file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Set/Basic.lean
|
Set.Subset.trans
|
[] |
[
363,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
363,
1
] |
Mathlib/CategoryTheory/Simple.lean
|
CategoryTheory.kernel_zero_of_nonzero_from_simple
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\nX Y : C\ninst✝¹ : Simple X\nf : X ⟶ Y\ninst✝ : HasKernel f\nw : f ≠ 0\n⊢ kernel.ι f = 0",
"tactic": "classical\n by_contra h\n haveI := isIso_of_mono_of_nonzero h\n exact w (eq_zero_of_epi_kernel f)"
},
{
"state_after": "C : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\nX Y : C\ninst✝¹ : Simple X\nf : X ⟶ Y\ninst✝ : HasKernel f\nw : f ≠ 0\nh : ¬kernel.ι f = 0\n⊢ False",
"state_before": "C : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\nX Y : C\ninst✝¹ : Simple X\nf : X ⟶ Y\ninst✝ : HasKernel f\nw : f ≠ 0\n⊢ kernel.ι f = 0",
"tactic": "by_contra h"
},
{
"state_after": "C : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\nX Y : C\ninst✝¹ : Simple X\nf : X ⟶ Y\ninst✝ : HasKernel f\nw : f ≠ 0\nh : ¬kernel.ι f = 0\nthis : IsIso (kernel.ι f)\n⊢ False",
"state_before": "C : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\nX Y : C\ninst✝¹ : Simple X\nf : X ⟶ Y\ninst✝ : HasKernel f\nw : f ≠ 0\nh : ¬kernel.ι f = 0\n⊢ False",
"tactic": "haveI := isIso_of_mono_of_nonzero h"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\nX Y : C\ninst✝¹ : Simple X\nf : X ⟶ Y\ninst✝ : HasKernel f\nw : f ≠ 0\nh : ¬kernel.ι f = 0\nthis : IsIso (kernel.ι f)\n⊢ False",
"tactic": "exact w (eq_zero_of_epi_kernel f)"
}
] |
[
93,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/CategoryTheory/Subobject/Limits.lean
|
CategoryTheory.Limits.imageSubobject_mono
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nX Y Z : C\nf✝ : X ⟶ Y\ninst✝¹ : HasImage f✝\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ (imageSubobjectIso f ≪≫ imageMonoIsoSource f ≪≫ (underlyingIso f).symm).hom ≫ arrow (Subobject.mk f) =\n arrow (imageSubobject f)",
"tactic": "simp"
}
] |
[
431,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.arg_le_pi_div_two_iff
|
[
{
"state_after": "case inl\nz : ℂ\nhre : 0 ≤ z.re\n⊢ arg z ≤ π / 2 ↔ 0 ≤ z.re ∨ z.im < 0\n\ncase inr\nz : ℂ\nhre : z.re < 0\n⊢ arg z ≤ π / 2 ↔ 0 ≤ z.re ∨ z.im < 0",
"state_before": "z : ℂ\n⊢ arg z ≤ π / 2 ↔ 0 ≤ z.re ∨ z.im < 0",
"tactic": "cases' le_or_lt 0 (re z) with hre hre"
},
{
"state_after": "case inr\nz : ℂ\nhre : z.re < 0\n⊢ arg z ≤ π / 2 ↔ z.im < 0",
"state_before": "case inr\nz : ℂ\nhre : z.re < 0\n⊢ arg z ≤ π / 2 ↔ 0 ≤ z.re ∨ z.im < 0",
"tactic": "simp only [hre.not_le, false_or_iff]"
},
{
"state_after": "case inr.inl\nz : ℂ\nhre : z.re < 0\nhim : 0 ≤ z.im\n⊢ arg z ≤ π / 2 ↔ z.im < 0\n\ncase inr.inr\nz : ℂ\nhre : z.re < 0\nhim : z.im < 0\n⊢ arg z ≤ π / 2 ↔ z.im < 0",
"state_before": "case inr\nz : ℂ\nhre : z.re < 0\n⊢ arg z ≤ π / 2 ↔ z.im < 0",
"tactic": "cases' le_or_lt 0 (im z) with him him"
},
{
"state_after": "no goals",
"state_before": "case inl\nz : ℂ\nhre : 0 ≤ z.re\n⊢ arg z ≤ π / 2 ↔ 0 ≤ z.re ∨ z.im < 0",
