file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Analysis/SpecialFunctions/Exponential.lean
|
hasDerivAt_exp_smul_const_of_mem_ball
|
[] |
[
358,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
ContinuousWithinAt.tendsto_nhdsWithin_image
|
[] |
[
559,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
557,
1
] |
Mathlib/Data/PNat/Interval.lean
|
PNat.card_fintype_Ioo
|
[
{
"state_after": "no goals",
"state_before": "a b : ℕ+\n⊢ Fintype.card ↑(Set.Ioo a b) = ↑b - ↑a - 1",
"tactic": "rw [← card_Ioo, Fintype.card_ofFinset]"
}
] |
[
112,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
IsDedekindDomain.inf_prime_pow_eq_prod
|
[
{
"state_after": "R : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nprime : ∀ (i : ι), i ∈ s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ≠ f j\nthis : DecidableEq ι := Classical.decEq ι\n⊢ (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i",
"state_before": "R : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nprime : ∀ (i : ι), i ∈ s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ≠ f j\n⊢ (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i",
"tactic": "letI := Classical.decEq ι"
},
{
"state_after": "R : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\n⊢ (∀ (i : ι), i ∈ s → Prime (f i)) →\n (∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ≠ f j) → (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i",
"state_before": "R : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nprime : ∀ (i : ι), i ∈ s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ≠ f j\nthis : DecidableEq ι := Classical.decEq ι\n⊢ (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i",
"tactic": "revert prime coprime"
},
{
"state_after": "case refine_1\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\n⊢ (∀ (i : ι), i ∈ ∅ → Prime (f i)) →\n (∀ (i : ι), i ∈ ∅ → ∀ (j : ι), j ∈ ∅ → i ≠ j → f i ≠ f j) → (Finset.inf ∅ fun i => f i ^ e i) = ∏ i in ∅, f i ^ e i\n\ncase refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n ((∀ (i : ι), i ∈ s → Prime (f i)) →\n (∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ≠ f j) →\n (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i) →\n (∀ (i : ι), i ∈ insert a s → Prime (f i)) →\n (∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j) →\n (Finset.inf (insert a s) fun i => f i ^ e i) = ∏ i in insert a s, f i ^ e i",
"state_before": "R : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\n⊢ (∀ (i : ι), i ∈ s → Prime (f i)) →\n (∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ≠ f j) → (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i",
"tactic": "refine s.induction ?_ ?_"
},
{
"state_after": "case refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nih :\n (∀ (i : ι), i ∈ s → Prime (f i)) →\n (∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ≠ f j) → (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\n⊢ (Finset.inf (insert a s) fun i => f i ^ e i) = ∏ i in insert a s, f i ^ e i",
"state_before": "case refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n ((∀ (i : ι), i ∈ s → Prime (f i)) →\n (∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ≠ f j) →\n (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i) →\n (∀ (i : ι), i ∈ insert a s → Prime (f i)) →\n (∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j) →\n (Finset.inf (insert a s) fun i => f i ^ e i) = ∏ i in insert a s, f i ^ e i",
"tactic": "intro a s ha ih prime coprime"
},
{
"state_after": "case refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\n⊢ (Finset.inf (insert a s) fun i => f i ^ e i) = ∏ i in insert a s, f i ^ e i",
"state_before": "case refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nih :\n (∀ (i : ι), i ∈ s → Prime (f i)) →\n (∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ≠ f j) → (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\n⊢ (Finset.inf (insert a s) fun i => f i ^ e i) = ∏ i in insert a s, f i ^ e i",
"tactic": "specialize\n ih (fun i hi => prime i (Finset.mem_insert_of_mem hi)) fun i hi j hj =>\n coprime i (Finset.mem_insert_of_mem hi) j (Finset.mem_insert_of_mem hj)"
},
{
"state_after": "case refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\n⊢ f a ^ e a ⊓ ∏ i in s, f i ^ e i = f a ^ e a * ∏ x in s, f x ^ e x",
"state_before": "case refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\n⊢ (Finset.inf (insert a s) fun i => f i ^ e i) = ∏ i in insert a s, f i ^ e i",
"tactic": "rw [Finset.inf_insert, Finset.prod_insert ha, ih]"
},
{
"state_after": "case refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\n⊢ ∀ (P : Ideal R), f a ^ e a ≤ P → ∏ x in s, f x ^ e x ≤ P → ¬IsPrime P",
"state_before": "case refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\n⊢ f a ^ e a ⊓ ∏ i in s, f i ^ e i = f a ^ e a * ∏ x in s, f x ^ e x",
"tactic": "refine' le_antisymm (Ideal.le_mul_of_no_prime_factors _ inf_le_left inf_le_right) Ideal.mul_le_inf"
},
{
"state_after": "case refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp : IsPrime P\n⊢ False",
"state_before": "case refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\n⊢ ∀ (P : Ideal R), f a ^ e a ≤ P → ∏ x in s, f x ^ e x ≤ P → ¬IsPrime P",
"tactic": "intro P hPa hPs hPp"
},
{
"state_after": "case refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝ : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this : IsPrime P\n⊢ False",
"state_before": "case refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp : IsPrime P\n⊢ False",
"tactic": "haveI := hPp"
},
{
"state_after": "case refine_2.intro.intro\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝ : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\n⊢ False",
"state_before": "case refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝ : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this : IsPrime P\n⊢ False",
"tactic": "obtain ⟨b, hb, hPb⟩ := Ideal.prod_le_prime.mp hPs"
},
{
"state_after": "case refine_2.intro.intro\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝¹ : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝ : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\nthis : IsPrime (f a)\n⊢ False",
"state_before": "case refine_2.intro.intro\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝ : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\n⊢ False",
"tactic": "haveI := Ideal.isPrime_of_prime (prime a (Finset.mem_insert_self a s))"
},
{
"state_after": "case refine_2.intro.intro\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝² : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝¹ : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\nthis✝ : IsPrime (f a)\nthis : IsPrime (f b)\n⊢ False",
"state_before": "case refine_2.intro.intro\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝¹ : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝ : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\nthis : IsPrime (f a)\n⊢ False",
"tactic": "haveI := Ideal.isPrime_of_prime (prime b (Finset.mem_insert_of_mem hb))"
},
{
"state_after": "case refine_2.intro.intro.refine_1\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝² : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝¹ : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\nthis✝ : IsPrime (f a)\nthis : IsPrime (f b)\n⊢ a ≠ b\n\ncase refine_2.intro.intro.refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝² : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝¹ : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\nthis✝ : IsPrime (f a)\nthis : IsPrime (f b)\n⊢ f a = f b",
"state_before": "case refine_2.intro.intro\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝² : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝¹ : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\nthis✝ : IsPrime (f a)\nthis : IsPrime (f b)\n⊢ False",
"tactic": "refine coprime a (Finset.mem_insert_self a s) b (Finset.mem_insert_of_mem hb) ?_ ?_"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis : DecidableEq ι := Classical.decEq ι\n⊢ (∀ (i : ι), i ∈ ∅ → Prime (f i)) →\n (∀ (i : ι), i ∈ ∅ → ∀ (j : ι), j ∈ ∅ → i ≠ j → f i ≠ f j) → (Finset.inf ∅ fun i => f i ^ e i) = ∏ i in ∅, f i ^ e i",
"tactic": "simp"
},
{
"state_after": "case refine_2.intro.intro.refine_1\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝² : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝¹ : IsPrime P\nthis✝ : IsPrime (f a)\nhb : a ∈ s\nhPb : f a ^ e a ≤ P\nthis : IsPrime (f a)\n⊢ False",
"state_before": "case refine_2.intro.intro.refine_1\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝² : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝¹ : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\nthis✝ : IsPrime (f a)\nthis : IsPrime (f b)\n⊢ a ≠ b",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case refine_2.intro.intro.refine_1\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝² : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝¹ : IsPrime P\nthis✝ : IsPrime (f a)\nhb : a ∈ s\nhPb : f a ^ e a ≤ P\nthis : IsPrime (f a)\n⊢ False",
"tactic": "contradiction"
},
{
"state_after": "case refine_2.intro.intro.refine_2.refine_1\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝² : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝¹ : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\nthis✝ : IsPrime (f a)\nthis : IsPrime (f b)\n⊢ f a ≠ ⊥\n\ncase refine_2.intro.intro.refine_2.refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝² : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝¹ : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\nthis✝ : IsPrime (f a)\nthis : IsPrime (f b)\n⊢ f b ≠ ⊥",
"state_before": "case refine_2.intro.intro.refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝² : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝¹ : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\nthis✝ : IsPrime (f a)\nthis : IsPrime (f b)\n⊢ f a = f b",
"tactic": "refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq IsDedekindDomain.dimensionLEOne ?_).mp\n (Ideal.le_of_pow_le_prime hPa)).trans\n ((Ring.DimensionLeOne.prime_le_prime_iff_eq IsDedekindDomain.dimensionLEOne ?_).mp\n (Ideal.le_of_pow_le_prime hPb)).symm"
},
{
"state_after": "case refine_2.intro.intro.refine_2.refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝² : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝¹ : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\nthis✝ : IsPrime (f a)\nthis : IsPrime (f b)\n⊢ f b ≠ ⊥",
"state_before": "case refine_2.intro.intro.refine_2.refine_1\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝² : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝¹ : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\nthis✝ : IsPrime (f a)\nthis : IsPrime (f b)\n⊢ f a ≠ ⊥\n\ncase refine_2.intro.intro.refine_2.refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝² : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝¹ : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\nthis✝ : IsPrime (f a)\nthis : IsPrime (f b)\n⊢ f b ≠ ⊥",
"tactic": "exact (prime a (Finset.mem_insert_self a s)).ne_zero"
},
{
"state_after": "no goals",
"state_before": "case refine_2.intro.intro.refine_2.refine_2\nR : Type u_2\nA : Type ?u.1194471\nK : Type ?u.1194474\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns✝ : Finset ι\nf : ι → Ideal R\ne : ι → ℕ\nthis✝² : DecidableEq ι := Classical.decEq ι\na : ι\ns : Finset ι\nha : ¬a ∈ s\nprime : ∀ (i : ι), i ∈ insert a s → Prime (f i)\ncoprime : ∀ (i : ι), i ∈ insert a s → ∀ (j : ι), j ∈ insert a s → i ≠ j → f i ≠ f j\nih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i\nP : Ideal R\nhPa : f a ^ e a ≤ P\nhPs : ∏ x in s, f x ^ e x ≤ P\nhPp this✝¹ : IsPrime P\nb : ι\nhb : b ∈ s\nhPb : f b ^ e b ≤ P\nthis✝ : IsPrime (f a)\nthis : IsPrime (f b)\n⊢ f b ≠ ⊥",
"tactic": "exact (prime b (Finset.mem_insert_of_mem hb)).ne_zero"
}
] |
[
1301,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1275,
1
] |
Mathlib/Dynamics/Ergodic/MeasurePreserving.lean
|
MeasureTheory.MeasurePreserving.sigmaFinite
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.271138\nδ : Type ?u.271141\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\ninst✝¹ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\nf : α → β\nhf : MeasurePreserving f\ninst✝ : SigmaFinite μb\n⊢ SigmaFinite (map f μa)",
"tactic": "rwa [hf.map_eq]"
}
] |
[
121,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
11
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.vecMul_add
|
[
{
"state_after": "case h\nl : Type ?u.857315\nm : Type u_1\nn : Type u_2\no : Type ?u.857324\nm' : o → Type ?u.857329\nn' : o → Type ?u.857334\nR : Type ?u.857337\nS : Type ?u.857340\nα : Type v\nβ : Type w\nγ : Type ?u.857347\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype m\nA B : Matrix m n α\nx : m → α\nx✝ : n\n⊢ vecMul x (A + B) x✝ = (vecMul x A + vecMul x B) x✝",
"state_before": "l : Type ?u.857315\nm : Type u_1\nn : Type u_2\no : Type ?u.857324\nm' : o → Type ?u.857329\nn' : o → Type ?u.857334\nR : Type ?u.857337\nS : Type ?u.857340\nα : Type v\nβ : Type w\nγ : Type ?u.857347\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype m\nA B : Matrix m n α\nx : m → α\n⊢ vecMul x (A + B) = vecMul x A + vecMul x B",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nl : Type ?u.857315\nm : Type u_1\nn : Type u_2\no : Type ?u.857324\nm' : o → Type ?u.857329\nn' : o → Type ?u.857334\nR : Type ?u.857337\nS : Type ?u.857340\nα : Type v\nβ : Type w\nγ : Type ?u.857347\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype m\nA B : Matrix m n α\nx : m → α\nx✝ : n\n⊢ vecMul x (A + B) x✝ = (vecMul x A + vecMul x B) x✝",
"tactic": "apply dotProduct_add"
}
] |
[
1763,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1760,
1
] |
src/lean/Init/Data/List/Basic.lean
|
List.cons_getElem_succ
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nas : List α\ni : Nat\nh : i + 1 < length (a :: as)\n⊢ getElem (a :: as) (i + 1) h = getElem as i (_ : i < length as)",
"tactic": "rfl"
}
] |
[
25,
6
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
24,
9
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
interior_mem_uniformity
|
[
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.91137\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\n⊢ interior s ∈ Filter.lift' (𝓤 α) interior",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.91137\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\n⊢ interior s ∈ 𝓤 α",
"tactic": "rw [uniformity_eq_uniformity_interior]"
},
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.91137\ninst✝ : UniformSpace α\ns : Set (α × α)\nhs : s ∈ 𝓤 α\n⊢ interior s ∈ Filter.lift' (𝓤 α) interior",
"tactic": "exact mem_lift' hs"
}
] |
[
1000,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
999,
1
] |
Mathlib/Order/BoundedOrder.lean
|
isMin_bot
|
[] |
[
321,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.ne_one
|
[] |
[
377,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
376,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Trivialization.coe_coe
|
[] |
[
349,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
348,
1
] |
Mathlib/CategoryTheory/Category/Cat.lean
|
CategoryTheory.Cat.comp_obj
|
[] |
[
101,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
measurable_findGreatest
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.52344\nγ : Type ?u.52347\nδ : Type ?u.52350\nδ' : Type ?u.52353\nι : Sort uι\ns t u : Set α\ninst✝¹ : MeasurableSpace α\np : α → ℕ → Prop\ninst✝ : (x : α) → DecidablePred (p x)\nN : ℕ\nhN : ∀ (k : ℕ), k ≤ N → MeasurableSet {x | p x k}\nk : ℕ\nhk : k ≤ N\n⊢ MeasurableSet {x | Nat.findGreatest (p x) N = k}",
