : b ∈ s\nh₀ : ∀ (b_1 : α), b_1 ∈ s → b_1 ≠ b → f b_1 = 1\nh₁ : ¬b ∈ s → f b = 1\nthis : ¬b ∈ s\n⊢ False",
"tactic": "exact this hb"
}
] |
[
806,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
801,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.coeff_X
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\ni : σ\n⊢ coeff (Finsupp.single i 1) (X i) = 1",
"tactic": "classical rw [coeff_X', if_pos rfl]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\ni : σ\n⊢ coeff (Finsupp.single i 1) (X i) = 1",
"tactic": "rw [coeff_X', if_pos rfl]"
}
] |
[
689,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
688,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
lt_mul_of_lt_of_one_lt_of_pos
|
[] |
[
813,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
811,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_eq_zero_iff
|
[] |
[
391,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
387,
1
] |
Mathlib/Algebra/Hom/Iterate.lean
|
MonoidHom.coe_pow
|
[] |
[
88,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Iso.preconnected_iff
|
[] |
[
1940,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1937,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.coe_sum
|
[] |
[
754,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
752,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.mk_div
|
[] |
[
756,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
754,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.coeff_zero_one
|
[] |
[
197,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Topology/Algebra/OpenSubgroup.lean
|
OpenSubgroup.comap_comap
|
[] |
[
301,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
1
] |
Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean
|
intervalIntegrable_inv_iff
|
[
{
"state_after": "no goals",
"state_before": "E : Type ?u.35816\nF : Type ?u.35819\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedAddCommGroup F\na b : ℝ\n⊢ IntervalIntegrable (fun x => x⁻¹) volume a b ↔ a = b ∨ ¬0 ∈ [[a, b]]",
"tactic": "simp only [← intervalIntegrable_sub_inv_iff, sub_zero]"
}
] |
[
178,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
1
] |
Mathlib/Topology/Order.lean
|
continuous_id_of_le
|
[] |
[
823,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
822,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
ContDiffOn.cosh
|
[] |
[
1112,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1110,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
le_of_mul_le_mul_right'
|
[] |
[
80,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/GroupTheory/Submonoid/Basic.lean
|
Submonoid.disjoint_def'
|
[] |
[
571,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
569,
1
] |
Mathlib/Topology/Constructions.lean
|
ContinuousAt.prod_map'
|
[] |
[
613,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
611,
1
] |
Mathlib/Data/Nat/Factors.lean
|
Nat.mem_factors_mul
|
[
{
"state_after": "a b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\np : ℕ\n⊢ Prime p ∧ p ∣ a * b ↔ Prime p ∧ (p ∣ a ∨ p ∣ b)",
"state_before": "a b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\np : ℕ\n⊢ p ∈ factors (a * b) ↔ p ∈ factors a ∨ p ∈ factors b",
"tactic": "rw [mem_factors (mul_ne_zero ha hb), mem_factors ha, mem_factors hb, ← and_or_left]"
},
{
"state_after": "no goals",
"state_before": "a b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\np : ℕ\n⊢ Prime p ∧ p ∣ a * b ↔ Prime p ∧ (p ∣ a ∨ p ∣ b)",
"tactic": "simpa only [and_congr_right_iff] using Prime.dvd_mul"
}
] |
[
261,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.coe_toRealHom
