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/Data/Sum/Basic.lean
|
Sum.isRight_map
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type u_2\nf : α → β\ng : γ → δ\nx : α ⊕ γ\n⊢ isRight (Sum.map f g x) = isRight x",
"tactic": "cases x <;> rfl"
}
] |
[
261,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/MeasureTheory/MeasurableSpaceDef.lean
|
MeasurableSet.empty
|
[] |
[
84,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Topology/Connected.lean
|
ConnectedComponents.quotientMap_coe
|
[] |
[
1469,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1468,
1
] |
Mathlib/Algebra/Order/WithZero.lean
|
mul_le_mul_right₀
|
[] |
[
232,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.le_iff_sign
|
[
{
"state_after": "case mp\nx y : EReal\n⊢ x ≤ y →\n ↑sign x < ↑sign y ∨\n ↑sign x = SignType.neg ∧ ↑sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n ↑sign x = SignType.zero ∧ ↑sign y = SignType.zero ∨\n ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y\n\ncase mpr\nx y : EReal\n⊢ ↑sign x < ↑sign y ∨\n ↑sign x = SignType.neg ∧ ↑sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n ↑sign x = SignType.zero ∧ ↑sign y = SignType.zero ∨\n ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y →\n x ≤ y",
"state_before": "x y : EReal\n⊢ x ≤ y ↔\n ↑sign x < ↑sign y ∨\n ↑sign x = SignType.neg ∧ ↑sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n ↑sign x = SignType.zero ∧ ↑sign y = SignType.zero ∨\n ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"tactic": "constructor"
},
{
"state_after": "case mp\nx y : EReal\nh : x ≤ y\n⊢ ↑sign x < ↑sign y ∨\n ↑sign x = SignType.neg ∧ ↑sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n ↑sign x = SignType.zero ∧ ↑sign y = SignType.zero ∨\n ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"state_before": "case mp\nx y : EReal\n⊢ x ≤ y →\n ↑sign x < ↑sign y ∨\n ↑sign x = SignType.neg ∧ ↑sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n ↑sign x = SignType.zero ∧ ↑sign y = SignType.zero ∨\n ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"tactic": "intro h"
},
{
"state_after": "case mp\nx y : EReal\nh : x ≤ y\nhs : ↑sign x = ↑sign y\n⊢ ↑sign x = SignType.neg ∧ ↑sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n ↑sign x = SignType.zero ∧ ↑sign y = SignType.zero ∨\n ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"state_before": "case mp\nx y : EReal\nh : x ≤ y\n⊢ ↑sign x < ↑sign y ∨\n ↑sign x = SignType.neg ∧ ↑sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n ↑sign x = SignType.zero ∧ ↑sign y = SignType.zero ∨\n ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"tactic": "refine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_)"
},
{
"state_after": "case mp\nx y : EReal\nh : ↑(↑sign x) * ↑(EReal.abs x) ≤ ↑(↑sign y) * ↑(EReal.abs y)\nhs : ↑sign x = ↑sign y\n⊢ ↑sign x = SignType.neg ∧ ↑sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n ↑sign x = SignType.zero ∧ ↑sign y = SignType.zero ∨\n ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"state_before": "case mp\nx y : EReal\nh : x ≤ y\nhs : ↑sign x = ↑sign y\n⊢ ↑sign x = SignType.neg ∧ ↑sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n ↑sign x = SignType.zero ∧ ↑sign y = SignType.zero ∨\n ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"tactic": "rw [← x.sign_mul_abs, ← y.sign_mul_abs] at h"
},
{
"state_after": "case mp.zero\nx y : EReal\nh : ↑SignType.zero * ↑(EReal.abs x) ≤ ↑SignType.zero * ↑(EReal.abs y)\nhs : ↑sign x = ↑sign y\nhy : ↑sign y = SignType.zero\n⊢ SignType.zero = SignType.neg ∧ SignType.zero = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n SignType.zero = SignType.zero ∧ SignType.zero = SignType.zero ∨\n SignType.zero = SignType.pos ∧ SignType.zero = SignType.pos ∧ EReal.abs x ≤ EReal.abs y\n\ncase mp.neg\nx y : EReal\nh : ↑SignType.neg * ↑(EReal.abs x) ≤ ↑SignType.neg * ↑(EReal.abs y)\nhs : ↑sign x = ↑sign y\nhy : ↑sign y = SignType.neg\n⊢ SignType.neg = SignType.neg ∧ SignType.neg = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n SignType.neg = SignType.zero ∧ SignType.neg = SignType.zero ∨\n SignType.neg = SignType.pos ∧ SignType.neg = SignType.pos ∧ EReal.abs x ≤ EReal.abs y\n\ncase mp.pos\nx y : EReal\nh : ↑SignType.pos * ↑(EReal.abs x) ≤ ↑SignType.pos * ↑(EReal.abs y)\nhs : ↑sign x = ↑sign y\nhy : ↑sign y = SignType.pos\n⊢ SignType.pos = SignType.neg ∧ SignType.pos = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n SignType.pos = SignType.zero ∧ SignType.pos = SignType.zero ∨\n SignType.pos = SignType.pos ∧ SignType.pos = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"state_before": "case mp\nx y : EReal\nh : ↑(↑sign x) * ↑(EReal.abs x) ≤ ↑(↑sign y) * ↑(EReal.abs y)\nhs : ↑sign x = ↑sign y\n⊢ ↑sign x = SignType.neg ∧ ↑sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n ↑sign x = SignType.zero ∧ ↑sign y = SignType.zero ∨\n ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"tactic": "cases hy : sign y <;> rw [hs, hy] at h ⊢"
},
{
"state_after": "no goals",
"state_before": "case mp.zero\nx y : EReal\nh : ↑SignType.zero * ↑(EReal.abs x) ≤ ↑SignType.zero * ↑(EReal.abs y)\nhs : ↑sign x = ↑sign y\nhy : ↑sign y = SignType.zero\n⊢ SignType.zero = SignType.neg ∧ SignType.zero = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n SignType.zero = SignType.zero ∧ SignType.zero = SignType.zero ∨\n SignType.zero = SignType.pos ∧ SignType.zero = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"tactic": "simp"
},
{
"state_after": "case mp.neg.h\nx y : EReal\nh : ↑SignType.neg * ↑(EReal.abs x) ≤ ↑SignType.neg * ↑(EReal.abs y)\nhs : ↑sign x = ↑sign y\nhy : ↑sign y = SignType.neg\n⊢ SignType.neg = SignType.neg ∧ SignType.neg = SignType.neg ∧ EReal.abs y ≤ EReal.abs x",
"state_before": "case mp.neg\nx y : EReal\nh : ↑SignType.neg * ↑(EReal.abs x) ≤ ↑SignType.neg * ↑(EReal.abs y)\nhs : ↑sign x = ↑sign y\nhy : ↑sign y = SignType.neg\n⊢ SignType.neg = SignType.neg ∧ SignType.neg = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n SignType.neg = SignType.zero ∧ SignType.neg = SignType.zero ∨\n SignType.neg = SignType.pos ∧ SignType.neg = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"tactic": "left"
},
{
"state_after": "no goals",
"state_before": "case mp.neg.h\nx y : EReal\nh : ↑SignType.neg * ↑(EReal.abs x) ≤ ↑SignType.neg * ↑(EReal.abs y)\nhs : ↑sign x = ↑sign y\nhy : ↑sign y = SignType.neg\n⊢ SignType.neg = SignType.neg ∧ SignType.neg = SignType.neg ∧ EReal.abs y ≤ EReal.abs x",
"tactic": "simpa using h"
},
{
"state_after": "case mp.pos.h\nx y : EReal\nh : ↑SignType.pos * ↑(EReal.abs x) ≤ ↑SignType.pos * ↑(EReal.abs y)\nhs : ↑sign x = ↑sign y\nhy : ↑sign y = SignType.pos\n⊢ SignType.pos = SignType.zero ∧ SignType.pos = SignType.zero ∨\n SignType.pos = SignType.pos ∧ SignType.pos = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"state_before": "case mp.pos\nx y : EReal\nh : ↑SignType.pos * ↑(EReal.abs x) ≤ ↑SignType.pos * ↑(EReal.abs y)\nhs : ↑sign x = ↑sign y\nhy : ↑sign y = SignType.pos\n⊢ SignType.pos = SignType.neg ∧ SignType.pos = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n SignType.pos = SignType.zero ∧ SignType.pos = SignType.zero ∨\n SignType.pos = SignType.pos ∧ SignType.pos = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"tactic": "right"
},
{
"state_after": "case mp.pos.h.h\nx y : EReal\nh : ↑SignType.pos * ↑(EReal.abs x) ≤ ↑SignType.pos * ↑(EReal.abs y)\nhs : ↑sign x = ↑sign y\nhy : ↑sign y = SignType.pos\n⊢ SignType.pos = SignType.pos ∧ SignType.pos = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"state_before": "case mp.pos.h\nx y : EReal\nh : ↑SignType.pos * ↑(EReal.abs x) ≤ ↑SignType.pos * ↑(EReal.abs y)\nhs : ↑sign x = ↑sign y\nhy : ↑sign y = SignType.pos\n⊢ SignType.pos = SignType.zero ∧ SignType.pos = SignType.zero ∨\n SignType.pos = SignType.pos ∧ SignType.pos = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"tactic": "right"
},
{
"state_after": "no goals",
"state_before": "case mp.pos.h.h\nx y : EReal\nh : ↑SignType.pos * ↑(EReal.abs x) ≤ ↑SignType.pos * ↑(EReal.abs y)\nhs : ↑sign x = ↑sign y\nhy : ↑sign y = SignType.pos\n⊢ SignType.pos = SignType.pos ∧ SignType.pos = SignType.pos ∧ EReal.abs x ≤ EReal.abs y",
"tactic": "simpa using h"
},
{
"state_after": "case mpr.inl\nx y : EReal\nh : ↑sign x < ↑sign y\n⊢ x ≤ y\n\ncase mpr.inr.inl\nx y : EReal\nh : ↑sign x = SignType.neg ∧ ↑sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x\n⊢ x ≤ y\n\ncase mpr.inr.inr.inl\nx y : EReal\nh : ↑sign x = SignType.zero ∧ ↑sign y = SignType.zero\n⊢ x ≤ y\n\ncase mpr.inr.inr.inr\nx y : EReal\nh : ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y\n⊢ x ≤ y",
"state_before": "case mpr\nx y : EReal\n⊢ ↑sign x < ↑sign y ∨\n ↑sign x = SignType.neg ∧ ↑sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨\n ↑sign x = SignType.zero ∧ ↑sign y = SignType.zero ∨\n ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y →\n x ≤ y",
"tactic": "rintro (h | h | h | h)"
},
{
"state_after": "no goals",
"state_before": "case mpr.inr.inl\nx y : EReal\nh : ↑sign x = SignType.neg ∧ ↑sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x\n⊢ x ≤ y\n\ncase mpr.inr.inr.inl\nx y : EReal\nh : ↑sign x = SignType.zero ∧ ↑sign y = SignType.zero\n⊢ x ≤ y\n\ncase mpr.inr.inr.inr\nx y : EReal\nh : ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y\n⊢ x ≤ y",
"tactic": "all_goals rw [← x.sign_mul_abs, ← y.sign_mul_abs]; simp [h]"
},
{
"state_after": "no goals",
"state_before": "case mpr.inl\nx y : EReal\nh : ↑sign x < ↑sign y\n⊢ x ≤ y",
"tactic": "exact (sign.monotone.reflect_lt h).le"
},
{
"state_after": "case mpr.inr.inr.inr\nx y : EReal\nh : ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y\n⊢ ↑(↑sign x) * ↑(EReal.abs x) ≤ ↑(↑sign y) * ↑(EReal.abs y)",
"state_before": "case mpr.inr.inr.inr\nx y : EReal\nh : ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y\n⊢ x ≤ y",
"tactic": "rw [← x.sign_mul_abs, ← y.sign_mul_abs]"
},
{
"state_after": "no goals",
"state_before": "case mpr.inr.inr.inr\nx y : EReal\nh : ↑sign x = SignType.pos ∧ ↑sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y\n⊢ ↑(↑sign x) * ↑(EReal.abs x) ≤ ↑(↑sign y) * ↑(EReal.abs y)",
"tactic": "simp [h]"
}
] |
[
1174,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1159,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.equivProdDfinsupp_add
|
[] |
[
1688,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1686,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.rank_adjoin_simple_eq_one_iff
|
[
{
"state_after": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\n⊢ Insert.insert ∅ α ⊆ ↑⊥ ↔ α ∈ ⊥",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\n⊢ Module.rank F { x // x ∈ F⟮α⟯ } = 1 ↔ α ∈ ⊥",
"tactic": "rw [rank_adjoin_eq_one_iff]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\nK L : IntermediateField F E\n⊢ Insert.insert ∅ α ⊆ ↑⊥ ↔ α ∈ ⊥",
"tactic": "exact Set.singleton_subset_iff"
}
] |
[
736,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
735,
1
] |
Mathlib/Topology/Instances/Irrational.lean
|
Irrational.eventually_forall_le_dist_cast_div
|
[
{
"state_after": "x : ℝ\nhx : Irrational x\nn : ℕ\nA : IsClosed (range fun m => (↑n)⁻¹ * ↑m)\n⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n)",
"state_before": "x : ℝ\nhx : Irrational x\nn : ℕ\n⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n)",
"tactic": "have A : IsClosed (range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ)) :=\n ((isClosedMap_smul₀ (n⁻¹ : ℝ)).comp Int.closedEmbedding_coe_real.isClosedMap).closed_range"
},
{
"state_after": "x : ℝ\nhx : Irrational x\nn : ℕ\nA : IsClosed (range fun m => (↑n)⁻¹ * ↑m)\nB : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m\n⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n)",
"state_before": "x : ℝ\nhx : Irrational x\nn : ℕ\nA : IsClosed (range fun m => (↑n)⁻¹ * ↑m)\n⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n)",
"tactic": "have B : x ∉ range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ) := by\n rintro ⟨m, rfl⟩\n simp at hx"
},
{
"state_after": "case intro.intro\nx : ℝ\nhx : Irrational x\nn : ℕ\nA : IsClosed (range fun m => (↑n)⁻¹ * ↑m)\nB : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m\nε : ℝ\nε0 : ε > 0\nhε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ\n⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n)",
"state_before": "x : ℝ\nhx : Irrational x\nn : ℕ\nA : IsClosed (range fun m => (↑n)⁻¹ * ↑m)\nB : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m\n⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n)",
"tactic": "rcases Metric.mem_nhds_iff.1 (A.isOpen_compl.mem_nhds B) with ⟨ε, ε0, hε⟩"
},
{
"state_after": "case intro.intro\nx : ℝ\nhx : Irrational x\nn : ℕ\nA : IsClosed (range fun m => (↑n)⁻¹ * ↑m)\nB : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m\nε : ℝ\nε0 : ε > 0\nhε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ\nδ : ℝ\nhδ : δ ≤ ε\nm : ℤ\nhlt : dist x (↑m / ↑n) < δ\n⊢ False",
"state_before": "case intro.intro\nx : ℝ\nhx : Irrational x\nn : ℕ\nA : IsClosed (range fun m => (↑n)⁻¹ * ↑m)\nB : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m\nε : ℝ\nε0 : ε > 0\nhε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ\n⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n)",
"tactic": "refine' (ge_mem_nhds ε0).mono fun δ hδ m => not_lt.1 fun hlt => _"
},
{
"state_after": "case intro.intro\nx : ℝ\nhx : Irrational x\nn : ℕ\nA : IsClosed (range fun m => (↑n)⁻¹ * ↑m)\nB : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m\nε : ℝ\nε0 : ε > 0\nhε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ\nδ : ℝ\nhδ : δ ≤ ε\nm : ℤ\nhlt : dist (↑m / ↑n) x < δ\n⊢ False",
"state_before": "case intro.intro\nx : ℝ\nhx : Irrational x\nn : ℕ\nA : IsClosed (range fun m => (↑n)⁻¹ * ↑m)\nB : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m\nε : ℝ\nε0 : ε > 0\nhε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ\nδ : ℝ\nhδ : δ ≤ ε\nm : ℤ\nhlt : dist x (↑m / ↑n) < δ\n⊢ False",
"tactic": "rw [dist_comm] at hlt"
},
{
"state_after": "case intro.intro\nx : ℝ\nhx : Irrational x\nn : ℕ\nA : IsClosed (range fun m => (↑n)⁻¹ * ↑m)\nB : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m\nε : ℝ\nε0 : ε > 0\nhε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ\nδ : ℝ\nhδ : δ ≤ ε\nm : ℤ\nhlt : dist (↑m / ↑n) x < δ\n⊢ (fun m => (↑n)⁻¹ * ↑m) m = ↑m / ↑n",
"state_before": "case intro.intro\nx : ℝ\nhx : Irrational x\nn : ℕ\nA : IsClosed (range fun m => (↑n)⁻¹ * ↑m)\nB : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m\nε : ℝ\nε0 : ε > 0\nhε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ\nδ : ℝ\nhδ : δ ≤ ε\nm : ℤ\nhlt : dist (↑m / ↑n) x < δ\n⊢ False",
"tactic": "refine' hε (ball_subset_ball hδ hlt) ⟨m, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nx : ℝ\nhx : Irrational x\nn : ℕ\nA : IsClosed (range fun m => (↑n)⁻¹ * ↑m)\nB : ¬x ∈ range fun m => (↑n)⁻¹ * ↑m\nε : ℝ\nε0 : ε > 0\nhε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ\nδ : ℝ\nhδ : δ ≤ ε\nm : ℤ\nhlt : dist (↑m / ↑n) x < δ\n⊢ (fun m => (↑n)⁻¹ * ↑m) m = ↑m / ↑n",
"tactic": "simp [div_eq_inv_mul]"
},
{
"state_after": "case intro\nn : ℕ\nA : IsClosed (range fun m => (↑n)⁻¹ * ↑m)\nm : ℤ\nhx : Irrational ((fun m => (↑n)⁻¹ * ↑m) m)\n⊢ False",