"tactic": "simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff]"
},
{
"state_after": "case inr.inl\nz : ℂ\nhre : z.re < 0\nhim : 0 ≤ z.im\n⊢ arg z ≤ π / 2 ↔ False",
"state_before": "case inr.inl\nz : ℂ\nhre : z.re < 0\nhim : 0 ≤ z.im\n⊢ arg z ≤ π / 2 ↔ z.im < 0",
"tactic": "simp only [him.not_lt]"
},
{
"state_after": "case inr.inl\nz : ℂ\nhre : z.re < 0\nhim : 0 ≤ z.im\n⊢ z.re ≠ 0\n\ncase inr.inl\nz : ℂ\nhre : z.re < 0\nhim : 0 ≤ z.im\n⊢ 0 < ↑abs z",
"state_before": "case inr.inl\nz : ℂ\nhre : z.re < 0\nhim : 0 ≤ z.im\n⊢ arg z ≤ π / 2 ↔ False",
"tactic": "rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub,\n Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ←\n _root_.abs_of_nonneg him, abs_im_lt_abs]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nz : ℂ\nhre : z.re < 0\nhim : 0 ≤ z.im\n⊢ z.re ≠ 0\n\ncase inr.inl\nz : ℂ\nhre : z.re < 0\nhim : 0 ≤ z.im\n⊢ 0 < ↑abs z",
"tactic": "exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne]"
},
{
"state_after": "case inr.inr\nz : ℂ\nhre : z.re < 0\nhim : z.im < 0\n⊢ arg z ≤ π / 2 ↔ True",
"state_before": "case inr.inr\nz : ℂ\nhre : z.re < 0\nhim : z.im < 0\n⊢ arg z ≤ π / 2 ↔ z.im < 0",
"tactic": "simp only [him]"
},
{
"state_after": "case inr.inr\nz : ℂ\nhre : z.re < 0\nhim : z.im < 0\n⊢ arcsin ((-z).im / ↑abs z) - π ≤ π / 2",
"state_before": "case inr.inr\nz : ℂ\nhre : z.re < 0\nhim : z.im < 0\n⊢ arg z ≤ π / 2 ↔ True",
"tactic": "rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nz : ℂ\nhre : z.re < 0\nhim : z.im < 0\n⊢ arcsin ((-z).im / ↑abs z) - π ≤ π / 2",
"tactic": "exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _)"
}
] |
[
341,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
329,
1
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean
|
Polynomial.prod_cyclotomic'_eq_X_pow_sub_one
|
[
{
"state_after": "K✝ : Type ?u.240784\ninst✝² : Field K✝\nK : Type u_1\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nζ : K\nn : ℕ\nhpos : 0 < n\nh : IsPrimitiveRoot ζ n\nhd : Set.PairwiseDisjoint ↑(Nat.divisors n) fun k => primitiveRoots k K\n⊢ ∏ i in Nat.divisors n, cyclotomic' i K = X ^ n - 1",
"state_before": "K✝ : Type ?u.240784\ninst✝² : Field K✝\nK : Type u_1\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nζ : K\nn : ℕ\nhpos : 0 < n\nh : IsPrimitiveRoot ζ n\n⊢ ∏ i in Nat.divisors n, cyclotomic' i K = X ^ n - 1",
"tactic": "have hd : (n.divisors : Set ℕ).PairwiseDisjoint fun k => primitiveRoots k K :=\n fun x _ y _ hne => IsPrimitiveRoot.disjoint hne"
},
{
"state_after": "no goals",
"state_before": "K✝ : Type ?u.240784\ninst✝² : Field K✝\nK : Type u_1\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nζ : K\nn : ℕ\nhpos : 0 < n\nh : IsPrimitiveRoot ζ n\nhd : Set.PairwiseDisjoint ↑(Nat.divisors n) fun k => primitiveRoots k K\n⊢ ∏ i in Nat.divisors n, cyclotomic' i K = X ^ n - 1",
"tactic": "simp only [X_pow_sub_one_eq_prod hpos h, cyclotomic', ← Finset.prod_biUnion hd,\n h.nthRoots_one_eq_biUnion_primitiveRoots]"
}
] |
[
