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/Set/Pointwise/SMul.lean
|
Set.op_smul_set_subset_mul
|
[] |
[
420,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
419,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
summable_geometric_iff_norm_lt_1
|
[
{
"state_after": "α : Type ?u.147129\nβ : Type ?u.147132\nι : Type ?u.147135\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : Summable fun n => ξ ^ n\n⊢ ‖ξ‖ < 1",
"state_before": "α : Type ?u.147129\nβ : Type ?u.147132\nι : Type ?u.147135\nK : Type u_1\ninst✝ : NormedField K\nξ : K\n⊢ (Summable fun n => ξ ^ n) ↔ ‖ξ‖ < 1",
"tactic": "refine' ⟨fun h ↦ _, summable_geometric_of_norm_lt_1⟩"
},
{
"state_after": "case intro\nα : Type ?u.147129\nβ : Type ?u.147132\nι : Type ?u.147135\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : Summable fun n => ξ ^ n\nk : ℕ\nhk : dist (ξ ^ k) 0 < 1\n⊢ ‖ξ‖ < 1",
"state_before": "α : Type ?u.147129\nβ : Type ?u.147132\nι : Type ?u.147135\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : Summable fun n => ξ ^ n\n⊢ ‖ξ‖ < 1",
"tactic": "obtain ⟨k : ℕ, hk : dist (ξ ^ k) 0 < 1⟩ :=\n (h.tendsto_cofinite_zero.eventually (ball_mem_nhds _ zero_lt_one)).exists"
},
{
"state_after": "case intro\nα : Type ?u.147129\nβ : Type ?u.147132\nι : Type ?u.147135\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : Summable fun n => ξ ^ n\nk : ℕ\nhk : ‖ξ‖ ^ k < 1\n⊢ ‖ξ‖ < 1",
"state_before": "case intro\nα : Type ?u.147129\nβ : Type ?u.147132\nι : Type ?u.147135\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : Summable fun n => ξ ^ n\nk : ℕ\nhk : dist (ξ ^ k) 0 < 1\n⊢ ‖ξ‖ < 1",
"tactic": "simp only [norm_pow, dist_zero_right] at hk"
},
{
"state_after": "case intro\nα : Type ?u.147129\nβ : Type ?u.147132\nι : Type ?u.147135\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : Summable fun n => ξ ^ n\nk : ℕ\nhk : ‖ξ‖ ^ k < 1 ^ k\n⊢ ‖ξ‖ < 1",
"state_before": "case intro\nα : Type ?u.147129\nβ : Type ?u.147132\nι : Type ?u.147135\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : Summable fun n => ξ ^ n\nk : ℕ\nhk : ‖ξ‖ ^ k < 1\n⊢ ‖ξ‖ < 1",
"tactic": "rw [← one_pow k] at hk"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type ?u.147129\nβ : Type ?u.147132\nι : Type ?u.147135\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : Summable fun n => ξ ^ n\nk : ℕ\nhk : ‖ξ‖ ^ k < 1 ^ k\n⊢ ‖ξ‖ < 1",
"tactic": "exact lt_of_pow_lt_pow _ zero_le_one hk"
}
] |
[
326,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.measurable_spanningSets
|
[] |
[
3473,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3471,
1
] |
Mathlib/Data/List/Card.lean
|
List.card_eq_of_equiv
|
[] |
[
173,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
|
coeff_charpoly_mem_ideal_pow
|
[
{
"state_after": "R : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\n⊢ coeff (det (charmatrix M)) k ∈ I ^ (Fintype.card n - k)",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\n⊢ coeff (charpoly M) k ∈ I ^ (Fintype.card n - k)",
"tactic": "delta charpoly"
},
{
"state_after": "R : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\n⊢ ∑ b : Equiv.Perm n, coeff (↑Equiv.Perm.sign b • ∏ i : n, charmatrix M (↑b i) i) k ∈ I ^ (Fintype.card n - k)",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\n⊢ coeff (det (charmatrix M)) k ∈ I ^ (Fintype.card n - k)",
"tactic": "rw [Matrix.det_apply, finset_sum_coeff]"
},
{
"state_after": "case h\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\n⊢ ∀ (c : Equiv.Perm n),\n c ∈ univ → coeff (↑Equiv.Perm.sign c • ∏ i : n, charmatrix M (↑c i) i) k ∈ I ^ (Fintype.card n - k)",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\n⊢ ∑ b : Equiv.Perm n, coeff (↑Equiv.Perm.sign b • ∏ i : n, charmatrix M (↑b i) i) k ∈ I ^ (Fintype.card n - k)",
"tactic": "apply sum_mem"
},
{
"state_after": "case h\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\n⊢ coeff (↑Equiv.Perm.sign c • ∏ i : n, charmatrix M (↑c i) i) k ∈ I ^ (Fintype.card n - k)",
"state_before": "case h\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\n⊢ ∀ (c : Equiv.Perm n),\n c ∈ univ → coeff (↑Equiv.Perm.sign c • ∏ i : n, charmatrix M (↑c i) i) k ∈ I ^ (Fintype.card n - k)",
"tactic": "rintro c -"
},
{
"state_after": "case h\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\n⊢ coeff (∏ i : n, charmatrix M (↑c i) i) k ∈ I ^ (Fintype.card n - k)",
"state_before": "case h\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\n⊢ coeff (↑Equiv.Perm.sign c • ∏ i : n, charmatrix M (↑c i) i) k ∈ I ^ (Fintype.card n - k)",
"tactic": "rw [coeff_smul, Submodule.smul_mem_iff']"
},
{
"state_after": "case h\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\n⊢ coeff (∏ i : n, charmatrix M (↑c i) i) k ∈ I ^ (Fintype.card n - k)",
"state_before": "case h\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\n⊢ coeff (∏ i : n, charmatrix M (↑c i) i) k ∈ I ^ (Fintype.card n - k)",
"tactic": "have : (∑ x : n, 1) = Fintype.card n := by rw [Finset.sum_const, card_univ, smul_eq_mul, mul_one]"
},
{
"state_after": "case h\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\n⊢ coeff (∏ i : n, charmatrix M (↑c i) i) k ∈ I ^ (∑ x : n, 1 - k)",
"state_before": "case h\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\n⊢ coeff (∏ i : n, charmatrix M (↑c i) i) k ∈ I ^ (Fintype.card n - k)",
"tactic": "rw [← this]"
},
{
"state_after": "case h.h\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\n⊢ ∀ (i : n), i ∈ univ → ∀ (k : ℕ), coeff (charmatrix M (↑c i) i) k ∈ I ^ (1 - k)",
"state_before": "case h\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\n⊢ coeff (∏ i : n, charmatrix M (↑c i) i) k ∈ I ^ (∑ x : n, 1 - k)",
"tactic": "apply coeff_prod_mem_ideal_pow_tsub"
},
{
"state_after": "case h.h.zero\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\ni : n\n⊢ coeff (charmatrix M (↑c i) i) Nat.zero ∈ I ^ (1 - Nat.zero)\n\ncase h.h.succ\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk✝ : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\ni : n\nk : ℕ\n⊢ coeff (charmatrix M (↑c i) i) (Nat.succ k) ∈ I ^ (1 - Nat.succ k)",
"state_before": "case h.h\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\n⊢ ∀ (i : n), i ∈ univ → ∀ (k : ℕ), coeff (charmatrix M (↑c i) i) k ∈ I ^ (1 - k)",
"tactic": "rintro i - (_ | k)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\n⊢ ∑ x : n, 1 = Fintype.card n",
"tactic": "rw [Finset.sum_const, card_univ, smul_eq_mul, mul_one]"
},
{
"state_after": "case h.h.zero\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\ni : n\n⊢ coeff (charmatrix M (↑c i) i) 0 ∈ I ^ (1 - 0)",
"state_before": "case h.h.zero\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\ni : n\n⊢ coeff (charmatrix M (↑c i) i) Nat.zero ∈ I ^ (1 - Nat.zero)",
"tactic": "rw [Nat.zero_eq]"
},
{
"state_after": "case h.h.zero\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\ni : n\n⊢ -M (↑c i) i ∈ I",
"state_before": "case h.h.zero\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\ni : n\n⊢ coeff (charmatrix M (↑c i) i) 0 ∈ I ^ (1 - 0)",
"tactic": "rw [tsub_zero, pow_one, charmatrix_apply, coeff_sub, coeff_X_mul_zero, coeff_C_zero, zero_sub]"
},
{
"state_after": "case h.h.zero.a\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\ni : n\n⊢ M (↑c i) i ∈ I",
"state_before": "case h.h.zero\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\ni : n\n⊢ -M (↑c i) i ∈ I",
"tactic": "apply neg_mem"
},
{
"state_after": "no goals",
"state_before": "case h.h.zero.a\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\ni : n\n⊢ M (↑c i) i ∈ I",
"tactic": "exact h (c i) i"
},
{
"state_after": "case h.h.succ\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk✝ : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\ni : n\nk : ℕ\n⊢ coeff (charmatrix M (↑c i) i) (1 + k) ∈ ⊤",
"state_before": "case h.h.succ\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk✝ : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\ni : n\nk : ℕ\n⊢ coeff (charmatrix M (↑c i) i) (Nat.succ k) ∈ I ^ (1 - Nat.succ k)",
"tactic": "rw [Nat.succ_eq_one_add, tsub_self_add, pow_zero, Ideal.one_eq_top]"
},
{
"state_after": "no goals",
"state_before": "case h.h.succ\nR : Type u\ninst✝⁴ : CommRing R\nn G : Type v\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα β : Type v\ninst✝¹ : DecidableEq α\nM : Matrix n n R\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nI : Ideal R\nh : ∀ (i j : n), M i j ∈ I\nk✝ : ℕ\nc : Equiv.Perm n\nthis : ∑ x : n, 1 = Fintype.card n\ni : n\nk : ℕ\n⊢ coeff (charmatrix M (↑c i) i) (1 + k) ∈ ⊤",
"tactic": "exact Submodule.mem_top"
}
] |
[
261,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.L1.dist_def
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1291582\nδ : Type ?u.1291585\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\n⊢ ENNReal.toReal (∫⁻ (a : α), ↑‖↑↑f a - ↑↑g a‖₊ ∂μ) = ENNReal.toReal (∫⁻ (a : α), edist (↑↑f a) (↑↑g a) ∂μ)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1291582\nδ : Type ?u.1291585\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\n⊢ dist f g = ENNReal.toReal (∫⁻ (a : α), edist (↑↑f a) (↑↑g a) ∂μ)",
"tactic": "simp [Lp.dist_def, snorm, snorm']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1291582\nδ : Type ?u.1291585\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : { x // x ∈ Lp β 1 }\n⊢ ENNReal.toReal (∫⁻ (a : α), ↑‖↑↑f a - ↑↑g a‖₊ ∂μ) = ENNReal.toReal (∫⁻ (a : α), edist (↑↑f a) (↑↑g a) ∂μ)",
"tactic": "simp [edist_eq_coe_nnnorm_sub]"
}
] |
[
1305,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1303,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
BilinForm.smul_apply
|
[] |
[
227,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Topology/LocallyConstant/Basic.lean
|
LocallyConstant.exists_eq_const
|
[
{
"state_after": "case inl.intro\nX : Type u_1\nY : Type u_2\nZ : Type ?u.24519\nα : Type ?u.24522\ninst✝² : TopologicalSpace X\ninst✝¹ : PreconnectedSpace X\ninst✝ : Nonempty Y\nf : LocallyConstant X Y\nx : X\n⊢ ∃ y, f = const X y\n\ncase inr\nX : Type u_1\nY : Type u_2\nZ : Type ?u.24519\nα : Type ?u.24522\ninst✝² : TopologicalSpace X\ninst✝¹ : PreconnectedSpace X\ninst✝ : Nonempty Y\nf : LocallyConstant X Y\nhX : ¬Nonempty X\n⊢ ∃ y, f = const X y",
"state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.24519\nα : Type ?u.24522\ninst✝² : TopologicalSpace X\ninst✝¹ : PreconnectedSpace X\ninst✝ : Nonempty Y\nf : LocallyConstant X Y\n⊢ ∃ y, f = const X y",
"tactic": "rcases Classical.em (Nonempty X) with (⟨⟨x⟩⟩ | hX)"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nX : Type u_1\nY : Type u_2\nZ : Type ?u.24519\nα : Type ?u.24522\ninst✝² : TopologicalSpace X\ninst✝¹ : PreconnectedSpace X\ninst✝ : Nonempty Y\nf : LocallyConstant X Y\nx : X\n⊢ ∃ y, f = const X y",
"tactic": "exact ⟨f x, f.eq_const x⟩"
},
{
"state_after": "no goals",
"state_before": "case inr\nX : Type u_1\nY : Type u_2\nZ : Type ?u.24519\nα : Type ?u.24522\ninst✝² : TopologicalSpace X\ninst✝¹ : PreconnectedSpace X\ninst✝ : Nonempty Y\nf : LocallyConstant X Y\nhX : ¬Nonempty X\n⊢ ∃ y, f = const X y",
"tactic": "exact ⟨Classical.arbitrary Y, ext fun x => (hX ⟨x⟩).elim⟩"
}
] |
[
400,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
isOpen_extChartAt_preimage'
|
[] |
[
1150,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1148,
1
] |
Mathlib/Logic/Basic.lean
|
dite_eq_right_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nβ : Sort ?u.34692\nσ : α → Sort ?u.34688\nf : α → β\nP Q : Prop\ninst✝¹ : Decidable P\ninst✝ : Decidable Q\na b c : α\nA : P → α\nB : ¬P → α\n⊢ (dite P A fun x => b) = b ↔ ∀ (h : P), A h = b",
"tactic": "by_cases P <;> simp [*, forall_prop_of_true, forall_prop_of_false]"
}
] |
[
1151,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1150,
9
] |
Mathlib/LinearAlgebra/Matrix/Determinant.lean
|
Matrix.det_isEmpty
|
[
{
"state_after": "no goals",
"state_before": "m : Type ?u.172043\nn : Type u_1\ninst✝⁵ : DecidableEq n\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\nR : Type v\ninst✝¹ : CommRing R\ninst✝ : IsEmpty n\nA : Matrix n n R\n⊢ det A = 1",