|
[] |
[
241,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
9
] |
Mathlib/Topology/Algebra/Module/WeakDual.lean
|
WeakBilin.eval_continuous
|
[] |
[
125,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.norm_integral_le_integral_norm
|
[
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.1238187\n𝕜 : Type ?u.1238190\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1240881\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → E\nle_ae : ∀ᵐ (a : α) ∂μ, 0 ≤ ‖f a‖\nh : ¬AEStronglyMeasurable f μ\n⊢ 0 ≤ ∫ (a : α), ‖f a‖ ∂μ",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1238187\n𝕜 : Type ?u.1238190\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1240881\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → E\nle_ae : ∀ᵐ (a : α) ∂μ, 0 ≤ ‖f a‖\nh : ¬AEStronglyMeasurable f μ\n⊢ ‖∫ (a : α), f a ∂μ‖ ≤ ∫ (a : α), ‖f a‖ ∂μ",
"tactic": "rw [integral_non_aestronglyMeasurable h, norm_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1238187\n𝕜 : Type ?u.1238190\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1240881\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf : α → E\nle_ae : ∀ᵐ (a : α) ∂μ, 0 ≤ ‖f a‖\nh : ¬AEStronglyMeasurable f μ\n⊢ 0 ≤ ∫ (a : α), ‖f a‖ ∂μ",
"tactic": "exact integral_nonneg_of_ae le_ae"
}
] |
[
1338,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1328,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
measurable_from_quotient
|
[] |
[
490,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
488,
1
] |
Mathlib/Data/Real/CauSeqCompletion.lean
|
CauSeq.Completion.ofRat_one
|
[] |
[
73,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.ofFractionRing_comp_algebraMap
|
[] |
[
913,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
911,
1
] |
Mathlib/Data/FunLike/Fintype.lean
|
FunLike.finite
|
[] |
[
70,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsField.localization_map_bijective
|
[
{
"state_after": "R✝ : Type ?u.3191178\ninst✝⁷ : CommRing R✝\nM✝ : Submonoid R✝\nS : Type ?u.3191370\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R✝ S\nP : Type ?u.3191607\ninst✝⁴ : CommRing P\nR : Type u_1\nRₘ : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing Rₘ\nM : Submonoid R\nhM : ¬0 ∈ M\nhR : IsField R\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\nthis : Field R := toField hR\n⊢ Bijective ↑(algebraMap R Rₘ)",
"state_before": "R✝ : Type ?u.3191178\ninst✝⁷ : CommRing R✝\nM✝ : Submonoid R✝\nS : Type ?u.3191370\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R✝ S\nP : Type ?u.3191607\ninst✝⁴ : CommRing P\nR : Type u_1\nRₘ : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing Rₘ\nM : Submonoid R\nhM : ¬0 ∈ M\nhR : IsField R\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\n⊢ Bijective ↑(algebraMap R Rₘ)",
"tactic": "letI := hR.toField"
},
{
"state_after": "R✝ : Type ?u.3191178\ninst✝⁷ : CommRing R✝\nM✝ : Submonoid R✝\nS : Type ?u.3191370\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R✝ S\nP : Type ?u.3191607\ninst✝⁴ : CommRing P\nR : Type u_1\nRₘ : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing Rₘ\nM : Submonoid R\nhR : IsField R\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\nthis : Field R := toField hR\nhM : M ≤ nonZeroDivisors R\n⊢ Bijective ↑(algebraMap R Rₘ)",
"state_before": "R✝ : Type ?u.3191178\ninst✝⁷ : CommRing R✝\nM✝ : Submonoid R✝\nS : Type ?u.3191370\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R✝ S\nP : Type ?u.3191607\ninst✝⁴ : CommRing P\nR : Type u_1\nRₘ : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing Rₘ\nM : Submonoid R\nhM : ¬0 ∈ M\nhR : IsField R\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\nthis : Field R := toField hR\n⊢ Bijective ↑(algebraMap R Rₘ)",