177,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.div_subset_div_left
|
[] |
[
690,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
689,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.card_omega
|
[] |
[
865,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
864,
1
] |
Std/Data/String/Lemmas.lean
|
Substring.ValidFor.nextn
|
[
{
"state_after": "no goals",
"state_before": "l m₁ m₂ r : List Char\nx✝¹ : Substring\nx✝ : ValidFor l (m₁ ++ m₂) r x✝¹\n⊢ Substring.nextn x✝¹ 0 { byteIdx := utf8Len m₁ } = { byteIdx := utf8Len m₁ + utf8Len (List.take 0 m₂) }",
"tactic": "simp [Substring.nextn]"
},
{
"state_after": "l m₁ m₂ r : List Char\ns : Substring\nh : ValidFor l (m₁ ++ m₂) r s\nn : Nat\n⊢ Substring.nextn s n (Substring.next s { byteIdx := utf8Len m₁ }) =\n { byteIdx := utf8Len m₁ + utf8Len (List.take (n + 1) m₂) }",
"state_before": "l m₁ m₂ r : List Char\ns : Substring\nh : ValidFor l (m₁ ++ m₂) r s\nn : Nat\n⊢ Substring.nextn s (n + 1) { byteIdx := utf8Len m₁ } = { byteIdx := utf8Len m₁ + utf8Len (List.take (n + 1) m₂) }",
"tactic": "simp [Substring.nextn]"
},
{
"state_after": "no goals",
"state_before": "l m₁ m₂ r : List Char\ns : Substring\nh : ValidFor l (m₁ ++ m₂) r s\nn : Nat\n⊢ Substring.nextn s n (Substring.next s { byteIdx := utf8Len m₁ }) =\n { byteIdx := utf8Len m₁ + utf8Len (List.take (n + 1) m₂) }",
"tactic": "match m₂ with\n| [] => simp at h; simp [h.next_stop, h.nextn_stop]\n| c::m₂ =>\n rw [h.next]\n have := @nextn l (m₁ ++ [c]) m₂ r s (by simp [h]) n\n simp at this; rw [this]; simp [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]"
},
{
"state_after": "l m₁ m₂ r : List Char\ns : Substring\nn : Nat\nh : ValidFor l m₁ r s\n⊢ Substring.nextn s n (Substring.next s { byteIdx := utf8Len m₁ }) =\n { byteIdx := utf8Len m₁ + utf8Len (List.take (n + 1) []) }",
"state_before": "l m₁ m₂ r : List Char\ns : Substring\nn : Nat\nh : ValidFor l (m₁ ++ []) r s\n⊢ Substring.nextn s n (Substring.next s { byteIdx := utf8Len m₁ }) =\n { byteIdx := utf8Len m₁ + utf8Len (List.take (n + 1) []) }",
"tactic": "simp at h"
},
{
"state_after": "no goals",
"state_before": "l m₁ m₂ r : List Char\ns : Substring\nn : Nat\nh : ValidFor l m₁ r s\n⊢ Substring.nextn s n (Substring.next s { byteIdx := utf8Len m₁ }) =\n { byteIdx := utf8Len m₁ + utf8Len (List.take (n + 1) []) }",
"tactic": "simp [h.next_stop, h.nextn_stop]"
},
{
"state_after": "l m₁ m₂✝ r : List Char\ns : Substring\nn : Nat\nc : Char\nm₂ : List Char\nh : ValidFor l (m₁ ++ c :: m₂) r s\n⊢ Substring.nextn s n { byteIdx := utf8Len m₁ + csize c } =\n { byteIdx := utf8Len m₁ + utf8Len (List.take (n + 1) (c :: m₂)) }",
"state_before": "l m₁ m₂✝ r : List Char\ns : Substring\nn : Nat\nc : Char\nm₂ : List Char\nh : ValidFor l (m₁ ++ c :: m₂) r s\n⊢ Substring.nextn s n (Substring.next s { byteIdx := utf8Len m₁ }) =\n { byteIdx := utf8Len m₁ + utf8Len (List.take (n + 1) (c :: m₂)) }",
"tactic": "rw [h.next]"
},
{