"state_before": "x : ℝ\nhx : Irrational x\nn : ℕ\nA : IsClosed (range fun m => (↑n)⁻¹ * ↑m)\n⊢ ¬x ∈ range fun m => (↑n)⁻¹ * ↑m",
"tactic": "rintro ⟨m, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nn : ℕ\nA : IsClosed (range fun m => (↑n)⁻¹ * ↑m)\nm : ℤ\nhx : Irrational ((fun m => (↑n)⁻¹ * ↑m) m)\n⊢ False",
"tactic": "simp at hx"
}
] |
[
90,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Topology/Constructions.lean
|
isOpen_sigma_fst_preimage
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.527967\nδ : Type ?u.527970\nε : Type ?u.527973\nζ : Type ?u.527976\nι : Type u_1\nκ : Type ?u.527982\nσ : ι → Type u_2\nτ : κ → Type ?u.527992\ninst✝² : (i : ι) → TopologicalSpace (σ i)\ninst✝¹ : (k : κ) → TopologicalSpace (τ k)\ninst✝ : TopologicalSpace α\ns : Set ι\n⊢ IsOpen (⋃ (i : ι) (_ : i ∈ s), Sigma.fst ⁻¹' {i})",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.527967\nδ : Type ?u.527970\nε : Type ?u.527973\nζ : Type ?u.527976\nι : Type u_1\nκ : Type ?u.527982\nσ : ι → Type u_2\nτ : κ → Type ?u.527992\ninst✝² : (i : ι) → TopologicalSpace (σ i)\ninst✝¹ : (k : κ) → TopologicalSpace (τ k)\ninst✝ : TopologicalSpace α\ns : Set ι\n⊢ IsOpen (Sigma.fst ⁻¹' s)",
"tactic": "rw [← biUnion_of_singleton s, preimage_iUnion₂]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.527967\nδ : Type ?u.527970\nε : Type ?u.527973\nζ : Type ?u.527976\nι : Type u_1\nκ : Type ?u.527982\nσ : ι → Type u_2\nτ : κ → Type ?u.527992\ninst✝² : (i : ι) → TopologicalSpace (σ i)\ninst✝¹ : (k : κ) → TopologicalSpace (τ k)\ninst✝ : TopologicalSpace α\ns : Set ι\n⊢ IsOpen (⋃ (i : ι) (_ : i ∈ s), range (Sigma.mk i))",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.527967\nδ : Type ?u.527970\nε : Type ?u.527973\nζ : Type ?u.527976\nι : Type u_1\nκ : Type ?u.527982\nσ : ι → Type u_2\nτ : κ → Type ?u.527992\ninst✝² : (i : ι) → TopologicalSpace (σ i)\ninst✝¹ : (k : κ) → TopologicalSpace (τ k)\ninst✝ : TopologicalSpace α\ns : Set ι\n⊢ IsOpen (⋃ (i : ι) (_ : i ∈ s), Sigma.fst ⁻¹' {i})",
"tactic": "simp only [← range_sigmaMk]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.527967\nδ : Type ?u.527970\nε : Type ?u.527973\nζ : Type ?u.527976\nι : Type u_1\nκ : Type ?u.527982\nσ : ι → Type u_2\nτ : κ → Type ?u.527992\ninst✝² : (i : ι) → TopologicalSpace (σ i)\ninst✝¹ : (k : κ) → TopologicalSpace (τ k)\ninst✝ : TopologicalSpace α\ns : Set ι\n⊢ IsOpen (⋃ (i : ι) (_ : i ∈ s), range (Sigma.mk i))",
"tactic": "exact isOpen_biUnion fun _ _ => isOpen_range_sigmaMk"
}
] |
[
1517,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1514,
1
] |
Mathlib/Data/Bitvec/Lemmas.lean
|
Bitvec.toFin_val
|
[
{
"state_after": "n : ℕ\nv : Bitvec n\n⊢ Bitvec.toNat v < 2 ^ n",
"state_before": "n : ℕ\nv : Bitvec n\n⊢ ↑(toFin v) = Bitvec.toNat v",
"tactic": "rw [toFin, Fin.coe_ofNat_eq_mod, Nat.mod_eq_of_lt]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nv : Bitvec n\n⊢ Bitvec.toNat v < 2 ^ n",
"tactic": "apply toNat_lt"
}
] |
[
152,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Order/Filter/Prod.lean
|
Filter.tendsto_snd
|
[] |
[
144,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/GroupTheory/GroupAction/Embedding.lean
|
Function.Embedding.smul_apply
|
[] |
[
43,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/Data/Setoid/Basic.lean
|
Setoid.mapOfSurjective_eq_map
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nr : Setoid α\nf : α → β\nh : ker f ≤ r\nhf : Surjective f\n⊢ map r f = EqvGen.Setoid Setoid.r",
"state_before": "α : Type u_1\nβ : Type u_2\nr : Setoid α\nf : α → β\nh : ker f ≤ r\nhf : Surjective f\n⊢ map r f = mapOfSurjective r f h hf",
"tactic": "rw [← eqvGen_of_setoid (mapOfSurjective r f h hf)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nr : Setoid α\nf : α → β\nh : ker f ≤ r\nhf : Surjective f\n⊢ map r f = EqvGen.Setoid Setoid.r",
"tactic": "rfl"
}
] |
[
385,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean
|
EuclideanGeometry.Sphere.secondInter_eq_lineMap
|
[] |
[
140,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.VectorMeasure.coe_zero
|
[] |
[
293,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.mk'_toAddSubmonoid
|
[] |
[
297,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.nat_casesOn'
|
[] |
[
593,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
591,
1
] |
Mathlib/Algebra/Order/Sub/Defs.lean
|
AddLECancellable.le_tsub_of_add_le_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.24546\ninst✝³ : Preorder α\ninst✝² : AddCommSemigroup α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nh : a + b ≤ c\n⊢ b + a ≤ c",
"tactic": "rwa [add_comm]"
}
] |
[
216,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
11
] |
Mathlib/Algebra/Tropical/Basic.lean
|
Tropical.trop_inf
|
[] |
[
284,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
Function.Surjective.lieModuleIsNilpotent
|
[
{
"state_after": "case mk.intro\nR : Type u\nL : Type v\nM : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁶ : LieRing L₂\ninst✝⁵ : LieAlgebra R L₂\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : Module R M₂\ninst✝² : LieRingModule L₂ M₂\ninst✝¹ : LieModule R L₂ M₂\nf : L →ₗ⁅R⁆ L₂\ng : M →ₗ[R] M₂\nhf : Surjective ↑f\nhg : Surjective ↑g\nhfg : ∀ (x : L) (m : M), ⁅↑f x, ↑g m⁆ = ↑g ⁅x, m⁆\ninst✝ : LieModule.IsNilpotent R L M\nk : ℕ\nhk : lowerCentralSeries R L M k = ⊥\n⊢ LieModule.IsNilpotent R L₂ M₂",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁶ : LieRing L₂\ninst✝⁵ : LieAlgebra R L₂\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : Module R M₂\ninst✝² : LieRingModule L₂ M₂\ninst✝¹ : LieModule R L₂ M₂\nf : L →ₗ⁅R⁆ L₂\ng : M →ₗ[R] M₂\nhf : Surjective ↑f\nhg : Surjective ↑g\nhfg : ∀ (x : L) (m : M), ⁅↑f x, ↑g m⁆ = ↑g ⁅x, m⁆\ninst✝ : LieModule.IsNilpotent R L M\n⊢ LieModule.IsNilpotent R L₂ M₂",
"tactic": "obtain ⟨k, hk⟩ := id (by infer_instance : IsNilpotent R L M)"
},
{
"state_after": "case mk.intro\nR : Type u\nL : Type v\nM : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁶ : LieRing L₂\ninst✝⁵ : LieAlgebra R L₂\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : Module R M₂\ninst✝² : LieRingModule L₂ M₂\ninst✝¹ : LieModule R L₂ M₂\nf : L →ₗ⁅R⁆ L₂\ng : M →ₗ[R] M₂\nhf : Surjective ↑f\nhg : Surjective ↑g\nhfg : ∀ (x : L) (m : M), ⁅↑f x, ↑g m⁆ = ↑g ⁅x, m⁆\ninst✝ : LieModule.IsNilpotent R L M\nk : ℕ\nhk : lowerCentralSeries R L M k = ⊥\n⊢ lowerCentralSeries R L₂ M₂ k = ⊥",
"state_before": "case mk.intro\nR : Type u\nL : Type v\nM : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁶ : LieRing L₂\ninst✝⁵ : LieAlgebra R L₂\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : Module R M₂\ninst✝² : LieRingModule L₂ M₂\ninst✝¹ : LieModule R L₂ M₂\nf : L →ₗ⁅R⁆ L₂\ng : M →ₗ[R] M₂\nhf : Surjective ↑f\nhg : Surjective ↑g\nhfg : ∀ (x : L) (m : M), ⁅↑f x, ↑g m⁆ = ↑g ⁅x, m⁆\ninst✝ : LieModule.IsNilpotent R L M\nk : ℕ\nhk : lowerCentralSeries R L M k = ⊥\n⊢ LieModule.IsNilpotent R L₂ M₂",
"tactic": "use k"
},
{
"state_after": "case mk.intro\nR : Type u\nL : Type v\nM : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁶ : LieRing L₂\ninst✝⁵ : LieAlgebra R L₂\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : Module R M₂\ninst✝² : LieRingModule L₂ M₂\ninst✝¹ : LieModule R L₂ M₂\nf : L →ₗ⁅R⁆ L₂\ng : M →ₗ[R] M₂\nhf : Surjective ↑f\nhg : Surjective ↑g\nhfg : ∀ (x : L) (m : M), ⁅↑f x, ↑g m⁆ = ↑g ⁅x, m⁆\ninst✝ : LieModule.IsNilpotent R L M\nk : ℕ\nhk : ↑(lowerCentralSeries R L M k) = ↑⊥\n⊢ ↑(lowerCentralSeries R L₂ M₂ k) = ↑⊥",
"state_before": "case mk.intro\nR : Type u\nL : Type v\nM : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁶ : LieRing L₂\ninst✝⁵ : LieAlgebra R L₂\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : Module R M₂\ninst✝² : LieRingModule L₂ M₂\ninst✝¹ : LieModule R L₂ M₂\nf : L →ₗ⁅R⁆ L₂\ng : M →ₗ[R] M₂\nhf : Surjective ↑f\nhg : Surjective ↑g\nhfg : ∀ (x : L) (m : M), ⁅↑f x, ↑g m⁆ = ↑g ⁅x, m⁆\ninst✝ : LieModule.IsNilpotent R L M\nk : ℕ\nhk : lowerCentralSeries R L M k = ⊥\n⊢ lowerCentralSeries R L₂ M₂ k = ⊥",
"tactic": "rw [← LieSubmodule.coe_toSubmodule_eq_iff] at hk ⊢"
},
{
"state_after": "no goals",
"state_before": "case mk.intro\nR : Type u\nL : Type v\nM : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁶ : LieRing L₂\ninst✝⁵ : LieAlgebra R L₂\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : Module R M₂\ninst✝² : LieRingModule L₂ M₂\ninst✝¹ : LieModule R L₂ M₂\nf : L →ₗ⁅R⁆ L₂\ng : M →ₗ[R] M₂\nhf : Surjective ↑f\nhg : Surjective ↑g\nhfg : ∀ (x : L) (m : M), ⁅↑f x, ↑g m⁆ = ↑g ⁅x, m⁆\ninst✝ : LieModule.IsNilpotent R L M\nk : ℕ\nhk : ↑(lowerCentralSeries R L M k) = ↑⊥\n⊢ ↑(lowerCentralSeries R L₂ M₂ k) = ↑⊥",
"tactic": "simp [← hf.lieModule_lcs_map_eq hg hfg k, hk]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁶ : LieRing L₂\ninst✝⁵ : LieAlgebra R L₂\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : Module R M₂\ninst✝² : LieRingModule L₂ M₂\ninst✝¹ : LieModule R L₂ M₂\nf : L →ₗ⁅R⁆ L₂\ng : M →ₗ[R] M₂\nhf : Surjective ↑f\nhg : Surjective ↑g\nhfg : ∀ (x : L) (m : M), ⁅↑f x, ↑g m⁆ = ↑g ⁅x, m⁆\ninst✝ : LieModule.IsNilpotent R L M\n⊢ LieModule.IsNilpotent R L M",
"tactic": "infer_instance"
}
] |
[
487,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.leadingCoeff_normalize
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizationMonoid R\np : R[X]\n⊢ leadingCoeff (↑normalize p) = ↑normalize (leadingCoeff p)",
"tactic": "simp"
}
] |
[
101,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.MeasurableSet.nullMeasurableSet_subtype_coe
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n⊢ NullMeasurableSet (Subtype.val '' t)",
"state_before": "α : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n⊢ NullMeasurableSet (Subtype.val '' t)",
"tactic": "rw [Subtype.instMeasurableSpace, comap_eq_generateFrom] at ht"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n⊢ ∀ (t : Set ↑s),\n t ∈ {t | ∃ s', MeasurableSet s' ∧ Subtype.val ⁻¹' s' = t} → (fun t => NullMeasurableSet (Subtype.val '' t)) t\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n⊢ (fun t => NullMeasurableSet (Subtype.val '' t)) ∅\n\ncase refine'_3\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n⊢ ∀ (t : Set ↑s),\n (fun t => NullMeasurableSet (Subtype.val '' t)) t → (fun t => NullMeasurableSet (Subtype.val '' t)) (tᶜ)\n\ncase refine'_4\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n⊢ ∀ (f : ℕ → Set ↑s),\n (∀ (n : ℕ), (fun t => NullMeasurableSet (Subtype.val '' t)) (f n)) →\n (fun t => NullMeasurableSet (Subtype.val '' t)) (⋃ (i : ℕ), f i)",
"state_before": "α : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n⊢ NullMeasurableSet (Subtype.val '' t)",
"tactic": "refine'\n generateFrom_induction (p := fun t : Set s => NullMeasurableSet ((↑) '' t) μ)\n { t : Set s | ∃ s' : Set α, MeasurableSet s' ∧ (↑) ⁻¹' s' = t } _ _ _ _ ht"
},
{
"state_after": "case refine'_1.intro.intro\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s'✝ t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\ns' : Set α\nhs' : MeasurableSet s'\n⊢ NullMeasurableSet (Subtype.val '' (Subtype.val ⁻¹' s'))",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n⊢ ∀ (t : Set ↑s),\n t ∈ {t | ∃ s', MeasurableSet s' ∧ Subtype.val ⁻¹' s' = t} → (fun t => NullMeasurableSet (Subtype.val '' t)) t",
"tactic": "rintro t' ⟨s', hs', rfl⟩"
},
{
"state_after": "case refine'_1.intro.intro\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s'✝ t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\ns' : Set α\nhs' : MeasurableSet s'\n⊢ NullMeasurableSet (s' ∩ s)",
"state_before": "case refine'_1.intro.intro\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s'✝ t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\ns' : Set α\nhs' : MeasurableSet s'\n⊢ NullMeasurableSet (Subtype.val '' (Subtype.val ⁻¹' s'))",
"tactic": "rw [Subtype.image_preimage_coe]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s'✝ t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\ns' : Set α\nhs' : MeasurableSet s'\n⊢ NullMeasurableSet (s' ∩ s)",
"tactic": "exact hs'.nullMeasurableSet.inter hs"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n⊢ (fun t => NullMeasurableSet (Subtype.val '' t)) ∅",
"tactic": "simp only [image_empty, nullMeasurableSet_empty]"
},
{
"state_after": "case refine'_3\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nt' : Set ↑s\n⊢ (fun t => NullMeasurableSet (Subtype.val '' t)) t' → (fun t => NullMeasurableSet (Subtype.val '' t)) (t'ᶜ)",
"state_before": "case refine'_3\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n⊢ ∀ (t : Set ↑s),\n (fun t => NullMeasurableSet (Subtype.val '' t)) t → (fun t => NullMeasurableSet (Subtype.val '' t)) (tᶜ)",
"tactic": "intro t'"
},
{
"state_after": "case refine'_3\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nt' : Set ↑s\n⊢ NullMeasurableSet (Subtype.val '' t') → NullMeasurableSet (s \\ (fun a => ↑a) '' t')",
"state_before": "case refine'_3\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nt' : Set ↑s\n⊢ (fun t => NullMeasurableSet (Subtype.val '' t)) t' → (fun t => NullMeasurableSet (Subtype.val '' t)) (t'ᶜ)",
"tactic": "simp only [← range_diff_image Subtype.coe_injective, Subtype.range_coe_subtype, setOf_mem_eq]"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nt' : Set ↑s\n⊢ NullMeasurableSet (Subtype.val '' t') → NullMeasurableSet (s \\ (fun a => ↑a) '' t')",
"tactic": "exact hs.diff"
},
{
"state_after": "case refine'_4\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nf : ℕ → Set ↑s\n⊢ (∀ (n : ℕ), (fun t => NullMeasurableSet (Subtype.val '' t)) (f n)) →\n (fun t => NullMeasurableSet (Subtype.val '' t)) (⋃ (i : ℕ), f i)",
"state_before": "case refine'_4\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\n⊢ ∀ (f : ℕ → Set ↑s),\n (∀ (n : ℕ), (fun t => NullMeasurableSet (Subtype.val '' t)) (f n)) →\n (fun t => NullMeasurableSet (Subtype.val '' t)) (⋃ (i : ℕ), f i)",
"tactic": "intro f"
},
{
"state_after": "case refine'_4\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nf : ℕ → Set ↑s\n⊢ (∀ (n : ℕ), NullMeasurableSet (Subtype.val '' f n)) → NullMeasurableSet (Subtype.val '' ⋃ (i : ℕ), f i)",
"state_before": "case refine'_4\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nf : ℕ → Set ↑s\n⊢ (∀ (n : ℕ), (fun t => NullMeasurableSet (Subtype.val '' t)) (f n)) →\n (fun t => NullMeasurableSet (Subtype.val '' t)) (⋃ (i : ℕ), f i)",
"tactic": "dsimp only []"
},
{
"state_after": "case refine'_4\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nf : ℕ → Set ↑s\n⊢ (∀ (n : ℕ), NullMeasurableSet (Subtype.val '' f n)) → NullMeasurableSet (⋃ (i : ℕ), Subtype.val '' f i)",
"state_before": "case refine'_4\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nf : ℕ → Set ↑s\n⊢ (∀ (n : ℕ), NullMeasurableSet (Subtype.val '' f n)) → NullMeasurableSet (Subtype.val '' ⋃ (i : ℕ), f i)",
"tactic": "rw [image_iUnion]"
},
{
"state_after": "no goals",
"state_before": "case refine'_4\nα : Type u_1\nβ : Type ?u.253013\nγ : Type ?u.253016\nδ : Type ?u.253019\nι : Type ?u.253022\nR : Type ?u.253025\nR' : Type ?u.253028\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nt : Set ↑s\nhs : NullMeasurableSet s\nht : MeasurableSet t\nf : ℕ → Set ↑s\n⊢ (∀ (n : ℕ), NullMeasurableSet (Subtype.val '' f n)) → NullMeasurableSet (⋃ (i : ℕ), Subtype.val '' f i)",
"tactic": "exact NullMeasurableSet.iUnion"