"state_before": "α : Type u_1\nβ : Type ?u.52344\nγ : Type ?u.52347\nδ : Type ?u.52350\nδ' : Type ?u.52353\nι : Sort uι\ns t u : Set α\ninst✝¹ : MeasurableSpace α\np : α → ℕ → Prop\ninst✝ : (x : α) → DecidablePred (p x)\nN : ℕ\nhN : ∀ (k : ℕ), k ≤ N → MeasurableSet {x | p x k}\n⊢ Measurable fun x => Nat.findGreatest (p x) N",
"tactic": "refine' measurable_findGreatest' fun k hk => _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.52344\nγ : Type ?u.52347\nδ : Type ?u.52350\nδ' : Type ?u.52353\nι : Sort uι\ns t u : Set α\ninst✝¹ : MeasurableSpace α\np : α → ℕ → Prop\ninst✝ : (x : α) → DecidablePred (p x)\nN : ℕ\nhN : ∀ (k : ℕ), k ≤ N → MeasurableSet {x | p x k}\nk : ℕ\nhk : k ≤ N\n⊢ MeasurableSet ({a | k ≤ N} ∩ ((⋂ (_ : k ≠ 0), {x | p x k}) ∩ ⋂ (i : ℕ) (_ : k < i) (_ : i ≤ N), {a | p a i}ᶜ))",
"state_before": "α : Type u_1\nβ : Type ?u.52344\nγ : Type ?u.52347\nδ : Type ?u.52350\nδ' : Type ?u.52353\nι : Sort uι\ns t u : Set α\ninst✝¹ : MeasurableSpace α\np : α → ℕ → Prop\ninst✝ : (x : α) → DecidablePred (p x)\nN : ℕ\nhN : ∀ (k : ℕ), k ≤ N → MeasurableSet {x | p x k}\nk : ℕ\nhk : k ≤ N\n⊢ MeasurableSet {x | Nat.findGreatest (p x) N = k}",
"tactic": "simp only [Nat.findGreatest_eq_iff, setOf_and, setOf_forall, ← compl_setOf]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.52344\nγ : Type ?u.52347\nδ : Type ?u.52350\nδ' : Type ?u.52353\nι : Sort uι\ns t u : Set α\ninst✝¹ : MeasurableSpace α\np : α → ℕ → Prop\ninst✝ : (x : α) → DecidablePred (p x)\nN : ℕ\nhN : ∀ (k : ℕ), k ≤ N → MeasurableSet {x | p x k}\nk : ℕ\nhk : k ≤ N\n⊢ MeasurableSet ({a | k ≤ N} ∩ ((⋂ (_ : k ≠ 0), {x | p x k}) ∩ ⋂ (i : ℕ) (_ : k < i) (_ : i ≤ N), {a | p a i}ᶜ))",
"tactic": "repeat' apply_rules [MeasurableSet.inter, MeasurableSet.const, MeasurableSet.iInter,\n MeasurableSet.compl, hN] <;> try intros"
},
{
"state_after": "no goals",
"state_before": "case h₂.h₂\nα : Type u_1\nβ : Type ?u.52344\nγ : Type ?u.52347\nδ : Type ?u.52350\nδ' : Type ?u.52353\nι : Sort uι\ns t u : Set α\ninst✝¹ : MeasurableSpace α\np : α → ℕ → Prop\ninst✝ : (x : α) → DecidablePred (p x)\nN : ℕ\nhN : ∀ (k : ℕ), k ≤ N → MeasurableSet {x | p x k}\nk : ℕ\nhk : k ≤ N\nb✝² : ℕ\nb✝¹ : k < b✝²\nb✝ : b✝² ≤ N\n⊢ MeasurableSet ({a | p a b✝²}ᶜ)",
"tactic": "apply_rules [MeasurableSet.inter, MeasurableSet.const, MeasurableSet.iInter,\nMeasurableSet.compl, hN] <;> try intros"
},
{
"state_after": "case h₂.h₂\nα : Type u_1\nβ : Type ?u.52344\nγ : Type ?u.52347\nδ : Type ?u.52350\nδ' : Type ?u.52353\nι : Sort uι\ns t u : Set α\ninst✝¹ : MeasurableSpace α\np : α → ℕ → Prop\ninst✝ : (x : α) → DecidablePred (p x)\nN : ℕ\nhN : ∀ (k : ℕ), k ≤ N → MeasurableSet {x | p x k}\nk : ℕ\nhk : k ≤ N\nb✝² : ℕ\nb✝¹ : k < b✝²\nb✝ : b✝² ≤ N\n⊢ MeasurableSet ({a | p a b✝²}ᶜ)",
"state_before": "case h₂.h₂\nα : Type u_1\nβ : Type ?u.52344\nγ : Type ?u.52347\nδ : Type ?u.52350\nδ' : Type ?u.52353\nι : Sort uι\ns t u : Set α\ninst✝¹ : MeasurableSpace α\np : α → ℕ → Prop\ninst✝ : (x : α) → DecidablePred (p x)\nN : ℕ\nhN : ∀ (k : ℕ), k ≤ N → MeasurableSet {x | p x k}\nk : ℕ\nhk : k ≤ N\nb✝¹ : ℕ\nb✝ : k < b✝¹\n⊢ b✝¹ ≤ N → MeasurableSet ({a | p a b✝¹}ᶜ)",
"tactic": "intros"
}
] |
[
450,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
445,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
coe_int_mem
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝¹ : Ring R\ninst✝ : SetLike S R\nhSR : SubringClass S R\ns : S\nn : ℤ\n⊢ ↑n ∈ s",
"tactic": "simp only [← zsmul_one, zsmul_mem, one_mem]"
}
] |
[
92,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/LinearAlgebra/Determinant.lean
|
LinearMap.det_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.899641\ninst✝¹² : CommRing R\nM✝ : Type ?u.899647\ninst✝¹¹ : AddCommGroup M✝\ninst✝¹⁰ : Module R M✝\nM' : Type ?u.900232\ninst✝⁹ : AddCommGroup M'\ninst✝⁸ : Module R M'\nι : Type ?u.900774\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : Fintype ι\ne : Basis ι R M✝\nA : Type ?u.901252\ninst✝⁵ : CommRing A\ninst✝⁴ : Module A M✝\nκ : Type ?u.901750\ninst✝³ : Fintype κ\n𝕜 : Type u_1\ninst✝² : Field 𝕜\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module 𝕜 M\n⊢ ↑LinearMap.det 0 = 0 ^ FiniteDimensional.finrank 𝕜 M",
"tactic": "simp only [← zero_smul 𝕜 (1 : M →ₗ[𝕜] M), det_smul, mul_one, MonoidHom.map_one]"
}
] |
[
289,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
|
LiouvilleWith.int_sub
|
[] |
[
303,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Topology/Maps.lean
|
ClosedEmbedding.isClosedMap
|
[] |
[
660,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
659,
1
] |
Mathlib/GroupTheory/Submonoid/Inverses.lean
|
Submonoid.mul_leftInvEquiv
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\nx : { x // x ∈ leftInv S }\n⊢ ↑x * ↑(↑(leftInvEquiv S hS) x) = 1",
"tactic": "simp only [leftInvEquiv_apply, fromCommLeftInv, mul_fromLeftInv]"
}
] |
[
204,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.div_zero
|
[
{
"state_after": "case zero\n\n⊢ div zero 0 = 0\n\ncase pos\na✝ : PosNum\n⊢ div (pos a✝) 0 = 0",
"state_before": "n : Num\n⊢ div n 0 = 0",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case zero\n\n⊢ div zero 0 = 0",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case pos\na✝ : PosNum\n⊢ div (pos a✝) 0 = 0",
"tactic": "simp [Num.div]"
}
] |
[
1647,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1643,
11
] |
Mathlib/Topology/Constructions.lean
|
isOpen_range_sigmaMk
|
[] |
[
1470,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1469,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.nat_coe_zmod_eq_iff
|
[
{
"state_after": "case mp\np n : ℕ\nz : ZMod p\ninst✝ : NeZero p\n⊢ ↑n = z → ∃ k, n = val z + p * k\n\ncase mpr\np n : ℕ\nz : ZMod p\ninst✝ : NeZero p\n⊢ (∃ k, n = val z + p * k) → ↑n = z",
"state_before": "p n : ℕ\nz : ZMod p\ninst✝ : NeZero p\n⊢ ↑n = z ↔ ∃ k, n = val z + p * k",
"tactic": "constructor"
},
{
"state_after": "case mp\np n : ℕ\ninst✝ : NeZero p\n⊢ ∃ k, n = val ↑n + p * k",
"state_before": "case mp\np n : ℕ\nz : ZMod p\ninst✝ : NeZero p\n⊢ ↑n = z → ∃ k, n = val z + p * k",
"tactic": "rintro rfl"
},
{
"state_after": "case mp\np n : ℕ\ninst✝ : NeZero p\n⊢ n = val ↑n + p * (n / p)",
"state_before": "case mp\np n : ℕ\ninst✝ : NeZero p\n⊢ ∃ k, n = val ↑n + p * k",
"tactic": "refine' ⟨n / p, _⟩"
},
{
"state_after": "no goals",
"state_before": "case mp\np n : ℕ\ninst✝ : NeZero p\n⊢ n = val ↑n + p * (n / p)",
"tactic": "rw [val_nat_cast, Nat.mod_add_div]"
},
{
"state_after": "case mpr.intro\np : ℕ\nz : ZMod p\ninst✝ : NeZero p\nk : ℕ\n⊢ ↑(val z + p * k) = z",
"state_before": "case mpr\np n : ℕ\nz : ZMod p\ninst✝ : NeZero p\n⊢ (∃ k, n = val z + p * k) → ↑n = z",
"tactic": "rintro ⟨k, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\np : ℕ\nz : ZMod p\ninst✝ : NeZero p\nk : ℕ\n⊢ ↑(val z + p * k) = z",
"tactic": "rw [Nat.cast_add, nat_cast_zmod_val, Nat.cast_mul, nat_cast_self, MulZeroClass.zero_mul,\n add_zero]"
}
] |
[
537,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/RingTheory/PowerBasis.lean
|
PowerBasis.minpolyGen_map
|
[
{
"state_after": "R : Type ?u.698148\nS : Type u_2\nT : Type ?u.698154\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : Ring S\ninst✝⁹ : Algebra R S\nA : Type u_1\nB : Type ?u.698460\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsDomain B\ninst✝⁵ : Algebra A B\nK : Type ?u.698882\ninst✝⁴ : Field K\nS' : Type u_3\ninst✝³ : CommRing S'\ninst✝² : Algebra R S'\ninst✝¹ : Algebra A S\ninst✝ : Algebra A S'\npb : PowerBasis A S\ne : S ≃ₐ[A] S'\n⊢ (X ^ pb.dim - Finset.sum Finset.univ fun i => ↑C (↑(↑(map pb e).basis.repr ((map pb e).gen ^ pb.dim)) i) * X ^ ↑i) =\n X ^ pb.dim - Finset.sum Finset.univ fun i => ↑C (↑(↑pb.basis.repr (pb.gen ^ pb.dim)) i) * X ^ ↑i",
"state_before": "R : Type ?u.698148\nS : Type u_2\nT : Type ?u.698154\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : Ring S\ninst✝⁹ : Algebra R S\nA : Type u_1\nB : Type ?u.698460\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsDomain B\ninst✝⁵ : Algebra A B\nK : Type ?u.698882\ninst✝⁴ : Field K\nS' : Type u_3\ninst✝³ : CommRing S'\ninst✝² : Algebra R S'\ninst✝¹ : Algebra A S\ninst✝ : Algebra A S'\npb : PowerBasis A S\ne : S ≃ₐ[A] S'\n⊢ minpolyGen (map pb e) = minpolyGen pb",
"tactic": "dsimp only [minpolyGen, map_dim]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.698148\nS : Type u_2\nT : Type ?u.698154\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : Ring S\ninst✝⁹ : Algebra R S\nA : Type u_1\nB : Type ?u.698460\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsDomain B\ninst✝⁵ : Algebra A B\nK : Type ?u.698882\ninst✝⁴ : Field K\nS' : Type u_3\ninst✝³ : CommRing S'\ninst✝² : Algebra R S'\ninst✝¹ : Algebra A S\ninst✝ : Algebra A S'\npb : PowerBasis A S\ne : S ≃ₐ[A] S'\n⊢ (X ^ pb.dim - Finset.sum Finset.univ fun i => ↑C (↑(↑(map pb e).basis.repr ((map pb e).gen ^ pb.dim)) i) * X ^ ↑i) =\n X ^ pb.dim - Finset.sum Finset.univ fun i => ↑C (↑(↑pb.basis.repr (pb.gen ^ pb.dim)) i) * X ^ ↑i",
"tactic": "simp only [LinearEquiv.trans_apply, map_basis, Basis.map_repr, map_gen,\n AlgEquiv.toLinearEquiv_apply, e.toLinearEquiv_symm, AlgEquiv.map_pow,\n AlgEquiv.symm_apply_apply, sub_right_inj]"
}
] |
[
482,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
476,
1
] |
Mathlib/Topology/LocallyConstant/Basic.lean
|
LocallyConstant.mulIndicator_apply_eq_if
|
[] |
[
550,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
548,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/NormalMono/Equalizers.lean
|
CategoryTheory.NormalMonoCategory.epi_of_zero_cancel
|
[] |
[
180,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
mul_lt_of_lt_one_of_lt
|
[] |
[
804,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
798,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.smul_apply
|
[] |
[
418,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
417,
1
] |
Std/Data/String/Lemmas.lean
|
String.Iterator.ValidFor.setCurr'
|
[
{
"state_after": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\n⊢ ValidFor l (List.modifyHead (fun x => c) r)\n (setCurr { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } } c)",
"state_before": "l r : List Char\nc : Char\nit : Iterator\nh : ValidFor l r it\n⊢ ValidFor l (List.modifyHead (fun x => c) r) (setCurr it c)",
"tactic": "cases h.out'"
},
{
"state_after": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\n⊢ ValidFor l\n (match r with\n | [] => []\n | a :: l => c :: l)\n { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }",
"state_before": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\n⊢ ValidFor l (List.modifyHead (fun x => c) r)\n (setCurr { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } } c)",
"tactic": "simp [Iterator.setCurr]"
},
{
"state_after": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\n⊢ { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.s.data =\n List.reverseAux l\n (match r with\n | [] => []\n | a :: l => c :: l)",
"state_before": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\n⊢ ValidFor l\n (match r with\n | [] => []\n | a :: l => c :: l)\n { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }",
"tactic": "refine .of_eq _ ?_ (by simp)"
},
{
"state_after": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\nthis :\n set { data := List.reverse l ++ r } { byteIdx := utf8Len (List.reverse l) } c =\n { data := List.reverse l ++ List.modifyHead (fun x => c) r }\n⊢ { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.s.data =\n List.reverseAux l\n (match r with\n | [] => []\n | a :: l => c :: l)",
"state_before": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\n⊢ { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.s.data =\n List.reverseAux l\n (match r with\n | [] => []\n | a :: l => c :: l)",
"tactic": "have := set_of_valid l.reverse r c"
},
{
"state_after": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\nthis :\n set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c =\n {\n data :=\n List.reverse l ++\n match r with\n | [] => []\n | a :: l => c :: l }\n⊢ { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.s.data =\n List.reverseAux l\n (match r with\n | [] => []\n | a :: l => c :: l)",
"state_before": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\nthis :\n set { data := List.reverse l ++ r } { byteIdx := utf8Len (List.reverse l) } c =\n { data := List.reverse l ++ List.modifyHead (fun x => c) r }\n⊢ { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.s.data =\n List.reverseAux l\n (match r with\n | [] => []\n | a :: l => c :: l)",
"tactic": "simp at this"
},
{
"state_after": "no goals",
"state_before": "case refl\nl r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\nthis :\n set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c =\n {\n data :=\n List.reverse l ++\n match r with\n | [] => []\n | a :: l => c :: l }\n⊢ { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.s.data =\n List.reverseAux l\n (match r with\n | [] => []\n | a :: l => c :: l)",
"tactic": "simp [List.reverseAux_eq, this]"
},
{
"state_after": "no goals",
"state_before": "l r : List Char\nc : Char\nh : ValidFor l r { s := { data := List.reverse l ++ r }, i := { byteIdx := utf8Len (List.reverse l) } }\n⊢ { s := set { data := List.reverse l ++ r } { byteIdx := utf8Len l } c, i := { byteIdx := utf8Len l } }.i.byteIdx =\n utf8Len l",
"tactic": "simp"
}
] |
[
582,
50
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
575,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.Red.cons_cons
|
[] |
[
250,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzWith.id
|
[] |
[
204,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
11
] |
Mathlib/Order/UpperLower/Basic.lean
|
Set.antitone_mem
|
[] |
[
272,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
1
] |
Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean
|
int_prod_range_nonneg
|
[
{
"state_after": "case intro\nm : ℤ\nn : ℕ\n⊢ 0 ≤ ∏ k in Finset.range (n + n), (m - ↑k)",
"state_before": "m : ℤ\nn : ℕ\nhn : Even n\n⊢ 0 ≤ ∏ k in Finset.range n, (m - ↑k)",
"tactic": "rcases hn with ⟨n, rfl⟩"
},
{
"state_after": "case intro.zero\nm : ℤ\n⊢ 0 ≤ ∏ k in Finset.range (Nat.zero + Nat.zero), (m - ↑k)\n\ncase intro.succ\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (n + n), (m - ↑k)\n⊢ 0 ≤ ∏ k in Finset.range (Nat.succ n + Nat.succ n), (m - ↑k)",
"state_before": "case intro\nm : ℤ\nn : ℕ\n⊢ 0 ≤ ∏ k in Finset.range (n + n), (m - ↑k)",
"tactic": "induction' n with n ihn"
},
{
"state_after": "case intro.succ\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\n⊢ 0 ≤ ∏ k in Finset.range (Nat.succ n + Nat.succ n), (m - ↑k)",
"state_before": "case intro.succ\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (n + n), (m - ↑k)\n⊢ 0 ≤ ∏ k in Finset.range (Nat.succ n + Nat.succ n), (m - ↑k)",
"tactic": "rw [← two_mul] at ihn"
},
{
"state_after": "case intro.succ\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\n⊢ 0 ≤ (∏ x in Finset.range ((1 + 1) * n), (m - ↑x)) * ((m - ↑((1 + 1) * n)) * (m - ↑((1 + 1) * n + 1)))",
"state_before": "case intro.succ\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\n⊢ 0 ≤ ∏ k in Finset.range (Nat.succ n + Nat.succ n), (m - ↑k)",
"tactic": "rw [← two_mul, Nat.succ_eq_add_one, mul_add, mul_one, ← one_add_one_eq_two, ← add_assoc,\n Finset.prod_range_succ, Finset.prod_range_succ, mul_assoc]"
},
{
"state_after": "case intro.succ\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\n⊢ 0 ≤ (m - ↑((1 + 1) * n)) * (m - ↑((1 + 1) * n + 1))",