"tactic": "simp [det_apply]"
}
] |
[
103,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/GroupTheory/Subsemigroup/Centralizer.lean
|
Set.centralizer_eq_top_iff_subset
|
[] |
[
136,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
symmetrizeRel_subset_self
|
[] |
[
222,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/Data/TypeVec.lean
|
TypeVec.Arrow.ext
|
[
{
"state_after": "n : ℕ\nα : TypeVec n\nβ : TypeVec n\nf g : α ⟹ β\nh : ∀ (i : Fin2 n), f i = g i\n⊢ f = g",
"state_before": "n : ℕ\nα : TypeVec n\nβ : TypeVec n\nf g : α ⟹ β\n⊢ (∀ (i : Fin2 n), f i = g i) → f = g",
"tactic": "intro h"
},
{
"state_after": "case h\nn : ℕ\nα : TypeVec n\nβ : TypeVec n\nf g : α ⟹ β\nh : ∀ (i : Fin2 n), f i = g i\ni : Fin2 n\n⊢ f i = g i",
"state_before": "n : ℕ\nα : TypeVec n\nβ : TypeVec n\nf g : α ⟹ β\nh : ∀ (i : Fin2 n), f i = g i\n⊢ f = g",
"tactic": "funext i"
},
{
"state_after": "no goals",
"state_before": "case h\nn : ℕ\nα : TypeVec n\nβ : TypeVec n\nf g : α ⟹ β\nh : ∀ (i : Fin2 n), f i = g i\ni : Fin2 n\n⊢ f i = g i",
"tactic": "apply h"
}
] |
[
65,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Logic/Function/Iterate.lean
|
Function.Commute.iterate_iterate_self
|
[] |
[
163,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean
|
CategoryTheory.Presieve.FamilyOfElements.IsAmalgamation.compPresheafMap
|
[
{
"state_after": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y✝ : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : P.obj X.op\nf : P ⟶ Q\nh : IsAmalgamation x t\nY : C\ng : Y ⟶ X\nhg : R g\n⊢ Q.map g.op (f.app X.op t) = FamilyOfElements.compPresheafMap f x g hg",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : P.obj X.op\nf : P ⟶ Q\nh : IsAmalgamation x t\n⊢ IsAmalgamation (FamilyOfElements.compPresheafMap f x) (f.app X.op t)",
"tactic": "intro Y g hg"
},
{
"state_after": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y✝ : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : P.obj X.op\nf : P ⟶ Q\nh : IsAmalgamation x t\nY : C\ng : Y ⟶ X\nhg : R g\n⊢ Q.map g.op (f.app X.op t) = f.app Y.op (x g hg)",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y✝ : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : P.obj X.op\nf : P ⟶ Q\nh : IsAmalgamation x t\nY : C\ng : Y ⟶ X\nhg : R g\n⊢ Q.map g.op (f.app X.op t) = FamilyOfElements.compPresheafMap f x g hg",
"tactic": "dsimp [FamilyOfElements.compPresheafMap]"
},
{
"state_after": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y✝ : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : P.obj X.op\nf : P ⟶ Q\nh : IsAmalgamation x t\nY : C\ng : Y ⟶ X\nhg : R g\n⊢ (f.app X.op ≫ Q.map g.op) t = f.app Y.op (x g hg)",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y✝ : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : P.obj X.op\nf : P ⟶ Q\nh : IsAmalgamation x t\nY : C\ng : Y ⟶ X\nhg : R g\n⊢ Q.map g.op (f.app X.op t) = f.app Y.op (x g hg)",
"tactic": "change (f.app _ ≫ Q.map _) _ = _"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y✝ : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : P.obj X.op\nf : P ⟶ Q\nh : IsAmalgamation x t\nY : C\ng : Y ⟶ X\nhg : R g\n⊢ (f.app X.op ≫ Q.map g.op) t = f.app Y.op (x g hg)",
"tactic": "rw [← f.naturality, types_comp_apply, h g hg]"
}
] |
[
388,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
IntervalIntegrable.neg
|
[] |
[
174,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
Sbtw.affineCombination_of_mem_affineSpan_pair
|
[
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nh : ∃ r, ∀ (i : ι), i ∈ s → w i = r * (w₂ i - w₁ i) + w₁ i\ni : ι\nhis : i ∈ s\nhs : Sbtw R (w₁ i) (w i) (w₂ i)\n⊢ Sbtw R (↑(Finset.affineCombination R s p) w₁) (↑(Finset.affineCombination R s p) w)\n (↑(Finset.affineCombination R s p) w₂)",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nh :\n ↑(Finset.affineCombination R s p) w ∈\n affineSpan R {↑(Finset.affineCombination R s p) w₁, ↑(Finset.affineCombination R s p) w₂}\ni : ι\nhis : i ∈ s\nhs : Sbtw R (w₁ i) (w i) (w₂ i)\n⊢ Sbtw R (↑(Finset.affineCombination R s p) w₁) (↑(Finset.affineCombination R s p) w)\n (↑(Finset.affineCombination R s p) w₂)",
"tactic": "rw [affineCombination_mem_affineSpan_pair ha hw hw₁ hw₂] at h"
},
{
"state_after": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nhs : Sbtw R (w₁ i) (w i) (w₂ i)\nr : R\nhr : ∀ (i : ι), i ∈ s → w i = r * (w₂ i - w₁ i) + w₁ i\n⊢ Sbtw R (↑(Finset.affineCombination R s p) w₁) (↑(Finset.affineCombination R s p) w)\n (↑(Finset.affineCombination R s p) w₂)",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\nh : ∃ r, ∀ (i : ι), i ∈ s → w i = r * (w₂ i - w₁ i) + w₁ i\ni : ι\nhis : i ∈ s\nhs : Sbtw R (w₁ i) (w i) (w₂ i)\n⊢ Sbtw R (↑(Finset.affineCombination R s p) w₁) (↑(Finset.affineCombination R s p) w)\n (↑(Finset.affineCombination R s p) w₂)",
"tactic": "rcases h with ⟨r, hr⟩"
},
{
"state_after": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nhs : Sbtw R (w₁ i) (w i) (w₂ i)\nr : R\nhr : ∀ (i : ι), i ∈ s → w i = r * (w₂ i - w₁ i) + w₁ i\n⊢ Sbtw R (↑(Finset.affineCombination R s p) w₁) (↑(Finset.affineCombination R s p) w)\n (↑(Finset.affineCombination R s p) w₂)",
"state_before": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nhs : Sbtw R (w₁ i) (w i) (w₂ i)\nr : R\nhr : ∀ (i : ι), i ∈ s → w i = r * (w₂ i - w₁ i) + w₁ i\n⊢ Sbtw R (↑(Finset.affineCombination R s p) w₁) (↑(Finset.affineCombination R s p) w)\n (↑(Finset.affineCombination R s p) w₂)",
"tactic": "dsimp only at hr"
},
{
"state_after": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = r * (w₂ i - w₁ i) + w₁ i\n⊢ Sbtw R (↑(Finset.affineCombination R s p) w₁) (↑(Finset.affineCombination R s p) w)\n (↑(Finset.affineCombination R s p) w₂)",
"state_before": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nhs : Sbtw R (w₁ i) (w i) (w₂ i)\nr : R\nhr : ∀ (i : ι), i ∈ s → w i = r * (w₂ i - w₁ i) + w₁ i\n⊢ Sbtw R (↑(Finset.affineCombination R s p) w₁) (↑(Finset.affineCombination R s p) w)\n (↑(Finset.affineCombination R s p) w₂)",
"tactic": "rw [hr i his, sbtw_mul_sub_add_iff] at hs"
},
{
"state_after": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\n⊢ Sbtw R (↑(Finset.affineCombination R s p) w₁) (↑(Finset.affineCombination R s p) w)\n (↑(Finset.affineCombination R s p) w₂)",
"state_before": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = r * (w₂ i - w₁ i) + w₁ i\n⊢ Sbtw R (↑(Finset.affineCombination R s p) w₁) (↑(Finset.affineCombination R s p) w)\n (↑(Finset.affineCombination R s p) w₂)",
"tactic": "change ∀ i ∈ s, w i = (r • (w₂ - w₁) + w₁) i at hr"
},
{
"state_after": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\n⊢ Sbtw R (↑(Finset.affineCombination R s p) w₁)\n (↑(Finset.affineCombination R s fun x => p x) fun i => (r • (w₂ - w₁) + w₁) i)\n (↑(Finset.affineCombination R s p) w₂)",
"state_before": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\n⊢ Sbtw R (↑(Finset.affineCombination R s p) w₁) (↑(Finset.affineCombination R s p) w)\n (↑(Finset.affineCombination R s p) w₂)",
"tactic": "rw [s.affineCombination_congr hr fun _ _ => rfl]"
},
{
"state_after": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\n⊢ Sbtw R (↑(Finset.affineCombination R s p) w₁)\n (↑(Finset.affineCombination R s fun x => p x) fun i => (r • (w₂ - w₁) + w₁) i)\n (↑(Finset.affineCombination R s p) w₂)",
"state_before": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\n⊢ Sbtw R (↑(Finset.affineCombination R s p) w₁)\n (↑(Finset.affineCombination R s fun x => p x) fun i => (r • (w₂ - w₁) + w₁) i)\n (↑(Finset.affineCombination R s p) w₂)",
"tactic": "dsimp only"
},
{
"state_after": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\n⊢ ↑(Finset.weightedVSub s p) (w₁ - w₂) ≠ 0",
"state_before": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\n⊢ Sbtw R (↑(Finset.affineCombination R s p) w₁)\n (↑(Finset.affineCombination R s fun x => p x) fun i => (r • (w₂ - w₁) + w₁) i)\n (↑(Finset.affineCombination R s p) w₂)",
"tactic": "rw [← s.weightedVSub_vadd_affineCombination, s.weightedVSub_const_smul,\n ← s.affineCombination_vsub, ← lineMap_apply, sbtw_lineMap_iff, and_iff_left hs.2,\n ← @vsub_ne_zero V, s.affineCombination_vsub]"
},
{
"state_after": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\nhz : ↑(Finset.weightedVSub s p) (w₁ - w₂) = 0\n⊢ False",
"state_before": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\n⊢ ↑(Finset.weightedVSub s p) (w₁ - w₂) ≠ 0",
"tactic": "intro hz"
},
{
"state_after": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\nhz : ↑(Finset.weightedVSub s p) (w₁ - w₂) = 0\nhw₁w₂ : ∑ i in s, (w₁ - w₂) i = 0\n⊢ False",
"state_before": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\nhz : ↑(Finset.weightedVSub s p) (w₁ - w₂) = 0\n⊢ False",
"tactic": "have hw₁w₂ : (∑ i in s, (w₁ - w₂) i) = 0 := by\n simp_rw [Pi.sub_apply, Finset.sum_sub_distrib, hw₁, hw₂, sub_self]"
},
{
"state_after": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\nhz : ↑(Finset.weightedVSub s p) (w₁ - w₂) = 0\nhw₁w₂ : ∑ i in s, (w₁ - w₂) i = 0\n⊢ w₁ i = w₂ i",
"state_before": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\nhz : ↑(Finset.weightedVSub s p) (w₁ - w₂) = 0\nhw₁w₂ : ∑ i in s, (w₁ - w₂) i = 0\n⊢ False",
"tactic": "refine' hs.1 _"
},
{
"state_after": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\nhz : ↑(Finset.weightedVSub s p) (w₁ - w₂) = 0\nhw₁w₂ : ∑ i in s, (w₁ - w₂) i = 0\nha' : (w₁ - w₂) i = 0\n⊢ w₁ i = w₂ i",
"state_before": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\nhz : ↑(Finset.weightedVSub s p) (w₁ - w₂) = 0\nhw₁w₂ : ∑ i in s, (w₁ - w₂) i = 0\n⊢ w₁ i = w₂ i",
"tactic": "have ha' := ha s (w₁ - w₂) hw₁w₂ hz i his"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\nhz : ↑(Finset.weightedVSub s p) (w₁ - w₂) = 0\nhw₁w₂ : ∑ i in s, (w₁ - w₂) i = 0\nha' : (w₁ - w₂) i = 0\n⊢ w₁ i = w₂ i",
"tactic": "rwa [Pi.sub_apply, sub_eq_zero] at ha'"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.333838\nP : Type u_4\nP' : Type ?u.333844\ninst✝⁸ : OrderedRing R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module R V\ninst✝⁵ : AddTorsor V P\ninst✝⁴ : AddCommGroup V'\ninst✝³ : Module R V'\ninst✝² : AddTorsor V' P'\ninst✝¹ : NoZeroDivisors R\ninst✝ : NoZeroSMulDivisors R V\nι : Type u_3\np : ι → P\nha : AffineIndependent R p\nw w₁ w₂ : ι → R\ns : Finset ι\nhw : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\ni : ι\nhis : i ∈ s\nr : R\nhs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1\nhr : ∀ (i : ι), i ∈ s → w i = (r • (w₂ - w₁) + w₁) i\nhz : ↑(Finset.weightedVSub s p) (w₁ - w₂) = 0\n⊢ ∑ i in s, (w₁ - w₂) i = 0",
"tactic": "simp_rw [Pi.sub_apply, Finset.sum_sub_distrib, hw₁, hw₂, sub_self]"
}
] |
[
550,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.sum_apply
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.976970\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : AddCommMonoid β\ninst✝ : LipschitzAdd β\nι : Type u_1\ns : Finset ι\nf : ι → α →ᵇ β\na : α\n⊢ ↑(∑ i in s, f i) a = ∑ i in s, ↑(f i) a",
"tactic": "simp"
}
] |
[
777,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
776,
1
] |
Mathlib/Topology/UniformSpace/Equicontinuity.lean
|
equicontinuous_iff_range
|
[] |
[
229,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.eq_X_add_C_of_degree_eq_one
|
[
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh : degree p = 1\n⊢ ↑C (coeff p 1) * X + ↑C (coeff p 0) = ↑C (coeff p One.one) * X + ↑C (coeff p 0)",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh : degree p = 1\n⊢ ↑C (coeff p 1) * X + ↑C (coeff p 0) = ↑C (leadingCoeff p) * X + ↑C (coeff p 0)",
"tactic": "simp only [leadingCoeff, natDegree_eq_of_degree_eq_some h]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh : degree p = 1\n⊢ ↑C (coeff p 1) * X + ↑C (coeff p 0) = ↑C (coeff p One.one) * X + ↑C (coeff p 0)",
"tactic": "rfl"
}
] |
[
455,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.restrict_iUnion
|
[] |
[
2169,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2167,
1
] |
Mathlib/Algebra/Ring/Prod.lean
|
NonUnitalRingHom.prod_unique