"state_after": "l m₁ m₂✝ r : List Char\ns : Substring\nn : Nat\nc : Char\nm₂ : List Char\nh : ValidFor l (m₁ ++ c :: m₂) r s\nthis :\n Substring.nextn s n { byteIdx := utf8Len (m₁ ++ [c]) } = { byteIdx := utf8Len (m₁ ++ [c]) + utf8Len (List.take n m₂) }\n⊢ Substring.nextn s n { byteIdx := utf8Len m₁ + csize c } =\n { byteIdx := utf8Len m₁ + utf8Len (List.take (n + 1) (c :: m₂)) }",
"state_before": "l m₁ m₂✝ r : List Char\ns : Substring\nn : Nat\nc : Char\nm₂ : List Char\nh : ValidFor l (m₁ ++ c :: m₂) r s\n⊢ Substring.nextn s n { byteIdx := utf8Len m₁ + csize c } =\n { byteIdx := utf8Len m₁ + utf8Len (List.take (n + 1) (c :: m₂)) }",
"tactic": "have := @nextn l (m₁ ++ [c]) m₂ r s (by simp [h]) n"
},
{
"state_after": "l m₁ m₂✝ r : List Char\ns : Substring\nn : Nat\nc : Char\nm₂ : List Char\nh : ValidFor l (m₁ ++ c :: m₂) r s\nthis :\n Substring.nextn s n { byteIdx := utf8Len m₁ + csize c } =\n { byteIdx := utf8Len m₁ + csize c + utf8Len (List.take n m₂) }\n⊢ Substring.nextn s n { byteIdx := utf8Len m₁ + csize c } =\n { byteIdx := utf8Len m₁ + utf8Len (List.take (n + 1) (c :: m₂)) }",
"state_before": "l m₁ m₂✝ r : List Char\ns : Substring\nn : Nat\nc : Char\nm₂ : List Char\nh : ValidFor l (m₁ ++ c :: m₂) r s\nthis :\n Substring.nextn s n { byteIdx := utf8Len (m₁ ++ [c]) } = { byteIdx := utf8Len (m₁ ++ [c]) + utf8Len (List.take n m₂) }\n⊢ Substring.nextn s n { byteIdx := utf8Len m₁ + csize c } =\n { byteIdx := utf8Len m₁ + utf8Len (List.take (n + 1) (c :: m₂)) }",
"tactic": "simp at this"
},
{
"state_after": "l m₁ m₂✝ r : List Char\ns : Substring\nn : Nat\nc : Char\nm₂ : List Char\nh : ValidFor l (m₁ ++ c :: m₂) r s\nthis :\n Substring.nextn s n { byteIdx := utf8Len m₁ + csize c } =\n { byteIdx := utf8Len m₁ + csize c + utf8Len (List.take n m₂) }\n⊢ { byteIdx := utf8Len m₁ + csize c + utf8Len (List.take n m₂) } =\n { byteIdx := utf8Len m₁ + utf8Len (List.take (n + 1) (c :: m₂)) }",
"state_before": "l m₁ m₂✝ r : List Char\ns : Substring\nn : Nat\nc : Char\nm₂ : List Char\nh : ValidFor l (m₁ ++ c :: m₂) r s\nthis :\n Substring.nextn s n { byteIdx := utf8Len m₁ + csize c } =\n { byteIdx := utf8Len m₁ + csize c + utf8Len (List.take n m₂) }\n⊢ Substring.nextn s n { byteIdx := utf8Len m₁ + csize c } =\n { byteIdx := utf8Len m₁ + utf8Len (List.take (n + 1) (c :: m₂)) }",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "l m₁ m₂✝ r : List Char\ns : Substring\nn : Nat\nc : Char\nm₂ : List Char\nh : ValidFor l (m₁ ++ c :: m₂) r s\nthis :\n Substring.nextn s n { byteIdx := utf8Len m₁ + csize c } =\n { byteIdx := utf8Len m₁ + csize c + utf8Len (List.take n m₂) }\n⊢ { byteIdx := utf8Len m₁ + csize c + utf8Len (List.take n m₂) } =\n { byteIdx := utf8Len m₁ + utf8Len (List.take (n + 1) (c :: m₂)) }",
"tactic": "simp [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]"
},
{
"state_after": "no goals",
"state_before": "l m₁ m₂✝ r : List Char\ns : Substring\nn : Nat\nc : Char\nm₂ : List Char\nh : ValidFor l (m₁ ++ c :: m₂) r s\n⊢ ValidFor l (m₁ ++ [c] ++ m₂) r s",
"tactic": "simp [h]"
}
] |
[
866,
85
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