"state_before": "case intro.succ\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\n⊢ 0 ≤ (∏ x in Finset.range ((1 + 1) * n), (m - ↑x)) * ((m - ↑((1 + 1) * n)) * (m - ↑((1 + 1) * n + 1)))",
"tactic": "refine' mul_nonneg ihn _"
},
{
"state_after": "case intro.succ\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\nk : ℕ\n⊢ 0 ≤ (m - ↑k) * (m - ↑(k + 1))",
"state_before": "case intro.succ\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\n⊢ 0 ≤ (m - ↑((1 + 1) * n)) * (m - ↑((1 + 1) * n + 1))",
"tactic": "generalize (1 + 1) * n = k"
},
{
"state_after": "case intro.succ.inl\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\nk : ℕ\nhmk : m ≤ ↑k\n⊢ 0 ≤ (m - ↑k) * (m - ↑(k + 1))\n\ncase intro.succ.inr\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\nk : ℕ\nhmk : ↑k < m\n⊢ 0 ≤ (m - ↑k) * (m - ↑(k + 1))",
"state_before": "case intro.succ\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\nk : ℕ\n⊢ 0 ≤ (m - ↑k) * (m - ↑(k + 1))",
"tactic": "cases' le_or_lt m k with hmk hmk"
},
{
"state_after": "no goals",
"state_before": "case intro.zero\nm : ℤ\n⊢ 0 ≤ ∏ k in Finset.range (Nat.zero + Nat.zero), (m - ↑k)",
"tactic": "simp"
},
{
"state_after": "case intro.succ.inl\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\nk : ℕ\nhmk : m ≤ ↑k\nthis : m ≤ ↑k + 1\n⊢ 0 ≤ (m - ↑k) * (m - ↑(k + 1))",
"state_before": "case intro.succ.inl\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\nk : ℕ\nhmk : m ≤ ↑k\n⊢ 0 ≤ (m - ↑k) * (m - ↑(k + 1))",
"tactic": "have : m ≤ k + 1 := hmk.trans (lt_add_one (k : ℤ)).le"
},
{
"state_after": "case intro.succ.inl.convert_2\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\nk : ℕ\nhmk : m ≤ ↑k\nthis : m ≤ ↑k + 1\n⊢ m - ↑(k + 1) ≤ 0",
"state_before": "case intro.succ.inl\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\nk : ℕ\nhmk : m ≤ ↑k\nthis : m ≤ ↑k + 1\n⊢ 0 ≤ (m - ↑k) * (m - ↑(k + 1))",
"tactic": "convert mul_nonneg_of_nonpos_of_nonpos (sub_nonpos_of_le hmk) _"
},
{
"state_after": "no goals",
"state_before": "case intro.succ.inl.convert_2\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\nk : ℕ\nhmk : m ≤ ↑k\nthis : m ≤ ↑k + 1\n⊢ m - ↑(k + 1) ≤ 0",
"tactic": "convert sub_nonpos_of_le this"
},
{
"state_after": "no goals",
"state_before": "case intro.succ.inr\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k in Finset.range (2 * n), (m - ↑k)\nk : ℕ\nhmk : ↑k < m\n⊢ 0 ≤ (m - ↑k) * (m - ↑(k + 1))",
"tactic": "exact mul_nonneg (sub_nonneg_of_le hmk.le) (sub_nonneg_of_le hmk)"
}
] |
[
89,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean
|
AlgebraicGeometry.StructureSheaf.openAlgebra_map
|
[] |
[
963,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
961,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
MulLECancellable.mul_le_iff_le_one_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.88936\ninst✝² : LE α\ninst✝¹ : CommMonoid α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b : α\nha : MulLECancellable a\n⊢ b * a ≤ a ↔ b ≤ 1",
"tactic": "rw [mul_comm, ha.mul_le_iff_le_one_right]"
}
] |
[
1672,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1670,
11
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.bit1_re
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.1751927\ninst✝ : IsROrC K\nz : K\n⊢ ↑re (bit1 z) = bit1 (↑re z)",
"tactic": "simp only [bit1, map_add, bit0_re, one_re]"
}
] |
[
184,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Data/List/Pairwise.lean
|
List.pwFilter_sublist
|
[
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.50704\nR S T : α → α → Prop\na : α\nl✝ : List α\ninst✝ : DecidableRel R\nx : α\nl : List α\nh : ∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ pwFilter R (x :: l) <+ x :: l\n\ncase neg\nα : Type u_1\nβ : Type ?u.50704\nR S T : α → α → Prop\na : α\nl✝ : List α\ninst✝ : DecidableRel R\nx : α\nl : List α\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ pwFilter R (x :: l) <+ x :: l",
"state_before": "α : Type u_1\nβ : Type ?u.50704\nR S T : α → α → Prop\na : α\nl✝ : List α\ninst✝ : DecidableRel R\nx : α\nl : List α\n⊢ pwFilter R (x :: l) <+ x :: l",
"tactic": "by_cases h : ∀ y ∈ pwFilter R l, R x y"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.50704\nR S T : α → α → Prop\na : α\nl✝ : List α\ninst✝ : DecidableRel R\nx : α\nl : List α\nh : ∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ x :: pwFilter R l <+ x :: l",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.50704\nR S T : α → α → Prop\na : α\nl✝ : List α\ninst✝ : DecidableRel R\nx : α\nl : List α\nh : ∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ pwFilter R (x :: l) <+ x :: l",
"tactic": "rw [pwFilter_cons_of_pos h]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.50704\nR S T : α → α → Prop\na : α\nl✝ : List α\ninst✝ : DecidableRel R\nx : α\nl : List α\nh : ∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ x :: pwFilter R l <+ x :: l",
"tactic": "exact (pwFilter_sublist l).cons_cons _"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.50704\nR S T : α → α → Prop\na : α\nl✝ : List α\ninst✝ : DecidableRel R\nx : α\nl : List α\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ pwFilter R l <+ x :: l",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.50704\nR S T : α → α → Prop\na : α\nl✝ : List α\ninst✝ : DecidableRel R\nx : α\nl : List α\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ pwFilter R (x :: l) <+ x :: l",
"tactic": "rw [pwFilter_cons_of_neg h]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.50704\nR S T : α → α → Prop\na : α\nl✝ : List α\ninst✝ : DecidableRel R\nx : α\nl : List α\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ pwFilter R l <+ x :: l",
"tactic": "exact sublist_cons_of_sublist _ (pwFilter_sublist l)"
}
] |
[
398,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
391,
1
] |
Mathlib/Algebra/Module/Zlattice.lean
|
Zspan.floor_eq_self_of_mem
|
[
{
"state_after": "E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\nh : m ∈ span ℤ (Set.range ↑b)\n⊢ ∀ (i : ι), ↑(↑b.repr ↑(floor b m)) i = ↑(↑b.repr m) i",
"state_before": "E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\nh : m ∈ span ℤ (Set.range ↑b)\n⊢ ↑(floor b m) = m",
"tactic": "apply b.ext_elem"
},
{
"state_after": "E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\nh : m ∈ span ℤ (Set.range ↑b)\n⊢ ∀ (i : ι), ↑⌊↑(↑b.repr m) i⌋ = ↑(↑b.repr m) i",
"state_before": "E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\nh : m ∈ span ℤ (Set.range ↑b)\n⊢ ∀ (i : ι), ↑(↑b.repr ↑(floor b m)) i = ↑(↑b.repr m) i",
"tactic": "simp_rw [repr_floor_apply b]"
},
{
"state_after": "E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\nh : m ∈ span ℤ (Set.range ↑b)\ni : ι\n⊢ ↑⌊↑(↑b.repr m) i⌋ = ↑(↑b.repr m) i",
"state_before": "E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\nh : m ∈ span ℤ (Set.range ↑b)\n⊢ ∀ (i : ι), ↑⌊↑(↑b.repr m) i⌋ = ↑(↑b.repr m) i",
"tactic": "intro i"
},
{
"state_after": "case intro\nE : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\nh : m ∈ span ℤ (Set.range ↑b)\ni : ι\nz : ℤ\nhz : ↑(algebraMap ℤ K) z = ↑(↑b.repr m) i\n⊢ ↑⌊↑(↑b.repr m) i⌋ = ↑(↑b.repr m) i",
"state_before": "E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\nh : m ∈ span ℤ (Set.range ↑b)\ni : ι\n⊢ ↑⌊↑(↑b.repr m) i⌋ = ↑(↑b.repr m) i",
"tactic": "obtain ⟨z, hz⟩ := (b.mem_span_iff_repr_mem ℤ _).mp h i"
},
{
"state_after": "case intro\nE : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\nh : m ∈ span ℤ (Set.range ↑b)\ni : ι\nz : ℤ\nhz : ↑(algebraMap ℤ K) z = ↑(↑b.repr m) i\n⊢ ↑⌊↑(algebraMap ℤ K) z⌋ = ↑(algebraMap ℤ K) z",
"state_before": "case intro\nE : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\nh : m ∈ span ℤ (Set.range ↑b)\ni : ι\nz : ℤ\nhz : ↑(algebraMap ℤ K) z = ↑(↑b.repr m) i\n⊢ ↑⌊↑(↑b.repr m) i⌋ = ↑(↑b.repr m) i",
"tactic": "rw [← hz]"
},
{
"state_after": "no goals",
"state_before": "case intro\nE : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\nh : m ∈ span ℤ (Set.range ↑b)\ni : ι\nz : ℤ\nhz : ↑(algebraMap ℤ K) z = ↑(↑b.repr m) i\n⊢ ↑⌊↑(algebraMap ℤ K) z⌋ = ↑(algebraMap ℤ K) z",
"tactic": "exact congr_arg (Int.cast : ℤ → K) (Int.floor_intCast z)"
}
] |
[
99,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/CategoryTheory/Idempotents/Basic.lean
|
CategoryTheory.Idempotents.isIdempotentComplete_iff_hasEqualizer_of_id_and_idempotent
|
[
{
"state_after": "case mp\nC : Type u_1\ninst✝ : Category C\n⊢ IsIdempotentComplete C → ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\n\ncase mpr\nC : Type u_1\ninst✝ : Category C\n⊢ (∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p) → IsIdempotentComplete C",
"state_before": "C : Type u_1\ninst✝ : Category C\n⊢ IsIdempotentComplete C ↔ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p",
"tactic": "constructor"
},
{
"state_after": "case mp\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\n⊢ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p",
"state_before": "case mp\nC : Type u_1\ninst✝ : Category C\n⊢ IsIdempotentComplete C → ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p",
"tactic": "intro"
},
{
"state_after": "case mp\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\n⊢ HasEqualizer (𝟙 X) p",
"state_before": "case mp\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\n⊢ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p",
"tactic": "intro X p hp"
},
{
"state_after": "case mp.intro.intro.intro.intro\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\n⊢ HasEqualizer (𝟙 X) p",
"state_before": "case mp\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\n⊢ HasEqualizer (𝟙 X) p",
"tactic": "rcases IsIdempotentComplete.idempotents_split X p hp with ⟨Y, i, e, ⟨h₁, h₂⟩⟩"
},
{
"state_after": "no goals",
"state_before": "C : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\n⊢ i ≫ 𝟙 X = i ≫ p",
"tactic": "rw [comp_id, ← h₂, ← assoc, h₁, id_comp]"
},
{
"state_after": "case create\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\n⊢ (s : Fork (𝟙 X) p) →\n { l //\n l ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s ∧\n ∀\n {m :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj\n WalkingParallelPair.zero},\n m ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s → m = l }",
"state_before": "C : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\n⊢ IsLimit (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p))",
"tactic": "apply Fork.IsLimit.mk'"
},
{
"state_after": "case create\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\n⊢ { l //\n l ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s ∧\n ∀\n {m :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj\n WalkingParallelPair.zero},\n m ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s → m = l }",
"state_before": "case create\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\n⊢ (s : Fork (𝟙 X) p) →\n { l //\n l ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s ∧\n ∀\n {m :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj\n WalkingParallelPair.zero},\n m ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s → m = l }",
"tactic": "intro s"
},
{
"state_after": "case create\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\n⊢ (Fork.ι s ≫ e) ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s ∧\n ∀\n {m :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj WalkingParallelPair.zero},\n m ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s → m = Fork.ι s ≫ e",
"state_before": "case create\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\n⊢ { l //\n l ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s ∧\n ∀\n {m :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj\n WalkingParallelPair.zero},\n m ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s → m = l }",
"tactic": "refine' ⟨s.ι ≫ e, _⟩"
},
{
"state_after": "case create.left\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\n⊢ (Fork.ι s ≫ e) ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s\n\ncase create.right\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\n⊢ ∀\n {m :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj WalkingParallelPair.zero},\n m ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s → m = Fork.ι s ≫ e",
"state_before": "case create\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\n⊢ (Fork.ι s ≫ e) ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s ∧\n ∀\n {m :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj WalkingParallelPair.zero},\n m ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s → m = Fork.ι s ≫ e",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case create.left\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\n⊢ (Fork.ι s ≫ e) ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s",
"tactic": "erw [assoc, h₂, ← Limits.Fork.condition s, comp_id]"
},
{
"state_after": "case create.right\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\nm :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj WalkingParallelPair.zero\nhm : m ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s\n⊢ m = Fork.ι s ≫ e",
"state_before": "case create.right\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\n⊢ ∀\n {m :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj WalkingParallelPair.zero},\n m ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s → m = Fork.ι s ≫ e",
"tactic": "intro m hm"
},
{
"state_after": "case create.right\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\nm :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj WalkingParallelPair.zero\nhm : m ≫ i = Fork.ι s\n⊢ m = Fork.ι s ≫ e",
"state_before": "case create.right\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\nm :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj WalkingParallelPair.zero\nhm : m ≫ Fork.ι (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)) = Fork.ι s\n⊢ m = Fork.ι s ≫ e",
"tactic": "rw [Fork.ι_ofι] at hm"
},
{
"state_after": "case create.right\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\nm :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj WalkingParallelPair.zero\nhm : m ≫ i = Fork.ι s\n⊢ m = (m ≫ i) ≫ e",
"state_before": "case create.right\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\nm :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj WalkingParallelPair.zero\nhm : m ≫ i = Fork.ι s\n⊢ m = Fork.ι s ≫ e",
"tactic": "rw [← hm]"
},
{
"state_after": "case create.right\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\nm :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj WalkingParallelPair.zero\nhm : m ≫ i = Fork.ι s\n⊢ m = m ≫ 𝟙 Y",
"state_before": "case create.right\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\nm :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj WalkingParallelPair.zero\nhm : m ≫ i = Fork.ι s\n⊢ m = (m ≫ i) ≫ e",
"tactic": "simp only [← hm, assoc, h₁]"
},
{
"state_after": "no goals",
"state_before": "case create.right\nC : Type u_1\ninst✝ : Category C\na✝ : IsIdempotentComplete C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nY : C\ni : Y ⟶ X\ne : X ⟶ Y\nh₁ : i ≫ e = 𝟙 Y\nh₂ : e ≫ i = p\ns : Fork (𝟙 X) p\nm :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero ⟶\n ((Functor.const WalkingParallelPair).obj (Fork.ofι i (_ : i ≫ 𝟙 X = i ≫ p)).pt).obj WalkingParallelPair.zero\nhm : m ≫ i = Fork.ι s\n⊢ m = m ≫ 𝟙 Y",
"tactic": "exact (comp_id m).symm"
},
{
"state_after": "case mpr\nC : Type u_1\ninst✝ : Category C\nh : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\n⊢ IsIdempotentComplete C",
"state_before": "case mpr\nC : Type u_1\ninst✝ : Category C\n⊢ (∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p) → IsIdempotentComplete C",
"tactic": "intro h"