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.22238\nβ : Type ?u.22241\nR : Type u_1\nR' : Type ?u.22247\nS : Type u_3\nS' : Type ?u.22253\nT : Type u_2\nT' : Type ?u.22259\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : NonUnitalNonAssocSemiring S\ninst✝ : NonUnitalNonAssocSemiring T\nf✝ : R →ₙ+* S\ng : R →ₙ+* T\nf : R →ₙ+* S × T\nx : R\n⊢ ↑(NonUnitalRingHom.prod (comp (fst S T) f) (comp (snd S T) f)) x = ↑f x",
"tactic": "simp only [prod_apply, coe_fst, coe_snd, comp_apply, Prod.mk.eta]"
}
] |
[
154,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/GroupTheory/Subgroup/Finite.lean
|
Subgroup.val_list_prod
|
[] |
[
94,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Data/SetLike/Basic.lean
|
SetLike.coe_set_eq
|
[] |
[
142,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
PosNum.cast_bit1
|
[] |
[
55,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Data/Fin/VecNotation.lean
|
Matrix.range_cons
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nm n o : ℕ\nm' : Type ?u.15284\nn' : Type ?u.15287\no' : Type ?u.15290\nx : α\nu : Fin n → α\ny : α\n⊢ y ∈ Set.range (vecCons x u) ↔ y ∈ {x} ∪ Set.range u",
"tactic": "simp [Fin.exists_fin_succ, eq_comm]"
}
] |
[
170,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Data/Multiset/Dedup.lean
|
Multiset.mem_dedup
|
[] |
[
44,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/Data/Multiset/Sum.lean
|
Multiset.card_disjSum
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\n⊢ ↑card (disjSum s t) = ↑card s + ↑card t",
"tactic": "rw [disjSum, card_add, card_map, card_map]"
}
] |
[
48,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/Topology/Homotopy/Basic.lean
|
ContinuousMap.HomotopyWith.coe_toContinuousMap
|
[] |
[
465,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
464,
1
] |
Mathlib/Algebra/Group/Units.lean
|
divp_inv
|
[] |
[
463,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/Topology/Bornology/Basic.lean
|
Bornology.isBounded_sUnion
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.8797\nα : Type u_1\nβ : Type ?u.8803\ns : Set α\ninst✝ : Bornology α\nS : Set (Set α)\nhs : Set.Finite S\n⊢ IsBounded (⋃₀ S) ↔ ∀ (s : Set α), s ∈ S → IsBounded s",
"tactic": "rw [sUnion_eq_biUnion, isBounded_biUnion hs]"
}
] |
[
295,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.Flat.mono
|
[] |
[
392,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
391,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean
|
expNegInvGlue.contDiff
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ∞\n⊢ ContDiff ℝ n expNegInvGlue",
"tactic": "simpa using contDiff_polynomial_eval_inv_mul 1"
}
] |
[
160,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
11
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean
|
FreeAbelianGroup.add_seq
|
[] |
[
278,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.eq_zero_of_forall_not_mem
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.16742\nγ : Type ?u.16745\ns : Multiset α\nl : List α\nH : ∀ (x : α), ¬x ∈ Quot.mk Setoid.r l\n⊢ Quot.mk Setoid.r [] = 0",
"state_before": "α : Type u_1\nβ : Type ?u.16742\nγ : Type ?u.16745\ns : Multiset α\nl : List α\nH : ∀ (x : α), ¬x ∈ Quot.mk Setoid.r l\n⊢ Quot.mk Setoid.r l = 0",
"tactic": "rw [eq_nil_iff_forall_not_mem.mpr H]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.16742\nγ : Type ?u.16745\ns : Multiset α\nl : List α\nH : ∀ (x : α), ¬x ∈ Quot.mk Setoid.r l\n⊢ Quot.mk Setoid.r [] = 0",
"tactic": "rfl"
}
] |
[
266,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.prod_eq_iff_prod_pow_moebius_eq_of_nonzero
|
[
{
"state_after": "case refine'_1\nR : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\n⊢ ∏ i in divisors n, f i = g n ↔\n (∏ i in divisors n, if h : 0 < i then Units.mk0 (f i) (_ : f i ≠ 0) else 1) =\n if h : 0 < n then Units.mk0 (g n) (_ : g n ≠ 0) else 1\n\ncase refine'_2\nR : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\n⊢ (∏ x in divisorsAntidiagonal n, (if h : 0 < x.snd then Units.mk0 (g x.snd) (_ : g x.snd ≠ 0) else 1) ^ ↑μ x.fst =\n if h : 0 < n then Units.mk0 (f n) (_ : f n ≠ 0) else 1) ↔\n ∏ x in divisorsAntidiagonal n, g x.snd ^ ↑μ x.fst = f n",
"state_before": "R : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\n⊢ (∀ (n : ℕ), 0 < n → ∏ i in divisors n, f i = g n) ↔\n ∀ (n : ℕ), 0 < n → ∏ x in divisorsAntidiagonal n, g x.snd ^ ↑μ x.fst = f n",
"tactic": "refine'\n Iff.trans\n (Iff.trans (forall_congr' fun n => _)\n (@prod_eq_iff_prod_pow_moebius_eq Rˣ _\n (fun n => if h : 0 < n then Units.mk0 (f n) (hf n h) else 1) fun n =>\n if h : 0 < n then Units.mk0 (g n) (hg n h) else 1))\n (forall_congr' fun n => _) <;>\n refine' imp_congr_right fun hn => _"
},
{
"state_after": "case refine'_1\nR : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\n⊢ ∏ i in divisors n, f i = g n ↔\n (∏ i in divisors n, if h : 0 < i then Units.mk0 (f i) (_ : f i ≠ 0) else 1) =\n if h : 0 < n then Units.mk0 (g n) (_ : g n ≠ 0) else 1",
"state_before": "case refine'_1\nR : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\n⊢ ∏ i in divisors n, f i = g n ↔\n (∏ i in divisors n, if h : 0 < i then Units.mk0 (f i) (_ : f i ≠ 0) else 1) =\n if h : 0 < n then Units.mk0 (g n) (_ : g n ≠ 0) else 1",
"tactic": "dsimp"
},
{
"state_after": "R : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\n⊢ ∀ (x : ℕ), x ∈ divisors n → f x = ↑(Units.coeHom R) (if h : 0 < x then Units.mk0 (f x) (_ : f x ≠ 0) else 1)",
"state_before": "case refine'_1\nR : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\n⊢ ∏ i in divisors n, f i = g n ↔\n (∏ i in divisors n, if h : 0 < i then Units.mk0 (f i) (_ : f i ≠ 0) else 1) =\n if h : 0 < n then Units.mk0 (g n) (_ : g n ≠ 0) else 1",
"tactic": "rw [dif_pos hn, ← Units.eq_iff, ← Units.coeHom_apply, map_prod, Units.val_mk0,\n prod_congr rfl _]"
},
{
"state_after": "R : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\nx : ℕ\nhx : x ∈ divisors n\n⊢ f x = ↑(Units.coeHom R) (if h : 0 < x then Units.mk0 (f x) (_ : f x ≠ 0) else 1)",
"state_before": "R : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\n⊢ ∀ (x : ℕ), x ∈ divisors n → f x = ↑(Units.coeHom R) (if h : 0 < x then Units.mk0 (f x) (_ : f x ≠ 0) else 1)",
"tactic": "intro x hx"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\nx : ℕ\nhx : x ∈ divisors n\n⊢ f x = ↑(Units.coeHom R) (if h : 0 < x then Units.mk0 (f x) (_ : f x ≠ 0) else 1)",
"tactic": "rw [dif_pos (Nat.pos_of_mem_divisors hx), Units.coeHom_apply, Units.val_mk0]"
},
{
"state_after": "case refine'_2\nR : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\n⊢ (∏ x in divisorsAntidiagonal n, (if h : 0 < x.snd then Units.mk0 (g x.snd) (_ : g x.snd ≠ 0) else 1) ^ ↑μ x.fst =\n if h : 0 < n then Units.mk0 (f n) (_ : f n ≠ 0) else 1) ↔\n ∏ x in divisorsAntidiagonal n, g x.snd ^ ↑μ x.fst = f n",
"state_before": "case refine'_2\nR : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\n⊢ (∏ x in divisorsAntidiagonal n, (if h : 0 < x.snd then Units.mk0 (g x.snd) (_ : g x.snd ≠ 0) else 1) ^ ↑μ x.fst =\n if h : 0 < n then Units.mk0 (f n) (_ : f n ≠ 0) else 1) ↔\n ∏ x in divisorsAntidiagonal n, g x.snd ^ ↑μ x.fst = f n",
"tactic": "dsimp"
},
{
"state_after": "R : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\n⊢ ∀ (x : ℕ × ℕ),\n x ∈ divisorsAntidiagonal n →\n ↑(Units.coeHom R) ((if h : 0 < x.snd then Units.mk0 (g x.snd) (_ : g x.snd ≠ 0) else 1) ^ ↑μ x.fst) =\n g x.snd ^ ↑μ x.fst",
"state_before": "case refine'_2\nR : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\n⊢ (∏ x in divisorsAntidiagonal n, (if h : 0 < x.snd then Units.mk0 (g x.snd) (_ : g x.snd ≠ 0) else 1) ^ ↑μ x.fst =\n if h : 0 < n then Units.mk0 (f n) (_ : f n ≠ 0) else 1) ↔\n ∏ x in divisorsAntidiagonal n, g x.snd ^ ↑μ x.fst = f n",
"tactic": "rw [dif_pos hn, ← Units.eq_iff, ← Units.coeHom_apply, map_prod, Units.val_mk0,\n prod_congr rfl _]"
},
{
"state_after": "R : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\nx : ℕ × ℕ\nhx : x ∈ divisorsAntidiagonal n\n⊢ ↑(Units.coeHom R) ((if h : 0 < x.snd then Units.mk0 (g x.snd) (_ : g x.snd ≠ 0) else 1) ^ ↑μ x.fst) =\n g x.snd ^ ↑μ x.fst",
"state_before": "R : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\n⊢ ∀ (x : ℕ × ℕ),\n x ∈ divisorsAntidiagonal n →\n ↑(Units.coeHom R) ((if h : 0 < x.snd then Units.mk0 (g x.snd) (_ : g x.snd ≠ 0) else 1) ^ ↑μ x.fst) =\n g x.snd ^ ↑μ x.fst",
"tactic": "intro x hx"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : 0 < n\nx : ℕ × ℕ\nhx : x ∈ divisorsAntidiagonal n\n⊢ ↑(Units.coeHom R) ((if h : 0 < x.snd then Units.mk0 (g x.snd) (_ : g x.snd ≠ 0) else 1) ^ ↑μ x.fst) =\n g x.snd ^ ↑μ x.fst",
"tactic": "rw [dif_pos (Nat.pos_of_mem_divisors (Nat.snd_mem_divisors_of_mem_antidiagonal hx)),\n Units.coeHom_apply, Units.val_zpow_eq_zpow_val, Units.val_mk0]"
}
] |
[
1164,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1142,
1
] |
Mathlib/Topology/Separation.lean
|
disjoint_nhds_pure
|
[] |
[
528,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.ofReal_ne_zero
|
[] |
[
204,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Probability/Independence/Basic.lean
|
ProbabilityTheory.indep_iff_forall_indepSet
|
[] |
[
642,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
637,
1
] |
Mathlib/Data/Real/ConjugateExponents.lean
|
Real.IsConjugateExponent.sub_one_mul_conj
|
[] |
[
84,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Algebra/Hom/Centroid.lean
|
CentroidHom.sub_apply
|
[] |
[
493,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
492,
1
] |
Mathlib/Analysis/Normed/Group/InfiniteSum.lean
|
summable_iff_vanishing_norm
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nα : Type ?u.2223\nE : Type u_1\nF : Type ?u.2229\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : CompleteSpace E\nf : ι → E\n⊢ Summable f ↔ ∀ (ε : ℝ), ε > 0 → ∃ s, ∀ (t : Finset ι), Disjoint t s → ‖∑ i in t, f i‖ < ε",
"tactic": "rw [summable_iff_cauchySeq_finset, cauchySeq_finset_iff_vanishing_norm]"
}
] |
[
54,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
|
UniformFun.hasBasis_nhds
|
[] |
[
340,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
11
] |
Mathlib/Data/Set/NAry.lean
|
Set.image2_image_right
|
[
{
"state_after": "case h\nα : Type u_2\nα' : Type ?u.34558\nβ : Type u_4\nβ' : Type ?u.34564\nγ : Type u_3\nγ' : Type ?u.34570\nδ : Type u_1\nδ' : Type ?u.34576\nε : Type ?u.34579\nε' : Type ?u.34582\nζ : Type ?u.34585\nζ' : Type ?u.34588\nν : Type ?u.34591\nf✝ f' : α → β → γ\ng✝ g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nf : α → γ → δ\ng : β → γ\nx✝ : δ\n⊢ x✝ ∈ image2 f s (g '' t) ↔ x✝ ∈ image2 (fun a b => f a (g b)) s t",
"state_before": "α : Type u_2\nα' : Type ?u.34558\nβ : Type u_4\nβ' : Type ?u.34564\nγ : Type u_3\nγ' : Type ?u.34570\nδ : Type u_1\nδ' : Type ?u.34576\nε : Type ?u.34579\nε' : Type ?u.34582\nζ : Type ?u.34585\nζ' : Type ?u.34588\nν : Type ?u.34591\nf✝ f' : α → β → γ\ng✝ g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nf : α → γ → δ\ng : β → γ\n⊢ image2 f s (g '' t) = image2 (fun a b => f a (g b)) s t",
"tactic": "ext"
},
{
"state_after": "case h.mp\nα : Type u_2\nα' : Type ?u.34558\nβ : Type u_4\nβ' : Type ?u.34564\nγ : Type u_3\nγ' : Type ?u.34570\nδ : Type u_1\nδ' : Type ?u.34576\nε : Type ?u.34579\nε' : Type ?u.34582\nζ : Type ?u.34585\nζ' : Type ?u.34588\nν : Type ?u.34591\nf✝ f' : α → β → γ\ng✝ g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nf : α → γ → δ\ng : β → γ\nx✝ : δ\n⊢ x✝ ∈ image2 f s (g '' t) → x✝ ∈ image2 (fun a b => f a (g b)) s t\n\ncase h.mpr\nα : Type u_2\nα' : Type ?u.34558\nβ : Type u_4\nβ' : Type ?u.34564\nγ : Type u_3\nγ' : Type ?u.34570\nδ : Type u_1\nδ' : Type ?u.34576\nε : Type ?u.34579\nε' : Type ?u.34582\nζ : Type ?u.34585\nζ' : Type ?u.34588\nν : Type ?u.34591\nf✝ f' : α → β → γ\ng✝ g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nf : α → γ → δ\ng : β → γ\nx✝ : δ\n⊢ x✝ ∈ image2 (fun a b => f a (g b)) s t → x✝ ∈ image2 f s (g '' t)",