856,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.map_id_mk'
|
[] |
[
632,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
630,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
contDiffOn_of_subsingleton
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.64561\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : Subsingleton F\n⊢ ContDiffOn 𝕜 n (fun x => 0) s",
"state_before": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.64561\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : Subsingleton F\n⊢ ContDiffOn 𝕜 n f s",
"tactic": "rw [Subsingleton.elim f fun _ => 0]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.64561\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : Subsingleton F\n⊢ ContDiffOn 𝕜 n (fun x => 0) s",
"tactic": "exact contDiffOn_const"
}
] |
[
126,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
|
HasFDerivWithinAt.uniqueDiffWithinAt
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\ns : Set E\nf' : E →L[𝕜] F\nx : E\nh : HasFDerivWithinAt f f' s x\nhs : UniqueDiffWithinAt 𝕜 s x\nh' : DenseRange ↑f'\n⊢ MapsTo ↑f' ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 (f '' s) (f x)))",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\ns : Set E\nf' : E →L[𝕜] F\nx : E\nh : HasFDerivWithinAt f f' s x\nhs : UniqueDiffWithinAt 𝕜 s x\nh' : DenseRange ↑f'\n⊢ UniqueDiffWithinAt 𝕜 (f '' s) (f x)",
"tactic": "refine' ⟨h'.dense_of_mapsTo f'.continuous hs.1 _, h.continuousWithinAt.mem_closure_image hs.2⟩"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\ns : Set E\nf' : E →L[𝕜] F\nx : E\nh : HasFDerivWithinAt f f' s x\nhs : UniqueDiffWithinAt 𝕜 s x\nh' : DenseRange ↑f'\n⊢ Submodule.span 𝕜 (tangentConeAt 𝕜 s x) ≤ Submodule.comap f' (Submodule.span 𝕜 (tangentConeAt 𝕜 (f '' s) (f x)))",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\ns : Set E\nf' : E →L[𝕜] F\nx : E\nh : HasFDerivWithinAt f f' s x\nhs : UniqueDiffWithinAt 𝕜 s x\nh' : DenseRange ↑f'\n⊢ MapsTo ↑f' ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 (f '' s) (f x)))",
"tactic": "show\n Submodule.span 𝕜 (tangentConeAt 𝕜 s x) ≤\n (Submodule.span 𝕜 (tangentConeAt 𝕜 (f '' s) (f x))).comap f'"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\ns : Set E\nf' : E →L[𝕜] F\nx : E\nh : HasFDerivWithinAt f f' s x\nhs : UniqueDiffWithinAt 𝕜 s x\nh' : DenseRange ↑f'\n⊢ tangentConeAt 𝕜 s x ⊆ ↑(Submodule.comap f' (Submodule.span 𝕜 (tangentConeAt 𝕜 (f '' s) (f x))))",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\ns : Set E\nf' : E →L[𝕜] F\nx : E\nh : HasFDerivWithinAt f f' s x\nhs : UniqueDiffWithinAt 𝕜 s x\nh' : DenseRange ↑f'\n⊢ Submodule.span 𝕜 (tangentConeAt 𝕜 s x) ≤ Submodule.comap f' (Submodule.span 𝕜 (tangentConeAt 𝕜 (f '' s) (f x)))",
"tactic": "rw [Submodule.span_le]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\ns : Set E\nf' : E →L[𝕜] F\nx : E\nh : HasFDerivWithinAt f f' s x\nhs : UniqueDiffWithinAt 𝕜 s x\nh' : DenseRange ↑f'\n⊢ tangentConeAt 𝕜 s x ⊆ ↑(Submodule.comap f' (Submodule.span 𝕜 (tangentConeAt 𝕜 (f '' s) (f x))))",