"tactic": "replace hM := le_nonZeroDivisors_of_noZeroDivisors hM"
},
{
"state_after": "R✝ : Type ?u.3191178\ninst✝⁷ : CommRing R✝\nM✝ : Submonoid R✝\nS : Type ?u.3191370\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R✝ S\nP : Type ?u.3191607\ninst✝⁴ : CommRing P\nR : Type u_1\nRₘ : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing Rₘ\nM : Submonoid R\nhR : IsField R\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\nthis : Field R := toField hR\nhM : M ≤ nonZeroDivisors R\nx : Rₘ\n⊢ ∃ a, ↑(algebraMap R Rₘ) a = x",
"state_before": "R✝ : Type ?u.3191178\ninst✝⁷ : CommRing R✝\nM✝ : Submonoid R✝\nS : Type ?u.3191370\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R✝ S\nP : Type ?u.3191607\ninst✝⁴ : CommRing P\nR : Type u_1\nRₘ : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing Rₘ\nM : Submonoid R\nhR : IsField R\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\nthis : Field R := toField hR\nhM : M ≤ nonZeroDivisors R\n⊢ Bijective ↑(algebraMap R Rₘ)",
"tactic": "refine' ⟨IsLocalization.injective _ hM, fun x => _⟩"
},
{
"state_after": "case intro.intro.mk\nR✝ : Type ?u.3191178\ninst✝⁷ : CommRing R✝\nM✝ : Submonoid R✝\nS : Type ?u.3191370\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R✝ S\nP : Type ?u.3191607\ninst✝⁴ : CommRing P\nR : Type u_1\nRₘ : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing Rₘ\nM : Submonoid R\nhR : IsField R\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\nthis : Field R := toField hR\nhM : M ≤ nonZeroDivisors R\nr m : R\nhm : m ∈ M\n⊢ ∃ a, ↑(algebraMap R Rₘ) a = mk' Rₘ r { val := m, property := hm }",
"state_before": "R✝ : Type ?u.3191178\ninst✝⁷ : CommRing R✝\nM✝ : Submonoid R✝\nS : Type ?u.3191370\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R✝ S\nP : Type ?u.3191607\ninst✝⁴ : CommRing P\nR : Type u_1\nRₘ : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing Rₘ\nM : Submonoid R\nhR : IsField R\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\nthis : Field R := toField hR\nhM : M ≤ nonZeroDivisors R\nx : Rₘ\n⊢ ∃ a, ↑(algebraMap R Rₘ) a = x",
"tactic": "obtain ⟨r, ⟨m, hm⟩, rfl⟩ := mk'_surjective M x"
},
{
"state_after": "case intro.intro.mk.intro\nR✝ : Type ?u.3191178\ninst✝⁷ : CommRing R✝\nM✝ : Submonoid R✝\nS : Type ?u.3191370\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R✝ S\nP : Type ?u.3191607\ninst✝⁴ : CommRing P\nR : Type u_1\nRₘ : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing Rₘ\nM : Submonoid R\nhR : IsField R\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\nthis : Field R := toField hR\nhM : M ≤ nonZeroDivisors R\nr m : R\nhm : m ∈ M\nn : R\nhn : m * n = 1\n⊢ ∃ a, ↑(algebraMap R Rₘ) a = mk' Rₘ r { val := m, property := hm }",
"state_before": "case intro.intro.mk\nR✝ : Type ?u.3191178\ninst✝⁷ : CommRing R✝\nM✝ : Submonoid R✝\nS : Type ?u.3191370\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R✝ S\nP : Type ?u.3191607\ninst✝⁴ : CommRing P\nR : Type u_1\nRₘ : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing Rₘ\nM : Submonoid R\nhR : IsField R\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\nthis : Field R := toField hR\nhM : M ≤ nonZeroDivisors R\nr m : R\nhm : m ∈ M\n⊢ ∃ a, ↑(algebraMap R Rₘ) a = mk' Rₘ r { val := m, property := hm }",