"state_before": "case h\nα : Type u_2\nα' : Type ?u.34558\nβ : Type u_4\nβ' : Type ?u.34564\nγ : Type u_3\nγ' : Type ?u.34570\nδ : Type u_1\nδ' : Type ?u.34576\nε : Type ?u.34579\nε' : Type ?u.34582\nζ : Type ?u.34585\nζ' : Type ?u.34588\nν : Type ?u.34591\nf✝ f' : α → β → γ\ng✝ g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nf : α → γ → δ\ng : β → γ\nx✝ : δ\n⊢ x✝ ∈ image2 f s (g '' t) ↔ x✝ ∈ image2 (fun a b => f a (g b)) s t",
"tactic": "constructor"
},
{
"state_after": "case h.mp.intro.intro.intro.intro.intro.intro\nα : Type u_2\nα' : Type ?u.34558\nβ : Type u_4\nβ' : Type ?u.34564\nγ : Type u_3\nγ' : Type ?u.34570\nδ : Type u_1\nδ' : Type ?u.34576\nε : Type ?u.34579\nε' : Type ?u.34582\nζ : Type ?u.34585\nζ' : Type ?u.34588\nν : Type ?u.34591\nf✝ f' : α → β → γ\ng✝ g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na✝ a' : α\nb✝ b' : β\nc c' : γ\nd d' : δ\nf : α → γ → δ\ng : β → γ\na : α\nha : a ∈ s\nb : β\nhb : b ∈ t\n⊢ f a (g b) ∈ image2 (fun a b => f a (g b)) s t",
"state_before": "case h.mp\nα : Type u_2\nα' : Type ?u.34558\nβ : Type u_4\nβ' : Type ?u.34564\nγ : Type u_3\nγ' : Type ?u.34570\nδ : Type u_1\nδ' : Type ?u.34576\nε : Type ?u.34579\nε' : Type ?u.34582\nζ : Type ?u.34585\nζ' : Type ?u.34588\nν : Type ?u.34591\nf✝ f' : α → β → γ\ng✝ g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nf : α → γ → δ\ng : β → γ\nx✝ : δ\n⊢ x✝ ∈ image2 f s (g '' t) → x✝ ∈ image2 (fun a b => f a (g b)) s t",
"tactic": "rintro ⟨a, _, ha, ⟨b, hb, rfl⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro.intro.intro.intro.intro.intro\nα : Type u_2\nα' : Type ?u.34558\nβ : Type u_4\nβ' : Type ?u.34564\nγ : Type u_3\nγ' : Type ?u.34570\nδ : Type u_1\nδ' : Type ?u.34576\nε : Type ?u.34579\nε' : Type ?u.34582\nζ : Type ?u.34585\nζ' : Type ?u.34588\nν : Type ?u.34591\nf✝ f' : α → β → γ\ng✝ g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na✝ a' : α\nb✝ b' : β\nc c' : γ\nd d' : δ\nf : α → γ → δ\ng : β → γ\na : α\nha : a ∈ s\nb : β\nhb : b ∈ t\n⊢ f a (g b) ∈ image2 (fun a b => f a (g b)) s t",
"tactic": "refine' ⟨a, b, ha, hb, rfl⟩"
},
{
"state_after": "case h.mpr.intro.intro.intro.intro\nα : Type u_2\nα' : Type ?u.34558\nβ : Type u_4\nβ' : Type ?u.34564\nγ : Type u_3\nγ' : Type ?u.34570\nδ : Type u_1\nδ' : Type ?u.34576\nε : Type ?u.34579\nε' : Type ?u.34582\nζ : Type ?u.34585\nζ' : Type ?u.34588\nν : Type ?u.34591\nf✝ f' : α → β → γ\ng✝ g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na✝ a' : α\nb✝ b' : β\nc c' : γ\nd d' : δ\nf : α → γ → δ\ng : β → γ\na : α\nb : β\nha : a ∈ s\nhb : b ∈ t\n⊢ (fun a b => f a (g b)) a b ∈ image2 f s (g '' t)",
"state_before": "case h.mpr\nα : Type u_2\nα' : Type ?u.34558\nβ : Type u_4\nβ' : Type ?u.34564\nγ : Type u_3\nγ' : Type ?u.34570\nδ : Type u_1\nδ' : Type ?u.34576\nε : Type ?u.34579\nε' : Type ?u.34582\nζ : Type ?u.34585\nζ' : Type ?u.34588\nν : Type ?u.34591\nf✝ f' : α → β → γ\ng✝ g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nf : α → γ → δ\ng : β → γ\nx✝ : δ\n⊢ x✝ ∈ image2 (fun a b => f a (g b)) s t → x✝ ∈ image2 f s (g '' t)",
"tactic": "rintro ⟨a, b, ha, hb, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro.intro.intro.intro\nα : Type u_2\nα' : Type ?u.34558\nβ : Type u_4\nβ' : Type ?u.34564\nγ : Type u_3\nγ' : Type ?u.34570\nδ : Type u_1\nδ' : Type ?u.34576\nε : Type ?u.34579\nε' : Type ?u.34582\nζ : Type ?u.34585\nζ' : Type ?u.34588\nν : Type ?u.34591\nf✝ f' : α → β → γ\ng✝ g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na✝ a' : α\nb✝ b' : β\nc c' : γ\nd d' : δ\nf : α → γ → δ\ng : β → γ\na : α\nb : β\nha : a ∈ s\nhb : b ∈ t\n⊢ (fun a b => f a (g b)) a b ∈ image2 f s (g '' t)",
"tactic": "refine' ⟨a, _, ha, ⟨b, hb, rfl⟩, rfl⟩"
}
] |
[
297,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
291,
1
] |
Mathlib/Data/List/Rdrop.lean
|
List.rdropWhile_concat_neg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\nl : List α\nn : ℕ\nx : α\nh : ¬p x = true\n⊢ rdropWhile p (l ++ [x]) = l ++ [x]",
"tactic": "rw [rdropWhile_concat, if_neg h]"
}
] |
[
120,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.liftRel_think_left
|
[] |
[
1205,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1201,
1
] |
Std/Data/Int/DivMod.lean
|
Int.natAbs_dvd_natAbs
|
[
{
"state_after": "a b : Int\nx✝ : natAbs a ∣ natAbs b\nk : Nat\nhk : natAbs b = natAbs a * k\n⊢ a ∣ b",
"state_before": "a b : Int\n⊢ natAbs a ∣ natAbs b ↔ a ∣ b",
"tactic": "refine ⟨fun ⟨k, hk⟩ => ?_, fun ⟨k, hk⟩ => ⟨natAbs k, hk.symm ▸ natAbs_mul a k⟩⟩"
},
{
"state_after": "a b : Int\nx✝ : natAbs a ∣ natAbs b\nk : Nat\nhk : b = a * ↑k ∨ b = -(a * ↑k)\n⊢ a ∣ b",
"state_before": "a b : Int\nx✝ : natAbs a ∣ natAbs b\nk : Nat\nhk : natAbs b = natAbs a * k\n⊢ a ∣ b",
"tactic": "rw [← natAbs_ofNat k, ← natAbs_mul, natAbs_eq_natAbs_iff] at hk"
},
{
"state_after": "case inl\na : Int\nk : Nat\nx✝ : natAbs a ∣ natAbs (a * ↑k)\n⊢ a ∣ a * ↑k\n\ncase inr\na : Int\nk : Nat\nx✝ : natAbs a ∣ natAbs (-(a * ↑k))\n⊢ a ∣ -(a * ↑k)",
"state_before": "a b : Int\nx✝ : natAbs a ∣ natAbs b\nk : Nat\nhk : b = a * ↑k ∨ b = -(a * ↑k)\n⊢ a ∣ b",
"tactic": "cases hk <;> subst b"
},
{
"state_after": "no goals",
"state_before": "case inl\na : Int\nk : Nat\nx✝ : natAbs a ∣ natAbs (a * ↑k)\n⊢ a ∣ a * ↑k",
"tactic": "apply Int.dvd_mul_right"
},
{
"state_after": "case inr\na : Int\nk : Nat\nx✝ : natAbs a ∣ natAbs (-(a * ↑k))\n⊢ a ∣ a * -↑k",
"state_before": "case inr\na : Int\nk : Nat\nx✝ : natAbs a ∣ natAbs (-(a * ↑k))\n⊢ a ∣ -(a * ↑k)",
"tactic": "rw [← Int.mul_neg]"
},
{
"state_after": "no goals",
"state_before": "case inr\na : Int\nk : Nat\nx✝ : natAbs a ∣ natAbs (-(a * ↑k))\n⊢ a ∣ a * -↑k",
"tactic": "apply Int.dvd_mul_right"
}
] |
[
649,
48
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
644,
9
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.inv_le_of_le_mul
|
[
{
"state_after": "no goals",
"state_before": "r p : ℝ≥0\nh : 1 ≤ r * p\n⊢ r⁻¹ ≤ p",
"tactic": "by_cases r = 0 <;> simp [*, inv_le]"
}
] |
[
786,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
785,
1
] |
Mathlib/Analysis/Convex/Combination.lean
|
Finset.centerMass_ite_eq
|
[
{
"state_after": "R : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i ∈ t\n⊢ ∑ i_1 in t, (if i = i_1 then 1 else 0) • z i_1 = z i\n\ncase hw\nR : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i ∈ t\n⊢ (∑ i_1 in t, if i = i_1 then 1 else 0) = 1",
"state_before": "R : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i ∈ t\n⊢ centerMass t (fun j => if i = j then 1 else 0) z = z i",
"tactic": "rw [Finset.centerMass_eq_of_sum_1]"
},
{
"state_after": "R : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i ∈ t\n⊢ ∑ i_1 in t, (if i = i_1 then 1 else 0) • z i_1 = ∑ j in t, if i = j then z i else 0\n\nR : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i ∈ t\n⊢ (∑ j in t, if i = j then z i else 0) = z i\n\ncase hw\nR : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i ∈ t\n⊢ (∑ i_1 in t, if i = i_1 then 1 else 0) = 1",
"state_before": "R : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i ∈ t\n⊢ ∑ i_1 in t, (if i = i_1 then 1 else 0) • z i_1 = z i\n\ncase hw\nR : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i ∈ t\n⊢ (∑ i_1 in t, if i = i_1 then 1 else 0) = 1",
"tactic": "trans ∑ j in t, if i = j then z i else 0"
},
{
"state_after": "case e_f.h\nR : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni✝ j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i✝ ∈ t\ni : ι\n⊢ (if i✝ = i then 1 else 0) • z i = if i✝ = i then z i✝ else 0",
"state_before": "R : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i ∈ t\n⊢ ∑ i_1 in t, (if i = i_1 then 1 else 0) • z i_1 = ∑ j in t, if i = j then z i else 0",
"tactic": "congr with i"
},
{
"state_after": "case e_f.h.inl\nR : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni✝ j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i✝ ∈ t\ni : ι\nh : i✝ = i\n⊢ 1 • z i = z i✝\n\ncase e_f.h.inr\nR : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni✝ j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i✝ ∈ t\ni : ι\nh : ¬i✝ = i\n⊢ 0 • z i = 0",
"state_before": "case e_f.h\nR : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni✝ j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i✝ ∈ t\ni : ι\n⊢ (if i✝ = i then 1 else 0) • z i = if i✝ = i then z i✝ else 0",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case e_f.h.inl\nR : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni✝ j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i✝ ∈ t\ni : ι\nh : i✝ = i\n⊢ 1 • z i = z i✝\n\ncase e_f.h.inr\nR : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni✝ j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i✝ ∈ t\ni : ι\nh : ¬i✝ = i\n⊢ 0 • z i = 0",
"tactic": "exacts [h ▸ one_smul _ _, zero_smul _ _]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i ∈ t\n⊢ (∑ j in t, if i = j then z i else 0) = z i",
"tactic": "rw [sum_ite_eq, if_pos hi]"
},
{
"state_after": "no goals",
"state_before": "case hw\nR : Type u_3\nE : Type u_2\nF : Type ?u.57939\nι : Type u_1\nι' : Type ?u.57945\nα : Type ?u.57948\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\nhi : i ∈ t\n⊢ (∑ i_1 in t, if i = i_1 then 1 else 0) = 1",
"tactic": "rw [sum_ite_eq, if_pos hi]"
}
] |
[
116,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean
|
CategoryTheory.Presieve.extend_agrees
|
[
{
"state_after": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y ⟶ X\nhf : R f\nh : ∃ h g, R g ∧ h ≫ g = f\n⊢ FamilyOfElements.sieveExtend x f (_ : f ∈ (generate R).arrows) = x f hf",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y ⟶ X\nhf : R f\n⊢ FamilyOfElements.sieveExtend x f (_ : f ∈ (generate R).arrows) = x f hf",
"tactic": "have h := (le_generate R Y hf).choose_spec"
},
{
"state_after": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y ⟶ X\nhf : R f\nh : ∃ h g, R g ∧ h ≫ g = f\n⊢ P.map (Exists.choose (_ : ∃ h g, R g ∧ h ≫ g = f)).op\n (x (Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h g, R g ∧ h ≫ g = f) ≫ g = f))\n (_ : R (Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h g, R g ∧ h ≫ g = f) ≫ g = f)))) =\n x f hf",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y ⟶ X\nhf : R f\nh : ∃ h g, R g ∧ h ≫ g = f\n⊢ FamilyOfElements.sieveExtend x f (_ : f ∈ (generate R).arrows) = x f hf",
"tactic": "unfold FamilyOfElements.sieveExtend"
},
{
"state_after": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y ⟶ X\nhf : R f\nh : ∃ h g, R g ∧ h ≫ g = f\n⊢ P.map (𝟙 Y).op (x f hf) = x f hf\n\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y ⟶ X\nhf : R f\nh : ∃ h g, R g ∧ h ≫ g = f\n⊢ Exists.choose h ≫ Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h g, R g ∧ h ≫ g = f) ≫ g = f) = 𝟙 Y ≫ f",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y ⟶ X\nhf : R f\nh : ∃ h g, R g ∧ h ≫ g = f\n⊢ P.map (Exists.choose (_ : ∃ h g, R g ∧ h ≫ g = f)).op\n (x (Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h g, R g ∧ h ≫ g = f) ≫ g = f))\n (_ : R (Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h g, R g ∧ h ≫ g = f) ≫ g = f)))) =\n x f hf",
"tactic": "rw [t h.choose (𝟙 _) _ hf _]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y ⟶ X\nhf : R f\nh : ∃ h g, R g ∧ h ≫ g = f\n⊢ P.map (𝟙 Y).op (x f hf) = x f hf",
"tactic": "simp"
},
{
"state_after": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y ⟶ X\nhf : R f\nh : ∃ h g, R g ∧ h ≫ g = f\n⊢ Exists.choose h ≫ Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h g, R g ∧ h ≫ g = f) ≫ g = f) = f",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y ⟶ X\nhf : R f\nh : ∃ h g, R g ∧ h ≫ g = f\n⊢ Exists.choose h ≫ Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h g, R g ∧ h ≫ g = f) ≫ g = f) = 𝟙 Y ≫ f",
"tactic": "rw [id_comp]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y ⟶ X\nhf : R f\nh : ∃ h g, R g ∧ h ≫ g = f\n⊢ Exists.choose h ≫ Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h g, R g ∧ h ≫ g = f) ≫ g = f) = f",
"tactic": "exact h.choose_spec.choose_spec.2"
}
] |
[
206,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean
|
FormalMultilinearSeries.order_eq_find