"tactic": "exact h.mapsTo_tangent_cone.mono Subset.rfl Submodule.subset_span"
}
] |
[
506,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
499,
1
] |
Mathlib/NumberTheory/Padics/PadicVal.lean
|
padicValNat_dvd_iff_le
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\na n : ℕ\nha : a ≠ 0\n⊢ p ^ n ∣ a ↔ n ≤ padicValNat p a",
"tactic": "rw [pow_dvd_iff_le_multiplicity, ← padicValNat_def' hp.out.ne_one ha.bot_lt, PartENat.coe_le_coe]"
}
] |
[
465,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
463,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.sub_le
|
[
{
"state_after": "no goals",
"state_before": "n m : Nat\n⊢ n - m ≤ n",
"tactic": "induction m with\n| zero => exact Nat.le_refl (n - 0)\n| succ m ih => apply Nat.le_trans (pred_le (n - m)) ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nn : Nat\n⊢ n - zero ≤ n",
"tactic": "exact Nat.le_refl (n - 0)"
},
{
"state_after": "no goals",
"state_before": "case succ\nn m : Nat\nih : n - m ≤ n\n⊢ n - succ m ≤ n",
"tactic": "apply Nat.le_trans (pred_le (n - m)) ih"
}
] |
[
234,
57
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
231,
1
] |
Mathlib/Algebra/Support.lean
|
Function.mulSupport_one_sub
|
[] |
[
439,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/Topology/Compactification/OnePoint.lean
|
OnePoint.not_inseparable_infty_coe
|
[] |
[
415,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.sum_apply'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nι : Type u_1\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : { x // x ∈ kernel α β }\ninst✝ : Countable ι\nκ : ι → { x // x ∈ kernel α β }\na : α\ns : Set β\nhs : MeasurableSet s\n⊢ ↑↑(↑(kernel.sum κ) a) s = ∑' (n : ι), ↑↑(↑(κ n) a) s",
"tactic": "rw [sum_apply κ a, Measure.sum_apply _ hs]"
}
] |
[
234,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
src/lean/Init/SizeOf.lean
|
sizeOf_default
|
[] |
[
47,
61
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
47,
9
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.map_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\nf : α → β\ns : Stream' α\n⊢ map f s = f (head s) :: map f (tail s)",
"tactic": "rw [← Stream'.eta (map f s), tail_map, head_map]"
}
] |
[
165,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
UpperSet.mem_Ici_iff
|
[] |
[
1097,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1096,
1
] |
Mathlib/Algebra/Lie/IdealOperations.lean
|
LieSubmodule.lieIdeal_oper_eq_linear_span'
|
[
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ Submodule.span R {m | ∃ x n, ⁅↑x, ↑n⁆ = m} = Submodule.span R {m | ∃ x, x ∈ I ∧ ∃ n, n ∈ N ∧ ⁅x, n⁆ = m}",
"state_before": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ ↑⁅I, N⁆ = Submodule.span R {m | ∃ x, x ∈ I ∧ ∃ n, n ∈ N ∧ ⁅x, n⁆ = m}",
"tactic": "rw [lieIdeal_oper_eq_linear_span]"
},
{
"state_after": "case e_s\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} = {m | ∃ x, x ∈ I ∧ ∃ n, n ∈ N ∧ ⁅x, n⁆ = m}",