}
] |
[
1407,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1391,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.mul_im
|
[] |
[
273,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.spanExt_hom_app_left
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nX Y Z X' Y' Z' : C\niX : X ≅ X'\niY : Y ≅ Y'\niZ : Z ≅ Z'\nf : X ⟶ Y\ng : X ⟶ Z\nf' : X' ⟶ Y'\ng' : X' ⟶ Z'\nwf : iX.hom ≫ f' = f ≫ iY.hom\nwg : iX.hom ≫ g' = g ≫ iZ.hom\n⊢ (spanExt iX iY iZ wf wg).hom.app WalkingSpan.left = iY.hom",
"tactic": "dsimp [spanExt]"
}
] |
[
476,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
475,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Filter.Tendsto.isBigO_one
|
[] |
[
1351,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1349,
1
] |
Mathlib/Topology/MetricSpace/Antilipschitz.lean
|
AntilipschitzWith.mul_le_dist
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.3257\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\nf : α → β\nhf : AntilipschitzWith K f\nx y : α\n⊢ ↑K⁻¹ * dist x y ≤ dist (f x) (f y)",
"tactic": "exact_mod_cast hf.mul_le_nndist x y"
}
] |
[
84,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
MeasurableEmbedding.measurable_comp_iff
|
[
{
"state_after": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.382740\nδ' : Type ?u.382743\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nH : Measurable (g ∘ f)\n⊢ Measurable f",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.382740\nδ' : Type ?u.382743\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\n⊢ Measurable (g ∘ f) ↔ Measurable f",
"tactic": "refine' ⟨fun H => _, hg.measurable.comp⟩"
},
{
"state_after": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.382740\nδ' : Type ?u.382743\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nH : Measurable (g ∘ f)\n⊢ Measurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f)",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.382740\nδ' : Type ?u.382743\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nH : Measurable (g ∘ f)\n⊢ Measurable f",
"tactic": "suffices Measurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f) by\n rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.382740\nδ' : Type ?u.382743\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nH : Measurable (g ∘ f)\n⊢ Measurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f)",
"tactic": "exact hg.measurable_rangeSplitting.comp H.subtype_mk"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.382740\nδ' : Type ?u.382743\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nH : Measurable (g ∘ f)\nthis : Measurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f)\n⊢ Measurable f",
"tactic": "rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this"
}
] |
[
1140,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1136,
1
] |
Mathlib/Data/Set/NAry.lean
|
Set.image2_subset_right
|
[] |
[
69,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
CharP.neg_one_ne_one
|
[
{
"state_after": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\n⊢ 2 ≠ 0",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\n⊢ -1 ≠ 1",
"tactic": "suffices (2 : R) ≠ 0 by\n intro h\n symm at h\n rw [← sub_eq_zero, sub_neg_eq_add] at h\n norm_num at h\n exact this h"
},
{
"state_after": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nh : 2 = 0\n⊢ False",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\n⊢ 2 ≠ 0",
"tactic": "intro h"
},
{
"state_after": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nh : ↑2 = 0\n⊢ False",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nh : 2 = 0\n⊢ False",
"tactic": "rw [show (2 : R) = (2 : ℕ) by norm_cast] at h"
},
{
"state_after": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nh : ↑2 = 0\nthis : p ∣ 2\n⊢ False",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nh : ↑2 = 0\n⊢ False",
"tactic": "have := (CharP.cast_eq_zero_iff R p 2).mp h"
},
{
"state_after": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nh : ↑2 = 0\nthis✝ : p ∣ 2\nthis : p ≤ 2\n⊢ False",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nh : ↑2 = 0\nthis : p ∣ 2\n⊢ False",
"tactic": "have := Nat.le_of_dvd (by decide) this"
},
{
"state_after": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : 2 < p\nh : ↑2 = 0\nthis✝ : p ∣ 2\nthis : p ≤ 2\n⊢ False",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nh : ↑2 = 0\nthis✝ : p ∣ 2\nthis : p ≤ 2\n⊢ False",
"tactic": "rw [fact_iff] at *"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : 2 < p\nh : ↑2 = 0\nthis✝ : p ∣ 2\nthis : p ≤ 2\n⊢ False",
"tactic": "linarith"
},
{
"state_after": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nthis : 2 ≠ 0\nh : -1 = 1\n⊢ False",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nthis : 2 ≠ 0\n⊢ -1 ≠ 1",
"tactic": "intro h"
},
{
"state_after": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nthis : 2 ≠ 0\nh : 1 = -1\n⊢ False",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nthis : 2 ≠ 0\nh : -1 = 1\n⊢ False",
"tactic": "symm at h"
},
{
"state_after": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nthis : 2 ≠ 0\nh✝ : 1 = -1\nh : 1 + 1 = 0\n⊢ False",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nthis : 2 ≠ 0\nh : 1 = -1\n⊢ False",
"tactic": "rw [← sub_eq_zero, sub_neg_eq_add] at h"
},
{
"state_after": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nthis : 2 ≠ 0\nh✝ : 1 = -1\nh : 2 = 0\n⊢ False",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nthis : 2 ≠ 0\nh✝ : 1 = -1\nh : 1 + 1 = 0\n⊢ False",
"tactic": "norm_num at h"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nthis : 2 ≠ 0\nh✝ : 1 = -1\nh : 2 = 0\n⊢ False",
"tactic": "exact this h"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nh : 2 = 0\n⊢ 2 = ↑2",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (2 < p)\nh : ↑2 = 0\nthis : p ∣ 2\n⊢ 0 < 2",
"tactic": "decide"
}
] |
[
308,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Data/Vector/Basic.lean
|
Vector.reverse_reverse
|
[
{
"state_after": "case mk\nn : ℕ\nα : Type u_1\nval✝ : List α\nproperty✝ : List.length val✝ = n\n⊢ reverse (reverse { val := val✝, property := property✝ }) = { val := val✝, property := property✝ }",
"state_before": "n : ℕ\nα : Type u_1\nv : Vector α n\n⊢ reverse (reverse v) = v",
"tactic": "cases v"
},
{
"state_after": "no goals",
"state_before": "case mk\nn : ℕ\nα : Type u_1\nval✝ : List α\nproperty✝ : List.length val✝ = n\n⊢ reverse (reverse { val := val✝, property := property✝ }) = { val := val✝, property := property✝ }",
"tactic": "simp [Vector.reverse]"
}
] |
[
253,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Algebra/AddTorsor.lean
|
vadd_vsub_vadd_cancel_left
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nP : Type u_2\ninst✝¹ : AddCommGroup G\ninst✝ : AddTorsor G P\nv : G\np1 p2 : P\n⊢ v +ᵥ p1 -ᵥ (v +ᵥ p2) = p1 -ᵥ p2",
"tactic": "rw [vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, add_sub_cancel']"
}
] |
[
263,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Algebra/BigOperators/Ring.lean
|
Finset.prod_natCast
|
[] |
[
220,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
Right.one_lt_mul_of_lt_of_lt
|
[] |
[
905,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
903,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean
|
CategoryTheory.Limits.IsTerminal.isSplitMono_from
|
[] |
[
174,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Data/Polynomial/Lifts.lean
|
Polynomial.mem_lifts_and_degree_eq
|
[
{
"state_after": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nhlifts : p ∈ lifts f\nd : ℕ\nhd : natDegree p = d\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"state_before": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nhlifts : p ∈ lifts f\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"tactic": "generalize hd : p.natDegree = d"
},
{
"state_after": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nd : ℕ\n⊢ ∀ {p : S[X]}, p ∈ lifts f → natDegree p = d → ∃ q, map f q = p ∧ degree q = degree p",
"state_before": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nhlifts : p ∈ lifts f\nd : ℕ\nhd : natDegree p = d\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"tactic": "revert hd p"
},
{
"state_after": "case h\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\n⊢ ∀ {p : S[X]}, p ∈ lifts f → natDegree p = n → ∃ q, map f q = p ∧ degree q = degree p",
"state_before": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nd : ℕ\n⊢ ∀ {p : S[X]}, p ∈ lifts f → natDegree p = d → ∃ q, map f q = p ∧ degree q = degree p",
"tactic": "induction' d using Nat.strong_induction_on with n hn"
},
{
"state_after": "case h\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"state_before": "case h\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\n⊢ ∀ {p : S[X]}, p ∈ lifts f → natDegree p = n → ∃ q, map f q = p ∧ degree q = degree p",
"tactic": "intros p hlifts hdeg"
},
{
"state_after": "case pos\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : eraseLead p = 0\n⊢ ∃ q, map f q = p ∧ degree q = degree p\n\ncase neg\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"state_before": "case h\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"tactic": "by_cases erase_zero : p.eraseLead = 0"
},
{
"state_after": "case neg\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"state_before": "case neg\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"tactic": "have deg_erase := Or.resolve_right (eraseLead_natDegree_lt_or_eraseLead_eq_zero p) erase_zero"
},
{
"state_after": "case neg\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p ≠ 0\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"state_before": "case neg\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"tactic": "have pzero : p ≠ 0 := by\n intro habs\n exfalso\n rw [habs, eraseLead_zero, eq_self_iff_true, not_true] at erase_zero\n exact erase_zero"
},
{
"state_after": "case neg\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"state_before": "case neg\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p ≠ 0\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"tactic": "have lead_zero : p.coeff p.natDegree ≠ 0 := by\n rw [← leadingCoeff, Ne.def, leadingCoeff_eq_zero] ; exact pzero"
},
{
"state_after": "case neg.intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"state_before": "case neg\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"tactic": "obtain ⟨lead, hlead⟩ :=\n monomial_mem_lifts_and_degree_eq\n (monomial_mem_lifts p.natDegree ((lifts_iff_coeff_lifts p).1 hlifts p.natDegree))"
},
{
"state_after": "case neg.intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"state_before": "case neg.intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"tactic": "have deg_lead : lead.degree = p.natDegree := by\n rw [hlead.2, ← C_mul_X_pow_eq_monomial, degree_C_mul_X_pow p.natDegree lead_zero]"
},
{
"state_after": "case neg.intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"state_before": "case neg.intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"tactic": "rw [hdeg] at deg_erase"
},
{
"state_after": "case neg.intro.intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"state_before": "case neg.intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"tactic": "obtain ⟨erase, herase⟩ :=\n hn p.eraseLead.natDegree deg_erase (erase_mem_lifts p.natDegree hlifts)\n (refl p.eraseLead.natDegree)"
},
{
"state_after": "case neg.intro.intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ map f (erase + lead) = p ∧ degree (erase + lead) = degree p",
"state_before": "case neg.intro.intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"tactic": "use erase + lead"
},
{
"state_after": "case neg.intro.intro.left\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ map f (erase + lead) = p\n\ncase neg.intro.intro.right\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ degree (erase + lead) = degree p",
"state_before": "case neg.intro.intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ map f (erase + lead) = p ∧ degree (erase + lead) = degree p",
"tactic": "constructor"
},
{
"state_after": "case neg.intro.intro.right\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ degree (erase + lead) = degree lead",
"state_before": "case neg.intro.intro.right\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ degree (erase + lead) = degree p",
"tactic": "rw [degree_eq_natDegree pzero, ←deg_lead]"
},
{
"state_after": "case neg.intro.intro.right.h\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ degree erase < degree lead",
"state_before": "case neg.intro.intro.right\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ degree (erase + lead) = degree lead",
"tactic": "apply degree_add_eq_right_of_degree_lt"
},
{
"state_after": "case neg.intro.intro.right.h\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ degree (Polynomial.erase (natDegree p) p) < degree p",
"state_before": "case neg.intro.intro.right.h\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ degree erase < degree lead",
"tactic": "rw [herase.2, deg_lead, ←degree_eq_natDegree pzero]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.right.h\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ degree (Polynomial.erase (natDegree p) p) < degree p",
"tactic": "exact degree_erase_lt pzero"
},
{
"state_after": "case pos\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : eraseLead p = 0\n⊢ ∃ q,\n map f q = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree q = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))",
"state_before": "case pos\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : eraseLead p = 0\n⊢ ∃ q, map f q = p ∧ degree q = degree p",
"tactic": "rw [← eraseLead_add_monomial_natDegree_leadingCoeff p, erase_zero, zero_add, leadingCoeff]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : eraseLead p = 0\n⊢ ∃ q,\n map f q = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree q = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))",
"tactic": "exact\n monomial_mem_lifts_and_degree_eq\n (monomial_mem_lifts p.natDegree ((lifts_iff_coeff_lifts p).1 hlifts p.natDegree))"
},
{
"state_after": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\nhabs : p = 0\n⊢ False",
"state_before": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\n⊢ p ≠ 0",
"tactic": "intro habs"
},
{
"state_after": "case h\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\nhabs : p = 0\n⊢ False",
"state_before": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\nhabs : p = 0\n⊢ False",
"tactic": "exfalso"
},
{
"state_after": "case h\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : False\ndeg_erase : natDegree (eraseLead p) < natDegree p\nhabs : p = 0\n⊢ False",
"state_before": "case h\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\nhabs : p = 0\n⊢ False",
"tactic": "rw [habs, eraseLead_zero, eq_self_iff_true, not_true] at erase_zero"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : False\ndeg_erase : natDegree (eraseLead p) < natDegree p\nhabs : p = 0\n⊢ False",
"tactic": "exact erase_zero"
},
{
"state_after": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p ≠ 0\n⊢ ¬p = 0",
"state_before": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p ≠ 0\n⊢ coeff p (natDegree p) ≠ 0",