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\n𝕜' : Type u'\nE : Type v\nF : Type w\nG : Type x\ninst✝¹¹ : CommRing 𝕜\nn : ℕ\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module 𝕜 E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : TopologicalAddGroup E\ninst✝⁶ : ContinuousConstSMul 𝕜 E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜 F\ninst✝³ : TopologicalSpace F\ninst✝² : TopologicalAddGroup F\ninst✝¹ : ContinuousConstSMul 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\ninst✝ : DecidablePred fun n => p n ≠ 0\nhp : ∃ n, p n ≠ 0\n⊢ order p = Nat.find hp",
"tactic": "convert Nat.sInf_def hp"
}
] |
[
230,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
WithTop.succ_coe_top
|
[] |
[
1059,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1058,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.EventuallyEq.eventually
|
[] |
[
1437,
4
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1435,
1
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
AffineIndependent.vectorSpan_eq_of_le_of_card_eq_finrank_add_one
|
[] |
[
229,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Algebra/Periodic.lean
|
Function.Periodic.smul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nf g : α → β\nc c₁ c₂ x : α\ninst✝¹ : Add α\ninst✝ : SMul γ β\nh : Periodic f c\na : γ\n⊢ Periodic (a • f) c",
"tactic": "simp_all"
}
] |
[
107,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
11
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.nsmul_apply
|
[] |
[
260,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
mul_inv_lt_iff_le_mul'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c d : α\n⊢ a * b⁻¹ < c ↔ a < b * c",
"tactic": "rw [← inv_mul_lt_iff_lt_mul, mul_comm]"
}
] |
[
560,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
559,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
Equiv.Perm.cycleType_prime_order
|
[
{
"state_after": "case refine_1\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nhσ : Nat.Prime (orderOf σ)\n⊢ ↑card (cycleType σ) = ↑card (cycleType σ) - 1 + 1\n\ncase refine_2\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nhσ : Nat.Prime (orderOf σ)\nn : ℕ\nhn : n ∈ cycleType σ\n⊢ n = orderOf σ",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nhσ : Nat.Prime (orderOf σ)\n⊢ ∃ n, cycleType σ = replicate (n + 1) (orderOf σ)",
"tactic": "refine ⟨Multiset.card σ.cycleType - 1, eq_replicate.2 ⟨?_, fun n hn ↦ ?_⟩⟩"
},
{
"state_after": "case refine_1\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nhσ : Nat.Prime (orderOf σ)\n⊢ 1 ≤ ↑card (cycleType σ)",
"state_before": "case refine_1\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nhσ : Nat.Prime (orderOf σ)\n⊢ ↑card (cycleType σ) = ↑card (cycleType σ) - 1 + 1",
"tactic": "rw [tsub_add_cancel_of_le]"
},
{
"state_after": "case refine_1\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nhσ : Nat.Prime (orderOf σ)\n⊢ ¬orderOf σ = 1",
"state_before": "case refine_1\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nhσ : Nat.Prime (orderOf σ)\n⊢ 1 ≤ ↑card (cycleType σ)",
"tactic": "rw [Nat.succ_le_iff, card_cycleType_pos, Ne.def, ← orderOf_eq_one_iff]"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nhσ : Nat.Prime (orderOf σ)\n⊢ ¬orderOf σ = 1",
"tactic": "exact hσ.ne_one"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nhσ : Nat.Prime (orderOf σ)\nn : ℕ\nhn : n ∈ cycleType σ\n⊢ n = orderOf σ",
"tactic": "exact (hσ.eq_one_or_self_of_dvd n (dvd_of_mem_cycleType hn)).resolve_left\n (one_lt_of_mem_cycleType hn).ne'"
}
] |
[
213,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.radical_pow
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type ?u.339982\ninst✝ : CommSemiring R\nI J K L : Ideal R\nn : ℕ\nH : n > 0\n⊢ ¬Nat.zero > 0",
"tactic": "decide"
},
{
"state_after": "R : Type u\nι : Type ?u.339982\ninst✝ : CommSemiring R\nI J K L : Ideal R\nn✝ : ℕ\nH✝¹ : n✝ > 0\nn : ℕ\nih : n > 0 → radical (I ^ n) = radical I\nH✝ : Nat.succ n > 0\nH : 0 < n\n⊢ radical (I * I ^ n) = radical I ⊓ radical (I ^ n)",
"state_before": "R : Type u\nι : Type ?u.339982\ninst✝ : CommSemiring R\nI J K L : Ideal R\nn✝ : ℕ\nH✝¹ : n✝ > 0\nn : ℕ\nih : n > 0 → radical (I ^ n) = radical I\nH✝ : Nat.succ n > 0\nH : 0 < n\n⊢ radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n)",
"tactic": "rw [pow_succ]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type ?u.339982\ninst✝ : CommSemiring R\nI J K L : Ideal R\nn✝ : ℕ\nH✝¹ : n✝ > 0\nn : ℕ\nih : n > 0 → radical (I ^ n) = radical I\nH✝ : Nat.succ n > 0\nH : 0 < n\n⊢ radical (I * I ^ n) = radical I ⊓ radical (I ^ n)",
"tactic": "exact radical_mul _ _"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type ?u.339982\ninst✝ : CommSemiring R\nI J K L : Ideal R\nn✝ : ℕ\nH✝¹ : n✝ > 0\nn : ℕ\nih : n > 0 → radical (I ^ n) = radical I\nH✝ : Nat.succ n > 0\nH : 0 < n\n⊢ radical I ⊓ radical (I ^ n) = radical I ⊓ radical I",
"tactic": "rw [ih H]"
}
] |
[
1020,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1007,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.prod_factorization_eq_prod_factors
|
[
{
"state_after": "n : ℕ\nβ : Type u_1\ninst✝ : CommMonoid β\nf : ℕ → β\n⊢ ∀ (x : ℕ), x ∈ toFinset (factors n) → (fun p x => f p) x (↑(factorization n) x) = f x",
"state_before": "n : ℕ\nβ : Type u_1\ninst✝ : CommMonoid β\nf : ℕ → β\n⊢ (Finsupp.prod (factorization n) fun p x => f p) = ∏ p in toFinset (factors n), f p",
"tactic": "apply prod_congr support_factorization"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nβ : Type u_1\ninst✝ : CommMonoid β\nf : ℕ → β\n⊢ ∀ (x : ℕ), x ∈ toFinset (factors n) → (fun p x => f p) x (↑(factorization n) x) = f x",
"tactic": "simp"
}
] |
[
241,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.cardFactors_multiset_prod
|
[
{
"state_after": "R : Type ?u.554754\ns : Multiset ℕ\n⊢ Multiset.prod s ≠ 0 → ↑Ω (Multiset.prod s) = Multiset.sum (Multiset.map (↑Ω) s)",
"state_before": "R : Type ?u.554754\ns : Multiset ℕ\nh0 : Multiset.prod s ≠ 0\n⊢ ↑Ω (Multiset.prod s) = Multiset.sum (Multiset.map (↑Ω) s)",
"tactic": "revert h0"
},
{
"state_after": "case refine_1\nR : Type ?u.554754\ns : Multiset ℕ\n⊢ Multiset.prod 0 ≠ 0 → ↑Ω (Multiset.prod 0) = Multiset.sum (Multiset.map (↑Ω) 0)\n\ncase refine_2\nR : Type ?u.554754\ns : Multiset ℕ\n⊢ ∀ ⦃a : ℕ⦄ {s : Multiset ℕ},\n (Multiset.prod s ≠ 0 → ↑Ω (Multiset.prod s) = Multiset.sum (Multiset.map (↑Ω) s)) →\n Multiset.prod (a ::ₘ s) ≠ 0 → ↑Ω (Multiset.prod (a ::ₘ s)) = Multiset.sum (Multiset.map (↑Ω) (a ::ₘ s))",
"state_before": "R : Type ?u.554754\ns : Multiset ℕ\n⊢ Multiset.prod s ≠ 0 → ↑Ω (Multiset.prod s) = Multiset.sum (Multiset.map (↑Ω) s)",
"tactic": "refine s.induction_on ?_ ?_"
},
{
"state_after": "case refine_2\nR : Type ?u.554754\ns : Multiset ℕ\na : ℕ\nt : Multiset ℕ\nh : Multiset.prod t ≠ 0 → ↑Ω (Multiset.prod t) = Multiset.sum (Multiset.map (↑Ω) t)\nh0 : Multiset.prod (a ::ₘ t) ≠ 0\n⊢ ↑Ω (Multiset.prod (a ::ₘ t)) = Multiset.sum (Multiset.map (↑Ω) (a ::ₘ t))",
"state_before": "case refine_2\nR : Type ?u.554754\ns : Multiset ℕ\n⊢ ∀ ⦃a : ℕ⦄ {s : Multiset ℕ},\n (Multiset.prod s ≠ 0 → ↑Ω (Multiset.prod s) = Multiset.sum (Multiset.map (↑Ω) s)) →\n Multiset.prod (a ::ₘ s) ≠ 0 → ↑Ω (Multiset.prod (a ::ₘ s)) = Multiset.sum (Multiset.map (↑Ω) (a ::ₘ s))",
"tactic": "intro a t h h0"
},
{
"state_after": "case refine_2\nR : Type ?u.554754\ns : Multiset ℕ\na : ℕ\nt : Multiset ℕ\nh : Multiset.prod t ≠ 0 → ↑Ω (Multiset.prod t) = Multiset.sum (Multiset.map (↑Ω) t)\nh0 : a ≠ 0 ∧ Multiset.prod t ≠ 0\n⊢ ↑Ω (Multiset.prod (a ::ₘ t)) = Multiset.sum (Multiset.map (↑Ω) (a ::ₘ t))",
"state_before": "case refine_2\nR : Type ?u.554754\ns : Multiset ℕ\na : ℕ\nt : Multiset ℕ\nh : Multiset.prod t ≠ 0 → ↑Ω (Multiset.prod t) = Multiset.sum (Multiset.map (↑Ω) t)\nh0 : Multiset.prod (a ::ₘ t) ≠ 0\n⊢ ↑Ω (Multiset.prod (a ::ₘ t)) = Multiset.sum (Multiset.map (↑Ω) (a ::ₘ t))",
"tactic": "rw [Multiset.prod_cons, mul_ne_zero_iff] at h0"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nR : Type ?u.554754\ns : Multiset ℕ\na : ℕ\nt : Multiset ℕ\nh : Multiset.prod t ≠ 0 → ↑Ω (Multiset.prod t) = Multiset.sum (Multiset.map (↑Ω) t)\nh0 : a ≠ 0 ∧ Multiset.prod t ≠ 0\n⊢ ↑Ω (Multiset.prod (a ::ₘ t)) = Multiset.sum (Multiset.map (↑Ω) (a ::ₘ t))",
"tactic": "simp [h0, cardFactors_mul, h]"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nR : Type ?u.554754\ns : Multiset ℕ\n⊢ Multiset.prod 0 ≠ 0 → ↑Ω (Multiset.prod 0) = Multiset.sum (Multiset.map (↑Ω) 0)",
"tactic": "simp"
}
] |
[
887,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
880,
1
] |
Mathlib/Data/Finite/Basic.lean
|
Finite.prod_right
|
[] |
[
81,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Algebra/GCDMonoid/Multiset.lean
|
Multiset.lcm_singleton
|
[] |
[
58,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/MeasureTheory/Integral/CircleIntegral.lean
|
circleIntegrable_iff
|
[
{
"state_after": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : R = 0\n⊢ CircleIntegrable f c R ↔ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)\n\ncase neg\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\n⊢ CircleIntegrable f c R ↔ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\n⊢ CircleIntegrable f c R ↔ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)",
"tactic": "by_cases h₀ : R = 0"
},
{
"state_after": "case neg\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)\n⊢ CircleIntegrable f c R",
"state_before": "case neg\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\n⊢ CircleIntegrable f c R ↔ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)",
"tactic": "refine' ⟨fun h => h.out, fun h => _⟩"
},
{
"state_after": "case neg\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π))\n⊢ IntegrableOn (fun θ => f (circleMap c R θ)) (Ι 0 (2 * π))",
"state_before": "case neg\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)\n⊢ CircleIntegrable f c R",
"tactic": "simp only [CircleIntegrable, intervalIntegrable_iff, deriv_circleMap] at h ⊢"
},
{
"state_after": "case neg.refine'_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π))\n⊢ AEStronglyMeasurable (fun θ => f (circleMap c R θ)) (Measure.restrict volume (Ι 0 (2 * π)))\n\ncase neg.refine'_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π))\n⊢ ∀ᵐ (a : ℝ) ∂Measure.restrict volume (Ι 0 (2 * π)),\n ‖f (circleMap c R a)‖ ≤ (Abs.abs R)⁻¹ * ‖(circleMap 0 R a * I) • f (circleMap c R a)‖",
"state_before": "case neg\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π))\n⊢ IntegrableOn (fun θ => f (circleMap c R θ)) (Ι 0 (2 * π))",
"tactic": "refine' (h.norm.const_mul (|R|)⁻¹).mono' _ _"
},
{
"state_after": "no goals",
"state_before": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : R = 0\n⊢ CircleIntegrable f c R ↔ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)",
"tactic": "simp [h₀, const]"
},
{
"state_after": "case neg.refine'_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π))\nH : ∀ {θ : ℝ}, circleMap 0 R θ * I ≠ 0\n⊢ AEStronglyMeasurable (fun θ => f (circleMap c R θ)) (Measure.restrict volume (Ι 0 (2 * π)))",
"state_before": "case neg.refine'_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π))\n⊢ AEStronglyMeasurable (fun θ => f (circleMap c R θ)) (Measure.restrict volume (Ι 0 (2 * π)))",
"tactic": "have H : ∀ {θ}, circleMap 0 R θ * I ≠ 0 := fun {θ} => by simp [h₀, I_ne_zero]"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π))\nH : ∀ {θ : ℝ}, circleMap 0 R θ * I ≠ 0\n⊢ AEStronglyMeasurable (fun θ => f (circleMap c R θ)) (Measure.restrict volume (Ι 0 (2 * π)))",
"tactic": "simpa only [inv_smul_smul₀ H]\n using ((continuous_circleMap 0 R).aestronglyMeasurable.mul_const\n I).aemeasurable.inv.aestronglyMeasurable.smul h.aestronglyMeasurable"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π))\nθ : ℝ\n⊢ circleMap 0 R θ * I ≠ 0",
"tactic": "simp [h₀, I_ne_zero]"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π))\n⊢ ∀ᵐ (a : ℝ) ∂Measure.restrict volume (Ι 0 (2 * π)),\n ‖f (circleMap c R a)‖ ≤ (Abs.abs R)⁻¹ * ‖(circleMap 0 R a * I) • f (circleMap c R a)‖",