"state_before": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ Submodule.span R {m | ∃ x n, ⁅↑x, ↑n⁆ = m} = Submodule.span R {m | ∃ x, x ∈ I ∧ ∃ n, n ∈ N ∧ ⁅x, n⁆ = m}",
"tactic": "congr"
},
{
"state_after": "case e_s.h\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nm : M\n⊢ m ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} ↔ m ∈ {m | ∃ x, x ∈ I ∧ ∃ n, n ∈ N ∧ ⁅x, n⁆ = m}",
"state_before": "case e_s\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} = {m | ∃ x, x ∈ I ∧ ∃ n, n ∈ N ∧ ⁅x, n⁆ = m}",
"tactic": "ext m"
},
{
"state_after": "case e_s.h.mp\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nm : M\n⊢ m ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} → m ∈ {m | ∃ x, x ∈ I ∧ ∃ n, n ∈ N ∧ ⁅x, n⁆ = m}\n\ncase e_s.h.mpr\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nm : M\n⊢ m ∈ {m | ∃ x, x ∈ I ∧ ∃ n, n ∈ N ∧ ⁅x, n⁆ = m} → m ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}",
"state_before": "case e_s.h\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nm : M\n⊢ m ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} ↔ m ∈ {m | ∃ x, x ∈ I ∧ ∃ n, n ∈ N ∧ ⁅x, n⁆ = m}",
"tactic": "constructor"
},
{
"state_after": "case e_s.h.mp.intro.mk.intro.mk\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nx : L\nhx : x ∈ I\nn : M\nhn : n ∈ N\n⊢ ⁅↑{ val := x, property := hx }, ↑{ val := n, property := hn }⁆ ∈ {m | ∃ x, x ∈ I ∧ ∃ n, n ∈ N ∧ ⁅x, n⁆ = m}",
"state_before": "case e_s.h.mp\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nm : M\n⊢ m ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} → m ∈ {m | ∃ x, x ∈ I ∧ ∃ n, n ∈ N ∧ ⁅x, n⁆ = m}",
"tactic": "rintro ⟨⟨x, hx⟩, ⟨n, hn⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case e_s.h.mp.intro.mk.intro.mk\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nx : L\nhx : x ∈ I\nn : M\nhn : n ∈ N\n⊢ ⁅↑{ val := x, property := hx }, ↑{ val := n, property := hn }⁆ ∈ {m | ∃ x, x ∈ I ∧ ∃ n, n ∈ N ∧ ⁅x, n⁆ = m}",
"tactic": "exact ⟨x, hx, n, hn, rfl⟩"
},
{
"state_after": "case e_s.h.mpr.intro.intro.intro.intro\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nx : L\nhx : x ∈ I\nn : M\nhn : n ∈ N\n⊢ ⁅x, n⁆ ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}",
"state_before": "case e_s.h.mpr\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nm : M\n⊢ m ∈ {m | ∃ x, x ∈ I ∧ ∃ n, n ∈ N ∧ ⁅x, n⁆ = m} → m ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}",
"tactic": "rintro ⟨x, hx, n, hn, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case e_s.h.mpr.intro.intro.intro.intro\nR : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\nx : L\nhx : x ∈ I\nn : M\nhn : n ∈ N\n⊢ ⁅x, n⁆ ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}",
"tactic": "exact ⟨⟨x, hx⟩, ⟨n, hn⟩, rfl⟩"
}
] |
[
100,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Combinatorics/SimpleGraph/Acyclic.lean
|
SimpleGraph.IsAcyclic.path_unique
|
[
{
"state_after": "case mk\nV : Type u\nG✝ G : SimpleGraph V\nh : IsAcyclic G\nv w : V\nq : Path G v w\np : Walk G v w\nhp : Walk.IsPath p\n⊢ { val := p, property := hp } = q",