},
{
"state_after": "case mpr\nC : Type u_1\ninst✝ : Category C\nh : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\n⊢ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → ∃ Y i e, i ≫ e = 𝟙 Y ∧ e ≫ i = p",
"state_before": "case mpr\nC : Type u_1\ninst✝ : Category C\nh : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\n⊢ IsIdempotentComplete C",
"tactic": "refine' ⟨_⟩"
},
{
"state_after": "case mpr\nC : Type u_1\ninst✝ : Category C\nh : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\nX : C\np : X ⟶ X\nhp : p ≫ p = p\n⊢ ∃ Y i e, i ≫ e = 𝟙 Y ∧ e ≫ i = p",
"state_before": "case mpr\nC : Type u_1\ninst✝ : Category C\nh : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\n⊢ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → ∃ Y i e, i ≫ e = 𝟙 Y ∧ e ≫ i = p",
"tactic": "intro X p hp"
},
{
"state_after": "case mpr\nC : Type u_1\ninst✝ : Category C\nh : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nthis : HasEqualizer (𝟙 X) p\n⊢ ∃ Y i e, i ≫ e = 𝟙 Y ∧ e ≫ i = p",
"state_before": "case mpr\nC : Type u_1\ninst✝ : Category C\nh : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\nX : C\np : X ⟶ X\nhp : p ≫ p = p\n⊢ ∃ Y i e, i ≫ e = 𝟙 Y ∧ e ≫ i = p",
"tactic": "haveI : HasEqualizer (𝟙 X) p := h X p hp"
},
{
"state_after": "case mpr\nC : Type u_1\ninst✝ : Category C\nh : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nthis : HasEqualizer (𝟙 X) p\n⊢ equalizer.ι (𝟙 X) p ≫ equalizer.lift p (_ : p ≫ 𝟙 X = p ≫ p) = 𝟙 (equalizer (𝟙 X) p)",
"state_before": "case mpr\nC : Type u_1\ninst✝ : Category C\nh : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nthis : HasEqualizer (𝟙 X) p\n⊢ ∃ Y i e, i ≫ e = 𝟙 Y ∧ e ≫ i = p",
"tactic": "refine' ⟨equalizer (𝟙 X) p, equalizer.ι (𝟙 X) p,\n equalizer.lift p (show p ≫ 𝟙 X = p ≫ p by rw [hp, comp_id]), _, equalizer.lift_ι _ _⟩"
},
{
"state_after": "case mpr.h\nC : Type u_1\ninst✝ : Category C\nh : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nthis : HasEqualizer (𝟙 X) p\n⊢ (equalizer.ι (𝟙 X) p ≫ equalizer.lift p (_ : p ≫ 𝟙 X = p ≫ p)) ≫ equalizer.ι (𝟙 X) p =\n 𝟙 (equalizer (𝟙 X) p) ≫ equalizer.ι (𝟙 X) p",
"state_before": "case mpr\nC : Type u_1\ninst✝ : Category C\nh : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nthis : HasEqualizer (𝟙 X) p\n⊢ equalizer.ι (𝟙 X) p ≫ equalizer.lift p (_ : p ≫ 𝟙 X = p ≫ p) = 𝟙 (equalizer (𝟙 X) p)",
"tactic": "apply equalizer.hom_ext"
},
{
"state_after": "case mpr.h\nC : Type u_1\ninst✝ : Category C\nh : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nthis : HasEqualizer (𝟙 X) p\n⊢ equalizer.ι (𝟙 X) p ≫ p = equalizer.ι (𝟙 X) p",
"state_before": "case mpr.h\nC : Type u_1\ninst✝ : Category C\nh : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nthis : HasEqualizer (𝟙 X) p\n⊢ (equalizer.ι (𝟙 X) p ≫ equalizer.lift p (_ : p ≫ 𝟙 X = p ≫ p)) ≫ equalizer.ι (𝟙 X) p =\n 𝟙 (equalizer (𝟙 X) p) ≫ equalizer.ι (𝟙 X) p",
"tactic": "simp only [assoc, limit.lift_π, Eq.ndrec, id_eq, eq_mpr_eq_cast, Fork.ofι_pt,\n Fork.ofι_π_app, id_comp]"
},
{
"state_after": "no goals",
"state_before": "case mpr.h\nC : Type u_1\ninst✝ : Category C\nh : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nthis : HasEqualizer (𝟙 X) p\n⊢ equalizer.ι (𝟙 X) p ≫ p = equalizer.ι (𝟙 X) p",
"tactic": "rw [← equalizer.condition, comp_id]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_1\ninst✝ : Category C\nh : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\nX : C\np : X ⟶ X\nhp : p ≫ p = p\nthis : HasEqualizer (𝟙 X) p\n⊢ p ≫ 𝟙 X = p ≫ p",
"tactic": "rw [hp, comp_id]"
}
] |
[
95,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean
|
finprod_mem_image'
|
[
{
"state_after": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : Set.Finite (s ∩ mulSupport (f ∘ g))\n⊢ (∏ᶠ (i : α) (_ : i ∈ g '' s), f i) = ∏ᶠ (j : β) (_ : j ∈ s), f (g j)\n\ncase neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : ¬Set.Finite (s ∩ mulSupport (f ∘ g))\n⊢ (∏ᶠ (i : α) (_ : i ∈ g '' s), f i) = ∏ᶠ (j : β) (_ : j ∈ s), f (g j)",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg : InjOn g (s ∩ mulSupport (f ∘ g))\n⊢ (∏ᶠ (i : α) (_ : i ∈ g '' s), f i) = ∏ᶠ (j : β) (_ : j ∈ s), f (g j)",
"tactic": "by_cases hs : (s ∩ mulSupport (f ∘ g)).Finite"
},
{
"state_after": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg✝ : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : Set.Finite (s ∩ mulSupport (f ∘ g))\nhg : ∀ (x : β), x ∈ Finite.toFinset hs → ∀ (y : β), y ∈ Finite.toFinset hs → g x = g y → x = y\n⊢ (∏ᶠ (i : α) (_ : i ∈ g '' s), f i) = ∏ᶠ (j : β) (_ : j ∈ s), f (g j)",
"state_before": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : Set.Finite (s ∩ mulSupport (f ∘ g))\n⊢ (∏ᶠ (i : α) (_ : i ∈ g '' s), f i) = ∏ᶠ (j : β) (_ : j ∈ s), f (g j)",
"tactic": "have hg : ∀ x ∈ hs.toFinset, ∀ y ∈ hs.toFinset, g x = g y → x = y := by\n simpa only [hs.mem_toFinset]"
},
{
"state_after": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg✝ : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : Set.Finite (s ∩ mulSupport (f ∘ g))\nhg : ∀ (x : β), x ∈ Finite.toFinset hs → ∀ (y : β), y ∈ Finite.toFinset hs → g x = g y → x = y\nthis : (∏ᶠ (i : β) (_ : i ∈ s), (f ∘ g) i) = ∏ i in Finite.toFinset hs, (f ∘ g) i\n⊢ (∏ᶠ (i : α) (_ : i ∈ g '' s), f i) = ∏ᶠ (j : β) (_ : j ∈ s), f (g j)",
"state_before": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg✝ : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : Set.Finite (s ∩ mulSupport (f ∘ g))\nhg : ∀ (x : β), x ∈ Finite.toFinset hs → ∀ (y : β), y ∈ Finite.toFinset hs → g x = g y → x = y\n⊢ (∏ᶠ (i : α) (_ : i ∈ g '' s), f i) = ∏ᶠ (j : β) (_ : j ∈ s), f (g j)",
"tactic": "have := finprod_mem_eq_prod (comp f g) hs"
},
{
"state_after": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg✝ : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : Set.Finite (s ∩ mulSupport (f ∘ g))\nhg : ∀ (x : β), x ∈ Finite.toFinset hs → ∀ (y : β), y ∈ Finite.toFinset hs → g x = g y → x = y\nthis : (∏ᶠ (i : β) (_ : i ∈ s), f (g i)) = ∏ i in Finite.toFinset hs, f (g i)\n⊢ (∏ᶠ (i : α) (_ : i ∈ g '' s), f i) = ∏ᶠ (j : β) (_ : j ∈ s), f (g j)",
"state_before": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg✝ : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : Set.Finite (s ∩ mulSupport (f ∘ g))\nhg : ∀ (x : β), x ∈ Finite.toFinset hs → ∀ (y : β), y ∈ Finite.toFinset hs → g x = g y → x = y\nthis : (∏ᶠ (i : β) (_ : i ∈ s), (f ∘ g) i) = ∏ i in Finite.toFinset hs, (f ∘ g) i\n⊢ (∏ᶠ (i : α) (_ : i ∈ g '' s), f i) = ∏ᶠ (j : β) (_ : j ∈ s), f (g j)",
"tactic": "unfold Function.comp at this"
},
{
"state_after": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg✝ : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : Set.Finite (s ∩ mulSupport (f ∘ g))\nhg : ∀ (x : β), x ∈ Finite.toFinset hs → ∀ (y : β), y ∈ Finite.toFinset hs → g x = g y → x = y\nthis : (∏ᶠ (i : β) (_ : i ∈ s), f (g i)) = ∏ i in Finite.toFinset hs, f (g i)\n⊢ (∏ᶠ (i : α) (_ : i ∈ g '' s), f i) = ∏ x in Finset.image (fun x => g x) (Finite.toFinset hs), f x",
"state_before": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg✝ : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : Set.Finite (s ∩ mulSupport (f ∘ g))\nhg : ∀ (x : β), x ∈ Finite.toFinset hs → ∀ (y : β), y ∈ Finite.toFinset hs → g x = g y → x = y\nthis : (∏ᶠ (i : β) (_ : i ∈ s), f (g i)) = ∏ i in Finite.toFinset hs, f (g i)\n⊢ (∏ᶠ (i : α) (_ : i ∈ g '' s), f i) = ∏ᶠ (j : β) (_ : j ∈ s), f (g j)",
"tactic": "rw [this, ← Finset.prod_image hg]"
},
{
"state_after": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg✝ : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : Set.Finite (s ∩ mulSupport (f ∘ g))\nhg : ∀ (x : β), x ∈ Finite.toFinset hs → ∀ (y : β), y ∈ Finite.toFinset hs → g x = g y → x = y\nthis : (∏ᶠ (i : β) (_ : i ∈ s), f (g i)) = ∏ i in Finite.toFinset hs, f (g i)\n⊢ g '' s ∩ mulSupport f = ↑(Finset.image (fun x => g x) (Finite.toFinset hs)) ∩ mulSupport f",
"state_before": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg✝ : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : Set.Finite (s ∩ mulSupport (f ∘ g))\nhg : ∀ (x : β), x ∈ Finite.toFinset hs → ∀ (y : β), y ∈ Finite.toFinset hs → g x = g y → x = y\nthis : (∏ᶠ (i : β) (_ : i ∈ s), f (g i)) = ∏ i in Finite.toFinset hs, f (g i)\n⊢ (∏ᶠ (i : α) (_ : i ∈ g '' s), f i) = ∏ x in Finset.image (fun x => g x) (Finite.toFinset hs), f x",
"tactic": "refine' finprod_mem_eq_prod_of_inter_mulSupport_eq f _"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg✝ : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : Set.Finite (s ∩ mulSupport (f ∘ g))\nhg : ∀ (x : β), x ∈ Finite.toFinset hs → ∀ (y : β), y ∈ Finite.toFinset hs → g x = g y → x = y\nthis : (∏ᶠ (i : β) (_ : i ∈ s), f (g i)) = ∏ i in Finite.toFinset hs, f (g i)\n⊢ g '' s ∩ mulSupport f = ↑(Finset.image (fun x => g x) (Finite.toFinset hs)) ∩ mulSupport f",
"tactic": "rw [Finset.coe_image, hs.coe_toFinset, ← image_inter_mulSupport_eq, inter_assoc, inter_self]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : Set.Finite (s ∩ mulSupport (f ∘ g))\n⊢ ∀ (x : β), x ∈ Finite.toFinset hs → ∀ (y : β), y ∈ Finite.toFinset hs → g x = g y → x = y",
"tactic": "simpa only [hs.mem_toFinset]"
},
{
"state_after": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : ¬Set.Finite (s ∩ mulSupport fun x => f (g x))\n⊢ (∏ᶠ (i : α) (_ : i ∈ g '' s), f i) = ∏ᶠ (j : β) (_ : j ∈ s), f (g j)",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : ¬Set.Finite (s ∩ mulSupport (f ∘ g))\n⊢ (∏ᶠ (i : α) (_ : i ∈ g '' s), f i) = ∏ᶠ (j : β) (_ : j ∈ s), f (g j)",
"tactic": "unfold Function.comp at hs"
},
{
"state_after": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : ¬Set.Finite (s ∩ mulSupport fun x => f (g x))\n⊢ Set.Infinite (g '' s ∩ mulSupport fun i => f i)",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : ¬Set.Finite (s ∩ mulSupport fun x => f (g x))\n⊢ (∏ᶠ (i : α) (_ : i ∈ g '' s), f i) = ∏ᶠ (j : β) (_ : j ∈ s), f (g j)",
"tactic": "rw [finprod_mem_eq_one_of_infinite hs, finprod_mem_eq_one_of_infinite]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nι : Type ?u.350982\nG : Type ?u.350985\nM : Type u_3\nN : Type ?u.350991\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g✝ : α → M\na b : α\ns✝ t : Set α\ns : Set β\ng : β → α\nhg : InjOn g (s ∩ mulSupport (f ∘ g))\nhs : ¬Set.Finite (s ∩ mulSupport fun x => f (g x))\n⊢ Set.Infinite (g '' s ∩ mulSupport fun i => f i)",
"tactic": "rwa [image_inter_mulSupport_eq, infinite_image_iff hg]"
}
] |
[
928,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
915,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean
|
CategoryTheory.Limits.hasStrictInitialObjects_of_initial_is_strict
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasInitial C\nh : ∀ (A : C) (f : A ⟶ ⊥_ C), IsIso f\nI A : C\nf : A ⟶ I\nhI : IsInitial I\nthis : IsIso (f ≫ IsInitial.to hI (⊥_ C))\n⊢ f ≫ IsInitial.to hI (⊥_ C) ≫ inv (f ≫ IsInitial.to hI (⊥_ C)) = 𝟙 A",
"tactic": "rw [← assoc, IsIso.hom_inv_id]"
}
] |
[
168,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Analysis/Calculus/LHopital.lean
|
HasDerivAt.lhopital_zero_nhds_right
|
[
{
"state_after": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∃ v, v ∈ 𝓝[Ioi a] a ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt f (f' y) y\nhgg' : ∃ v, v ∈ 𝓝[Ioi a] a ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt g (g' y) y\nhg' : ∃ v, v ∈ 𝓝[Ioi a] a ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in 𝓝[Ioi a] a, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝[Ioi a] a, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝[Ioi a] a, g' x ≠ 0\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"tactic": "rw [eventually_iff_exists_mem] at *"
},
{
"state_after": "case intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhgg' : ∃ v, v ∈ 𝓝[Ioi a] a ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt g (g' y) y\nhg' : ∃ v, v ∈ 𝓝[Ioi a] a ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∃ v, v ∈ 𝓝[Ioi a] a ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt f (f' y) y\nhgg' : ∃ v, v ∈ 𝓝[Ioi a] a ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt g (g' y) y\nhg' : ∃ v, v ∈ 𝓝[Ioi a] a ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"tactic": "rcases hff' with ⟨s₁, hs₁, hff'⟩"
},
{
"state_after": "case intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhg' : ∃ v, v ∈ 𝓝[Ioi a] a ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"state_before": "case intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhgg' : ∃ v, v ∈ 𝓝[Ioi a] a ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt g (g' y) y\nhg' : ∃ v, v ∈ 𝓝[Ioi a] a ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"tactic": "rcases hgg' with ⟨s₂, hs₂, hgg'⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ 𝓝[Ioi a] a\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"state_before": "case intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhg' : ∃ v, v ∈ 𝓝[Ioi a] a ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"tactic": "rcases hg' with ⟨s₃, hs₃, hg'⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ 𝓝[Ioi a] a\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"state_before": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ 𝓝[Ioi a] a\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"tactic": "let s := s₁ ∩ s₂ ∩ s₃"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ 𝓝[Ioi a] a\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nhs : s ∈ 𝓝[Ioi a] a\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"state_before": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ 𝓝[Ioi a] a\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"tactic": "have hs : s ∈ 𝓝[>] a := inter_mem (inter_mem hs₁ hs₂) hs₃"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ 𝓝[Ioi a] a\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nhs : ∃ u, u ∈ Ioi a ∧ Ioo a u ⊆ s\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"state_before": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ 𝓝[Ioi a] a\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nhs : s ∈ 𝓝[Ioi a] a\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"tactic": "rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset] at hs"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ 𝓝[Ioi a] a\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nu : ℝ\nhau : u ∈ Ioi a\nhu : Ioo a u ⊆ s\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"state_before": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ 𝓝[Ioi a] a\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nhs : ∃ u, u ∈ Ioi a ∧ Ioo a u ⊆ s\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"tactic": "rcases hs with ⟨u, hau, hu⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ 𝓝[Ioi a] a\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nu : ℝ\nhau : u ∈ Ioi a\nhu : Ioo a u ⊆ s\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"tactic": "refine' lhopital_zero_right_on_Ioo hau _ _ _ hfa hga hdiv <;> intro x hx <;> apply_assumption <;>\n first | exact (hu hx).1.1 | exact (hu hx).1.2 | exact (hu hx).2"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.a\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ 𝓝[Ioi a] a\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nu : ℝ\nhau : u ∈ Ioi a\nhu : Ioo a u ⊆ s\nx : ℝ\nhx : x ∈ Ioo a u\n⊢ x ∈ s₃",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.a\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ 𝓝[Ioi a] a\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nu : ℝ\nhau : u ∈ Ioi a\nhu : Ioo a u ⊆ s\nx : ℝ\nhx : x ∈ Ioo a u\n⊢ x ∈ s₃",