"tactic": "rw [← leadingCoeff, Ne.def, leadingCoeff_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p ≠ 0\n⊢ ¬p = 0",
"tactic": "exact pzero"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < natDegree p\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\n⊢ degree lead = ↑(natDegree p)",
"tactic": "rw [hlead.2, ← C_mul_X_pow_eq_monomial, degree_C_mul_X_pow p.natDegree lead_zero]"
},
{
"state_after": "case neg.intro.intro.left\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ Polynomial.erase (natDegree p) p + ↑(monomial (natDegree p)) (coeff p (natDegree p)) = p",
"state_before": "case neg.intro.intro.left\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ map f (erase + lead) = p",
"tactic": "simp only [hlead, herase, Polynomial.map_add]"
},
{
"state_after": "case neg.intro.intro.left\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ eraseLead p + ↑(monomial (natDegree p)) (leadingCoeff p) = p",
"state_before": "case neg.intro.intro.left\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ Polynomial.erase (natDegree p) p + ↑(monomial (natDegree p)) (coeff p (natDegree p)) = p",
"tactic": "rw [←eraseLead, ←leadingCoeff]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.left\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\nhn : ∀ (m : ℕ), m < n → ∀ {p : S[X]}, p ∈ lifts f → natDegree p = m → ∃ q, map f q = p ∧ degree q = degree p\np : S[X]\nhlifts : p ∈ lifts f\nhdeg : natDegree p = n\nerase_zero : ¬eraseLead p = 0\ndeg_erase : natDegree (eraseLead p) < n\npzero : p ≠ 0\nlead_zero : coeff p (natDegree p) ≠ 0\nlead : R[X]\nhlead :\n map f lead = ↑(monomial (natDegree p)) (coeff p (natDegree p)) ∧\n degree lead = degree (↑(monomial (natDegree p)) (coeff p (natDegree p)))\ndeg_lead : degree lead = ↑(natDegree p)\nerase : R[X]\nherase : map f erase = Polynomial.erase (natDegree p) p ∧ degree erase = degree (Polynomial.erase (natDegree p) p)\n⊢ eraseLead p + ↑(monomial (natDegree p)) (leadingCoeff p) = p",
"tactic": "rw [eraseLead_add_monomial_natDegree_leadingCoeff p]"
}
] |
[
205,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/CategoryTheory/Closed/Monoidal.lean
|
CategoryTheory.MonoidalClosed.coev_app_comp_pre_app
|
[] |
[
271,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/Data/Finset/MulAntidiagonal.lean
|
Finset.mulAntidiagonal_min_mul_min
|
[
{
"state_after": "case a.mk\nα✝ : Type ?u.14971\ninst✝¹ : OrderedCancelCommMonoid α✝\ns✝ t✝ : Set α✝\nhs✝ : Set.IsPwo s✝\nht✝ : Set.IsPwo t✝\na✝ : α✝\nu : Set α✝\nhu : Set.IsPwo u\nx : α✝ × α✝\nα : Type u_1\ninst✝ : LinearOrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\na b : α\n⊢ (a, b) ∈ mulAntidiagonal (_ : Set.IsPwo s) (_ : Set.IsPwo t) (Set.IsWf.min hs hns * Set.IsWf.min ht hnt) ↔\n (a, b) ∈ {(Set.IsWf.min hs hns, Set.IsWf.min ht hnt)}",
"state_before": "α✝ : Type ?u.14971\ninst✝¹ : OrderedCancelCommMonoid α✝\ns✝ t✝ : Set α✝\nhs✝ : Set.IsPwo s✝\nht✝ : Set.IsPwo t✝\na : α✝\nu : Set α✝\nhu : Set.IsPwo u\nx : α✝ × α✝\nα : Type u_1\ninst✝ : LinearOrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\n⊢ mulAntidiagonal (_ : Set.IsPwo s) (_ : Set.IsPwo t) (Set.IsWf.min hs hns * Set.IsWf.min ht hnt) =\n {(Set.IsWf.min hs hns, Set.IsWf.min ht hnt)}",
"tactic": "ext ⟨a, b⟩"
},
{
"state_after": "case a.mk\nα✝ : Type ?u.14971\ninst✝¹ : OrderedCancelCommMonoid α✝\ns✝ t✝ : Set α✝\nhs✝ : Set.IsPwo s✝\nht✝ : Set.IsPwo t✝\na✝ : α✝\nu : Set α✝\nhu : Set.IsPwo u\nx : α✝ × α✝\nα : Type u_1\ninst✝ : LinearOrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\na b : α\n⊢ a ∈ s ∧ b ∈ t ∧ a * b = Set.IsWf.min hs hns * Set.IsWf.min ht hnt ↔ a = Set.IsWf.min hs hns ∧ b = Set.IsWf.min ht hnt",
"state_before": "case a.mk\nα✝ : Type ?u.14971\ninst✝¹ : OrderedCancelCommMonoid α✝\ns✝ t✝ : Set α✝\nhs✝ : Set.IsPwo s✝\nht✝ : Set.IsPwo t✝\na✝ : α✝\nu : Set α✝\nhu : Set.IsPwo u\nx : α✝ × α✝\nα : Type u_1\ninst✝ : LinearOrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\na b : α\n⊢ (a, b) ∈ mulAntidiagonal (_ : Set.IsPwo s) (_ : Set.IsPwo t) (Set.IsWf.min hs hns * Set.IsWf.min ht hnt) ↔\n (a, b) ∈ {(Set.IsWf.min hs hns, Set.IsWf.min ht hnt)}",
"tactic": "simp only [mem_mulAntidiagonal, mem_singleton, Prod.ext_iff]"
},
{
"state_after": "case a.mk.mp\nα✝ : Type ?u.14971\ninst✝¹ : OrderedCancelCommMonoid α✝\ns✝ t✝ : Set α✝\nhs✝ : Set.IsPwo s✝\nht✝ : Set.IsPwo t✝\na✝ : α✝\nu : Set α✝\nhu : Set.IsPwo u\nx : α✝ × α✝\nα : Type u_1\ninst✝ : LinearOrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\na b : α\n⊢ a ∈ s ∧ b ∈ t ∧ a * b = Set.IsWf.min hs hns * Set.IsWf.min ht hnt → a = Set.IsWf.min hs hns ∧ b = Set.IsWf.min ht hnt\n\ncase a.mk.mpr\nα✝ : Type ?u.14971\ninst✝¹ : OrderedCancelCommMonoid α✝\ns✝ t✝ : Set α✝\nhs✝ : Set.IsPwo s✝\nht✝ : Set.IsPwo t✝\na✝ : α✝\nu : Set α✝\nhu : Set.IsPwo u\nx : α✝ × α✝\nα : Type u_1\ninst✝ : LinearOrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\na b : α\n⊢ a = Set.IsWf.min hs hns ∧ b = Set.IsWf.min ht hnt → a ∈ s ∧ b ∈ t ∧ a * b = Set.IsWf.min hs hns * Set.IsWf.min ht hnt",
"state_before": "case a.mk\nα✝ : Type ?u.14971\ninst✝¹ : OrderedCancelCommMonoid α✝\ns✝ t✝ : Set α✝\nhs✝ : Set.IsPwo s✝\nht✝ : Set.IsPwo t✝\na✝ : α✝\nu : Set α✝\nhu : Set.IsPwo u\nx : α✝ × α✝\nα : Type u_1\ninst✝ : LinearOrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\na b : α\n⊢ a ∈ s ∧ b ∈ t ∧ a * b = Set.IsWf.min hs hns * Set.IsWf.min ht hnt ↔ a = Set.IsWf.min hs hns ∧ b = Set.IsWf.min ht hnt",
"tactic": "constructor"
},
{
"state_after": "case a.mk.mp.intro.intro\nα✝ : Type ?u.14971\ninst✝¹ : OrderedCancelCommMonoid α✝\ns✝ t✝ : Set α✝\nhs✝ : Set.IsPwo s✝\nht✝ : Set.IsPwo t✝\na✝ : α✝\nu : Set α✝\nhu : Set.IsPwo u\nx : α✝ × α✝\nα : Type u_1\ninst✝ : LinearOrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\na b : α\nhas : a ∈ s\nhat : b ∈ t\nhst : a * b = Set.IsWf.min hs hns * Set.IsWf.min ht hnt\n⊢ a = Set.IsWf.min hs hns ∧ b = Set.IsWf.min ht hnt",
"state_before": "case a.mk.mp\nα✝ : Type ?u.14971\ninst✝¹ : OrderedCancelCommMonoid α✝\ns✝ t✝ : Set α✝\nhs✝ : Set.IsPwo s✝\nht✝ : Set.IsPwo t✝\na✝ : α✝\nu : Set α✝\nhu : Set.IsPwo u\nx : α✝ × α✝\nα : Type u_1\ninst✝ : LinearOrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\na b : α\n⊢ a ∈ s ∧ b ∈ t ∧ a * b = Set.IsWf.min hs hns * Set.IsWf.min ht hnt → a = Set.IsWf.min hs hns ∧ b = Set.IsWf.min ht hnt",
"tactic": "rintro ⟨has, hat, hst⟩"
},
{
"state_after": "case a.mk.mp.intro.intro\nα✝ : Type ?u.14971\ninst✝¹ : OrderedCancelCommMonoid α✝\ns✝ t✝ : Set α✝\nhs✝ : Set.IsPwo s✝\nht✝ : Set.IsPwo t✝\na : α✝\nu : Set α✝\nhu : Set.IsPwo u\nx : α✝ × α✝\nα : Type u_1\ninst✝ : LinearOrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\nb : α\nhat : b ∈ t\nhas : Set.IsWf.min hs hns ∈ s\nhst : Set.IsWf.min hs hns * b = Set.IsWf.min hs hns * Set.IsWf.min ht hnt\n⊢ Set.IsWf.min hs hns = Set.IsWf.min hs hns ∧ b = Set.IsWf.min ht hnt",
"state_before": "case a.mk.mp.intro.intro\nα✝ : Type ?u.14971\ninst✝¹ : OrderedCancelCommMonoid α✝\ns✝ t✝ : Set α✝\nhs✝ : Set.IsPwo s✝\nht✝ : Set.IsPwo t✝\na✝ : α✝\nu : Set α✝\nhu : Set.IsPwo u\nx : α✝ × α✝\nα : Type u_1\ninst✝ : LinearOrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\na b : α\nhas : a ∈ s\nhat : b ∈ t\nhst : a * b = Set.IsWf.min hs hns * Set.IsWf.min ht hnt\n⊢ a = Set.IsWf.min hs hns ∧ b = Set.IsWf.min ht hnt",
"tactic": "obtain rfl :=\n (hs.min_le hns has).eq_of_not_lt fun hlt =>\n (mul_lt_mul_of_lt_of_le hlt <| ht.min_le hnt hat).ne' hst"
},
{
"state_after": "no goals",
"state_before": "case a.mk.mp.intro.intro\nα✝ : Type ?u.14971\ninst✝¹ : OrderedCancelCommMonoid α✝\ns✝ t✝ : Set α✝\nhs✝ : Set.IsPwo s✝\nht✝ : Set.IsPwo t✝\na : α✝\nu : Set α✝\nhu : Set.IsPwo u\nx : α✝ × α✝\nα : Type u_1\ninst✝ : LinearOrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\nb : α\nhat : b ∈ t\nhas : Set.IsWf.min hs hns ∈ s\nhst : Set.IsWf.min hs hns * b = Set.IsWf.min hs hns * Set.IsWf.min ht hnt\n⊢ Set.IsWf.min hs hns = Set.IsWf.min hs hns ∧ b = Set.IsWf.min ht hnt",
"tactic": "exact ⟨rfl, mul_left_cancel hst⟩"
},
{
"state_after": "case a.mk.mpr.intro\nα✝ : Type ?u.14971\ninst✝¹ : OrderedCancelCommMonoid α✝\ns✝ t✝ : Set α✝\nhs✝ : Set.IsPwo s✝\nht✝ : Set.IsPwo t✝\na : α✝\nu : Set α✝\nhu : Set.IsPwo u\nx : α✝ × α✝\nα : Type u_1\ninst✝ : LinearOrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\n⊢ Set.IsWf.min hs hns ∈ s ∧\n Set.IsWf.min ht hnt ∈ t ∧ Set.IsWf.min hs hns * Set.IsWf.min ht hnt = Set.IsWf.min hs hns * Set.IsWf.min ht hnt",
"state_before": "case a.mk.mpr\nα✝ : Type ?u.14971\ninst✝¹ : OrderedCancelCommMonoid α✝\ns✝ t✝ : Set α✝\nhs✝ : Set.IsPwo s✝\nht✝ : Set.IsPwo t✝\na✝ : α✝\nu : Set α✝\nhu : Set.IsPwo u\nx : α✝ × α✝\nα : Type u_1\ninst✝ : LinearOrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\na b : α\n⊢ a = Set.IsWf.min hs hns ∧ b = Set.IsWf.min ht hnt → a ∈ s ∧ b ∈ t ∧ a * b = Set.IsWf.min hs hns * Set.IsWf.min ht hnt",
"tactic": "rintro ⟨rfl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case a.mk.mpr.intro\nα✝ : Type ?u.14971\ninst✝¹ : OrderedCancelCommMonoid α✝\ns✝ t✝ : Set α✝\nhs✝ : Set.IsPwo s✝\nht✝ : Set.IsPwo t✝\na : α✝\nu : Set α✝\nhu : Set.IsPwo u\nx : α✝ × α✝\nα : Type u_1\ninst✝ : LinearOrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\n⊢ Set.IsWf.min hs hns ∈ s ∧\n Set.IsWf.min ht hnt ∈ t ∧ Set.IsWf.min hs hns * Set.IsWf.min ht hnt = Set.IsWf.min hs hns * Set.IsWf.min ht hnt",
"tactic": "exact ⟨hs.min_mem _, ht.min_mem _, rfl⟩"
}
] |
[
130,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.measurable
|
[] |
[
146,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
11
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.some_eq_coe
|
[] |
[
152,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
9
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
Summable.even_add_odd
|
[] |
[
387,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
Std/Data/Array/Init/Lemmas.lean
|
Array.mkEmpty_eq
|
[] |
[
22,
61
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
22,
9
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.mem_Iio_iff
|
[] |
[
1195,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1194,
1
] |
Mathlib/CategoryTheory/Filtered.lean
|
CategoryTheory.IsFiltered.sup_exists
|
[
{
"state_after": "case empty\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\n⊢ ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ ∅ → f ≫ T mY = T mX\n\ncase insert\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nh' : (X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y)\nH' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nnmf : ¬h' ∈ H'\nh'' :\n ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T mY = T mX\n⊢ ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ insert h' H' →\n f ≫ T mY = T mX",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\n⊢ ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H → f ≫ T mY = T mX",
"tactic": "induction' H using Finset.induction with h' H' nmf h''"
},
{
"state_after": "case empty.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nS : C\nf : ∀ {X : C}, X ∈ O → _root_.Nonempty (X ⟶ S)\n⊢ ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ ∅ → f ≫ T mY = T mX",
"state_before": "case empty\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\n⊢ ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ ∅ → f ≫ T mY = T mX",
"tactic": "obtain ⟨S, f⟩ := sup_objs_exists O"
},
{
"state_after": "no goals",
"state_before": "case empty.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nS : C\nf : ∀ {X : C}, X ∈ O → _root_.Nonempty (X ⟶ S)\n⊢ ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ ∅ → f ≫ T mY = T mX",
"tactic": "refine' ⟨S, fun mX => (f mX).some, by rintro - - - - - ⟨⟩⟩"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nS : C\nf : ∀ {X : C}, X ∈ O → _root_.Nonempty (X ⟶ S)\n⊢ ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f_1 : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f_1 } } } } ∈ ∅ →\n f_1 ≫ (fun {X} mX => Nonempty.some (_ : _root_.Nonempty (X ⟶ S))) mY =\n (fun {X} mX => Nonempty.some (_ : _root_.Nonempty (X ⟶ S))) mX",
"tactic": "rintro - - - - - ⟨⟩"
},
{
"state_after": "case insert.mk.mk.mk.mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nh'' :\n ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T mY = T mX\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\n⊢ ∃ S T,\n ∀ {X_1 Y_1 : C} (mX_1 : X_1 ∈ O) (mY_1 : Y_1 ∈ O) {f_1 : X_1 ⟶ Y_1},\n { fst := X_1, snd := { fst := Y_1, snd := { fst := mX_1, snd := { fst := mY_1, snd := f_1 } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H' →\n f_1 ≫ T mY_1 = T mX_1",
"state_before": "case insert\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nh' : (X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y)\nH' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nnmf : ¬h' ∈ H'\nh'' :\n ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T mY = T mX\n⊢ ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ insert h' H' →\n f ≫ T mY = T mX",
"tactic": "obtain ⟨X, Y, mX, mY, f⟩ := h'"
},
{
"state_after": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\n⊢ ∃ S T,\n ∀ {X_1 Y_1 : C} (mX_1 : X_1 ∈ O) (mY_1 : Y_1 ∈ O) {f_1 : X_1 ⟶ Y_1},\n { fst := X_1, snd := { fst := Y_1, snd := { fst := mX_1, snd := { fst := mY_1, snd := f_1 } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H' →\n f_1 ≫ T mY_1 = T mX_1",
"state_before": "case insert.mk.mk.mk.mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nh'' :\n ∃ S T,\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T mY = T mX\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\n⊢ ∃ S T,\n ∀ {X_1 Y_1 : C} (mX_1 : X_1 ∈ O) (mY_1 : Y_1 ∈ O) {f_1 : X_1 ⟶ Y_1},\n { fst := X_1, snd := { fst := Y_1, snd := { fst := mX_1, snd := { fst := mY_1, snd := f_1 } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H' →\n f_1 ≫ T mY_1 = T mX_1",
"tactic": "obtain ⟨S', T', w'⟩ := h''"
},
{
"state_after": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\n⊢ ∀ {X_1 Y_1 : C} (mX_1 : X_1 ∈ O) (mY_1 : Y_1 ∈ O) {f_1 : X_1 ⟶ Y_1},\n { fst := X_1, snd := { fst := Y_1, snd := { fst := mX_1, snd := { fst := mY_1, snd := f_1 } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H' →\n f_1 ≫ (fun {X_2} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mY_1 =\n (fun {X_2} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX_1",
"state_before": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\n⊢ ∃ S T,\n ∀ {X_1 Y_1 : C} (mX_1 : X_1 ∈ O) (mY_1 : Y_1 ∈ O) {f_1 : X_1 ⟶ Y_1},\n { fst := X_1, snd := { fst := Y_1, snd := { fst := mX_1, snd := { fst := mY_1, snd := f_1 } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H' →\n f_1 ≫ T mY_1 = T mX_1",
"tactic": "refine' ⟨coeq (f ≫ T' mY) (T' mX), fun mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX), _⟩"
},
{
"state_after": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ f' ≫ (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mY' =\n (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'",
"state_before": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\n⊢ ∀ {X_1 Y_1 : C} (mX_1 : X_1 ∈ O) (mY_1 : Y_1 ∈ O) {f_1 : X_1 ⟶ Y_1},\n { fst := X_1, snd := { fst := Y_1, snd := { fst := mX_1, snd := { fst := mY_1, snd := f_1 } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H' →\n f_1 ≫ (fun {X_2} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mY_1 =\n (fun {X_2} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX_1",