"tactic": "simp [norm_smul, h₀]"
}
] |
[
288,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
RingHom.one_def
|
[] |
[
730,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
729,
1
] |
Mathlib/Topology/Algebra/GroupWithZero.lean
|
Filter.Tendsto.inv₀
|
[] |
[
119,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
InnerProductGeometry.angle_add_le_pi_div_two_of_inner_eq_zero
|
[
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\n⊢ 0 ≤ ‖x‖ / ‖x + y‖",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\n⊢ angle x (x + y) ≤ π / 2",
"tactic": "rw [angle_add_eq_arccos_of_inner_eq_zero h, Real.arccos_le_pi_div_two]"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\n⊢ 0 ≤ ‖x‖ / ‖x + y‖",
"tactic": "exact div_nonneg (norm_nonneg _) (norm_nonneg _)"
}
] |
[
126,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Topology/ContinuousFunction/Basic.lean
|
ContinuousMap.toFun_eq_coe
|
[] |
[
100,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/Data/Int/Parity.lean
|
Int.odd_sub'
|
[
{
"state_after": "no goals",
"state_before": "m n : ℤ\n⊢ Odd (m - n) ↔ (Odd n ↔ Even m)",
"tactic": "rw [odd_iff_not_even, even_sub, not_iff, not_iff_comm, odd_iff_not_even]"
}
] |
[
193,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.subset_insert_iff_of_not_mem
|
[] |
[
1169,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1168,
1
] |
Mathlib/CategoryTheory/Abelian/Homology.lean
|
homology.π'_ι
|
[
{
"state_after": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\n⊢ (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫\n ((homologyIsoCokernelLift f g w).hom ≫ Abelian.homologyCToK f g w) ≫ kernel.ι (cokernel.desc f g w) =\n kernel.ι g ≫ cokernel.π f",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\n⊢ π' f g w ≫ ι f g w = kernel.ι g ≫ cokernel.π f",
"tactic": "dsimp [π', ι, homologyIsoKernelDesc]"
},
{
"state_after": "no goals",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\n⊢ (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫\n ((homologyIsoCokernelLift f g w).hom ≫ Abelian.homologyCToK f g w) ≫ kernel.ι (cokernel.desc f g w) =\n kernel.ι g ≫ cokernel.π f",
"tactic": "simp"
}
] |
[
193,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.factorization_one
|
[
{
"state_after": "no goals",
"state_before": "⊢ factorization 1 = 0",
"tactic": "simp [factorization]"
}
] |
[
127,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
contDiffWithinAt_nat
|
[] |
[
427,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
StrictConvexOn.lt_on_open_segment'
|
[
{
"state_after": "case h₁.h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.484521\nα : Type ?u.484524\nβ : Type u_3\nι : Type ?u.484530\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : StrictConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ f x ≤ max (f x) (f y)\n\ncase h₂.h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.484521\nα : Type ?u.484524\nβ : Type u_3\nι : Type ?u.484530\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : StrictConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ f y ≤ max (f x) (f y)",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.484521\nα : Type ?u.484524\nβ : Type u_3\nι : Type ?u.484530\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : StrictConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • f x + b • f y ≤ a • max (f x) (f y) + b • max (f x) (f y)",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case h₁.h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.484521\nα : Type ?u.484524\nβ : Type u_3\nι : Type ?u.484530\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : StrictConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ f x ≤ max (f x) (f y)",
"tactic": "apply le_max_left"
},
{
"state_after": "no goals",
"state_before": "case h₂.h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.484521\nα : Type ?u.484524\nβ : Type u_3\nι : Type ?u.484530\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : StrictConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ f y ≤ max (f x) (f y)",
"tactic": "apply le_max_right"
}
] |
[
675,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
666,
1
] |
Mathlib/MeasureTheory/Decomposition/Jordan.lean
|
MeasureTheory.SignedMeasure.of_diff_eq_zero_of_symmDiff_eq_zero_positive
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u ∆ v) = 0\n⊢ ↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"state_before": "α : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ ↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0",
"tactic": "rw [restrict_le_restrict_iff] at hsu hsv"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u ∆ v) = 0\na : ↑0 (u \\ v) ≤ ↑s (u \\ v)\n⊢ ↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"state_before": "α : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u ∆ v) = 0\n⊢ ↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"tactic": "have a := hsu (hu.diff hv) (u.diff_subset v)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u ∆ v) = 0\na : ↑0 (u \\ v) ≤ ↑s (u \\ v)\nb : ↑0 (v \\ u) ≤ ↑s (v \\ u)\n⊢ ↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"state_before": "α : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u ∆ v) = 0\na : ↑0 (u \\ v) ≤ ↑s (u \\ v)\n⊢ ↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"tactic": "have b := hsv (hv.diff hu) (v.diff_subset u)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u \\ v) + ↑s (v \\ u) = 0\na : ↑0 (u \\ v) ≤ ↑s (u \\ v)\nb : ↑0 (v \\ u) ≤ ↑s (v \\ u)\n⊢ ↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"state_before": "α : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u ∆ v) = 0\na : ↑0 (u \\ v) ≤ ↑s (u \\ v)\nb : ↑0 (v \\ u) ≤ ↑s (v \\ u)\n⊢ ↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"tactic": "erw [of_union (Set.disjoint_of_subset_left (u.diff_subset v) disjoint_sdiff_self_right)\n (hu.diff hv) (hv.diff hu)] at hs"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u \\ v) + ↑s (v \\ u) = 0\na : 0 ≤ ↑s (u \\ v)\nb : 0 ≤ ↑s (v \\ u)\n⊢ ↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"state_before": "α : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u \\ v) + ↑s (v \\ u) = 0\na : ↑0 (u \\ v) ≤ ↑s (u \\ v)\nb : ↑0 (v \\ u) ≤ ↑s (v \\ u)\n⊢ ↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"tactic": "rw [zero_apply] at a b"
},
{
"state_after": "case left\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u \\ v) + ↑s (v \\ u) = 0\na : 0 ≤ ↑s (u \\ v)\nb : 0 ≤ ↑s (v \\ u)\n⊢ ↑s (u \\ v) = 0\n\ncase right\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u \\ v) + ↑s (v \\ u) = 0\na : 0 ≤ ↑s (u \\ v)\nb : 0 ≤ ↑s (v \\ u)\n⊢ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"state_before": "α : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u \\ v) + ↑s (v \\ u) = 0\na : 0 ≤ ↑s (u \\ v)\nb : 0 ≤ ↑s (v \\ u)\n⊢ ↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case left\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u \\ v) + ↑s (v \\ u) = 0\na : 0 ≤ ↑s (u \\ v)\nb : 0 ≤ ↑s (v \\ u)\n⊢ ↑s (u \\ v) = 0\n\ncase right\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j\nhs : ↑s (u \\ v) + ↑s (v \\ u) = 0\na : 0 ≤ ↑s (u \\ v)\nb : 0 ≤ ↑s (v \\ u)\n⊢ ↑s (v \\ u) = 0\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet v\n\ncase hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"tactic": "all_goals first | linarith | infer_instance | assumption"
},
{
"state_after": "no goals",
"state_before": "case hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"tactic": "first | linarith | infer_instance | assumption"
},
{
"state_after": "no goals",
"state_before": "case hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"tactic": "linarith"
},
{
"state_after": "case hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"state_before": "case hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"tactic": "infer_instance"
},
{
"state_after": "no goals",
"state_before": "case hi\nα : Type u_1\nβ : Type ?u.53280\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ MeasurableSet u",
"tactic": "assumption"
}
] |
[
317,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
308,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.image_preimage_inter
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.52437\nι : Sort ?u.52440\nι' : Sort ?u.52443\nf✝ : α → β\ns✝ t✝ : Set α\nf : α → β\ns : Set α\nt : Set β\n⊢ f '' (f ⁻¹' t ∩ s) = t ∩ f '' s",
"tactic": "simp only [inter_comm, image_inter_preimage]"
}
] |
[
516,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/Algebra/ContinuedFractions/Translations.lean
|
GeneralizedContinuedFraction.terminatedAt_iff_part_denom_none
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ng : GeneralizedContinuedFraction α\nn : ℕ\n⊢ TerminatedAt g n ↔ Stream'.Seq.get? (partialDenominators g) n = none",
"tactic": "rw [terminatedAt_iff_s_none, part_denom_none_iff_s_none]"
}
] |
[
56,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/CategoryTheory/Limits/Types.lean
|
CategoryTheory.Limits.Types.Limit.lift_π_apply
|
[] |
[
197,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/CategoryTheory/Sites/Closed.lean
|
CategoryTheory.classifier_isSheaf
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\n⊢ Presieve.IsSheafFor (Functor.closedSieves J₁) S.arrows",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\n⊢ Presieve.IsSheaf J₁ (Functor.closedSieves J₁)",
"tactic": "intro X S hS"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\n⊢ Presieve.IsSeparatedFor (Functor.closedSieves J₁) S.arrows ∧\n ∀ (x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows),\n Presieve.FamilyOfElements.Compatible x → ∃ t, Presieve.FamilyOfElements.IsAmalgamation x t",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\n⊢ Presieve.IsSheafFor (Functor.closedSieves J₁) S.arrows",
"tactic": "rw [← Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]"
},
{
"state_after": "case refine'_1\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\n⊢ Presieve.IsSeparatedFor (Functor.closedSieves J₁) S.arrows\n\ncase refine'_2\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\n⊢ ∀ (x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows),\n Presieve.FamilyOfElements.Compatible x → ∃ t, Presieve.FamilyOfElements.IsAmalgamation x t",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\n⊢ Presieve.IsSeparatedFor (Functor.closedSieves J₁) S.arrows ∧\n ∀ (x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows),\n Presieve.FamilyOfElements.Compatible x → ∃ t, Presieve.FamilyOfElements.IsAmalgamation x t",
"tactic": "refine' ⟨_, _⟩"
},
{
"state_after": "case refine'_1.mk.mk\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\n⊢ { val := M, property := hM } = { val := N, property := hN }",
"state_before": "case refine'_1\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\n⊢ Presieve.IsSeparatedFor (Functor.closedSieves J₁) S.arrows",
"tactic": "rintro x ⟨M, hM⟩ ⟨N, hN⟩ hM₂ hN₂"
},
{
"state_after": "case refine'_1.mk.mk\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\n⊢ { val := M, property := hM } = { val := N, property := hN }",
"state_before": "case refine'_1.mk.mk\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\n⊢ { val := M, property := hM } = { val := N, property := hN }",
"tactic": "simp only [Functor.closedSieves_obj]"
},
{
"state_after": "case refine'_1.mk.mk.a.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\n⊢ ∀ (f : Y ⟶ X.op.unop), (↑{ val := M, property := hM }).arrows f ↔ (↑{ val := N, property := hN }).arrows f",
"state_before": "case refine'_1.mk.mk\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\n⊢ { val := M, property := hM } = { val := N, property := hN }",
"tactic": "ext Y"
},
{
"state_after": "case refine'_1.mk.mk.a.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\n⊢ (↑{ val := M, property := hM }).arrows f ↔ (↑{ val := N, property := hN }).arrows f",
"state_before": "case refine'_1.mk.mk.a.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\n⊢ ∀ (f : Y ⟶ X.op.unop), (↑{ val := M, property := hM }).arrows f ↔ (↑{ val := N, property := hN }).arrows f",