"tactic": "exact (hu hx).1.1"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.a\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ 𝓝[Ioi a] a\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nu : ℝ\nhau : u ∈ Ioi a\nhu : Ioo a u ⊆ s\nx : ℝ\nhx : x ∈ Ioo a u\n⊢ x ∈ s₃",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.a\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ 𝓝[Ioi a] a\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nu : ℝ\nhau : u ∈ Ioi a\nhu : Ioo a u ⊆ s\nx : ℝ\nhx : x ∈ Ioo a u\n⊢ x ∈ s₃",
"tactic": "exact (hu hx).1.2"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.a\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Ioi a] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[Ioi a] a\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[Ioi a] a\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ 𝓝[Ioi a] a\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nu : ℝ\nhau : u ∈ Ioi a\nhu : Ioo a u ⊆ s\nx : ℝ\nhx : x ∈ Ioo a u\n⊢ x ∈ s₃",
"tactic": "exact (hu hx).2"
}
] |
[
299,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/Algebra/GroupPower/Order.lean
|
abs_lt_of_sq_lt_sq
|
[
{
"state_after": "no goals",
"state_before": "β : Type ?u.263157\nA : Type ?u.263160\nG : Type ?u.263163\nM : Type ?u.263166\nR : Type u_1\ninst✝ : LinearOrderedRing R\nx y : R\nh : x ^ 2 < y ^ 2\nhy : 0 ≤ y\n⊢ abs x < y",
"tactic": "rwa [← abs_of_nonneg hy, ← sq_lt_sq]"
}
] |
[
710,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
709,
1
] |
Mathlib/Algebra/Order/WithZero.lean
|
one_le_mul₀
|
[] |
[
133,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/Data/Matrix/Notation.lean
|
Matrix.vecMulVec_empty
|
[] |
[
337,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/CategoryTheory/Closed/Monoidal.lean
|
CategoryTheory.MonoidalClosed.uncurry_injective
|
[] |
[
233,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/Data/List/Rotate.lean
|
List.length_rotate
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nl : List α\nn : ℕ\n⊢ length (rotate l n) = length l",
"tactic": "rw [rotate_eq_rotate', length_rotate']"
}
] |
[
133,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/Analysis/LocallyConvex/Bounded.lean
|
NormedSpace.isVonNBounded_closedBall
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕜' : Type ?u.241289\nE : Type u_2\nE' : Type ?u.241295\nF : Type ?u.241298\nι : Type ?u.241301\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\n⊢ r < r + 1",
"tactic": "linarith"
}
] |
[
284,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/Computability/Reduce.lean
|
Computable.eqv
|
[] |
[
225,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/Analysis/Calculus/LocalExtr.lean
|
IsLocalMax.deriv_eq_zero
|
[] |
[
248,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.integral_hasFDerivAt_of_tendsto_ae
|
[] |
[
712,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
707,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.coe_inf
|
[] |
[
513,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
512,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
Set.MapsTo.closure_of_continuousOn
|
[] |
[
744,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
742,
1
] |
Mathlib/LinearAlgebra/Ray.lean
|
ray_pos_smul
|
[] |
[
282,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometry.congr_fun
|
[] |
[
199,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
11
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
Polynomial.constantCoeff_coe
|
[] |
[
2605,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2604,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.map_cons
|
[] |
[
1177,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1176,
1
] |
Mathlib/Data/Vector.lean
|
Vector.toList_mk
|
[] |
[
216,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Topology/Spectral/Hom.lean
|
IsSpectralMap.comp
|
[] |
[
63,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
contDiffAt_snd
|
[] |
[
822,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
821,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.card_Ioc_eq_card_Icc_sub_one
|
[] |
[
649,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
648,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
measurable_edist_left
|
[] |
[
1641,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1640,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_pow_atTop
|
[] |
[
892,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
891,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.nonempty_cons
|
[] |
[
866,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
865,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.PartrecToTM2.trNum_natEnd
|
[] |
[
1463,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1462,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.filter_cons_of_neg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.209744\nγ : Type ?u.209747\np : α → Prop\ninst✝ : DecidablePred p\na : α\ns : Multiset α\nl : List α\nh : ¬p a\n⊢ ¬decide (p a) = true",
"tactic": "simpa using h"
}
] |
[
1974,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1973,
1
] |
Mathlib/Topology/Algebra/Star.lean
|
Filter.Tendsto.star
|
[] |
[
55,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.coe_inf
|
[] |
[
688,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
687,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
dist_le_coe
|
[] |
[
366,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.zpowers_eq
|
[
{
"state_after": "case a\nM : Type ?u.3298392\nN : Type ?u.3298395\nG : Type ?u.3298398\nR : Type u_1\nS : Type ?u.3298404\nF : Type ?u.3298407\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nk : ℕ+\nζ : Rˣ\nh : IsPrimitiveRoot ζ ↑k\n⊢ ↑(Subgroup.zpowers ζ) = ↑(rootsOfUnity k R)",
"state_before": "M : Type ?u.3298392\nN : Type ?u.3298395\nG : Type ?u.3298398\nR : Type u_1\nS : Type ?u.3298404\nF : Type ?u.3298407\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nk : ℕ+\nζ : Rˣ\nh : IsPrimitiveRoot ζ ↑k\n⊢ Subgroup.zpowers ζ = rootsOfUnity k R",
"tactic": "apply SetLike.coe_injective"
},
{
"state_after": "case a\nM : Type ?u.3298392\nN : Type ?u.3298395\nG : Type ?u.3298398\nR : Type u_1\nS : Type ?u.3298404\nF✝ : Type ?u.3298407\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nk : ℕ+\nζ : Rˣ\nh : IsPrimitiveRoot ζ ↑k\nF : Fintype { x // x ∈ Subgroup.zpowers ζ }\n⊢ ↑(Subgroup.zpowers ζ) = ↑(rootsOfUnity k R)",
"state_before": "case a\nM : Type ?u.3298392\nN : Type ?u.3298395\nG : Type ?u.3298398\nR : Type u_1\nS : Type ?u.3298404\nF : Type ?u.3298407\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nk : ℕ+\nζ : Rˣ\nh : IsPrimitiveRoot ζ ↑k\n⊢ ↑(Subgroup.zpowers ζ) = ↑(rootsOfUnity k R)",
"tactic": "haveI F : Fintype (Subgroup.zpowers ζ) := Fintype.ofEquiv _ h.zmodEquivZpowers.toEquiv"
},
{
"state_after": "case a\nM : Type ?u.3298392\nN : Type ?u.3298395\nG : Type ?u.3298398\nR : Type u_1\nS : Type ?u.3298404\nF✝ : Type ?u.3298407\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nk : ℕ+\nζ : Rˣ\nh : IsPrimitiveRoot ζ ↑k\nF : Fintype { x // x ∈ Subgroup.zpowers ζ }\n⊢ Fintype.card ↑↑(rootsOfUnity k R) ≤ Fintype.card ↑↑(Subgroup.zpowers ζ)",
"state_before": "case a\nM : Type ?u.3298392\nN : Type ?u.3298395\nG : Type ?u.3298398\nR : Type u_1\nS : Type ?u.3298404\nF✝ : Type ?u.3298407\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nk : ℕ+\nζ : Rˣ\nh : IsPrimitiveRoot ζ ↑k\nF : Fintype { x // x ∈ Subgroup.zpowers ζ }\n⊢ ↑(Subgroup.zpowers ζ) = ↑(rootsOfUnity k R)",
"tactic": "refine'\n @Set.eq_of_subset_of_card_le Rˣ (Subgroup.zpowers ζ) (rootsOfUnity k R) F\n (rootsOfUnity.fintype R k)\n (Subgroup.zpowers_le_of_mem <| show ζ ∈ rootsOfUnity k R from h.pow_eq_one) _"
},
{
"state_after": "no goals",
"state_before": "case a\nM : Type ?u.3298392\nN : Type ?u.3298395\nG : Type ?u.3298398\nR : Type u_1\nS : Type ?u.3298404\nF✝ : Type ?u.3298407\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk✝ l : ℕ\ninst✝¹ : CommRing R\nζ✝ : Rˣ\nh✝ : IsPrimitiveRoot ζ✝ k✝\ninst✝ : IsDomain R\nk : ℕ+\nζ : Rˣ\nh : IsPrimitiveRoot ζ ↑k\nF : Fintype { x // x ∈ Subgroup.zpowers ζ }\n⊢ Fintype.card ↑↑(rootsOfUnity k R) ≤ Fintype.card ↑↑(Subgroup.zpowers ζ)",
"tactic": "calc\n Fintype.card (rootsOfUnity k R) ≤ k := card_rootsOfUnity R k\n _ = Fintype.card (ZMod k) := (ZMod.card k).symm\n _ = Fintype.card (Subgroup.zpowers ζ) := Fintype.card_congr h.zmodEquivZpowers.toEquiv"
}
] |
[
753,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
742,
1
] |
Mathlib/Data/Set/Intervals/Infinite.lean
|
Set.Ioi_infinite
|
[] |
[
95,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Analysis/MeanInequalities.lean
|
NNReal.inner_le_Lp_mul_Lp_of_norm_eq_zero
|
[
{
"state_after": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ∑ i in s, f i ^ p = 0\n⊢ ∀ (x : ι), x ∈ s → f x = 0 ∨ g x = 0",
"state_before": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ∑ i in s, f i ^ p = 0\n⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q)",
"tactic": "simp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, MulZeroClass.zero_mul,\n inv_eq_zero, Ne.def, not_false_iff, le_zero_iff, mul_eq_zero]"
},
{
"state_after": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ∑ i in s, f i ^ p = 0\ni : ι\nhis : i ∈ s\n⊢ f i = 0 ∨ g i = 0",
"state_before": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ∑ i in s, f i ^ p = 0\n⊢ ∀ (x : ι), x ∈ s → f x = 0 ∨ g x = 0",
"tactic": "intro i his"
},
{
"state_after": "case h\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ∑ i in s, f i ^ p = 0\ni : ι\nhis : i ∈ s\n⊢ f i = 0",
"state_before": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ∑ i in s, f i ^ p = 0\ni : ι\nhis : i ∈ s\n⊢ f i = 0 ∨ g i = 0",
"tactic": "left"
},
{
"state_after": "case h\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ∀ (x : ι), x ∈ s → f x ^ p = 0\ni : ι\nhis : i ∈ s\n⊢ f i = 0",
"state_before": "case h\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ∑ i in s, f i ^ p = 0\ni : ι\nhis : i ∈ s\n⊢ f i = 0",
"tactic": "rw [sum_eq_zero_iff] at hf"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : Real.IsConjugateExponent p q\nhf : ∀ (x : ι), x ∈ s → f x ^ p = 0\ni : ι\nhis : i ∈ s\n⊢ f i = 0",
"tactic": "exact (rpow_eq_zero_iff.mp (hf i his)).left"
}
] |
[
346,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
9
] |
Mathlib/CategoryTheory/Iso.lean
|
CategoryTheory.isIso_of_comp_hom_eq_id
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\nX Y Z : C\ng : X ⟶ Y\ninst✝ : IsIso g\nf : Y ⟶ X\nh : f ≫ g = 𝟙 Y\n⊢ IsIso (inv g)",
"state_before": "C : Type u\ninst✝¹ : Category C\nX Y Z : C\ng : X ⟶ Y\ninst✝ : IsIso g\nf : Y ⟶ X\nh : f ≫ g = 𝟙 Y\n⊢ IsIso f",
"tactic": "rw [(comp_hom_eq_id _).mp h]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nX Y Z : C\ng : X ⟶ Y\ninst✝ : IsIso g\nf : Y ⟶ X\nh : f ≫ g = 𝟙 Y\n⊢ IsIso (inv g)",
"tactic": "infer_instance"
}
] |
[
504,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
1
] |
Mathlib/Logic/Function/Conjugate.lean
|
Function.Semiconj₂.comp_eq
|
[] |
[
154,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
11
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
Filter.EventuallyEq.differentiableAt_iff
|
[] |
[
854,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
852,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.ext_abs_arg
|
[
{
"state_after": "no goals",
"state_before": "x y : ℂ\nh₁ : ↑abs x = ↑abs y\nh₂ : arg x = arg y\n⊢ x = y",
"tactic": "rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂]"
}
] |
[
127,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.add_left_neg_equiv
|
[] |
[
1686,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1685,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Cardinal.mk_subset_mk_lt_cof
|
[
{
"state_after": "case inl\nα✝ : Type ?u.117471\nr : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) = 0\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) }) = (#α)\n\ncase inr\nα✝ : Type ?u.117471\nr : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) }) = (#α)",
"state_before": "α✝ : Type ?u.117471\nr : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) }) = (#α)",
"tactic": "rcases eq_or_ne (#α) 0 with (ha | ha)"
},
{
"state_after": "case inr\nα✝ : Type ?u.117471\nr : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) }) = (#α)",
"state_before": "case inr\nα✝ : Type ?u.117471\nr : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) }) = (#α)",
"tactic": "have h' : IsStrongLimit (#α) := ⟨ha, h⟩"
},
{
"state_after": "case inr.intro.intro\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) }) = (#α)",
"state_before": "case inr\nα✝ : Type ?u.117471\nr : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) }) = (#α)",
"tactic": "rcases ord_eq α with ⟨r, wo, hr⟩"
},
{
"state_after": "case inr.intro.intro\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) }) = (#α)",
"state_before": "case inr.intro.intro\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) }) = (#α)",
"tactic": "haveI := wo"
},
{
"state_after": "case inr.intro.intro.a\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) }) ≤ (#α)\n\ncase inr.intro.intro.a\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\n⊢ (#α) ≤ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) })",
"state_before": "case inr.intro.intro\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) }) = (#α)",
"tactic": "apply le_antisymm"
},
{
"state_after": "case inl\nα✝ : Type ?u.117471\nr : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) = 0\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord 0) }) = 0",
"state_before": "case inl\nα✝ : Type ?u.117471\nr : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) = 0\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) }) = (#α)",
"tactic": "rw [ha]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα✝ : Type ?u.117471\nr : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) = 0\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord 0) }) = 0",
"tactic": "simp [fun s => (Cardinal.zero_le s).not_lt]"
},
{
"state_after": "case inr.intro.intro.a\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) }) ≤ (#{ s // Bounded r s })",
"state_before": "case inr.intro.intro.a\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) }) ≤ (#α)",
"tactic": "conv_rhs => rw [← mk_bounded_subset h hr]"
},
{
"state_after": "case inr.intro.intro.a.h\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\n⊢ (fun x =>\n Quotient.liftOn₂ (#↑x) (Ordinal.cof (ord (#α))) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 ∧\n ¬Quotient.liftOn₂ (Ordinal.cof (ord (#α))) (#↑x) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1) ⊆\n fun x => ∃ a, ∀ (b : α), b ∈ x → r b a",
"state_before": "case inr.intro.intro.a\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\n⊢ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) }) ≤ (#{ s // Bounded r s })",
"tactic": "apply mk_le_mk_of_subset"
},
{
"state_after": "case inr.intro.intro.a.h\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\ns : Set α\nhs :\n s ∈ fun x =>\n Quotient.liftOn₂ (#↑x) (Ordinal.cof (ord (#α))) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 ∧\n ¬Quotient.liftOn₂ (Ordinal.cof (ord (#α))) (#↑x) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1\n⊢ s ∈ fun x => ∃ a, ∀ (b : α), b ∈ x → r b a",
"state_before": "case inr.intro.intro.a.h\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\n⊢ (fun x =>\n Quotient.liftOn₂ (#↑x) (Ordinal.cof (ord (#α))) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 ∧\n ¬Quotient.liftOn₂ (Ordinal.cof (ord (#α))) (#↑x) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1) ⊆\n fun x => ∃ a, ∀ (b : α), b ∈ x → r b a",