"tactic": "intro X' Y' mX' mY' f' mf'"
},
{
"state_after": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ (f' ≫ T' mY') ≫ coeqHom (f ≫ T' mY) (T' mX) = (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'",
"state_before": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ f' ≫ (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mY' =\n (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'",
"tactic": "rw [← Category.assoc]"
},
{
"state_after": "case pos\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : X = X' ∧ Y = Y'\n⊢ (f' ≫ T' mY') ≫ coeqHom (f ≫ T' mY) (T' mX) = (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'\n\ncase neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : ¬(X = X' ∧ Y = Y')\n⊢ (f' ≫ T' mY') ≫ coeqHom (f ≫ T' mY) (T' mX) = (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'",
"state_before": "case insert.mk.mk.mk.mk.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ (f' ≫ T' mY') ≫ coeqHom (f ≫ T' mY) (T' mX) = (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'",
"tactic": "by_cases h : X = X' ∧ Y = Y'"
},
{
"state_after": "case pos.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ (f' ≫ T' mY') ≫ coeqHom (f ≫ T' mY) (T' mX) = (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'",
"state_before": "case pos\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : X = X' ∧ Y = Y'\n⊢ (f' ≫ T' mY') ≫ coeqHom (f ≫ T' mY) (T' mX) = (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'",
"tactic": "rcases h with ⟨rfl, rfl⟩"
},
{
"state_after": "case pos\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nhf : f = f'\n⊢ (f' ≫ T' mY') ≫ coeqHom (f ≫ T' mY) (T' mX) = (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'\n\ncase neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nhf : ¬f = f'\n⊢ (f' ≫ T' mY') ≫ coeqHom (f ≫ T' mY) (T' mX) = (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'",
"state_before": "case pos.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ (f' ≫ T' mY') ≫ coeqHom (f ≫ T' mY) (T' mX) = (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'",
"tactic": "by_cases hf : f = f'"
},
{
"state_after": "case pos\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ (f ≫ T' mY') ≫ coeqHom (f ≫ T' mY) (T' mX) = (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'",
"state_before": "case pos\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nhf : f = f'\n⊢ (f' ≫ T' mY') ≫ coeqHom (f ≫ T' mY) (T' mX) = (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'",
"tactic": "subst hf"
},
{
"state_after": "no goals",
"state_before": "case pos\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ (f ≫ T' mY') ≫ coeqHom (f ≫ T' mY) (T' mX) = (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'",
"tactic": "apply coeq_condition"
},
{
"state_after": "case neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nhf : ¬f = f'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"state_before": "case neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nhf : ¬f = f'\n⊢ (f' ≫ T' mY') ≫ coeqHom (f ≫ T' mY) (T' mX) = (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'",
"tactic": "rw [@w' _ _ mX mY f']"
},
{
"state_after": "case neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : f' = f ∨ { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"state_before": "case neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nhf : ¬f = f'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"tactic": "simp only [Finset.mem_insert, PSigma.mk.injEq, heq_eq_eq, true_and] at mf'"
},
{
"state_after": "case neg.inl\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : f' = f\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'\n\ncase neg.inr\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"state_before": "case neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : f' = f ∨ { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"tactic": "rcases mf' with mf' | mf'"
},
{
"state_after": "case neg.inr\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"state_before": "case neg.inl\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : f' = f\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'\n\ncase neg.inr\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"tactic": ". exfalso\n exact hf mf'.symm"
},
{
"state_after": "no goals",
"state_before": "case neg.inr\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"tactic": ". exact mf'"
},
{
"state_after": "case neg.inl.h\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : f' = f\n⊢ False",
"state_before": "case neg.inl\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : f' = f\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case neg.inl.h\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : f' = f\n⊢ False",
"tactic": "exact hf mf'.symm"
},
{
"state_after": "no goals",
"state_before": "case neg.inr\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmX' : X ∈ O\nmY' : Y ∈ O\nf' : X ⟶ Y\nhf : ¬f = f'\nmf' : { fst := X, snd := { fst := Y, snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'\n⊢ { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f' } } } } ∈ H'",
"tactic": "exact mf'"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : ¬(X = X' ∧ Y = Y')\n⊢ { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'",
"state_before": "case neg\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : ¬(X = X' ∧ Y = Y')\n⊢ (f' ≫ T' mY') ≫ coeqHom (f ≫ T' mY) (T' mX) = (fun {X_1} mZ => T' mZ ≫ coeqHom (f ≫ T' mY) (T' mX)) mX'",
"tactic": "rw [@w' _ _ mX' mY' f' _]"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : ¬(X = X' ∧ Y = Y')\n⊢ { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ≠\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } }",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : ¬(X = X' ∧ Y = Y')\n⊢ { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈ H'",
"tactic": "apply Finset.mem_of_mem_insert_of_ne mf'"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } =\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } }\n⊢ X = X' ∧ Y = Y'",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh : ¬(X = X' ∧ Y = Y')\n⊢ { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ≠\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } }",
"tactic": "contrapose! h"
},
{
"state_after": "case refl\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ X = X ∧ Y = Y",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nX' Y' : C\nmX' : X' ∈ O\nmY' : Y' ∈ O\nf' : X' ⟶ Y'\nmf' :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\nh :\n { fst := X', snd := { fst := Y', snd := { fst := mX', snd := { fst := mY', snd := f' } } } } =\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } }\n⊢ X = X' ∧ Y = Y'",
"tactic": "obtain ⟨rfl, h⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case refl\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFiltered C\nO : Finset C\nH H' : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\nX Y : C\nmX : X ∈ O\nmY : Y ∈ O\nf : X ⟶ Y\nnmf : ¬{ fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H'\nS' : C\nT' : {X : C} → X ∈ O → (X ⟶ S')\nw' :\n ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈ H' → f ≫ T' mY = T' mX\nmf' :\n { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } ∈\n insert { fst := X, snd := { fst := Y, snd := { fst := mX, snd := { fst := mY, snd := f } } } } H'\n⊢ X = X ∧ Y = Y",
"tactic": "trivial"
}
] |
[
255,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.inter_congr_left
|
[] |
[
1803,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1802,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.mapsTo_empty
|
[] |
[
404,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/Data/List/Pairwise.lean
|
List.Pairwise.filter
|
[] |
[
230,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/LinearAlgebra/Finrank.lean
|
FiniteDimensional.finrank_zero_of_subsingleton
|
[
{
"state_after": "K : Type u\nV : Type v\ninst✝⁶ : Ring K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : Nontrivial K\ninst✝ : NoZeroSMulDivisors K V\nh : Subsingleton V\nh0 : ¬finrank K V = 0\n⊢ False",
"state_before": "K : Type u\nV : Type v\ninst✝⁶ : Ring K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : Nontrivial K\ninst✝ : NoZeroSMulDivisors K V\nh : Subsingleton V\n⊢ finrank K V = 0",
"tactic": "by_contra h0"
},
{
"state_after": "case mk.intro.intro\nK : Type u\nV : Type v\ninst✝⁶ : Ring K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : Nontrivial K\ninst✝ : NoZeroSMulDivisors K V\nh : Subsingleton V\nh0 : ¬finrank K V = 0\nx y : V\nhxy : x ≠ y\n⊢ False",
"state_before": "K : Type u\nV : Type v\ninst✝⁶ : Ring K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : Nontrivial K\ninst✝ : NoZeroSMulDivisors K V\nh : Subsingleton V\nh0 : ¬finrank K V = 0\n⊢ False",
"tactic": "obtain ⟨x, y, hxy⟩ := nontrivial_of_finrank_pos (Nat.pos_of_ne_zero h0)"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro\nK : Type u\nV : Type v\ninst✝⁶ : Ring K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : Nontrivial K\ninst✝ : NoZeroSMulDivisors K V\nh : Subsingleton V\nh0 : ¬finrank K V = 0\nx y : V\nhxy : x ≠ y\n⊢ False",
"tactic": "exact hxy (Subsingleton.elim _ _)"
}
] |
[
116,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/GroupTheory/Subsemigroup/Basic.lean
|
Subsemigroup.closure_eq
|
[] |
[
416,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
415,
1
] |
Mathlib/Data/Multiset/Sum.lean
|
Multiset.disjSum_lt_disjSum_of_le_of_lt
|
[] |
[
95,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/RingTheory/PrincipalIdealDomain.lean
|
Ideal.IsPrincipal.of_comap
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\nS : Type u_1\nN : Type ?u.185417\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Ring S\ninst✝¹ : Module R M\ninst✝ : Module R N\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\nhI : IsPrincipal (comap f I)\n⊢ I = Submodule.span S {↑f (IsPrincipal.generator (comap f I))}",
"tactic": "rw [Ideal.submodule_span_eq, ← Set.image_singleton, ← Ideal.map_span,\n Ideal.span_singleton_generator, Ideal.map_comap_of_surjective f hf]"
}
] |
[
339,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Topology/Sheaves/Sheafify.lean
|
TopCat.Presheaf.stalkToFiber_surjective
|
[
{
"state_after": "case w\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\n⊢ ∀ (t : stalk F x), ∃ U f x_1, f { val := x, property := (_ : x ∈ U.obj) } = t",
"state_before": "X : TopCat\nF : Presheaf (Type v) X\nx : ↑X\n⊢ Function.Surjective (stalkToFiber F x)",
"tactic": "apply TopCat.stalkToFiber_surjective"
},
{
"state_after": "case w\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nt : stalk F x\n⊢ ∃ U f x_1, f { val := x, property := (_ : x ∈ U.obj) } = t",
"state_before": "case w\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\n⊢ ∀ (t : stalk F x), ∃ U f x_1, f { val := x, property := (_ : x ∈ U.obj) } = t",
"tactic": "intro t"
},
{
"state_after": "case w.intro.intro.intro\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nU : Opens ↑X\nm : x ∈ U\ns : (CategoryTheory.forget (Type v)).obj (F.obj U.op)\n⊢ ∃ U_1 f x_1,\n f { val := x, property := (_ : x ∈ U_1.obj) } =\n (CategoryTheory.forget (Type v)).map (germ F { val := x, property := m }) s",
"state_before": "case w\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nt : stalk F x\n⊢ ∃ U f x_1, f { val := x, property := (_ : x ∈ U.obj) } = t",
"tactic": "obtain ⟨U, m, s, rfl⟩ := F.germ_exist _ t"
},
{
"state_after": "case w.intro.intro.intro\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nU : Opens ↑X\nm : x ∈ U\ns : (CategoryTheory.forget (Type v)).obj (F.obj U.op)\n⊢ ∃ f x_1,\n f { val := x, property := (_ : x ∈ { obj := U, property := m }.obj) } =\n (CategoryTheory.forget (Type v)).map (germ F { val := x, property := m }) s",
"state_before": "case w.intro.intro.intro\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nU : Opens ↑X\nm : x ∈ U\ns : (CategoryTheory.forget (Type v)).obj (F.obj U.op)\n⊢ ∃ U_1 f x_1,\n f { val := x, property := (_ : x ∈ U_1.obj) } =\n (CategoryTheory.forget (Type v)).map (germ F { val := x, property := m }) s",
"tactic": "use ⟨U, m⟩"
},
{
"state_after": "case w.intro.intro.intro.w\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nU : Opens ↑X\nm : x ∈ U\ns : (CategoryTheory.forget (Type v)).obj (F.obj U.op)\n⊢ (y : { x_1 // x_1 ∈ { obj := U, property := m }.obj }) → stalk F ↑y\n\ncase w.intro.intro.intro.h\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nU : Opens ↑X\nm : x ∈ U\ns : (CategoryTheory.forget (Type v)).obj (F.obj U.op)\n⊢ ∃ x_1,\n ?w.intro.intro.intro.w { val := x, property := (_ : x ∈ { obj := U, property := m }.obj) } =\n (CategoryTheory.forget (Type v)).map (germ F { val := x, property := m }) s",
"state_before": "case w.intro.intro.intro\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nU : Opens ↑X\nm : x ∈ U\ns : (CategoryTheory.forget (Type v)).obj (F.obj U.op)\n⊢ ∃ f x_1,\n f { val := x, property := (_ : x ∈ { obj := U, property := m }.obj) } =\n (CategoryTheory.forget (Type v)).map (germ F { val := x, property := m }) s",
"tactic": "fconstructor"
},
{
"state_after": "no goals",
"state_before": "case w.intro.intro.intro.w\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nU : Opens ↑X\nm : x ∈ U\ns : (CategoryTheory.forget (Type v)).obj (F.obj U.op)\n⊢ (y : { x_1 // x_1 ∈ { obj := U, property := m }.obj }) → stalk F ↑y",
"tactic": "exact fun y => F.germ y s"
},
{
"state_after": "no goals",
"state_before": "case w.intro.intro.intro.h\nX : TopCat\nF : Presheaf (Type v) X\nx : ↑X\nU : Opens ↑X\nm : x ∈ U\ns : (CategoryTheory.forget (Type v)).obj (F.obj U.op)\n⊢ ∃ x_1,\n germ F { val := x, property := (_ : x ∈ { obj := U, property := m }.obj) } s =\n (CategoryTheory.forget (Type v)).map (germ F { val := x, property := m }) s",
"tactic": "exact ⟨PrelocalPredicate.sheafifyOf ⟨s, fun _ => rfl⟩, rfl⟩"
}
] |
[
104,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
expSeries_div_hasSum_exp_of_mem_ball
|
[
{
"state_after": "𝕂 : Type u_2\n𝔸 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedDivisionRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ HasSum (fun n => ↑(expSeries 𝕂 𝔸 n) fun x_1 => x) (exp 𝕂 x)",
"state_before": "𝕂 : Type u_2\n𝔸 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedDivisionRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ HasSum (fun n => x ^ n / ↑n !) (exp 𝕂 x)",
"tactic": "rw [← expSeries_apply_eq_div' (𝕂 := 𝕂) x]"
},
{
"state_after": "no goals",
"state_before": "𝕂 : Type u_2\n𝔸 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedDivisionRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nx : 𝔸\nhx : x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\n⊢ HasSum (fun n => ↑(expSeries 𝕂 𝔸 n) fun x_1 => x) (exp 𝕂 x)",
"tactic": "exact expSeries_hasSum_exp_of_mem_ball x hx"
}
] |
[
354,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.isLocalization_iff_of_ringEquiv
|
[] |
[
775,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
772,
1
] |
Mathlib/Data/QPF/Univariate/Basic.lean
|
Qpf.Cofix.dest_corec