"tactic": "intro f"
},
{
"state_after": "case refine'_1.mk.mk.a.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\n⊢ M.arrows f ↔ N.arrows f",
"state_before": "case refine'_1.mk.mk.a.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\n⊢ (↑{ val := M, property := hM }).arrows f ↔ (↑{ val := N, property := hN }).arrows f",
"tactic": "dsimp only [Subtype.coe_mk]"
},
{
"state_after": "case refine'_1.mk.mk.a.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\n⊢ GrothendieckTopology.Covers J₁ M f ↔ GrothendieckTopology.Covers J₁ N f",
"state_before": "case refine'_1.mk.mk.a.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\n⊢ M.arrows f ↔ N.arrows f",
"tactic": "rw [← J₁.covers_iff_mem_of_isClosed hM, ← J₁.covers_iff_mem_of_isClosed hN]"
},
{
"state_after": "case refine'_1.mk.mk.a.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\n⊢ GrothendieckTopology.Covers J₁ M f ↔ GrothendieckTopology.Covers J₁ N f",
"state_before": "case refine'_1.mk.mk.a.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\n⊢ GrothendieckTopology.Covers J₁ M f ↔ GrothendieckTopology.Covers J₁ N f",
"tactic": "have q : ∀ ⦃Z : C⦄ (g : Z ⟶ X) (_ : S g), M.pullback g = N.pullback g :=\n fun Z g hg => congr_arg Subtype.val ((hM₂ g hg).trans (hN₂ g hg).symm)"
},
{
"state_after": "case refine'_1.mk.mk.a.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\n⊢ GrothendieckTopology.Covers J₁ M f ↔ GrothendieckTopology.Covers J₁ N f",
"state_before": "case refine'_1.mk.mk.a.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\n⊢ GrothendieckTopology.Covers J₁ M f ↔ GrothendieckTopology.Covers J₁ N f",
"tactic": "have MSNS : M ⊓ S = N ⊓ S := by\n ext Z\n intro g\n rw [Sieve.inter_apply, Sieve.inter_apply]\n simp only [and_comm]\n apply and_congr_right\n intro hg\n rw [Sieve.pullback_eq_top_iff_mem, Sieve.pullback_eq_top_iff_mem, q g hg]"
},
{
"state_after": "case refine'_1.mk.mk.a.h.mp\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\n⊢ GrothendieckTopology.Covers J₁ M f → GrothendieckTopology.Covers J₁ N f\n\ncase refine'_1.mk.mk.a.h.mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\n⊢ GrothendieckTopology.Covers J₁ N f → GrothendieckTopology.Covers J₁ M f",
"state_before": "case refine'_1.mk.mk.a.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\n⊢ GrothendieckTopology.Covers J₁ M f ↔ GrothendieckTopology.Covers J₁ N f",
"tactic": "constructor"
},
{
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"state_before": "C : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\n⊢ M ⊓ S = N ⊓ S",
"tactic": "ext Z"
},
{
"state_after": "case h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nZ : C\ng : Z ⟶ X.op.unop\n⊢ (M ⊓ S).arrows g ↔ (N ⊓ S).arrows g",
"state_before": "case h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nZ : C\n⊢ ∀ (f : Z ⟶ X.op.unop), (M ⊓ S).arrows f ↔ (N ⊓ S).arrows f",
"tactic": "intro g"
},
{
"state_after": "case h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nZ : C\ng : Z ⟶ X.op.unop\n⊢ M.arrows g ∧ S.arrows g ↔ N.arrows g ∧ S.arrows g",
"state_before": "case h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nZ : C\ng : Z ⟶ X.op.unop\n⊢ (M ⊓ S).arrows g ↔ (N ⊓ S).arrows g",
"tactic": "rw [Sieve.inter_apply, Sieve.inter_apply]"
},
{
"state_after": "case h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nZ : C\ng : Z ⟶ X.op.unop\n⊢ S.arrows g ∧ M.arrows g ↔ S.arrows g ∧ N.arrows g",
"state_before": "case h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nZ : C\ng : Z ⟶ X.op.unop\n⊢ M.arrows g ∧ S.arrows g ↔ N.arrows g ∧ S.arrows g",
"tactic": "simp only [and_comm]"
},
{
"state_after": "case h.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nZ : C\ng : Z ⟶ X.op.unop\n⊢ S.arrows g → (M.arrows g ↔ N.arrows g)",
"state_before": "case h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nZ : C\ng : Z ⟶ X.op.unop\n⊢ S.arrows g ∧ M.arrows g ↔ S.arrows g ∧ N.arrows g",
"tactic": "apply and_congr_right"
},
{
"state_after": "case h.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nZ : C\ng : Z ⟶ X.op.unop\nhg : S.arrows g\n⊢ M.arrows g ↔ N.arrows g",
"state_before": "case h.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nZ : C\ng : Z ⟶ X.op.unop\n⊢ S.arrows g → (M.arrows g ↔ N.arrows g)",
"tactic": "intro hg"
},
{
"state_after": "no goals",
"state_before": "case h.h\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nZ : C\ng : Z ⟶ X.op.unop\nhg : S.arrows g\n⊢ M.arrows g ↔ N.arrows g",
"tactic": "rw [Sieve.pullback_eq_top_iff_mem, Sieve.pullback_eq_top_iff_mem, q g hg]"
},
{
"state_after": "case refine'_1.mk.mk.a.h.mp\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ M f\n⊢ GrothendieckTopology.Covers J₁ N f",
"state_before": "case refine'_1.mk.mk.a.h.mp\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\n⊢ GrothendieckTopology.Covers J₁ M f → GrothendieckTopology.Covers J₁ N f",
"tactic": "intro hf"
},
{
"state_after": "case refine'_1.mk.mk.a.h.mp\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ M f\n⊢ Sieve.pullback f N ∈ GrothendieckTopology.sieves J₁ Y",
"state_before": "case refine'_1.mk.mk.a.h.mp\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ M f\n⊢ GrothendieckTopology.Covers J₁ N f",
"tactic": "rw [J₁.covers_iff]"
},
{
"state_after": "case refine'_1.mk.mk.a.h.mp\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ M f\n⊢ Sieve.pullback f (N ⊓ ?m.20451) ∈ GrothendieckTopology.sieves J₁ Y\n\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ M f\n⊢ Sieve X.op.unop",
"state_before": "case refine'_1.mk.mk.a.h.mp\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ M f\n⊢ Sieve.pullback f N ∈ GrothendieckTopology.sieves J₁ Y",
"tactic": "apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)"
},
{
"state_after": "case refine'_1.mk.mk.a.h.mp\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ M f\n⊢ Sieve.pullback f (M ⊓ S) ∈ GrothendieckTopology.sieves J₁ Y",
"state_before": "case refine'_1.mk.mk.a.h.mp\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ M f\n⊢ Sieve.pullback f (N ⊓ ?m.20451) ∈ GrothendieckTopology.sieves J₁ Y\n\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ M f\n⊢ Sieve X.op.unop",
"tactic": "rw [← MSNS]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.mk.mk.a.h.mp\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ M f\n⊢ Sieve.pullback f (M ⊓ S) ∈ GrothendieckTopology.sieves J₁ Y",
"tactic": "apply J₁.arrow_intersect f M S hf (J₁.pullback_stable _ hS)"
},
{
"state_after": "case refine'_1.mk.mk.a.h.mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ N f\n⊢ GrothendieckTopology.Covers J₁ M f",
"state_before": "case refine'_1.mk.mk.a.h.mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\n⊢ GrothendieckTopology.Covers J₁ N f → GrothendieckTopology.Covers J₁ M f",
"tactic": "intro hf"
},
{
"state_after": "case refine'_1.mk.mk.a.h.mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ N f\n⊢ Sieve.pullback f M ∈ GrothendieckTopology.sieves J₁ Y",
"state_before": "case refine'_1.mk.mk.a.h.mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ N f\n⊢ GrothendieckTopology.Covers J₁ M f",
"tactic": "rw [J₁.covers_iff]"
},
{
"state_after": "case refine'_1.mk.mk.a.h.mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ N f\n⊢ Sieve.pullback f (M ⊓ ?m.20658) ∈ GrothendieckTopology.sieves J₁ Y\n\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ N f\n⊢ Sieve X.op.unop",
"state_before": "case refine'_1.mk.mk.a.h.mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ N f\n⊢ Sieve.pullback f M ∈ GrothendieckTopology.sieves J₁ Y",
"tactic": "apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)"
},
{
"state_after": "case refine'_1.mk.mk.a.h.mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ N f\n⊢ Sieve.pullback f (N ⊓ S) ∈ GrothendieckTopology.sieves J₁ Y",
"state_before": "case refine'_1.mk.mk.a.h.mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ N f\n⊢ Sieve.pullback f (M ⊓ ?m.20658) ∈ GrothendieckTopology.sieves J₁ Y\n\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ N f\n⊢ Sieve X.op.unop",
"tactic": "rw [MSNS]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.mk.mk.a.h.mpr\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nM : Sieve X.op.unop\nhM : GrothendieckTopology.IsClosed J₁ M\nN : Sieve X.op.unop\nhN : GrothendieckTopology.IsClosed J₁ N\nhM₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := M, property := hM }\nhN₂ : Presieve.FamilyOfElements.IsAmalgamation x { val := N, property := hN }\nY : C\nf : Y ⟶ X.op.unop\nq : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N\nMSNS : M ⊓ S = N ⊓ S\nhf : GrothendieckTopology.Covers J₁ N f\n⊢ Sieve.pullback f (N ⊓ S) ∈ GrothendieckTopology.sieves J₁ Y",
"tactic": "apply J₁.arrow_intersect f N S hf (J₁.pullback_stable _ hS)"
},
{
"state_after": "case refine'_2\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\n⊢ ∃ t, Presieve.FamilyOfElements.IsAmalgamation x t",
"state_before": "case refine'_2\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\n⊢ ∀ (x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows),\n Presieve.FamilyOfElements.Compatible x → ∃ t, Presieve.FamilyOfElements.IsAmalgamation x t",
"tactic": "intro x hx"
},
{
"state_after": "case refine'_2\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\n⊢ ∃ t, Presieve.FamilyOfElements.IsAmalgamation x t",
"state_before": "case refine'_2\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\n⊢ ∃ t, Presieve.FamilyOfElements.IsAmalgamation x t",
"tactic": "rw [Presieve.compatible_iff_sieveCompatible] at hx"
},
{
"state_after": "case refine'_2\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\n⊢ ∃ t, Presieve.FamilyOfElements.IsAmalgamation x t",
"state_before": "case refine'_2\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\n⊢ ∃ t, Presieve.FamilyOfElements.IsAmalgamation x t",
"tactic": "let M := Sieve.bind S fun Y f hf => (x f hf).1"
},
{
"state_after": "case refine'_2\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nthis : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)\n⊢ Presieve.FamilyOfElements.IsAmalgamation x\n { val := GrothendieckTopology.close J₁ M,\n property := (_ : GrothendieckTopology.IsClosed J₁ (GrothendieckTopology.close J₁ M)) }",
"state_before": "case refine'_2\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nthis : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)\n⊢ ∃ t, Presieve.FamilyOfElements.IsAmalgamation x t",
"tactic": "refine' ⟨⟨_, J₁.close_isClosed M⟩, _⟩"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ Sieve.pullback f M = ↑(x f hf)",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\n⊢ ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)",
"tactic": "intro Y f hf"
},
{
"state_after": "case a\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ Sieve.pullback f M ≤ ↑(x f hf)\n\ncase a\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ ↑(x f hf) ≤ Sieve.pullback f M",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ Sieve.pullback f M = ↑(x f hf)",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\nu : Z ⟶ Y\nW : C\ng : Z ⟶ W\nf' : W ⟶ X\nhf' : S.arrows f'\nhg : (↑(x f' hf')).arrows g\nc : g ≫ f' = u ≫ f\n⊢ (↑(x f hf)).arrows u",
"state_before": "case a\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ Sieve.pullback f M ≤ ↑(x f hf)",
"tactic": "rintro Z u ⟨W, g, f', hf', hg : (x f' hf').1 _, c⟩"
},
{
"state_after": "case a.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\nu : Z ⟶ Y\nW : C\ng : Z ⟶ W\nf' : W ⟶ X\nhf' : S.arrows f'\nhg : (↑(x f' hf')).arrows g\nc : g ≫ f' = u ≫ f\n⊢ ↑(x (u ≫ f) (_ : S.arrows (u ≫ f))) = ⊤",
"state_before": "case a.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\nu : Z ⟶ Y\nW : C\ng : Z ⟶ W\nf' : W ⟶ X\nhf' : S.arrows f'\nhg : (↑(x f' hf')).arrows g\nc : g ≫ f' = u ≫ f\n⊢ (↑(x f hf)).arrows u",
"tactic": "rw [Sieve.pullback_eq_top_iff_mem,\n ← show (x (u ≫ f) _).1 = (x f hf).1.pullback u from congr_arg Subtype.val (hx f u hf)]"
},
{