"tactic": "intro s hs"
},
{
"state_after": "case inr.intro.intro.a.h\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\ns : Set α\nhs :\n s ∈ fun x =>\n Quotient.liftOn₂ (#↑x) (Ordinal.cof (type r)) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 ∧\n ¬Quotient.liftOn₂ (Ordinal.cof (type r)) (#↑x) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1\n⊢ s ∈ fun x => ∃ a, ∀ (b : α), b ∈ x → r b a",
"state_before": "case inr.intro.intro.a.h\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\ns : Set α\nhs :\n s ∈ fun x =>\n Quotient.liftOn₂ (#↑x) (Ordinal.cof (ord (#α))) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 ∧\n ¬Quotient.liftOn₂ (Ordinal.cof (ord (#α))) (#↑x) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1\n⊢ s ∈ fun x => ∃ a, ∀ (b : α), b ∈ x → r b a",
"tactic": "rw [hr] at hs"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.a.h\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\ns : Set α\nhs :\n s ∈ fun x =>\n Quotient.liftOn₂ (#↑x) (Ordinal.cof (type r)) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 ∧\n ¬Quotient.liftOn₂ (Ordinal.cof (type r)) (#↑x) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1\n⊢ s ∈ fun x => ∃ a, ∀ (b : α), b ∈ x → r b a",
"tactic": "exact lt_cof_type hs"
},
{
"state_after": "case inr.intro.intro.a.refine'_1\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\nx : α\n⊢ (#↑{x}) < Ordinal.cof (ord (#α))\n\ncase inr.intro.intro.a.refine'_2\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\n⊢ Injective fun x => { val := {x}, property := (_ : (#↑{x}) < Ordinal.cof (ord (#α))) }",
"state_before": "case inr.intro.intro.a\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\n⊢ (#α) ≤ (#{ s // (#↑s) < Ordinal.cof (ord (#α)) })",
"tactic": "refine' @mk_le_of_injective α _ (fun x => Subtype.mk {x} _) _"
},
{
"state_after": "case inr.intro.intro.a.refine'_1\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\nx : α\n⊢ 1 < Ordinal.cof (ord (#α))",
"state_before": "case inr.intro.intro.a.refine'_1\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\nx : α\n⊢ (#↑{x}) < Ordinal.cof (ord (#α))",
"tactic": "rw [mk_singleton]"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.a.refine'_1\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\nx : α\n⊢ 1 < Ordinal.cof (ord (#α))",
"tactic": "exact one_lt_aleph0.trans_le (aleph0_le_cof.2 (ord_isLimit h'.isLimit.aleph0_le))"
},
{
"state_after": "case inr.intro.intro.a.refine'_2\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\na b : α\nhab :\n (fun x => { val := {x}, property := (_ : (#↑{x}) < Ordinal.cof (ord (#α))) }) a =\n (fun x => { val := {x}, property := (_ : (#↑{x}) < Ordinal.cof (ord (#α))) }) b\n⊢ a = b",
"state_before": "case inr.intro.intro.a.refine'_2\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\n⊢ Injective fun x => { val := {x}, property := (_ : (#↑{x}) < Ordinal.cof (ord (#α))) }",
"tactic": "intro a b hab"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.a.refine'_2\nα✝ : Type ?u.117471\nr✝ : α✝ → α✝ → Prop\nα : Type u_1\nh : ∀ (x : Cardinal), x < (#α) → 2 ^ x < (#α)\nha : (#α) ≠ 0\nh' : IsStrongLimit (#α)\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : ord (#α) = type r\nthis : IsWellOrder α r\na b : α\nhab :\n (fun x => { val := {x}, property := (_ : (#↑{x}) < Ordinal.cof (ord (#α))) }) a =\n (fun x => { val := {x}, property := (_ : (#↑{x}) < Ordinal.cof (ord (#α))) }) b\n⊢ a = b",
"tactic": "simpa [singleton_eq_singleton_iff] using hab"
}
] |
[
946,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
928,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.Subsingleton.aestronglyMeasurable'
|
[] |
[
1171,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1169,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
MulHom.comp_assoc
|
[] |
[
1143,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1142,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.norm_mkPiAlgebraFin
|
[
{
"state_after": "case zero\n𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁷ : Fintype ι\ninst✝¹⁶ : Fintype ι'\ninst✝¹⁵ : NontriviallyNormedField 𝕜\ninst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedSpace 𝕜 G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝⁴ : NormedRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : NormOneClass A\nEi : Fin (Nat.succ Nat.zero) → Type wEi\ninst✝¹ : (i : Fin (Nat.succ Nat.zero)) → NormedAddCommGroup (Ei i)\ninst✝ : (i : Fin (Nat.succ Nat.zero)) → NormedSpace 𝕜 (Ei i)\n⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 Nat.zero A‖ = 1\n\ncase succ\n𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁷ : Fintype ι\ninst✝¹⁶ : Fintype ι'\ninst✝¹⁵ : NontriviallyNormedField 𝕜\ninst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedSpace 𝕜 G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝⁴ : NormedRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : NormOneClass A\nn✝ : ℕ\nEi : Fin (Nat.succ (Nat.succ n✝)) → Type wEi\ninst✝¹ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedAddCommGroup (Ei i)\ninst✝ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedSpace 𝕜 (Ei i)\n⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ n✝) A‖ = 1",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁷ : Fintype ι\ninst✝¹⁶ : Fintype ι'\ninst✝¹⁵ : NontriviallyNormedField 𝕜\ninst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\ninst✝⁴ : NormedAddCommGroup G'\ninst✝³ : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : NormOneClass A\n⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ = 1",
"tactic": "cases n"
},
{
"state_after": "case zero\n𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁷ : Fintype ι\ninst✝¹⁶ : Fintype ι'\ninst✝¹⁵ : NontriviallyNormedField 𝕜\ninst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedSpace 𝕜 G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝⁴ : NormedRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : NormOneClass A\nEi : Fin (Nat.succ Nat.zero) → Type wEi\ninst✝¹ : (i : Fin (Nat.succ Nat.zero)) → NormedAddCommGroup (Ei i)\ninst✝ : (i : Fin (Nat.succ Nat.zero)) → NormedSpace 𝕜 (Ei i)\n⊢ ‖1‖ = 1",
"state_before": "case zero\n𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁷ : Fintype ι\ninst✝¹⁶ : Fintype ι'\ninst✝¹⁵ : NontriviallyNormedField 𝕜\ninst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedSpace 𝕜 G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝⁴ : NormedRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : NormOneClass A\nEi : Fin (Nat.succ Nat.zero) → Type wEi\ninst✝¹ : (i : Fin (Nat.succ Nat.zero)) → NormedAddCommGroup (Ei i)\ninst✝ : (i : Fin (Nat.succ Nat.zero)) → NormedSpace 𝕜 (Ei i)\n⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 Nat.zero A‖ = 1",
"tactic": "rw [norm_mkPiAlgebraFin_zero]"
},
{
"state_after": "no goals",
"state_before": "case zero\n𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁷ : Fintype ι\ninst✝¹⁶ : Fintype ι'\ninst✝¹⁵ : NontriviallyNormedField 𝕜\ninst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedSpace 𝕜 G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝⁴ : NormedRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : NormOneClass A\nEi : Fin (Nat.succ Nat.zero) → Type wEi\ninst✝¹ : (i : Fin (Nat.succ Nat.zero)) → NormedAddCommGroup (Ei i)\ninst✝ : (i : Fin (Nat.succ Nat.zero)) → NormedSpace 𝕜 (Ei i)\n⊢ ‖1‖ = 1",
"tactic": "simp"
},
{
"state_after": "case succ\n𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁷ : Fintype ι\ninst✝¹⁶ : Fintype ι'\ninst✝¹⁵ : NontriviallyNormedField 𝕜\ninst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedSpace 𝕜 G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝⁴ : NormedRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : NormOneClass A\nn✝ : ℕ\nEi : Fin (Nat.succ (Nat.succ n✝)) → Type wEi\ninst✝¹ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedAddCommGroup (Ei i)\ninst✝ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedSpace 𝕜 (Ei i)\n⊢ 1 ≤ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ n✝) A‖",
"state_before": "case succ\n𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁷ : Fintype ι\ninst✝¹⁶ : Fintype ι'\ninst✝¹⁵ : NontriviallyNormedField 𝕜\ninst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedSpace 𝕜 G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝⁴ : NormedRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : NormOneClass A\nn✝ : ℕ\nEi : Fin (Nat.succ (Nat.succ n✝)) → Type wEi\ninst✝¹ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedAddCommGroup (Ei i)\ninst✝ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedSpace 𝕜 (Ei i)\n⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ n✝) A‖ = 1",
"tactic": "refine' le_antisymm norm_mkPiAlgebraFin_succ_le _"
},
{
"state_after": "case h.e'_3\n𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁷ : Fintype ι\ninst✝¹⁶ : Fintype ι'\ninst✝¹⁵ : NontriviallyNormedField 𝕜\ninst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedSpace 𝕜 G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝⁴ : NormedRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : NormOneClass A\nn✝ : ℕ\nEi : Fin (Nat.succ (Nat.succ n✝)) → Type wEi\ninst✝¹ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedAddCommGroup (Ei i)\ninst✝ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedSpace 𝕜 (Ei i)\n⊢ 1 = ‖↑(ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ n✝) A) fun x => 1‖ / ∏ i : Fin (Nat.succ n✝), ‖1‖",
"state_before": "case succ\n𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁷ : Fintype ι\ninst✝¹⁶ : Fintype ι'\ninst✝¹⁵ : NontriviallyNormedField 𝕜\ninst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedSpace 𝕜 G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝⁴ : NormedRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : NormOneClass A\nn✝ : ℕ\nEi : Fin (Nat.succ (Nat.succ n✝)) → Type wEi\ninst✝¹ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedAddCommGroup (Ei i)\ninst✝ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedSpace 𝕜 (Ei i)\n⊢ 1 ≤ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ n✝) A‖",
"tactic": "convert ratio_le_op_norm (ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ _) A)\n fun _ => 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\n𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁷ : Fintype ι\ninst✝¹⁶ : Fintype ι'\ninst✝¹⁵ : NontriviallyNormedField 𝕜\ninst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedSpace 𝕜 G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝⁴ : NormedRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : NormOneClass A\nn✝ : ℕ\nEi : Fin (Nat.succ (Nat.succ n✝)) → Type wEi\ninst✝¹ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedAddCommGroup (Ei i)\ninst✝ : (i : Fin (Nat.succ (Nat.succ n✝))) → NormedSpace 𝕜 (Ei i)\n⊢ 1 = ‖↑(ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (Nat.succ n✝) A) fun x => 1‖ / ∏ i : Fin (Nat.succ n✝), ‖1‖",
"tactic": "simp"
}
] |
[
877,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
869,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
round_eq
|
[
{
"state_after": "F : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\n⊢ (if fract x < 1 / 2 then ⌊x⌋ else ⌈x⌉) = ⌊x + 1 / 2⌋",
"state_before": "F : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\n⊢ round x = ⌊x + 1 / 2⌋",
"tactic": "simp_rw [round, (by simp only [lt_div_iff', two_pos] : 2 * fract x < 1 ↔ fract x < 1 / 2)]"
},
{
"state_after": "case inl\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : fract x < 1 / 2\n⊢ (if fract x < 1 / 2 then ⌊x⌋ else ⌈x⌉) = ⌊x + 1 / 2⌋\n\ncase inr\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\n⊢ (if fract x < 1 / 2 then ⌊x⌋ else ⌈x⌉) = ⌊x + 1 / 2⌋",
"state_before": "F : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\n⊢ (if fract x < 1 / 2 then ⌊x⌋ else ⌈x⌉) = ⌊x + 1 / 2⌋",
"tactic": "cases' lt_or_le (fract x) (1 / 2) with hx hx"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\n⊢ 2 * fract x < 1 ↔ fract x < 1 / 2",
"tactic": "simp only [lt_div_iff', two_pos]"
},
{
"state_after": "case inl\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : fract x < 1 / 2\n⊢ (if fract x < 1 / 2 then ⌊x⌋ else ⌈x⌉) = ⌊x⌋ + ⌊fract x + 1 / 2⌋",
"state_before": "case inl\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : fract x < 1 / 2\n⊢ (if fract x < 1 / 2 then ⌊x⌋ else ⌈x⌉) = ⌊x + 1 / 2⌋",
"tactic": "conv_rhs => rw [← fract_add_floor x, add_assoc, add_left_comm, floor_int_add]"
},
{
"state_after": "case inl\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : fract x < 1 / 2\n⊢ 0 ≤ fract x + 1 / 2 ∧ fract x + 1 / 2 < 1",
"state_before": "case inl\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : fract x < 1 / 2\n⊢ (if fract x < 1 / 2 then ⌊x⌋ else ⌈x⌉) = ⌊x⌋ + ⌊fract x + 1 / 2⌋",
"tactic": "rw [if_pos hx, self_eq_add_right, floor_eq_iff, cast_zero, zero_add]"
},
{
"state_after": "case inl.left\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : fract x < 1 / 2\n⊢ 0 ≤ fract x + 1 / 2\n\ncase inl.right\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : fract x < 1 / 2\n⊢ fract x + 1 / 2 < 1",
"state_before": "case inl\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : fract x < 1 / 2\n⊢ 0 ≤ fract x + 1 / 2 ∧ fract x + 1 / 2 < 1",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case inl.left\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : fract x < 1 / 2\n⊢ 0 ≤ fract x + 1 / 2",
"tactic": "linarith [fract_nonneg x]"
},
{
"state_after": "no goals",
"state_before": "case inl.right\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : fract x < 1 / 2\n⊢ fract x + 1 / 2 < 1",
"tactic": "linarith"
},
{
"state_after": "case inr\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\nthis : ⌊fract x + 1 / 2⌋ = 1\n⊢ 0 < fract x ∧ fract x ≤ 1",
"state_before": "case inr\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\nthis : ⌊fract x + 1 / 2⌋ = 1\n⊢ (if fract x < 1 / 2 then ⌊x⌋ else ⌈x⌉) = ⌊x + 1 / 2⌋",
"tactic": "rw [if_neg (not_lt.mpr hx), ← fract_add_floor x, add_assoc, add_left_comm, floor_int_add,\n ceil_add_int, add_comm _ ⌊x⌋, add_right_inj, ceil_eq_iff, this, cast_one, sub_self]"
},
{
"state_after": "case inr.left\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\nthis : ⌊fract x + 1 / 2⌋ = 1\n⊢ 0 < fract x\n\ncase inr.right\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\nthis : ⌊fract x + 1 / 2⌋ = 1\n⊢ fract x ≤ 1",
"state_before": "case inr\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\nthis : ⌊fract x + 1 / 2⌋ = 1\n⊢ 0 < fract x ∧ fract x ≤ 1",
"tactic": "constructor"
},
{
"state_after": "F : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\n⊢ ↑1 ≤ fract x + 1 / 2 ∧ fract x + 1 / 2 < ↑1 + 1",
"state_before": "F : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\n⊢ ⌊fract x + 1 / 2⌋ = 1",
"tactic": "rw [floor_eq_iff]"
},
{
"state_after": "case left\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\n⊢ ↑1 ≤ fract x + 1 / 2\n\ncase right\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\n⊢ fract x + 1 / 2 < ↑1 + 1",
"state_before": "F : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\n⊢ ↑1 ≤ fract x + 1 / 2 ∧ fract x + 1 / 2 < ↑1 + 1",
"tactic": "constructor"
},
{
"state_after": "case left\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\n⊢ 1 ≤ fract x + 1 / 2",
"state_before": "case left\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\n⊢ ↑1 ≤ fract x + 1 / 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case left\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\n⊢ 1 ≤ fract x + 1 / 2",