"tactic": "obtain ⟨n, hn⟩ := hR.mul_inv_cancel (nonZeroDivisors.ne_zero <| hM hm)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.mk.intro\nR✝ : Type ?u.3191178\ninst✝⁷ : CommRing R✝\nM✝ : Submonoid R✝\nS : Type ?u.3191370\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R✝ S\nP : Type ?u.3191607\ninst✝⁴ : CommRing P\nR : Type u_1\nRₘ : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing Rₘ\nM : Submonoid R\nhR : IsField R\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\nthis : Field R := toField hR\nhM : M ≤ nonZeroDivisors R\nr m : R\nhm : m ∈ M\nn : R\nhn : m * n = 1\n⊢ ∃ a, ↑(algebraMap R Rₘ) a = mk' Rₘ r { val := m, property := hm }",
"tactic": "exact ⟨r * n, by erw [eq_mk'_iff_mul_eq, ← map_mul, mul_assoc, _root_.mul_comm n, hn, mul_one]⟩"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.3191178\ninst✝⁷ : CommRing R✝\nM✝ : Submonoid R✝\nS : Type ?u.3191370\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R✝ S\nP : Type ?u.3191607\ninst✝⁴ : CommRing P\nR : Type u_1\nRₘ : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing Rₘ\nM : Submonoid R\nhR : IsField R\ninst✝¹ : Algebra R Rₘ\ninst✝ : IsLocalization M Rₘ\nthis : Field R := toField hR\nhM : M ≤ nonZeroDivisors R\nr m : R\nhm : m ∈ M\nn : R\nhn : m * n = 1\n⊢ ↑(algebraMap R Rₘ) (r * n) = mk' Rₘ r { val := m, property := hm }",
"tactic": "erw [eq_mk'_iff_mul_eq, ← map_mul, mul_assoc, _root_.mul_comm n, hn, mul_one]"
}
] |
[
1279,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1271,
1
] |
Mathlib/Logic/Encodable/Basic.lean
|
Encodable.encode_sigma_val
|
[] |
[
374,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/FieldTheory/Minpoly/Field.lean
|
minpoly.root
|
[
{
"state_after": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx✝ x : B\nhx : IsIntegral A x\ny : A\nh : IsRoot (minpoly A x) y\nkey : minpoly A x = X - ↑C y\n⊢ ↑(algebraMap A B) y = x",
"state_before": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx✝ x : B\nhx : IsIntegral A x\ny : A\nh : IsRoot (minpoly A x) y\n⊢ ↑(algebraMap A B) y = x",
"tactic": "have key : minpoly A x = X - C y := eq_of_monic_of_associated (monic hx) (monic_X_sub_C y)\n (associated_of_dvd_dvd ((irreducible_X_sub_C y).dvd_symm (irreducible hx) (dvd_iff_isRoot.2 h))\n (dvd_iff_isRoot.2 h))"
},
{
"state_after": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx✝ x : B\nhx : IsIntegral A x\ny : A\nh : IsRoot (minpoly A x) y\nkey : minpoly A x = X - ↑C y\nthis : ↑(Polynomial.aeval x) (minpoly A x) = 0\n⊢ ↑(algebraMap A B) y = x",
"state_before": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx✝ x : B\nhx : IsIntegral A x\ny : A\nh : IsRoot (minpoly A x) y\nkey : minpoly A x = X - ↑C y\n⊢ ↑(algebraMap A B) y = x",
"tactic": "have := aeval A x"
},
{
"state_after": "no goals",
"state_before": "A : Type u_1\nB : Type u_2\ninst✝³ : Field A\ninst✝² : Ring B\ninst✝¹ : IsDomain B\ninst✝ : Algebra A B\nx✝ x : B\nhx : IsIntegral A x\ny : A\nh : IsRoot (minpoly A x) y\nkey : minpoly A x = X - ↑C y\nthis : ↑(Polynomial.aeval x) (minpoly A x) = 0\n⊢ ↑(algebraMap A B) y = x",
"tactic": "rwa [key, AlgHom.map_sub, aeval_X, aeval_C, sub_eq_zero, eq_comm] at this"
}
] |
[
260,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Analysis/BoxIntegral/Box/Basic.lean
|
BoxIntegral.Box.le_iff_bounds
|
[] |
[
174,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Topology/Bases.lean
|
TopologicalSpace.empty_nmem_countableBasis
|
[] |
[
634,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
633,
1
] |
Mathlib/Order/PFilter.lean
|
Order.PFilter.inf_mem
|
[] |
[
163,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
one_div_one
|
[] |
[
347,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
346,
1
] |
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