|
[
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : α → F α\nx : α\n⊢ Quot.lift (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x))\n (_ :\n ∀ (x y : PFunctor.M (P F)),\n Mcongr x y →\n (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) x =\n (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) y)\n (Quot.mk Mcongr (corecF g x)) =\n corec g <$> g x",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : α → F α\nx : α\n⊢ dest (corec g x) = corec g <$> g x",
"tactic": "conv =>\n lhs\n rw [Cofix.dest, Cofix.corec];"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : α → F α\nx : α\n⊢ Quot.mk Mcongr <$> abs (PFunctor.M.dest (corecF g x)) = corec g <$> g x",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : α → F α\nx : α\n⊢ Quot.lift (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x))\n (_ :\n ∀ (x y : PFunctor.M (P F)),\n Mcongr x y →\n (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) x =\n (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) y)\n (Quot.mk Mcongr (corecF g x)) =\n corec g <$> g x",
"tactic": "dsimp"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : α → F α\nx : α\n⊢ (Quot.mk Mcongr ∘ corecF g) <$> g x = corec g <$> g x",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : α → F α\nx : α\n⊢ Quot.mk Mcongr <$> abs (PFunctor.M.dest (corecF g x)) = corec g <$> g x",
"tactic": "rw [corecF_eq, abs_map, abs_repr, ← comp_map]"
},
{
"state_after": "no goals",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : α → F α\nx : α\n⊢ (Quot.mk Mcongr ∘ corecF g) <$> g x = corec g <$> g x",
"tactic": "rfl"
}
] |
[
431,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
425,
1
] |
Mathlib/Data/Nat/Squarefree.lean
|
Nat.squarefree_iff_nodup_factors
|
[
{
"state_after": "n : ℕ\nh0 : n ≠ 0\n⊢ Multiset.Nodup ↑(factors n) ↔ List.Nodup (factors n)",
"state_before": "n : ℕ\nh0 : n ≠ 0\n⊢ Squarefree n ↔ List.Nodup (factors n)",
"tactic": "rw [UniqueFactorizationMonoid.squarefree_iff_nodup_normalizedFactors h0, Nat.factors_eq]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nh0 : n ≠ 0\n⊢ Multiset.Nodup ↑(factors n) ↔ List.Nodup (factors n)",
"tactic": "simp"
}
] |
[
34,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
32,
1
] |
Mathlib/RepresentationTheory/Action.lean
|
Action.comp_hom
|
[] |
[
160,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Algebra/Algebra/Basic.lean
|
Algebra.smul_def
|
[] |
[
353,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.X_pow_order_dvd
|
[
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : (order φ).Dom\n⊢ φ = X ^ Part.get (order φ) h * mk fun n => ↑(coeff R (n + Part.get (order φ) h)) φ",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : (order φ).Dom\n⊢ X ^ Part.get (order φ) h ∣ φ",
"tactic": "refine' ⟨PowerSeries.mk fun n => coeff R (n + (order φ).get h) φ, _⟩"
},
{
"state_after": "case h\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : (order φ).Dom\nn : ℕ\n⊢ ↑(coeff R n) φ = ↑(coeff R n) (X ^ Part.get (order φ) h * mk fun n => ↑(coeff R (n + Part.get (order φ) h)) φ)",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : (order φ).Dom\n⊢ φ = X ^ Part.get (order φ) h * mk fun n => ↑(coeff R (n + Part.get (order φ) h)) φ",
"tactic": "ext n"
},
{
"state_after": "case h\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : (order φ).Dom\nn : ℕ\n⊢ ↑(coeff R n) φ =\n ∑ x in if Part.get (order φ) h ≤ n then {(Part.get (order φ) h, n - Part.get (order φ) h)} else ∅,\n ↑(coeff R (x.snd + Part.get (order φ) h)) φ",
"state_before": "case h\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : (order φ).Dom\nn : ℕ\n⊢ ↑(coeff R n) φ = ↑(coeff R n) (X ^ Part.get (order φ) h * mk fun n => ↑(coeff R (n + Part.get (order φ) h)) φ)",
"tactic": "simp only [coeff_mul, coeff_X_pow, coeff_mk, boole_mul, Finset.sum_ite,\n Finset.Nat.filter_fst_eq_antidiagonal, Finset.sum_const_zero, add_zero]"
},
{
"state_after": "case h.inl\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : (order φ).Dom\nn : ℕ\nhn : Part.get (order φ) h ≤ n\n⊢ ↑(coeff R n) φ =\n ∑ x in {(Part.get (order φ) h, n - Part.get (order φ) h)}, ↑(coeff R (x.snd + Part.get (order φ) h)) φ\n\ncase h.inr\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : (order φ).Dom\nn : ℕ\nhn : ¬Part.get (order φ) h ≤ n\n⊢ ↑(coeff R n) φ = ∑ x in ∅, ↑(coeff R (x.snd + Part.get (order φ) h)) φ",
"state_before": "case h\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : (order φ).Dom\nn : ℕ\n⊢ ↑(coeff R n) φ =\n ∑ x in if Part.get (order φ) h ≤ n then {(Part.get (order φ) h, n - Part.get (order φ) h)} else ∅,\n ↑(coeff R (x.snd + Part.get (order φ) h)) φ",
"tactic": "split_ifs with hn"
},
{
"state_after": "no goals",
"state_before": "case h.inl\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : (order φ).Dom\nn : ℕ\nhn : Part.get (order φ) h ≤ n\n⊢ ↑(coeff R n) φ =\n ∑ x in {(Part.get (order φ) h, n - Part.get (order φ) h)}, ↑(coeff R (x.snd + Part.get (order φ) h)) φ",
"tactic": "simp [tsub_add_cancel_of_le hn]"
},
{
"state_after": "case h.inr\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : (order φ).Dom\nn : ℕ\nhn : ¬Part.get (order φ) h ≤ n\n⊢ ↑(coeff R n) φ = 0",
"state_before": "case h.inr\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : (order φ).Dom\nn : ℕ\nhn : ¬Part.get (order φ) h ≤ n\n⊢ ↑(coeff R n) φ = ∑ x in ∅, ↑(coeff R (x.snd + Part.get (order φ) h)) φ",
"tactic": "simp only [Finset.sum_empty]"
},
{
"state_after": "case h.inr\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : (order φ).Dom\nn : ℕ\nhn : ¬Part.get (order φ) h ≤ n\n⊢ ↑n < order φ",
"state_before": "case h.inr\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : (order φ).Dom\nn : ℕ\nhn : ¬Part.get (order φ) h ≤ n\n⊢ ↑(coeff R n) φ = 0",
"tactic": "refine' coeff_of_lt_order _ _"
},
{
"state_after": "no goals",
"state_before": "case h.inr\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nh : (order φ).Dom\nn : ℕ\nhn : ¬Part.get (order φ) h ≤ n\n⊢ ↑n < order φ",
"tactic": "simpa [PartENat.coe_lt_iff] using fun _ => hn"
}
] |
[
2450,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2441,
1
] |
Mathlib/CategoryTheory/Subobject/MonoOver.lean
|
CategoryTheory.MonoOver.map_obj_left
|
[] |
[
270,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/Order/Cover.lean
|
Wcovby.rfl
|
[] |
[
58,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/RingTheory/Localization/Submodule.lean
|
IsLocalization.coeSubmodule_injective
|
[] |
[
127,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/MeasureTheory/Group/Integration.lean
|
MeasureTheory.integral_smul_eq_self
|
[
{
"state_after": "𝕜 : Type ?u.84709\nM : Type ?u.84712\nα : Type u_1\nG : Type u_2\nE : Type u_3\nF : Type ?u.84724\ninst✝⁹ : MeasurableSpace G\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\ninst✝⁵ : NormedAddCommGroup F\nμ✝ : Measure G\nf✝ : G → E\ng✝ : G\ninst✝⁴ : Group G\ninst✝³ : MeasurableSpace α\ninst✝² : MulAction G α\ninst✝¹ : MeasurableSMul G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nf : α → E\ng : G\nh : MeasurableEmbedding fun x => g • x\n⊢ (∫ (x : α), f (g • x) ∂μ) = ∫ (x : α), f x ∂μ",
"state_before": "𝕜 : Type ?u.84709\nM : Type ?u.84712\nα : Type u_1\nG : Type u_2\nE : Type u_3\nF : Type ?u.84724\ninst✝⁹ : MeasurableSpace G\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\ninst✝⁵ : NormedAddCommGroup F\nμ✝ : Measure G\nf✝ : G → E\ng✝ : G\ninst✝⁴ : Group G\ninst✝³ : MeasurableSpace α\ninst✝² : MulAction G α\ninst✝¹ : MeasurableSMul G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nf : α → E\ng : G\n⊢ (∫ (x : α), f (g • x) ∂μ) = ∫ (x : α), f x ∂μ",
"tactic": "have h : MeasurableEmbedding fun x : α => g • x := (MeasurableEquiv.smul g).measurableEmbedding"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.84709\nM : Type ?u.84712\nα : Type u_1\nG : Type u_2\nE : Type u_3\nF : Type ?u.84724\ninst✝⁹ : MeasurableSpace G\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\ninst✝⁵ : NormedAddCommGroup F\nμ✝ : Measure G\nf✝ : G → E\ng✝ : G\ninst✝⁴ : Group G\ninst✝³ : MeasurableSpace α\ninst✝² : MulAction G α\ninst✝¹ : MeasurableSMul G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nf : α → E\ng : G\nh : MeasurableEmbedding fun x => g • x\n⊢ (∫ (x : α), f (g • x) ∂μ) = ∫ (x : α), f x ∂μ",
"tactic": "rw [← h.integral_map, map_smul]"
}
] |
[
208,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean
|
IsCoercive.unique_continuousLinearEquivOfBilin
|
[] |
[
128,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Algebra/Algebra/Basic.lean
|
Algebra.algebraMap_ofSubring_apply
|
[] |
[
547,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
545,
1
] |
Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean
|
TrivSqZeroExt.fst_exp
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\nR : Type u_1\nM : Type u_2\ninst✝¹⁵ : IsROrC 𝕜\ninst✝¹⁴ : NormedCommRing R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : NormedAlgebra 𝕜 R\ninst✝¹¹ : Module R M\ninst✝¹⁰ : Module Rᵐᵒᵖ M\ninst✝⁹ : IsCentralScalar R M\ninst✝⁸ : Module 𝕜 M\ninst✝⁷ : IsScalarTower 𝕜 R M\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : TopologicalRing R\ninst✝⁴ : TopologicalAddGroup M\ninst✝³ : ContinuousSMul R M\ninst✝² : CompleteSpace R\ninst✝¹ : T2Space R\ninst✝ : T2Space M\nx : tsze R M\n⊢ fst (exp 𝕜 x) = exp 𝕜 (fst x)",
"tactic": "rw [exp_def, fst_add, fst_inl, fst_inr, add_zero]"
}
] |
[
153,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Data/Finset/Fold.lean
|
Finset.fold_max_add
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.73169\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ninst✝² : LinearOrder β\nc : β\ninst✝¹ : Add β\ninst✝ : CovariantClass β β (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nn : WithBot β\ns : Finset α\n⊢ fold max ⊥ (fun x => ↑(f x) + n) s = fold max ⊥ (WithBot.some ∘ f) s + n",
"tactic": "classical\n induction' s using Finset.induction_on with a s _ ih <;> simp [*, max_add_add_right]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.73169\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ninst✝² : LinearOrder β\nc : β\ninst✝¹ : Add β\ninst✝ : CovariantClass β β (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nn : WithBot β\ns : Finset α\n⊢ fold max ⊥ (fun x => ↑(f x) + n) s = fold max ⊥ (WithBot.some ∘ f) s + n",
"tactic": "induction' s using Finset.induction_on with a s _ ih <;> simp [*, max_add_add_right]"
}
] |
[
269,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.affineCombination_subtype_eq_filter
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.365709\ns₂ : Finset ι₂\nw : ι → k\np : ι → P\npred : ι → Prop\ninst✝ : DecidablePred pred\n⊢ (↑(affineCombination k (Finset.subtype pred s) fun i => p ↑i) fun i => w ↑i) =\n ↑(affineCombination k (filter pred s) p) w",
"tactic": "rw [affineCombination_apply, affineCombination_apply, weightedVSubOfPoint_subtype_eq_filter]"
}
] |
[
563,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
559,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
gcd_dvd_gcd_mul_left
|
[] |
[
484,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/MeasureTheory/Function/UniformIntegrable.lean
|
MeasureTheory.unifIntegrable_const
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf g : α → β\nhp : 1 ≤ p\nhp_ne_top : p ≠ ⊤\nhg : Memℒp g p\nε : ℝ\nhε : 0 < ε\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((fun x => g) i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf g : α → β\nhp : 1 ≤ p\nhp_ne_top : p ≠ ⊤\nhg : Memℒp g p\n⊢ UnifIntegrable (fun x => g) p μ",
"tactic": "intro ε hε"
},
{
"state_after": "case intro.intro\nα : Type u_2\nβ : Type u_1\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf g : α → β\nhp : 1 ≤ p\nhp_ne_top : p ≠ ⊤\nhg : Memℒp g p\nε : ℝ\nhε : 0 < ε\nδ : ℝ\nhδ_pos : 0 < δ\nhgδ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s g) p μ ≤ ENNReal.ofReal ε\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((fun x => g) i)) p μ ≤ ENNReal.ofReal ε",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf g : α → β\nhp : 1 ≤ p\nhp_ne_top : p ≠ ⊤\nhg : Memℒp g p\nε : ℝ\nhε : 0 < ε\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((fun x => g) i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "obtain ⟨δ, hδ_pos, hgδ⟩ := hg.snorm_indicator_le μ hp hp_ne_top hε"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type u_1\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf g : α → β\nhp : 1 ≤ p\nhp_ne_top : p ≠ ⊤\nhg : Memℒp g p\nε : ℝ\nhε : 0 < ε\nδ : ℝ\nhδ_pos : 0 < δ\nhgδ : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s g) p μ ≤ ENNReal.ofReal ε\n⊢ ∃ δ x,\n ∀ (i : ι) (s : Set α),\n MeasurableSet s → ↑↑μ s ≤ ENNReal.ofReal δ → snorm (Set.indicator s ((fun x => g) i)) p μ ≤ ENNReal.ofReal ε",
"tactic": "exact ⟨δ, hδ_pos, fun _ => hgδ⟩"
}
] |
[
409,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/Combinatorics/Pigeonhole.lean
|
Fintype.exists_le_sum_fiber_of_nsmul_le_sum
|
[] |
[
350,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
346,
1
] |
Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean
|
Real.measurable_arcsin
|
[] |
[
64,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean
|
Real.logb_inv
|
[
{
"state_after": "no goals",
"state_before": "b x✝ y x : ℝ\n⊢ logb b x⁻¹ = -logb b x",
"tactic": "simp [logb, neg_div]"
}
] |
[
77,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Algebra/Order/Field/Power.lean
|
zpow_two_nonneg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na✝ b c d : α\nn : ℤ\na : α\n⊢ 0 ≤ a ^ 2",
"tactic": "convert zpow_bit0_nonneg a 1"
}
] |
[
134,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.coe_zero
|
[] |
[
270,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.trim_trim
|
[] |
[
1667,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1666,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
FiniteDimensional.finrank_pos
|
[] |
[
337,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
IsCompl.compl_eq
|
[] |
[
860,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
859,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean
|
CauchySeq.comp_injective
|
[] |
[
209,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Data/List/Basic.lean
|
List.mem_pure
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.23772\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u_1\nx y : α\n⊢ x ∈ [y] ↔ x = y",
"tactic": "simp"
}
] |
[
525,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
524,
1
] |
Mathlib/Data/Nat/Order/Lemmas.lean
|
Nat.mul_div_eq_iff_dvd
|
[
{
"state_after": "no goals",
"state_before": "a b m n✝ k n d : ℕ\n⊢ d * (n / d) = n ↔ d * (n / d) = d * (n / d) + n % d",
"tactic": "rw [div_add_mod]"
},
{
"state_after": "no goals",
"state_before": "a b m n✝ k n d : ℕ\n⊢ d * (n / d) = d * (n / d) + n % d ↔ d ∣ n",
"tactic": "rw [self_eq_add_right, dvd_iff_mod_eq_zero]"
}
] |
[
65,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/RingTheory/AlgebraicIndependent.lean
|
AlgebraicIndependent.aeval_comp_mvPolynomialOptionEquivPolynomialAdjoin
|
[
{