"tactic": "linarith"
},
{
"state_after": "case right\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\n⊢ fract x + 1 / 2 < 2",
"state_before": "case right\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\n⊢ fract x + 1 / 2 < ↑1 + 1",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case right\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\n⊢ fract x + 1 / 2 < 2",
"tactic": "linarith [fract_lt_one x]"
},
{
"state_after": "no goals",
"state_before": "case inr.left\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\nthis : ⌊fract x + 1 / 2⌋ = 1\n⊢ 0 < fract x",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "case inr.right\nF : Type ?u.270515\nα : Type u_1\nβ : Type ?u.270521\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : α\nhx : 1 / 2 ≤ fract x\nthis : ⌊fract x + 1 / 2⌋ = 1\n⊢ fract x ≤ 1",
"tactic": "linarith [fract_lt_one x]"
}
] |
[
1449,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1430,
1
] |
Mathlib/Data/List/Pairwise.lean
|
List.forall_mem_pwFilter
|
[
{
"state_after": "case nil\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na : α\n⊢ (∀ (b : α), b ∈ pwFilter R [] → R a b) → ∀ (b : α), b ∈ [] → R a b\n\ncase cons\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\n⊢ (∀ (b : α), b ∈ pwFilter R (x :: l) → R a b) → ∀ (b : α), b ∈ x :: l → R a b",
"state_before": "α : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na : α\nl : List α\n⊢ (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b",
"tactic": "induction' l with x l IH"
},
{
"state_after": "case cons\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\n⊢ (∀ (b : α), b ∈ pwFilter R (x :: l) → R a b) → R a x ∧ ∀ (x : α), x ∈ l → R a x",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\n⊢ (∀ (b : α), b ∈ pwFilter R (x :: l) → R a b) → ∀ (b : α), b ∈ x :: l → R a b",
"tactic": "simp only [forall_mem_cons]"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ (∀ (b : α), b ∈ pwFilter R (x :: l) → R a b) → R a x ∧ ∀ (x : α), x ∈ l → R a x\n\ncase neg\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ (∀ (b : α), b ∈ pwFilter R (x :: l) → R a b) → R a x ∧ ∀ (x : α), x ∈ l → R a x",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\n⊢ (∀ (b : α), b ∈ pwFilter R (x :: l) → R a b) → R a x ∧ ∀ (x : α), x ∈ l → R a x",
"tactic": "by_cases h : ∀ y ∈ pwFilter R l, R x y <;> dsimp at h"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na : α\n⊢ (∀ (b : α), b ∈ pwFilter R [] → R a b) → ∀ (b : α), b ∈ [] → R a b",
"tactic": "exact fun _ _ h => (not_mem_nil _ h).elim"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ R a x → (∀ (x : α), x ∈ pwFilter R l → R a x) → R a x ∧ ∀ (x : α), x ∈ l → R a x",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ (∀ (b : α), b ∈ pwFilter R (x :: l) → R a b) → R a x ∧ ∀ (x : α), x ∈ l → R a x",
"tactic": "simp only [pwFilter_cons_of_pos h, forall_mem_cons, and_imp]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ R a x → (∀ (x : α), x ∈ pwFilter R l → R a x) → R a x ∧ ∀ (x : α), x ∈ l → R a x",
"tactic": "exact fun r H => ⟨r, IH H⟩"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ (∀ (b : α), b ∈ pwFilter R l → R a b) → R a x ∧ ∀ (x : α), x ∈ l → R a x",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ (∀ (b : α), b ∈ pwFilter R (x :: l) → R a b) → R a x ∧ ∀ (x : α), x ∈ l → R a x",
"tactic": "rw [pwFilter_cons_of_neg h]"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\nH : ∀ (b : α), b ∈ pwFilter R l → R a b\n⊢ R a x",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\n⊢ (∀ (b : α), b ∈ pwFilter R l → R a b) → R a x ∧ ∀ (x : α), x ∈ l → R a x",
"tactic": "refine' fun H => ⟨_, IH H⟩"
},
{
"state_after": "case neg.none\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\nH : ∀ (b : α), b ∈ pwFilter R l → R a b\ne : find? (fun y => decide ¬R x y) (pwFilter R l) = none\n⊢ R a x\n\ncase neg.some\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\nH : ∀ (b : α), b ∈ pwFilter R l → R a b\nk : α\ne : find? (fun y => decide ¬R x y) (pwFilter R l) = some k\n⊢ R a x",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\nH : ∀ (b : α), b ∈ pwFilter R l → R a b\n⊢ R a x",
"tactic": "cases' e : find? (fun y => ¬R x y) (pwFilter R l) with k"
},
{
"state_after": "case neg.none\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\nH : ∀ (b : α), b ∈ pwFilter R l → R a b\ne : find? (fun y => decide ¬R x y) (pwFilter R l) = none\n⊢ ∀ (x_1 : α), x_1 ∈ pwFilter R l → ¬(decide ¬R x x_1) = true → R x x_1",
"state_before": "case neg.none\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\nH : ∀ (b : α), b ∈ pwFilter R l → R a b\ne : find? (fun y => decide ¬R x y) (pwFilter R l) = none\n⊢ R a x",
"tactic": "refine' h.elim (BAll.imp_right _ (find?_eq_none.1 e))"
},
{
"state_after": "no goals",
"state_before": "case neg.none\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\nH : ∀ (b : α), b ∈ pwFilter R l → R a b\ne : find? (fun y => decide ¬R x y) (pwFilter R l) = none\n⊢ ∀ (x_1 : α), x_1 ∈ pwFilter R l → ¬(decide ¬R x x_1) = true → R x x_1",
"tactic": "exact fun y _ => by simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\nH : ∀ (b : α), b ∈ pwFilter R l → R a b\ne : find? (fun y => decide ¬R x y) (pwFilter R l) = none\ny : α\nx✝ : y ∈ pwFilter R l\n⊢ ¬(decide ¬R x y) = true → R x y",
"tactic": "simp"
},
{
"state_after": "case neg.some\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\nH : ∀ (b : α), b ∈ pwFilter R l → R a b\nk : α\ne : find? (fun y => decide ¬R x y) (pwFilter R l) = some k\nthis : (decide ¬R x k) = true\n⊢ R a x",
"state_before": "case neg.some\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\nH : ∀ (b : α), b ∈ pwFilter R l → R a b\nk : α\ne : find? (fun y => decide ¬R x y) (pwFilter R l) = some k\n⊢ R a x",
"tactic": "have := find?_some e"
},
{
"state_after": "no goals",
"state_before": "case neg.some\nα : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\nH : ∀ (b : α), b ∈ pwFilter R l → R a b\nk : α\ne : find? (fun y => decide ¬R x y) (pwFilter R l) = some k\nthis : (decide ¬R x k) = true\n⊢ R a x",
"tactic": "exact (neg_trans (H k (find?_mem e))).resolve_right (by simpa)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.53307\nR S T : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝ : DecidableRel R\nneg_trans : ∀ {x y z : α}, R x z → R x y ∨ R y z\na x : α\nl : List α\nIH : (∀ (b : α), b ∈ pwFilter R l → R a b) → ∀ (b : α), b ∈ l → R a b\nh : ¬∀ (y : α), y ∈ pwFilter R l → R x y\nH : ∀ (b : α), b ∈ pwFilter R l → R a b\nk : α\ne : find? (fun y => decide ¬R x y) (pwFilter R l) = some k\nthis : (decide ¬R x k) = true\n⊢ ¬R x k",
"tactic": "simpa"
}
] |
[
448,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/Algebra/Algebra/Basic.lean
|
algebraMap_rat_rat
|
[] |
[
756,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
756,
9
] |
Mathlib/GroupTheory/Index.lean
|
Subgroup.nat_card_dvd_of_injective
|
[
{
"state_after": "G✝ : Type ?u.97421\ninst✝² : Group G✝\nH✝ K L : Subgroup G✝\nG : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nf : G →* H\nhf : Function.Injective ↑f\n⊢ Nat.card { x // x ∈ MonoidHom.range f } ∣ Nat.card H",
"state_before": "G✝ : Type ?u.97421\ninst✝² : Group G✝\nH✝ K L : Subgroup G✝\nG : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nf : G →* H\nhf : Function.Injective ↑f\n⊢ Nat.card G ∣ Nat.card H",
"tactic": "rw [Nat.card_congr (MonoidHom.ofInjective hf).toEquiv]"
},
{
"state_after": "no goals",
"state_before": "G✝ : Type ?u.97421\ninst✝² : Group G✝\nH✝ K L : Subgroup G✝\nG : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nf : G →* H\nhf : Function.Injective ↑f\n⊢ Nat.card { x // x ∈ MonoidHom.range f } ∣ Nat.card H",
"tactic": "exact Dvd.intro f.range.index f.range.card_mul_index"
}
] |
[
302,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.Nontrivial.of_polynomial_ne
|
[] |
[
778,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
777,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.colimit.pre_pre
|
[
{
"state_after": "case w\nJ : Type u₁\ninst✝⁶ : Category J\nK : Type u₂\ninst✝⁵ : Category K\nC : Type u\ninst✝⁴ : Category C\nF : J ⥤ C\ninst✝³ : HasColimit F\nE : K ⥤ J\ninst✝² : HasColimit (E ⋙ F)\nL : Type u₃\ninst✝¹ : Category L\nD : L ⥤ K\ninst✝ h : HasColimit (D ⋙ E ⋙ F)\nj : L\n⊢ ι (D ⋙ E ⋙ F) j ≫ pre (E ⋙ F) D ≫ pre F E = ι (D ⋙ E ⋙ F) j ≫ pre F (D ⋙ E)",
"state_before": "J : Type u₁\ninst✝⁶ : Category J\nK : Type u₂\ninst✝⁵ : Category K\nC : Type u\ninst✝⁴ : Category C\nF : J ⥤ C\ninst✝³ : HasColimit F\nE : K ⥤ J\ninst✝² : HasColimit (E ⋙ F)\nL : Type u₃\ninst✝¹ : Category L\nD : L ⥤ K\ninst✝ h : HasColimit (D ⋙ E ⋙ F)\n⊢ pre (E ⋙ F) D ≫ pre F E = pre F (D ⋙ E)",
"tactic": "ext j"
},
{
"state_after": "case w\nJ : Type u₁\ninst✝⁶ : Category J\nK : Type u₂\ninst✝⁵ : Category K\nC : Type u\ninst✝⁴ : Category C\nF : J ⥤ C\ninst✝³ : HasColimit F\nE : K ⥤ J\ninst✝² : HasColimit (E ⋙ F)\nL : Type u₃\ninst✝¹ : Category L\nD : L ⥤ K\ninst✝ h : HasColimit (D ⋙ E ⋙ F)\nj : L\n⊢ ι F (E.obj (D.obj j)) = ι (D ⋙ E ⋙ F) j ≫ pre F (D ⋙ E)",
"state_before": "case w\nJ : Type u₁\ninst✝⁶ : Category J\nK : Type u₂\ninst✝⁵ : Category K\nC : Type u\ninst✝⁴ : Category C\nF : J ⥤ C\ninst✝³ : HasColimit F\nE : K ⥤ J\ninst✝² : HasColimit (E ⋙ F)\nL : Type u₃\ninst✝¹ : Category L\nD : L ⥤ K\ninst✝ h : HasColimit (D ⋙ E ⋙ F)\nj : L\n⊢ ι (D ⋙ E ⋙ F) j ≫ pre (E ⋙ F) D ≫ pre F E = ι (D ⋙ E ⋙ F) j ≫ pre F (D ⋙ E)",
"tactic": "rw [← assoc, colimit.ι_pre, colimit.ι_pre]"
},
{
"state_after": "case w\nJ : Type u₁\ninst✝⁶ : Category J\nK : Type u₂\ninst✝⁵ : Category K\nC : Type u\ninst✝⁴ : Category C\nF : J ⥤ C\ninst✝³ : HasColimit F\nE : K ⥤ J\ninst✝² : HasColimit (E ⋙ F)\nL : Type u₃\ninst✝¹ : Category L\nD : L ⥤ K\ninst✝ h : HasColimit (D ⋙ E ⋙ F)\nj : L\nthis : HasColimit ((D ⋙ E) ⋙ F)\n⊢ ι F (E.obj (D.obj j)) = ι (D ⋙ E ⋙ F) j ≫ pre F (D ⋙ E)",
"state_before": "case w\nJ : Type u₁\ninst✝⁶ : Category J\nK : Type u₂\ninst✝⁵ : Category K\nC : Type u\ninst✝⁴ : Category C\nF : J ⥤ C\ninst✝³ : HasColimit F\nE : K ⥤ J\ninst✝² : HasColimit (E ⋙ F)\nL : Type u₃\ninst✝¹ : Category L\nD : L ⥤ K\ninst✝ h : HasColimit (D ⋙ E ⋙ F)\nj : L\n⊢ ι F (E.obj (D.obj j)) = ι (D ⋙ E ⋙ F) j ≫ pre F (D ⋙ E)",
"tactic": "haveI : HasColimit ((D ⋙ E) ⋙ F) := h"
},
{
"state_after": "no goals",
"state_before": "case w\nJ : Type u₁\ninst✝⁶ : Category J\nK : Type u₂\ninst✝⁵ : Category K\nC : Type u\ninst✝⁴ : Category C\nF : J ⥤ C\ninst✝³ : HasColimit F\nE : K ⥤ J\ninst✝² : HasColimit (E ⋙ F)\nL : Type u₃\ninst✝¹ : Category L\nD : L ⥤ K\ninst✝ h : HasColimit (D ⋙ E ⋙ F)\nj : L\nthis : HasColimit ((D ⋙ E) ⋙ F)\n⊢ ι F (E.obj (D.obj j)) = ι (D ⋙ E ⋙ F) j ≫ pre F (D ⋙ E)",
"tactic": "exact (colimit.ι_pre F (D ⋙ E) j).symm"
}
] |
[
996,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
990,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
ContinuousWithinAt.mono_of_mem
|
[] |
[
703,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
701,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.le_normalizer_map
|
[
{
"state_after": "G : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx✝ : N\n⊢ ∀ (x : G),\n (∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H) →\n ↑f x = x✝ → ∀ (h : N), (∃ x, x ∈ H ∧ ↑f x = h) ↔ ∃ x, x ∈ H ∧ ↑f x = x✝ * h * x✝⁻¹",
"state_before": "G : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx✝ : N\n⊢ x✝ ∈ map f (normalizer H) → x✝ ∈ normalizer (map f H)",
"tactic": "simp only [and_imp, exists_prop, mem_map, exists_imp, mem_normalizer_iff]"
},
{
"state_after": "G : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\nn : N\n⊢ (∃ x, x ∈ H ∧ ↑f x = n) ↔ ∃ x_1, x_1 ∈ H ∧ ↑f x_1 = ↑f x * n * (↑f x)⁻¹",
"state_before": "G : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx✝ : N\n⊢ ∀ (x : G),\n (∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H) →\n ↑f x = x✝ → ∀ (h : N), (∃ x, x ∈ H ∧ ↑f x = h) ↔ ∃ x, x ∈ H ∧ ↑f x = x✝ * h * x✝⁻¹",
"tactic": "rintro x hx rfl n"
},
{
"state_after": "case mp\nG : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\nn : N\n⊢ (∃ x, x ∈ H ∧ ↑f x = n) → ∃ x_1, x_1 ∈ H ∧ ↑f x_1 = ↑f x * n * (↑f x)⁻¹\n\ncase mpr\nG : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\nn : N\n⊢ (∃ x_1, x_1 ∈ H ∧ ↑f x_1 = ↑f x * n * (↑f x)⁻¹) → ∃ x, x ∈ H ∧ ↑f x = n",
"state_before": "G : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\nn : N\n⊢ (∃ x, x ∈ H ∧ ↑f x = n) ↔ ∃ x_1, x_1 ∈ H ∧ ↑f x_1 = ↑f x * n * (↑f x)⁻¹",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro\nG : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\ny : G\nhy : y ∈ H\n⊢ ∃ x_1, x_1 ∈ H ∧ ↑f x_1 = ↑f x * ↑f y * (↑f x)⁻¹",
"state_before": "case mp\nG : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\nn : N\n⊢ (∃ x, x ∈ H ∧ ↑f x = n) → ∃ x_1, x_1 ∈ H ∧ ↑f x_1 = ↑f x * n * (↑f x)⁻¹",
"tactic": "rintro ⟨y, hy, rfl⟩"
},
{
"state_after": "case mp.intro.intro\nG : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\ny : G\nhy : y ∈ H\n⊢ ↑f (x * y * x⁻¹) = ↑f x * ↑f y * (↑f x)⁻¹",
"state_before": "case mp.intro.intro\nG : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\ny : G\nhy : y ∈ H\n⊢ ∃ x_1, x_1 ∈ H ∧ ↑f x_1 = ↑f x * ↑f y * (↑f x)⁻¹",
"tactic": "use x * y * x⁻¹, (hx y).1 hy"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro\nG : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\ny : G\nhy : y ∈ H\n⊢ ↑f (x * y * x⁻¹) = ↑f x * ↑f y * (↑f x)⁻¹",
"tactic": "simp"
},
{
"state_after": "case mpr.intro.intro\nG : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\nn : N\ny : G\nhyH : y ∈ H\nhy : ↑f y = ↑f x * n * (↑f x)⁻¹\n⊢ ∃ x, x ∈ H ∧ ↑f x = n",
"state_before": "case mpr\nG : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\nn : N\n⊢ (∃ x_1, x_1 ∈ H ∧ ↑f x_1 = ↑f x * n * (↑f x)⁻¹) → ∃ x, x ∈ H ∧ ↑f x = n",
"tactic": "rintro ⟨y, hyH, hy⟩"
},
{
"state_after": "case mpr.intro.intro\nG : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\nn : N\ny : G\nhyH : y ∈ H\nhy : ↑f y = ↑f x * n * (↑f x)⁻¹\n⊢ x⁻¹ * y * x ∈ H ∧ ↑f (x⁻¹ * y * x) = n",
"state_before": "case mpr.intro.intro\nG : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\nn : N\ny : G\nhyH : y ∈ H\nhy : ↑f y = ↑f x * n * (↑f x)⁻¹\n⊢ ∃ x, x ∈ H ∧ ↑f x = n",
"tactic": "use x⁻¹ * y * x"
},
{
"state_after": "case mpr.intro.intro\nG : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\nn : N\ny : G\nhyH : y ∈ H\nhy : ↑f y = ↑f x * n * (↑f x)⁻¹\n⊢ x * (x⁻¹ * y * x) * x⁻¹ ∈ H ∧ ↑f (x⁻¹ * y * x) = n",
"state_before": "case mpr.intro.intro\nG : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\nn : N\ny : G\nhyH : y ∈ H\nhy : ↑f y = ↑f x * n * (↑f x)⁻¹\n⊢ x⁻¹ * y * x ∈ H ∧ ↑f (x⁻¹ * y * x) = n",
"tactic": "rw [hx]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\nG : Type u_1\nG' : Type ?u.395731\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.395740\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type u_2\ninst✝ : Group N\nf : G →* N\nx : G\nhx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H\nn : N\ny : G\nhyH : y ∈ H\nhy : ↑f y = ↑f x * n * (↑f x)⁻¹\n⊢ x * (x⁻¹ * y * x) * x⁻¹ ∈ H ∧ ↑f (x⁻¹ * y * x) = n",
"tactic": "simp [hy, hyH, mul_assoc]"
}
] |
[
2232,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2222,
1
] |
Mathlib/Tactic/NormNum/Prime.lean
|
Mathlib.Meta.NormNum.isNat_minFac_1
|
[] |
[
103,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.left_mem_Ico
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.4228\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\n⊢ a ∈ Ico a b ↔ a < b",
"tactic": "simp only [mem_Ico, true_and_iff, le_refl]"
}
] |
[
129,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Logic/Equiv/Fin.lean
|