"state_after": "case a.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\nu : Z ⟶ Y\nW : C\ng : Z ⟶ W\nf' : W ⟶ X\nhf' : S.arrows f'\nhg : (↑(x f' hf')).arrows g\nc : g ≫ f' = u ≫ f\n⊢ ↑(x (g ≫ f') (_ : S.arrows (g ≫ f'))) = ⊤",
"state_before": "case a.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\nu : Z ⟶ Y\nW : C\ng : Z ⟶ W\nf' : W ⟶ X\nhf' : S.arrows f'\nhg : (↑(x f' hf')).arrows g\nc : g ≫ f' = u ≫ f\n⊢ ↑(x (u ≫ f) (_ : S.arrows (u ≫ f))) = ⊤",
"tactic": "conv_lhs => congr; congr; rw [← c]"
},
{
"state_after": "case a.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\nu : Z ⟶ Y\nW : C\ng : Z ⟶ W\nf' : W ⟶ X\nhf' : S.arrows f'\nhg : (↑(x f' hf')).arrows g\nc : g ≫ f' = u ≫ f\n⊢ ↑((Functor.closedSieves J₁).map g.op (x f' hf')) = ⊤",
"state_before": "case a.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\nu : Z ⟶ Y\nW : C\ng : Z ⟶ W\nf' : W ⟶ X\nhf' : S.arrows f'\nhg : (↑(x f' hf')).arrows g\nc : g ≫ f' = u ≫ f\n⊢ ↑(x (g ≫ f') (_ : S.arrows (g ≫ f'))) = ⊤",
"tactic": "rw [show (x (g ≫ f') _).1 = _ from congr_arg Subtype.val (hx f' g hf')]"
},
{
"state_after": "no goals",
"state_before": "case a.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\nZ : C\nu : Z ⟶ Y\nW : C\ng : Z ⟶ W\nf' : W ⟶ X\nhf' : S.arrows f'\nhg : (↑(x f' hf')).arrows g\nc : g ≫ f' = u ≫ f\n⊢ ↑((Functor.closedSieves J₁).map g.op (x f' hf')) = ⊤",
"tactic": "apply Sieve.pullback_eq_top_of_mem _ hg"
},
{
"state_after": "no goals",
"state_before": "case a\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ ↑(x f hf) ≤ Sieve.pullback f M",
"tactic": "apply Sieve.le_pullback_bind S fun Y f hf => (x f hf).1"
},
{
"state_after": "case refine'_2\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nthis : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ (Functor.closedSieves J₁).map f.op\n { val := GrothendieckTopology.close J₁ M,\n property := (_ : GrothendieckTopology.IsClosed J₁ (GrothendieckTopology.close J₁ M)) } =\n x f hf",
"state_before": "case refine'_2\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nthis : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)\n⊢ Presieve.FamilyOfElements.IsAmalgamation x\n { val := GrothendieckTopology.close J₁ M,\n property := (_ : GrothendieckTopology.IsClosed J₁ (GrothendieckTopology.close J₁ M)) }",
"tactic": "intro Y f hf"
},
{
"state_after": "case refine'_2\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nthis : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ (Functor.closedSieves J₁).map f.op\n { val := GrothendieckTopology.close J₁ (Sieve.bind S.arrows fun Y f hf => ↑(x f hf)),\n property := (_ : GrothendieckTopology.IsClosed J₁ (GrothendieckTopology.close J₁ M)) } =\n x f hf",
"state_before": "case refine'_2\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nthis : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ (Functor.closedSieves J₁).map f.op\n { val := GrothendieckTopology.close J₁ M,\n property := (_ : GrothendieckTopology.IsClosed J₁ (GrothendieckTopology.close J₁ M)) } =\n x f hf",
"tactic": "simp only [Functor.closedSieves_obj]"
},
{
"state_after": "case refine'_2.a\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nthis : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ ↑((Functor.closedSieves J₁).map f.op\n { val := GrothendieckTopology.close J₁ (Sieve.bind S.arrows fun Y f hf => ↑(x f hf)),\n property := (_ : GrothendieckTopology.IsClosed J₁ (GrothendieckTopology.close J₁ M)) }) =\n ↑(x f hf)",
"state_before": "case refine'_2\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nthis : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ (Functor.closedSieves J₁).map f.op\n { val := GrothendieckTopology.close J₁ (Sieve.bind S.arrows fun Y f hf => ↑(x f hf)),\n property := (_ : GrothendieckTopology.IsClosed J₁ (GrothendieckTopology.close J₁ M)) } =\n x f hf",
"tactic": "ext1"
},
{
"state_after": "case refine'_2.a\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nthis : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ Sieve.pullback f (GrothendieckTopology.close J₁ (Sieve.bind S.arrows fun Y f hf => ↑(x f hf))) = ↑(x f hf)",
"state_before": "case refine'_2.a\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nthis : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ ↑((Functor.closedSieves J₁).map f.op\n { val := GrothendieckTopology.close J₁ (Sieve.bind S.arrows fun Y f hf => ↑(x f hf)),\n property := (_ : GrothendieckTopology.IsClosed J₁ (GrothendieckTopology.close J₁ M)) }) =\n ↑(x f hf)",
"tactic": "dsimp"
},
{
"state_after": "case refine'_2.a\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nthis : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ GrothendieckTopology.close J₁ ↑(x f hf) = ↑(x f hf)",
"state_before": "case refine'_2.a\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nthis : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ Sieve.pullback f (GrothendieckTopology.close J₁ (Sieve.bind S.arrows fun Y f hf => ↑(x f hf))) = ↑(x f hf)",
"tactic": "rw [← J₁.pullback_close, this _ hf]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.a\nC : Type u\ninst✝ : Category C\nJ₁ J₂ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows\nhx : Presieve.FamilyOfElements.SieveCompatible x\nM : Sieve X := Sieve.bind S.arrows fun Y f hf => ↑(x f hf)\nthis : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ GrothendieckTopology.close J₁ ↑(x f hf) = ↑(x f hf)",
"tactic": "apply le_antisymm (J₁.le_close_of_isClosed le_rfl (x f hf).2) (J₁.le_close _)"
}
] |
[
247,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/GroupTheory/Congruence.lean
|
Con.conGen_mono
|
[] |
[
539,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
538,
1
] |
Std/Data/Rat/Lemmas.lean
|
Rat.maybeNormalize_eq_normalize
|
[
{
"state_after": "num : Int\nden g : Nat\nden_nz : den / g ≠ 0\nreduced : Nat.coprime (Int.natAbs (Int.div num ↑g)) (den / g)\nhn : ↑g ∣ num\nhd : g ∣ den\n⊢ normalize (num / ↑g) (den / g) = normalize num den",
"state_before": "num : Int\nden g : Nat\nden_nz : den / g ≠ 0\nreduced : Nat.coprime (Int.natAbs (Int.div num ↑g)) (den / g)\nhn : ↑g ∣ num\nhd : g ∣ den\n⊢ maybeNormalize num den g den_nz reduced = normalize num den",
"tactic": "simp only [maybeNormalize_eq, mk_eq_normalize, Int.div_eq_ediv_of_dvd hn]"
},
{
"state_after": "num : Int\nden g : Nat\nden_nz : den / g ≠ 0\nreduced : Nat.coprime (Int.natAbs (Int.div num ↑g)) (den / g)\nhn : ↑g ∣ num\nhd : g ∣ den\nthis : g ≠ 0\n⊢ normalize num (den / g * g) = normalize num den",
"state_before": "num : Int\nden g : Nat\nden_nz : den / g ≠ 0\nreduced : Nat.coprime (Int.natAbs (Int.div num ↑g)) (den / g)\nhn : ↑g ∣ num\nhd : g ∣ den\nthis : g ≠ 0\n⊢ normalize (num / ↑g) (den / g) = normalize num den",
"tactic": "rw [← normalize_mul_right _ this, Int.ediv_mul_cancel hn]"
},
{
"state_after": "case e_den\nnum : Int\nden g : Nat\nden_nz : den / g ≠ 0\nreduced : Nat.coprime (Int.natAbs (Int.div num ↑g)) (den / g)\nhn : ↑g ∣ num\nhd : g ∣ den\nthis : g ≠ 0\n⊢ den / g * g = den",
"state_before": "num : Int\nden g : Nat\nden_nz : den / g ≠ 0\nreduced : Nat.coprime (Int.natAbs (Int.div num ↑g)) (den / g)\nhn : ↑g ∣ num\nhd : g ∣ den\nthis : g ≠ 0\n⊢ normalize num (den / g * g) = normalize num den",
"tactic": "congr 1"
},
{
"state_after": "no goals",
"state_before": "case e_den\nnum : Int\nden g : Nat\nden_nz : den / g ≠ 0\nreduced : Nat.coprime (Int.natAbs (Int.div num ↑g)) (den / g)\nhn : ↑g ∣ num\nhd : g ∣ den\nthis : g ≠ 0\n⊢ den / g * g = den",
"tactic": "exact Nat.div_mul_cancel hd"
}
] |
[
66,
39
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
60,
1
] |
Mathlib/Order/Hom/Bounded.lean
|
BotHom.coe_inf
|
[] |
[
527,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
526,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.Cofork.IsColimit.π_desc
|
[] |
[
437,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
436,
1
] |
Mathlib/Topology/Basic.lean
|
mem_closure_iff_nhds_basis'
|
[
{
"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\na : α\np : ι → Prop\ns : ι → Set α\nh : HasBasis (𝓝 a) p s\nt : Set α\n⊢ (∀ ⦃i : ι⦄, p i → ∀ ⦃j : Unit⦄, True → Set.Nonempty (s i ∩ t)) ↔ ∀ (i : ι), p i → Set.Nonempty (s i ∩ t)",
"tactic": "simp only [exists_prop, forall_const]"
}
] |
[
1349,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1346,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.sub_lt
|
[] |
[
242,
33
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
236,
1
] |
Mathlib/Data/Complex/Module.lean
|
finrank_real_of_complex
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℂ E\n⊢ FiniteDimensional.finrank ℝ E = 2 * FiniteDimensional.finrank ℂ E",
"tactic": "rw [← FiniteDimensional.finrank_mul_finrank ℝ ℂ E, Complex.finrank_real_complex]"
}
] |
[
245,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Std/Data/List/Lemmas.lean
|
List.diff_erase
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nl₁ l₂ : List α\na : α\n⊢ List.erase (List.diff l₁ l₂) a = List.diff (List.erase l₁ a) l₂",
"tactic": "rw [← diff_cons_right, diff_cons]"
}
] |
[
1501,
36
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1500,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.inv_insert
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.7176\nα : Type u_1\nβ : Type ?u.7182\nγ : Type ?u.7185\ninst✝ : InvolutiveInv α\ns✝ t : Set α\na✝ a : α\ns : Set α\n⊢ (insert a s)⁻¹ = insert a⁻¹ s⁻¹",
"tactic": "rw [insert_eq, union_inv, inv_singleton, insert_eq]"
}
] |
[
286,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.map_X
|
[] |
[
703,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
702,
1
] |
Mathlib/LinearAlgebra/AffineSpace/MidpointZero.lean
|
homothety_one_half
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Field k\ninst✝³ : CharZero k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AddTorsor V P\na b : P\n⊢ ↑(homothety a (1 / 2)) b = midpoint k a b",
"tactic": "rw [one_div, homothety_inv_two]"
}
] |
[
50,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
1
] |
Mathlib/Algebra/Algebra/Spectrum.lean
|
spectrum.zero_not_mem_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\n⊢ ¬0 ∈ σ a ↔ IsUnit a",
"tactic": "rw [zero_mem_iff, Classical.not_not]"
}
] |
[
124,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Topology/Order/Basic.lean
|
csInf_mem_closure
|
[] |
[
2780,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2779,
1
] |
Mathlib/Data/Real/ENatENNReal.lean
|
ENat.toENNReal_lt
|
[] |
[
73,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
left_eq_mul₀
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.12127\nM₀ : Type u_1\nG₀ : Type ?u.12133\nM₀' : Type ?u.12136\nG₀' : Type ?u.12139\nF : Type ?u.12142\nF' : Type ?u.12145\ninst✝ : CancelMonoidWithZero M₀\na b c : M₀\nha : a ≠ 0\n⊢ a = a * b ↔ b = 1",
"tactic": "rw [eq_comm, mul_eq_left₀ ha]"
}
] |
[
212,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Tactic/Ring/RingNF.lean
|
Mathlib.Tactic.RingNF.mul_neg
|
[
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.74049\ninst✝¹ : CommSemiring R✝\nR : Type u_1\ninst✝ : Ring R\na b : R\n⊢ a * -b = -(a * b)",
"tactic": "simp"
}
] |
[
112,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Algebra/DirectSum/Ring.lean
|
DirectSum.of_mul_of
|
[] |
[
218,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.arg_of_re_nonneg
|
[] |
[
277,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Mathlib/Data/Set/Intervals/WithBotTop.lean
|
WithTop.preimage_coe_Ico_top
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : PartialOrder α\na b : α\n⊢ some ⁻¹' Ico ↑a ⊤ = Ici a",
"tactic": "simp [← Ici_inter_Iio]"
}
] |
[
82,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.length_toList
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.12605\ns✝ t : Finset α\nf : α → β\nn : ℕ\ns : Finset α\n⊢ List.length (toList s) = card s",
"tactic": "rw [toList, ← Multiset.coe_card, Multiset.coe_toList, card_def]"
}
] |
[
216,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
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