"state_after": "case refine'_1\nι : Type u_1\nι' : Type ?u.1231476\nR : Type u_2\nK : Type ?u.1231482\nA : Type u_3\nA' : Type ?u.1231488\nA'' : Type ?u.1231491\nV : Type u\nV' : Type ?u.1231496\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing A'\ninst✝³ : CommRing A''\ninst✝² : Algebra R A\ninst✝¹ : Algebra R A'\ninst✝ : Algebra R A''\na✝ b : R\nhx : AlgebraicIndependent R x\na : A\n⊢ ∀ (r : R),\n ↑(Polynomial.aeval a) (↑(mvPolynomialOptionEquivPolynomialAdjoin hx) (↑C r)) =\n ↑(aeval fun o => Option.elim o a x) (↑C r)\n\ncase refine'_2\nι : Type u_1\nι' : Type ?u.1231476\nR : Type u_2\nK : Type ?u.1231482\nA : Type u_3\nA' : Type ?u.1231488\nA'' : Type ?u.1231491\nV : Type u\nV' : Type ?u.1231496\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing A'\ninst✝³ : CommRing A''\ninst✝² : Algebra R A\ninst✝¹ : Algebra R A'\ninst✝ : Algebra R A''\na✝ b : R\nhx : AlgebraicIndependent R x\na : A\n⊢ ∀ (i : Option ι),\n ↑(Polynomial.aeval a) (↑(mvPolynomialOptionEquivPolynomialAdjoin hx) (X i)) =\n ↑(aeval fun o => Option.elim o a x) (X i)",
"state_before": "ι : Type u_1\nι' : Type ?u.1231476\nR : Type u_2\nK : Type ?u.1231482\nA : Type u_3\nA' : Type ?u.1231488\nA'' : Type ?u.1231491\nV : Type u\nV' : Type ?u.1231496\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing A'\ninst✝³ : CommRing A''\ninst✝² : Algebra R A\ninst✝¹ : Algebra R A'\ninst✝ : Algebra R A''\na✝ b : R\nhx : AlgebraicIndependent R x\na : A\n⊢ RingHom.comp (↑(Polynomial.aeval a)) (RingEquiv.toRingHom (mvPolynomialOptionEquivPolynomialAdjoin hx)) =\n ↑(aeval fun o => Option.elim o a x)",
"tactic": "refine' MvPolynomial.ringHom_ext _ _ <;>\n simp only [RingHom.comp_apply, RingEquiv.toRingHom_eq_coe, RingEquiv.coe_toRingHom,\n AlgHom.coe_toRingHom, AlgHom.coe_toRingHom]"
},
{
"state_after": "case refine'_1\nι : Type u_1\nι' : Type ?u.1231476\nR : Type u_2\nK : Type ?u.1231482\nA : Type u_3\nA' : Type ?u.1231488\nA'' : Type ?u.1231491\nV : Type u\nV' : Type ?u.1231496\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing A'\ninst✝³ : CommRing A''\ninst✝² : Algebra R A\ninst✝¹ : Algebra R A'\ninst✝ : Algebra R A''\na✝ b : R\nhx : AlgebraicIndependent R x\na : A\nr : R\n⊢ ↑(Polynomial.aeval a) (↑(mvPolynomialOptionEquivPolynomialAdjoin hx) (↑C r)) =\n ↑(aeval fun o => Option.elim o a x) (↑C r)",
"state_before": "case refine'_1\nι : Type u_1\nι' : Type ?u.1231476\nR : Type u_2\nK : Type ?u.1231482\nA : Type u_3\nA' : Type ?u.1231488\nA'' : Type ?u.1231491\nV : Type u\nV' : Type ?u.1231496\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing A'\ninst✝³ : CommRing A''\ninst✝² : Algebra R A\ninst✝¹ : Algebra R A'\ninst✝ : Algebra R A''\na✝ b : R\nhx : AlgebraicIndependent R x\na : A\n⊢ ∀ (r : R),\n ↑(Polynomial.aeval a) (↑(mvPolynomialOptionEquivPolynomialAdjoin hx) (↑C r)) =\n ↑(aeval fun o => Option.elim o a x) (↑C r)",
"tactic": "intro r"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nι : Type u_1\nι' : Type ?u.1231476\nR : Type u_2\nK : Type ?u.1231482\nA : Type u_3\nA' : Type ?u.1231488\nA'' : Type ?u.1231491\nV : Type u\nV' : Type ?u.1231496\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing A'\ninst✝³ : CommRing A''\ninst✝² : Algebra R A\ninst✝¹ : Algebra R A'\ninst✝ : Algebra R A''\na✝ b : R\nhx : AlgebraicIndependent R x\na : A\nr : R\n⊢ ↑(Polynomial.aeval a) (↑(mvPolynomialOptionEquivPolynomialAdjoin hx) (↑C r)) =\n ↑(aeval fun o => Option.elim o a x) (↑C r)",
"tactic": "rw [hx.mvPolynomialOptionEquivPolynomialAdjoin_C, aeval_C, Polynomial.aeval_C,\n IsScalarTower.algebraMap_apply R (adjoin R (range x)) A]"
},
{
"state_after": "case refine'_2.none\nι : Type u_1\nι' : Type ?u.1231476\nR : Type u_2\nK : Type ?u.1231482\nA : Type u_3\nA' : Type ?u.1231488\nA'' : Type ?u.1231491\nV : Type u\nV' : Type ?u.1231496\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing A'\ninst✝³ : CommRing A''\ninst✝² : Algebra R A\ninst✝¹ : Algebra R A'\ninst✝ : Algebra R A''\na✝ b : R\nhx : AlgebraicIndependent R x\na : A\n⊢ ↑(Polynomial.aeval a) (↑(mvPolynomialOptionEquivPolynomialAdjoin hx) (X none)) =\n ↑(aeval fun o => Option.elim o a x) (X none)\n\ncase refine'_2.some\nι : Type u_1\nι' : Type ?u.1231476\nR : Type u_2\nK : Type ?u.1231482\nA : Type u_3\nA' : Type ?u.1231488\nA'' : Type ?u.1231491\nV : Type u\nV' : Type ?u.1231496\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing A'\ninst✝³ : CommRing A''\ninst✝² : Algebra R A\ninst✝¹ : Algebra R A'\ninst✝ : Algebra R A''\na✝ b : R\nhx : AlgebraicIndependent R x\na : A\ni : ι\n⊢ ↑(Polynomial.aeval a) (↑(mvPolynomialOptionEquivPolynomialAdjoin hx) (X (some i))) =\n ↑(aeval fun o => Option.elim o a x) (X (some i))",
"state_before": "case refine'_2\nι : Type u_1\nι' : Type ?u.1231476\nR : Type u_2\nK : Type ?u.1231482\nA : Type u_3\nA' : Type ?u.1231488\nA'' : Type ?u.1231491\nV : Type u\nV' : Type ?u.1231496\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing A'\ninst✝³ : CommRing A''\ninst✝² : Algebra R A\ninst✝¹ : Algebra R A'\ninst✝ : Algebra R A''\na✝ b : R\nhx : AlgebraicIndependent R x\na : A\n⊢ ∀ (i : Option ι),\n ↑(Polynomial.aeval a) (↑(mvPolynomialOptionEquivPolynomialAdjoin hx) (X i)) =\n ↑(aeval fun o => Option.elim o a x) (X i)",
"tactic": "rintro (⟨⟩ | ⟨i⟩)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.none\nι : Type u_1\nι' : Type ?u.1231476\nR : Type u_2\nK : Type ?u.1231482\nA : Type u_3\nA' : Type ?u.1231488\nA'' : Type ?u.1231491\nV : Type u\nV' : Type ?u.1231496\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing A'\ninst✝³ : CommRing A''\ninst✝² : Algebra R A\ninst✝¹ : Algebra R A'\ninst✝ : Algebra R A''\na✝ b : R\nhx : AlgebraicIndependent R x\na : A\n⊢ ↑(Polynomial.aeval a) (↑(mvPolynomialOptionEquivPolynomialAdjoin hx) (X none)) =\n ↑(aeval fun o => Option.elim o a x) (X none)",
"tactic": "rw [hx.mvPolynomialOptionEquivPolynomialAdjoin_X_none, aeval_X, Polynomial.aeval_X,\n Option.elim]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.some\nι : Type u_1\nι' : Type ?u.1231476\nR : Type u_2\nK : Type ?u.1231482\nA : Type u_3\nA' : Type ?u.1231488\nA'' : Type ?u.1231491\nV : Type u\nV' : Type ?u.1231496\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing A'\ninst✝³ : CommRing A''\ninst✝² : Algebra R A\ninst✝¹ : Algebra R A'\ninst✝ : Algebra R A''\na✝ b : R\nhx : AlgebraicIndependent R x\na : A\ni : ι\n⊢ ↑(Polynomial.aeval a) (↑(mvPolynomialOptionEquivPolynomialAdjoin hx) (X (some i))) =\n ↑(aeval fun o => Option.elim o a x) (X (some i))",
"tactic": "rw [hx.mvPolynomialOptionEquivPolynomialAdjoin_X_some, Polynomial.aeval_C,\n hx.algebraMap_aevalEquiv, aeval_X, aeval_X, Option.elim]"
}
] |
[
487,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
470,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.mul_le_mul_of_nonneg_left
|
[
{
"state_after": "no goals",
"state_before": "a b c : Int\nh₁ : a ≤ b\nh₂ : 0 ≤ c\n⊢ c * a ≤ c * b",
"tactic": "if hba : b ≤ a then rw [Int.le_antisymm hba h₁]; apply Int.le_refl else\nif hc0 : c ≤ 0 then simp [Int.le_antisymm hc0 h₂, Int.zero_mul] else\nexact Int.le_of_lt <| Int.mul_lt_mul_of_pos_left\n (Int.lt_iff_le_not_le.2 ⟨h₁, hba⟩) (Int.lt_iff_le_not_le.2 ⟨h₂, hc0⟩)"
},
{
"state_after": "a b c : Int\nh₁ : a ≤ b\nh₂ : 0 ≤ c\nhba : b ≤ a\n⊢ c * a ≤ c * a",
"state_before": "a b c : Int\nh₁ : a ≤ b\nh₂ : 0 ≤ c\nhba : b ≤ a\n⊢ c * a ≤ c * b",
"tactic": "rw [Int.le_antisymm hba h₁]"
},
{
"state_after": "no goals",
"state_before": "a b c : Int\nh₁ : a ≤ b\nh₂ : 0 ≤ c\nhba : b ≤ a\n⊢ c * a ≤ c * a",
"tactic": "apply Int.le_refl"
},
{
"state_after": "no goals",
"state_before": "a b c : Int\nh₁ : a ≤ b\nh₂ : 0 ≤ c\nhba : ¬b ≤ a\n⊢ c * a ≤ c * b",
"tactic": "if hc0 : c ≤ 0 then simp [Int.le_antisymm hc0 h₂, Int.zero_mul] else\nexact Int.le_of_lt <| Int.mul_lt_mul_of_pos_left\n (Int.lt_iff_le_not_le.2 ⟨h₁, hba⟩) (Int.lt_iff_le_not_le.2 ⟨h₂, hc0⟩)"
},
{
"state_after": "no goals",
"state_before": "a b c : Int\nh₁ : a ≤ b\nh₂ : 0 ≤ c\nhba : ¬b ≤ a\nhc0 : c ≤ 0\n⊢ c * a ≤ c * b",
"tactic": "simp [Int.le_antisymm hc0 h₂, Int.zero_mul]"
},
{
"state_after": "no goals",
"state_before": "a b c : Int\nh₁ : a ≤ b\nh₂ : 0 ≤ c\nhba : ¬b ≤ a\nhc0 : ¬c ≤ 0\n⊢ c * a ≤ c * b",
"tactic": "exact Int.le_of_lt <| Int.mul_lt_mul_of_pos_left\n (Int.lt_iff_le_not_le.2 ⟨h₁, hba⟩) (Int.lt_iff_le_not_le.2 ⟨h₂, hc0⟩)"
}
] |
[
1168,
74
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1163,
11
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.map_denom_ne_zero
|
[] |
[
1342,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1340,
1
] |
Mathlib/NumberTheory/LucasLehmer.lean
|
lucas_lehmer_sufficiency
|
[
{
"state_after": "p : ℕ\nw : 1 < p\np' : ℕ := p - 2\n⊢ LucasLehmerTest p → Nat.Prime (mersenne p)",
"state_before": "p : ℕ\nw : 1 < p\n⊢ LucasLehmerTest p → Nat.Prime (mersenne p)",
"tactic": "let p' := p - 2"
},
{
"state_after": "p : ℕ\nw : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\n⊢ LucasLehmerTest p → Nat.Prime (mersenne p)",
"state_before": "p : ℕ\nw : 1 < p\np' : ℕ := p - 2\n⊢ LucasLehmerTest p → Nat.Prime (mersenne p)",
"tactic": "have z : p = p' + 2 := (tsub_eq_iff_eq_add_of_le w.nat_succ_le).mp rfl"
},
{
"state_after": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\n⊢ LucasLehmerTest p → Nat.Prime (mersenne p)",
"state_before": "p : ℕ\nw : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\n⊢ LucasLehmerTest p → Nat.Prime (mersenne p)",
"tactic": "have w : 1 < p' + 2 := Nat.lt_of_sub_eq_succ rfl"
},
{
"state_after": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\n⊢ ¬Nat.Prime (mersenne p) → ¬LucasLehmerTest p",
"state_before": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\n⊢ LucasLehmerTest p → Nat.Prime (mersenne p)",
"tactic": "contrapose"
},
{
"state_after": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : ¬Nat.Prime (mersenne p)\nt : LucasLehmerTest p\n⊢ False",
"state_before": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\n⊢ ¬Nat.Prime (mersenne p) → ¬LucasLehmerTest p",
"tactic": "intro a t"
},
{
"state_after": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : ¬Nat.Prime (mersenne (p' + 2))\nt : LucasLehmerTest p\n⊢ False",
"state_before": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : ¬Nat.Prime (mersenne p)\nt : LucasLehmerTest p\n⊢ False",
"tactic": "rw [z] at a"
},
{
"state_after": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : ¬Nat.Prime (mersenne (p' + 2))\nt : LucasLehmerTest (p' + 2)\n⊢ False",
"state_before": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : ¬Nat.Prime (mersenne (p' + 2))\nt : LucasLehmerTest p\n⊢ False",
"tactic": "rw [z] at t"
},
{
"state_after": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : ¬Nat.Prime (mersenne (p' + 2))\nt : LucasLehmerTest (p' + 2)\nh₁ : 2 ^ (p' + 2) < ↑(q (p' + 2)) ^ 2\n⊢ False",
"state_before": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : ¬Nat.Prime (mersenne (p' + 2))\nt : LucasLehmerTest (p' + 2)\n⊢ False",
"tactic": "have h₁ := order_ineq p' t"
},
{
"state_after": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : ¬Nat.Prime (mersenne (p' + 2))\nt : LucasLehmerTest (p' + 2)\nh₁ : 2 ^ (p' + 2) < ↑(q (p' + 2)) ^ 2\nh₂ : Nat.minFac (mersenne (p' + 2)) ^ 2 ≤ mersenne (p' + 2)\n⊢ False",
"state_before": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : ¬Nat.Prime (mersenne (p' + 2))\nt : LucasLehmerTest (p' + 2)\nh₁ : 2 ^ (p' + 2) < ↑(q (p' + 2)) ^ 2\n⊢ False",
"tactic": "have h₂ := Nat.minFac_sq_le_self (mersenne_pos (Nat.lt_of_succ_lt w)) a"
},
{
"state_after": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : ¬Nat.Prime (mersenne (p' + 2))\nt : LucasLehmerTest (p' + 2)\nh₁ : 2 ^ (p' + 2) < ↑(q (p' + 2)) ^ 2\nh₂ : Nat.minFac (mersenne (p' + 2)) ^ 2 ≤ mersenne (p' + 2)\nh : 2 ^ (p' + 2) < mersenne (p' + 2)\n⊢ False",
"state_before": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : ¬Nat.Prime (mersenne (p' + 2))\nt : LucasLehmerTest (p' + 2)\nh₁ : 2 ^ (p' + 2) < ↑(q (p' + 2)) ^ 2\nh₂ : Nat.minFac (mersenne (p' + 2)) ^ 2 ≤ mersenne (p' + 2)\n⊢ False",
"tactic": "have h := lt_of_lt_of_le h₁ h₂"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nw✝ : 1 < p\np' : ℕ := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : ¬Nat.Prime (mersenne (p' + 2))\nt : LucasLehmerTest (p' + 2)\nh₁ : 2 ^ (p' + 2) < ↑(q (p' + 2)) ^ 2\nh₂ : Nat.minFac (mersenne (p' + 2)) ^ 2 ≤ mersenne (p' + 2)\nh : 2 ^ (p' + 2) < mersenne (p' + 2)\n⊢ False",
"tactic": "exact not_lt_of_ge (Nat.sub_le _ _) h"
}
] |
[
517,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
1
] |
Mathlib/Order/SupIndep.lean
|
CompleteLattice.Independent.comp'
|
[
{
"state_after": "α : Type u_3\nβ : Type ?u.52493\nι✝ : Type ?u.52496\nι'✝ : Type ?u.52499\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt✝ : ι✝ → α\nht✝ : Independent t✝\nι : Sort u_1\nι' : Sort u_2\nt : ι → α\nf : ι' → ι\nht : Independent (t ∘ f)\nhf : Surjective f\ni : ι\n⊢ Disjoint (t i) (⨆ (j : ι) (_ : j ≠ i), t j)",
"state_before": "α : Type u_3\nβ : Type ?u.52493\nι✝ : Type ?u.52496\nι'✝ : Type ?u.52499\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt✝ : ι✝ → α\nht✝ : Independent t✝\nι : Sort u_1\nι' : Sort u_2\nt : ι → α\nf : ι' → ι\nht : Independent (t ∘ f)\nhf : Surjective f\n⊢ Independent t",
"tactic": "intro i"
},
{
"state_after": "case intro\nα : Type u_3\nβ : Type ?u.52493\nι✝ : Type ?u.52496\nι'✝ : Type ?u.52499\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt✝ : ι✝ → α\nht✝ : Independent t✝\nι : Sort u_1\nι' : Sort u_2\nt : ι → α\nf : ι' → ι\nht : Independent (t ∘ f)\nhf : Surjective f\ni' : ι'\n⊢ Disjoint (t (f i')) (⨆ (j : ι) (_ : j ≠ f i'), t j)",
"state_before": "α : Type u_3\nβ : Type ?u.52493\nι✝ : Type ?u.52496\nι'✝ : Type ?u.52499\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt✝ : ι✝ → α\nht✝ : Independent t✝\nι : Sort u_1\nι' : Sort u_2\nt : ι → α\nf : ι' → ι\nht : Independent (t ∘ f)\nhf : Surjective f\ni : ι\n⊢ Disjoint (t i) (⨆ (j : ι) (_ : j ≠ i), t j)",
"tactic": "obtain ⟨i', rfl⟩ := hf i"
},
{
"state_after": "case intro\nα : Type u_3\nβ : Type ?u.52493\nι✝ : Type ?u.52496\nι'✝ : Type ?u.52499\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt✝ : ι✝ → α\nht✝ : Independent t✝\nι : Sort u_1\nι' : Sort u_2\nt : ι → α\nf : ι' → ι\nht : Independent (t ∘ f)\nhf : Surjective f\ni' : ι'\n⊢ Disjoint (t (f i')) (⨆ (x : ι') (_ : f x ≠ f i'), t (f x))",
"state_before": "case intro\nα : Type u_3\nβ : Type ?u.52493\nι✝ : Type ?u.52496\nι'✝ : Type ?u.52499\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt✝ : ι✝ → α\nht✝ : Independent t✝\nι : Sort u_1\nι' : Sort u_2\nt : ι → α\nf : ι' → ι\nht : Independent (t ∘ f)\nhf : Surjective f\ni' : ι'\n⊢ Disjoint (t (f i')) (⨆ (j : ι) (_ : j ≠ f i'), t j)",
"tactic": "rw [← hf.iSup_comp]"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_3\nβ : Type ?u.52493\nι✝ : Type ?u.52496\nι'✝ : Type ?u.52499\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt✝ : ι✝ → α\nht✝ : Independent t✝\nι : Sort u_1\nι' : Sort u_2\nt : ι → α\nf : ι' → ι\nht : Independent (t ∘ f)\nhf : Surjective f\ni' : ι'\n⊢ Disjoint (t (f i')) (⨆ (x : ι') (_ : f x ≠ f i'), t (f x))",
"tactic": "exact (ht i').mono_right (biSup_mono fun j' hij => mt (congr_arg f) hij)"
}
] |
[
303,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.aeval_eq_zero
|
[] |
[
1550,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1548,
1
] |
Mathlib/Topology/Basic.lean
|
acc_iff_cluster
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\nx : α\nF : Filter α\n⊢ AccPt x F ↔ ClusterPt x (𝓟 ({x}ᶜ) ⊓ F)",