finCongr_apply_mk
|
[] |
[
114,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
9
] |
Mathlib/Topology/GDelta.lean
|
IsClosed.isGδ
|
[
{
"state_after": "case intro.intro.mk\nα✝ : Type ?u.10695\nβ : Type ?u.10698\nγ : Type ?u.10701\nι : Type ?u.10704\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : IsCountablyGenerated (uniformity α)\ns : Set α\nhs : IsClosed s\nU : ℕ → Set (α × α)\nhUo : ∀ (i : ℕ), U i ∈ uniformity α ∧ IsOpen (U i)\nhU : HasBasis (uniformity α) (fun x => True) fun i => id (U i)\n⊢ IsGδ s",
"state_before": "α✝ : Type ?u.10695\nβ : Type ?u.10698\nγ : Type ?u.10701\nι : Type ?u.10704\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : IsCountablyGenerated (uniformity α)\ns : Set α\nhs : IsClosed s\n⊢ IsGδ s",
"tactic": "rcases(@uniformity_hasBasis_open α _).exists_antitone_subbasis with ⟨U, hUo, hU, -⟩"
},
{
"state_after": "case intro.intro.mk\nα✝ : Type ?u.10695\nβ : Type ?u.10698\nγ : Type ?u.10701\nι : Type ?u.10704\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : IsCountablyGenerated (uniformity α)\ns : Set α\nhs : IsClosed s\nU : ℕ → Set (α × α)\nhUo : ∀ (i : ℕ), U i ∈ uniformity α ∧ IsOpen (U i)\nhU : HasBasis (uniformity α) (fun x => True) fun i => id (U i)\n⊢ IsGδ (⋂ (i : ℕ) (_ : True), ⋃ (x : α) (_ : x ∈ s), UniformSpace.ball x (id (U i)))",
"state_before": "case intro.intro.mk\nα✝ : Type ?u.10695\nβ : Type ?u.10698\nγ : Type ?u.10701\nι : Type ?u.10704\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : IsCountablyGenerated (uniformity α)\ns : Set α\nhs : IsClosed s\nU : ℕ → Set (α × α)\nhUo : ∀ (i : ℕ), U i ∈ uniformity α ∧ IsOpen (U i)\nhU : HasBasis (uniformity α) (fun x => True) fun i => id (U i)\n⊢ IsGδ s",
"tactic": "rw [← hs.closure_eq, ← hU.biInter_biUnion_ball]"
},
{
"state_after": "case intro.intro.mk\nα✝ : Type ?u.10695\nβ : Type ?u.10698\nγ : Type ?u.10701\nι : Type ?u.10704\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : IsCountablyGenerated (uniformity α)\ns : Set α\nhs : IsClosed s\nU : ℕ → Set (α × α)\nhUo : ∀ (i : ℕ), U i ∈ uniformity α ∧ IsOpen (U i)\nhU : HasBasis (uniformity α) (fun x => True) fun i => id (U i)\nn : ℕ\nx✝ : n ∈ fun i => True\n⊢ IsOpen (⋃ (x : α) (_ : x ∈ s), UniformSpace.ball x (id (U n)))",
"state_before": "case intro.intro.mk\nα✝ : Type ?u.10695\nβ : Type ?u.10698\nγ : Type ?u.10701\nι : Type ?u.10704\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : IsCountablyGenerated (uniformity α)\ns : Set α\nhs : IsClosed s\nU : ℕ → Set (α × α)\nhUo : ∀ (i : ℕ), U i ∈ uniformity α ∧ IsOpen (U i)\nhU : HasBasis (uniformity α) (fun x => True) fun i => id (U i)\n⊢ IsGδ (⋂ (i : ℕ) (_ : True), ⋃ (x : α) (_ : x ∈ s), UniformSpace.ball x (id (U i)))",
"tactic": "refine' isGδ_biInter (to_countable _) fun n _ => IsOpen.isGδ _"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.mk\nα✝ : Type ?u.10695\nβ : Type ?u.10698\nγ : Type ?u.10701\nι : Type ?u.10704\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : UniformSpace α\ninst✝ : IsCountablyGenerated (uniformity α)\ns : Set α\nhs : IsClosed s\nU : ℕ → Set (α × α)\nhUo : ∀ (i : ℕ), U i ∈ uniformity α ∧ IsOpen (U i)\nhU : HasBasis (uniformity α) (fun x => True) fun i => id (U i)\nn : ℕ\nx✝ : n ∈ fun i => True\n⊢ IsOpen (⋃ (x : α) (_ : x ∈ s), UniformSpace.ball x (id (U n)))",
"tactic": "exact isOpen_biUnion fun x _ => UniformSpace.isOpen_ball _ (hUo _).2"
}
] |
[
133,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean
|
LiouvilleNumber.aux_calc
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 ≤ 1 / m ^ (n + 1)!",
"tactic": "positivity"
},
{
"state_after": "n : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 2 * m ^ (n ! * n) ≤ 1 * m ^ (n ! * (n + 1))\n\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 < m ^ (n ! * (n + 1))\n\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 < m ^ (n ! * n)",
"state_before": "n : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 2 / m ^ (n ! * (n + 1)) ≤ 1 / m ^ (n ! * n)",
"tactic": "apply (div_le_div_iff _ _).mpr"
},
{
"state_after": "n : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 2 * m ^ (n ! * n) ≤ m ^ n ! * m ^ (n ! * n)\n\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 < m ^ (n ! * (n + 1))\n\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 < m ^ (n ! * n)",
"state_before": "n : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 2 * m ^ (n ! * n) ≤ 1 * m ^ (n ! * (n + 1))\n\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 < m ^ (n ! * (n + 1))\n\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 < m ^ (n ! * n)",
"tactic": "conv_rhs => rw [one_mul, mul_add, pow_add, mul_one, pow_mul, mul_comm, ← pow_mul]"
},
{
"state_after": "case refine'_1\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 < m ^ (n ! * n)\n\ncase refine'_2\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 2 ≤ m ^ n !\n\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 < m ^ (n ! * (n + 1))\n\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 < m ^ (n ! * n)",
"state_before": "n : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 2 * m ^ (n ! * n) ≤ m ^ n ! * m ^ (n ! * n)\n\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 < m ^ (n ! * (n + 1))\n\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 < m ^ (n ! * n)",
"tactic": "refine' (mul_le_mul_right _).mpr _"
},
{
"state_after": "case refine'_2\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 2 ≤ m ^ n !",
"state_before": "case refine'_1\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 < m ^ (n ! * n)\n\ncase refine'_2\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 2 ≤ m ^ n !\n\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 < m ^ (n ! * (n + 1))\n\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 < m ^ (n ! * n)",
"tactic": "any_goals exact pow_pos (zero_lt_two.trans_le hm) _"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nn : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 2 ≤ m ^ n !",
"tactic": "exact _root_.trans (_root_.trans hm (pow_one _).symm.le)\n (pow_mono (one_le_two.trans hm) n.factorial_pos)"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 0 < m ^ (n ! * n)",
"tactic": "exact pow_pos (zero_lt_two.trans_le hm) _"
}
] |
[
162,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Topology/Basic.lean
|
IsClosed.inter
|
[
{
"state_after": "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\nh₁ : IsOpen (s₁ᶜ)\nh₂ : IsOpen (s₂ᶜ)\n⊢ IsOpen ((s₁ ∩ s₂)ᶜ)",
"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\nh₁ : IsClosed s₁\nh₂ : IsClosed s₂\n⊢ IsClosed (s₁ ∩ s₂)",
"tactic": "rw [← isOpen_compl_iff] at *"
},
{
"state_after": "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\nh₁ : IsOpen (s₁ᶜ)\nh₂ : IsOpen (s₂ᶜ)\n⊢ IsOpen (s₁ᶜ ∪ s₂ᶜ)",
"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\nh₁ : IsOpen (s₁ᶜ)\nh₂ : IsOpen (s₂ᶜ)\n⊢ IsOpen ((s₁ ∩ s₂)ᶜ)",
"tactic": "rw [compl_inter]"
},
{
"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\nh₁ : IsOpen (s₁ᶜ)\nh₂ : IsOpen (s₂ᶜ)\n⊢ IsOpen (s₁ᶜ ∪ s₂ᶜ)",
"tactic": "exact IsOpen.union h₁ h₂"
}
] |
[
246,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.one_pos
|
[] |
[
376,
43
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
376,
1
] |
Mathlib/Order/JordanHolder.lean
|
CompositionSeries.mem_snoc
|
[
{
"state_after": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx y : X\nhsat : IsMaximal (top s) x\n⊢ y ∈ range (Fin.snoc s.series x) ↔ y ∈ range s.series ∨ y = x",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx y : X\nhsat : IsMaximal (top s) x\n⊢ y ∈ snoc s x hsat ↔ y ∈ s ∨ y = x",
"tactic": "simp only [snoc, mem_def]"
},
{
"state_after": "case mp\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx y : X\nhsat : IsMaximal (top s) x\n⊢ y ∈ range (Fin.snoc s.series x) → y ∈ range s.series ∨ y = x\n\ncase mpr\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx y : X\nhsat : IsMaximal (top s) x\n⊢ y ∈ range s.series ∨ y = x → y ∈ range (Fin.snoc s.series x)",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx y : X\nhsat : IsMaximal (top s) x\n⊢ y ∈ range (Fin.snoc s.series x) ↔ y ∈ range s.series ∨ y = x",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx : X\nhsat : IsMaximal (top s) x\ni : Fin (s.length + 1 + 1)\n⊢ Fin.snoc s.series x i ∈ range s.series ∨ Fin.snoc s.series x i = x",
"state_before": "case mp\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx y : X\nhsat : IsMaximal (top s) x\n⊢ y ∈ range (Fin.snoc s.series x) → y ∈ range s.series ∨ y = x",
"tactic": "rintro ⟨i, rfl⟩"
},
{
"state_after": "case mp.intro.refine'_1\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx : X\nhsat : IsMaximal (top s) x\ni : Fin (s.length + 1 + 1)\n⊢ Fin.snoc s.series x (Fin.last (s.length + 1)) ∈ range s.series ∨ Fin.snoc s.series x (Fin.last (s.length + 1)) = x\n\ncase mp.intro.refine'_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx : X\nhsat : IsMaximal (top s) x\ni✝ : Fin (s.length + 1 + 1)\ni : Fin (s.length + 1)\n⊢ Fin.snoc s.series x (↑Fin.castSucc i) ∈ range s.series ∨ Fin.snoc s.series x (↑Fin.castSucc i) = x",
"state_before": "case mp.intro\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx : X\nhsat : IsMaximal (top s) x\ni : Fin (s.length + 1 + 1)\n⊢ Fin.snoc s.series x i ∈ range s.series ∨ Fin.snoc s.series x i = x",
"tactic": "refine' Fin.lastCases _ (fun i => _) i"
},
{
"state_after": "case mp.intro.refine'_1.h\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx : X\nhsat : IsMaximal (top s) x\ni : Fin (s.length + 1 + 1)\n⊢ Fin.snoc s.series x (Fin.last (s.length + 1)) = x",
"state_before": "case mp.intro.refine'_1\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx : X\nhsat : IsMaximal (top s) x\ni : Fin (s.length + 1 + 1)\n⊢ Fin.snoc s.series x (Fin.last (s.length + 1)) ∈ range s.series ∨ Fin.snoc s.series x (Fin.last (s.length + 1)) = x",
"tactic": "right"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.refine'_1.h\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx : X\nhsat : IsMaximal (top s) x\ni : Fin (s.length + 1 + 1)\n⊢ Fin.snoc s.series x (Fin.last (s.length + 1)) = x",
"tactic": "simp"
},
{
"state_after": "case mp.intro.refine'_2.h\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx : X\nhsat : IsMaximal (top s) x\ni✝ : Fin (s.length + 1 + 1)\ni : Fin (s.length + 1)\n⊢ Fin.snoc s.series x (↑Fin.castSucc i) ∈ range s.series",
"state_before": "case mp.intro.refine'_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx : X\nhsat : IsMaximal (top s) x\ni✝ : Fin (s.length + 1 + 1)\ni : Fin (s.length + 1)\n⊢ Fin.snoc s.series x (↑Fin.castSucc i) ∈ range s.series ∨ Fin.snoc s.series x (↑Fin.castSucc i) = x",
"tactic": "left"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.refine'_2.h\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx : X\nhsat : IsMaximal (top s) x\ni✝ : Fin (s.length + 1 + 1)\ni : Fin (s.length + 1)\n⊢ Fin.snoc s.series x (↑Fin.castSucc i) ∈ range s.series",
"tactic": "simp"
},
{
"state_after": "case mpr\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx y : X\nhsat : IsMaximal (top s) x\nh : y ∈ range s.series ∨ y = x\n⊢ y ∈ range (Fin.snoc s.series x)",
"state_before": "case mpr\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx y : X\nhsat : IsMaximal (top s) x\n⊢ y ∈ range s.series ∨ y = x → y ∈ range (Fin.snoc s.series x)",
"tactic": "intro h"
},
{
"state_after": "case mpr.inl.intro\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx : X\nhsat : IsMaximal (top s) x\ni : Fin (s.length + 1)\n⊢ series s i ∈ range (Fin.snoc s.series x)\n\ncase mpr.inr\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ny : X\nhsat : IsMaximal (top s) y\n⊢ y ∈ range (Fin.snoc s.series y)",
"state_before": "case mpr\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx y : X\nhsat : IsMaximal (top s) x\nh : y ∈ range s.series ∨ y = x\n⊢ y ∈ range (Fin.snoc s.series x)",
"tactic": "rcases h with (⟨i, rfl⟩ | rfl)"
},
{
"state_after": "case mpr.inl.intro\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx : X\nhsat : IsMaximal (top s) x\ni : Fin (s.length + 1)\n⊢ Fin.snoc s.series x (↑Fin.castSucc i) = series s i",
"state_before": "case mpr.inl.intro\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx : X\nhsat : IsMaximal (top s) x\ni : Fin (s.length + 1)\n⊢ series s i ∈ range (Fin.snoc s.series x)",
"tactic": "use Fin.castSucc i"
},
{
"state_after": "no goals",
"state_before": "case mpr.inl.intro\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx : X\nhsat : IsMaximal (top s) x\ni : Fin (s.length + 1)\n⊢ Fin.snoc s.series x (↑Fin.castSucc i) = series s i",
"tactic": "simp"
},
{
"state_after": "case mpr.inr\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ny : X\nhsat : IsMaximal (top s) y\n⊢ Fin.snoc s.series y (Fin.last (s.length + 1)) = y",
"state_before": "case mpr.inr\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ny : X\nhsat : IsMaximal (top s) y\n⊢ y ∈ range (Fin.snoc s.series y)",
"tactic": "use Fin.last _"
},
{
"state_after": "no goals",
"state_before": "case mpr.inr\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ny : X\nhsat : IsMaximal (top s) y\n⊢ Fin.snoc s.series y (Fin.last (s.length + 1)) = y",
"tactic": "simp"
}
] |
[
595,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/Topology/Homotopy/Basic.lean
|
ContinuousMap.Homotopy.extend_apply_of_mem_I
|
[] |
[
194,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Algebra/DirectSum/Decomposition.lean
|
DirectSum.decompose_add
|
[] |
[
157,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Control/Functor.lean
|
Functor.Comp.functor_comp_id
|
[] |
[
229,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.mem_Ioc
|
[] |
[
146,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Control/Traversable/Basic.lean
|
ApplicativeTransformation.ext
|
[
{
"state_after": "case h\nF : Type u → Type v\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\nG : Type u → Type w\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nη η' : ApplicativeTransformation F G\nh : ∀ (α : Type u) (x : F α), (fun {α} => app η α) x = (fun {α} => app η' α) x\n⊢ (fun {α} => app η α) = fun {α} => app η' α",
"state_before": "F : Type u → Type v\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\nG : Type u → Type w\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nη η' : ApplicativeTransformation F G\nh : ∀ (α : Type u) (x : F α), (fun {α} => app η α) x = (fun {α} => app η' α) x\n⊢ η = η'",
"tactic": "apply coe_inj"
},
{
"state_after": "case h.h\nF : Type u → Type v\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\nG : Type u → Type w\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nη η' : ApplicativeTransformation F G\nh : ∀ (α : Type u) (x : F α), (fun {α} => app η α) x = (fun {α} => app η' α) x\nα : Type u\n⊢ app η α = app η' α",
"state_before": "case h\nF : Type u → Type v\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\nG : Type u → Type w\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nη η' : ApplicativeTransformation F G\nh : ∀ (α : Type u) (x : F α), (fun {α} => app η α) x = (fun {α} => app η' α) x\n⊢ (fun {α} => app η α) = fun {α} => app η' α",
"tactic": "ext1 α"
},
{
"state_after": "no goals",
"state_before": "case h.h\nF : Type u → Type v\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\nG : Type u → Type w\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nη η' : ApplicativeTransformation F G\nh : ∀ (α : Type u) (x : F α), (fun {α} => app η α) x = (fun {α} => app η' α) x\nα : Type u\n⊢ app η α = app η' α",
"tactic": "exact funext (h α)"
}
] |
[
130,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.