"tactic": "rw [AccPt, nhdsWithin, ClusterPt, inf_assoc]"
}
] |
[
1173,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1172,
1
] |
Mathlib/LinearAlgebra/Matrix/MvPolynomial.lean
|
Matrix.mvPolynomialX_apply
|
[] |
[
44,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/Algebra/Module/Submodule/Pointwise.lean
|
Submodule.smul_mem_pointwise_smul
|
[] |
[
231,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
1
] |
Mathlib/Data/SetLike/Basic.lean
|
SetLike.coe_injective
|
[] |
[
137,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.inter_subset_inter_right
|
[] |
[
1019,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1018,
1
] |
Mathlib/CategoryTheory/Sites/Subsheaf.lean
|
CategoryTheory.GrothendieckTopology.Subpresheaf.eq_sheafify
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\n⊢ sheafify J G ≤ G",
"state_before": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\n⊢ G = sheafify J G",
"tactic": "apply (G.le_sheafify J).antisymm"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj (sheafify J G) U\n⊢ s ∈ obj G U",
"state_before": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\n⊢ sheafify J G ≤ 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\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj (sheafify J G) U\n⊢ ↑(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =\n s",
"state_before": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj (sheafify J G) U\n⊢ s ∈ obj G U",
"tactic": "suffices ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).1 = s by\n rw [← this]\n exact ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).2"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj (sheafify J G) U\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ U.unop⦄,\n (sieveOfSection G s).arrows f →\n F.map f.op\n ↑(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s)\n (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =\n F.map f.op s",
"state_before": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj (sheafify J G) U\n⊢ ↑(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =\n s",
"tactic": "apply (h _ hs).isSeparatedFor.ext"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj (sheafify J G) U\nV : C\ni : V ⟶ U.unop\nhi : (sieveOfSection G s).arrows i\n⊢ F.map i.op\n ↑(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =\n F.map i.op s",
"state_before": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj (sheafify J G) U\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ U.unop⦄,\n (sieveOfSection G s).arrows f →\n F.map f.op\n ↑(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s)\n (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =\n F.map f.op s",
"tactic": "intro V i hi"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj (sheafify J G) U\nV : C\ni : V ⟶ U.unop\nhi : (sieveOfSection G s).arrows i\n⊢ F.map i.op\n ↑(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =\n F.map i.op s",
"tactic": "exact (congr_arg Subtype.val ((hG _ hs).valid_glue (G.family_of_elements_compatible s) _ hi) : _)"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj (sheafify J G) U\nthis :\n ↑(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =\n s\n⊢ ↑(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) ∈\n obj G U",
"state_before": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj (sheafify J G) U\nthis :\n ↑(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =\n s\n⊢ s ∈ obj G U",
"tactic": "rw [← this]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F\nhG : Presieve.IsSheaf J (toPresheaf G)\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ obj (sheafify J G) U\nthis :\n ↑(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) =\n s\n⊢ ↑(Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor (toPresheaf G) (sieveOfSection G s).arrows)\n (familyOfElementsOfSection G s) (_ : Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s))) ∈\n obj G U",
"tactic": "exact ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).2"
}
] |
[
215,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Data/Real/Pointwise.lean
|
Real.smul_iInf_of_nonneg
|
[] |
[
52,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Analysis/Normed/Group/Seminorm.lean
|
GroupSeminorm.coe_lt_coe
|
[] |
[
243,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.withDensity_apply
|
[] |
[
1551,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1549,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.mk_finsupp_lift_of_infinite'
|
[
{
"state_after": "case inl\nα : Type u\nβ : Type v\ninst✝² : Nonempty α\ninst✝¹ : Zero β\ninst✝ : Infinite β\nval✝ : Fintype α\n⊢ (#α →₀ β) = max (lift (#α)) (lift (#β))\n\ncase inr\nα : Type u\nβ : Type v\ninst✝² : Nonempty α\ninst✝¹ : Zero β\ninst✝ : Infinite β\nval✝ : Infinite α\n⊢ (#α →₀ β) = max (lift (#α)) (lift (#β))",
"state_before": "α : Type u\nβ : Type v\ninst✝² : Nonempty α\ninst✝¹ : Zero β\ninst✝ : Infinite β\n⊢ (#α →₀ β) = max (lift (#α)) (lift (#β))",
"tactic": "cases fintypeOrInfinite α"
},
{
"state_after": "case inl\nα : Type u\nβ : Type v\ninst✝² : Nonempty α\ninst✝¹ : Zero β\ninst✝ : Infinite β\nval✝ : Fintype α\n⊢ lift (#β) ^ Fintype.card α = max (lift (#α)) (lift (#β))",
"state_before": "case inl\nα : Type u\nβ : Type v\ninst✝² : Nonempty α\ninst✝¹ : Zero β\ninst✝ : Infinite β\nval✝ : Fintype α\n⊢ (#α →₀ β) = max (lift (#α)) (lift (#β))",
"tactic": "rw [mk_finsupp_lift_of_fintype]"
},
{
"state_after": "case inl\nα : Type u\nβ : Type v\ninst✝² : Nonempty α\ninst✝¹ : Zero β\ninst✝ : Infinite β\nval✝ : Fintype α\nthis : ℵ₀ ≤ lift (#β)\n⊢ lift (#β) ^ Fintype.card α = max (lift (#α)) (lift (#β))",
"state_before": "case inl\nα : Type u\nβ : Type v\ninst✝² : Nonempty α\ninst✝¹ : Zero β\ninst✝ : Infinite β\nval✝ : Fintype α\n⊢ lift (#β) ^ Fintype.card α = max (lift (#α)) (lift (#β))",
"tactic": "have : ℵ₀ ≤ (#β).lift := aleph0_le_lift.2 (aleph0_le_mk β)"
},
{
"state_after": "case inl\nα : Type u\nβ : Type v\ninst✝² : Nonempty α\ninst✝¹ : Zero β\ninst✝ : Infinite β\nval✝ : Fintype α\nthis : ℵ₀ ≤ lift (#β)\n⊢ 1 ≤ Fintype.card α\n\nα : Type u\nβ : Type v\ninst✝² : Nonempty α\ninst✝¹ : Zero β\ninst✝ : Infinite β\nval✝ : Fintype α\nthis : ℵ₀ ≤ lift (#β)\n⊢ lift (#α) ≤ ℵ₀",
"state_before": "case inl\nα : Type u\nβ : Type v\ninst✝² : Nonempty α\ninst✝¹ : Zero β\ninst✝ : Infinite β\nval✝ : Fintype α\nthis : ℵ₀ ≤ lift (#β)\n⊢ lift (#β) ^ Fintype.card α = max (lift (#α)) (lift (#β))",
"tactic": "rw [max_eq_right (le_trans _ this), power_nat_eq this]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u\nβ : Type v\ninst✝² : Nonempty α\ninst✝¹ : Zero β\ninst✝ : Infinite β\nval✝ : Fintype α\nthis : ℵ₀ ≤ lift (#β)\n⊢ 1 ≤ Fintype.card α\n\nα : Type u\nβ : Type v\ninst✝² : Nonempty α\ninst✝¹ : Zero β\ninst✝ : Infinite β\nval✝ : Fintype α\nthis : ℵ₀ ≤ lift (#β)\n⊢ lift (#α) ≤ ℵ₀",
"tactic": "exacts [Fintype.card_pos, lift_le_aleph0.2 (lt_aleph0_of_finite _).le]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u\nβ : Type v\ninst✝² : Nonempty α\ninst✝¹ : Zero β\ninst✝ : Infinite β\nval✝ : Infinite α\n⊢ (#α →₀ β) = max (lift (#α)) (lift (#β))",
"tactic": "apply mk_finsupp_lift_of_infinite"
}
] |
[
1061,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1054,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieSubmodule.add_eq_sup
|
[] |
[
468,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
467,
1
] |
Mathlib/Topology/Constructions.lean
|
pi_generateFrom_eq
|
[
{
"state_after": "case refine_1\nα : Type u\nβ : Type v\nγ : Type ?u.341773\nδ : Type ?u.341776\nε : Type ?u.341779\nζ : Type ?u.341782\nι : Type u_2\nπ✝ : ι → Type ?u.341790\nκ : Type ?u.341793\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π✝ i)\nf : α → (i : ι) → π✝ i\nπ : ι → Type u_1\ng : (a : ι) → Set (Set (π a))\n⊢ Pi.topologicalSpace ≤ generateFrom {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ g a) ∧ t = Set.pi (↑i) s}\n\ncase refine_2\nα : Type u\nβ : Type v\nγ : Type ?u.341773\nδ : Type ?u.341776\nε : Type ?u.341779\nζ : Type ?u.341782\nι : Type u_2\nπ✝ : ι → Type ?u.341790\nκ : Type ?u.341793\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π✝ i)\nf : α → (i : ι) → π✝ i\nπ : ι → Type u_1\ng : (a : ι) → Set (Set (π a))\n⊢ generateFrom {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ g a) ∧ t = Set.pi (↑i) s} ≤ Pi.topologicalSpace",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.341773\nδ : Type ?u.341776\nε : Type ?u.341779\nζ : Type ?u.341782\nι : Type u_2\nπ✝ : ι → Type ?u.341790\nκ : Type ?u.341793\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π✝ i)\nf : α → (i : ι) → π✝ i\nπ : ι → Type u_1\ng : (a : ι) → Set (Set (π a))\n⊢ Pi.topologicalSpace = generateFrom {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ g a) ∧ t = Set.pi (↑i) s}",
"tactic": "refine le_antisymm ?_ ?_"
},
{
"state_after": "case refine_1.h\nα : Type u\nβ : Type v\nγ : Type ?u.341773\nδ : Type ?u.341776\nε : Type ?u.341779\nζ : Type ?u.341782\nι : Type u_2\nπ✝ : ι → Type ?u.341790\nκ : Type ?u.341793\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π✝ i)\nf : α → (i : ι) → π✝ i\nπ : ι → Type u_1\ng : (a : ι) → Set (Set (π a))\n⊢ ∀ (s : Set ((a : ι) → π a)), s ∈ {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ g a) ∧ t = Set.pi (↑i) s} → IsOpen s",
"state_before": "case refine_1\nα : Type u\nβ : Type v\nγ : Type ?u.341773\nδ : Type ?u.341776\nε : Type ?u.341779\nζ : Type ?u.341782\nι : Type u_2\nπ✝ : ι → Type ?u.341790\nκ : Type ?u.341793\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π✝ i)\nf : α → (i : ι) → π✝ i\nπ : ι → Type u_1\ng : (a : ι) → Set (Set (π a))\n⊢ Pi.topologicalSpace ≤ generateFrom {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ g a) ∧ t = Set.pi (↑i) s}",
"tactic": "apply le_generateFrom"
},
{
"state_after": "case refine_1.h.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type ?u.341773\nδ : Type ?u.341776\nε : Type ?u.341779\nζ : Type ?u.341782\nι : Type u_2\nπ✝ : ι → Type ?u.341790\nκ : Type ?u.341793\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π✝ i)\nf : α → (i : ι) → π✝ i\nπ : ι → Type u_1\ng : (a : ι) → Set (Set (π a))\ns : (a : ι) → Set (π a)\ni : Finset ι\nhi : ∀ (a : ι), a ∈ i → s a ∈ g a\n⊢ IsOpen (Set.pi (↑i) s)",
"state_before": "case refine_1.h\nα : Type u\nβ : Type v\nγ : Type ?u.341773\nδ : Type ?u.341776\nε : Type ?u.341779\nζ : Type ?u.341782\nι : Type u_2\nπ✝ : ι → Type ?u.341790\nκ : Type ?u.341793\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π✝ i)\nf : α → (i : ι) → π✝ i\nπ : ι → Type u_1\ng : (a : ι) → Set (Set (π a))\n⊢ ∀ (s : Set ((a : ι) → π a)), s ∈ {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ g a) ∧ t = Set.pi (↑i) s} → IsOpen s",
"tactic": "rintro _ ⟨s, i, hi, rfl⟩"
},
{
"state_after": "case refine_1.h.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type ?u.341773\nδ : Type ?u.341776\nε : Type ?u.341779\nζ : Type ?u.341782\nι : Type u_2\nπ✝ : ι → Type ?u.341790\nκ : Type ?u.341793\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π✝ i)\nf : α → (i : ι) → π✝ i\nπ : ι → Type u_1\ng : (a : ι) → Set (Set (π a))\ns : (a : ι) → Set (π a)\ni : Finset ι\nhi : ∀ (a : ι), a ∈ i → s a ∈ g a\nthis : (a : ι) → TopologicalSpace (π a) := fun a => generateFrom (g a)\n⊢ IsOpen (Set.pi (↑i) s)",
"state_before": "case refine_1.h.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type ?u.341773\nδ : Type ?u.341776\nε : Type ?u.341779\nζ : Type ?u.341782\nι : Type u_2\nπ✝ : ι → Type ?u.341790\nκ : Type ?u.341793\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π✝ i)\nf : α → (i : ι) → π✝ i\nπ : ι → Type u_1\ng : (a : ι) → Set (Set (π a))\ns : (a : ι) → Set (π a)\ni : Finset ι\nhi : ∀ (a : ι), a ∈ i → s a ∈ g a\n⊢ IsOpen (Set.pi (↑i) s)",
"tactic": "letI := fun a => generateFrom (g a)"
},
{
"state_after": "no goals",
"state_before": "case refine_1.h.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type ?u.341773\nδ : Type ?u.341776\nε : Type ?u.341779\nζ : Type ?u.341782\nι : Type u_2\nπ✝ : ι → Type ?u.341790\nκ : Type ?u.341793\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π✝ i)\nf : α → (i : ι) → π✝ i\nπ : ι → Type u_1\ng : (a : ι) → Set (Set (π a))\ns : (a : ι) → Set (π a)\ni : Finset ι\nhi : ∀ (a : ι), a ∈ i → s a ∈ g a\nthis : (a : ι) → TopologicalSpace (π a) := fun a => generateFrom (g a)\n⊢ IsOpen (Set.pi (↑i) s)",
"tactic": "exact isOpen_set_pi i.finite_toSet (fun a ha => GenerateOpen.basic _ (hi a ha))"
},
{
"state_after": "case refine_2\nα : Type u\nβ : Type v\nγ : Type ?u.341773\nδ : Type ?u.341776\nε : Type ?u.341779\nζ : Type ?u.341782\nι : Type u_2\nπ✝ : ι → Type ?u.341790\nκ : Type ?u.341793\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π✝ i)\nf : α → (i : ι) → π✝ i\nπ : ι → Type u_1\ng : (a : ι) → Set (Set (π a))\na : ι\ns : Set (π a)\nhs : s ∈ g a\n⊢ IsOpen s",
"state_before": "case refine_2\nα : Type u\nβ : Type v\nγ : Type ?u.341773\nδ : Type ?u.341776\nε : Type ?u.341779\nζ : Type ?u.341782\nι : Type u_2\nπ✝ : ι → Type ?u.341790\nκ : Type ?u.341793\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π✝ i)\nf : α → (i : ι) → π✝ i\nπ : ι → Type u_1\ng : (a : ι) → Set (Set (π a))\n⊢ generateFrom {t | ∃ s i, (∀ (a : ι), a ∈ i → s a ∈ g a) ∧ t = Set.pi (↑i) s} ≤ Pi.topologicalSpace",
"tactic": "refine le_iInf fun a => coinduced_le_iff_le_induced.1 <| le_generateFrom fun s hs => ?_"
},
{
"state_after": "case refine_2\nα : Type u\nβ : Type v\nγ : Type ?u.341773\nδ : Type ?u.341776\nε : Type ?u.341779\nζ : Type ?u.341782\nι : Type u_2\nπ✝ : ι → Type ?u.341790\nκ : Type ?u.341793\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π✝ i)\nf : α → (i : ι) → π✝ i\nπ : ι → Type u_1\ng : (a : ι) → Set (Set (π a))\na : ι\ns : Set (π a)\nhs : s ∈ g a\n⊢ (∀ (a_1 : ι), a_1 ∈ {a} → update (fun a => univ) a s a_1 ∈ g a_1) ∧\n (fun f => f a) ⁻¹' s = Set.pi (↑{a}) (update (fun a => univ) a s)",
"state_before": "case refine_2\nα : Type u\nβ : Type v\nγ : Type ?u.341773\nδ : Type ?u.341776\nε : Type ?u.341779\nζ : Type ?u.341782\nι : Type u_2\nπ✝ : ι → Type ?u.341790\nκ : Type ?u.341793\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π✝ i)\nf : α → (i : ι) → π✝ i\nπ : ι → Type u_1\ng : (a : ι) → Set (Set (π a))\na : ι\ns : Set (π a)\nhs : s ∈ g a\n⊢ IsOpen s",
"tactic": "refine GenerateOpen.basic _ ⟨update (fun a => univ) a s, {a}, ?_⟩"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nα : Type u\nβ : Type v\nγ : Type ?u.341773\nδ : Type ?u.341776\nε : Type ?u.341779\nζ : Type ?u.341782\nι : Type u_2\nπ✝ : ι → Type ?u.341790\nκ : Type ?u.341793\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π✝ i)\nf : α → (i : ι) → π✝ i\nπ : ι → Type u_1\ng : (a : ι) → Set (Set (π a))\na : ι\ns : Set (π a)\nhs : s ∈ g a\n⊢ (∀ (a_1 : ι), a_1 ∈ {a} → update (fun a => univ) a s a_1 ∈ g a_1) ∧\n (fun f => f a) ⁻¹' s = Set.pi (↑{a}) (update (fun a => univ) a s)",
"tactic": "simp [hs]"
}
] |
[
1379,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1368,
1
] |
Mathlib/Data/List/Cycle.lean
|
Cycle.coe_toFinset
|
[] |
[
810,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
809,
1
] |
Mathlib/Combinatorics/Additive/SalemSpencer.lean
|
mulSalemSpencer_pi
|
[] |
[
113,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
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