"state_before": "V : Type u\nG✝ G : SimpleGraph V\nh : IsAcyclic G\nv w : V\np q : Path G v w\n⊢ p = q",
"tactic": "obtain ⟨p, hp⟩ := p"
},
{
"state_after": "case mk.mk\nV : Type u\nG✝ G : SimpleGraph V\nh : IsAcyclic G\nv w : V\np : Walk G v w\nhp : Walk.IsPath p\nq : Walk G v w\nhq : Walk.IsPath q\n⊢ { val := p, property := hp } = { val := q, property := hq }",
"state_before": "case mk\nV : Type u\nG✝ G : SimpleGraph V\nh : IsAcyclic G\nv w : V\nq : Path G v w\np : Walk G v w\nhp : Walk.IsPath p\n⊢ { val := p, property := hp } = q",
"tactic": "obtain ⟨q, hq⟩ := q"
},
{
"state_after": "case mk.mk\nV : Type u\nG✝ G : SimpleGraph V\nh : IsAcyclic G\nv w : V\np : Walk G v w\nhp : Walk.IsPath p\nq : Walk G v w\nhq : Walk.IsPath q\n⊢ p = q",
"state_before": "case mk.mk\nV : Type u\nG✝ G : SimpleGraph V\nh : IsAcyclic G\nv w : V\np : Walk G v w\nhp : Walk.IsPath p\nq : Walk G v w\nhq : Walk.IsPath q\n⊢ { val := p, property := hp } = { val := q, property := hq }",
"tactic": "rw [Subtype.mk.injEq]"
},
{
"state_after": "case mk.mk.nil.refl\nV : Type u\nG✝ G : SimpleGraph V\nh : IsAcyclic G\nv w u✝ : V\nhp hq : Walk.IsPath Walk.nil\n⊢ Walk.nil = Walk.nil",
"state_before": "case mk.mk.nil\nV : Type u\nG✝ G : SimpleGraph V\nh : IsAcyclic G\nv w u✝ : V\nhp : Walk.IsPath Walk.nil\nq : Walk G u✝ u✝\nhq : Walk.IsPath q\n⊢ Walk.nil = q",
"tactic": "cases (Walk.isPath_iff_eq_nil _).mp hq"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.nil.refl\nV : Type u\nG✝ G : SimpleGraph V\nh : IsAcyclic G\nv w u✝ : V\nhp hq : Walk.IsPath Walk.nil\n⊢ Walk.nil = Walk.nil",
"tactic": "rfl"
},
{
"state_after": "case mk.mk.cons\nV : Type u\nG✝ G : SimpleGraph V\nh : ∀ ⦃v w : V⦄, Adj G v w → IsBridge G (Quotient.mk (Sym2.Rel.setoid V) (v, w))\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\n⊢ Walk.cons ph p = q",
"state_before": "case mk.mk.cons\nV : Type u\nG✝ G : SimpleGraph V\nh : IsAcyclic G\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\n⊢ Walk.cons ph p = q",
"tactic": "rw [isAcyclic_iff_forall_adj_isBridge] at h"
},
{
"state_after": "case mk.mk.cons\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\nh : IsBridge G (Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝))\n⊢ Walk.cons ph p = q",
"state_before": "case mk.mk.cons\nV : Type u\nG✝ G : SimpleGraph V\nh : ∀ ⦃v w : V⦄, Adj G v w → IsBridge G (Quotient.mk (Sym2.Rel.setoid V) (v, w))\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\n⊢ Walk.cons ph p = q",
"tactic": "specialize h ph"
},
{
"state_after": "case mk.mk.cons\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\nh : Adj G u✝ v✝ ∧ ∀ (p : Walk G u✝ v✝), Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝) ∈ Walk.edges p\n⊢ Walk.cons ph p = q",
"state_before": "case mk.mk.cons\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\nh : IsBridge G (Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝))\n⊢ Walk.cons ph p = q",
"tactic": "rw [isBridge_iff_adj_and_forall_walk_mem_edges] at h"
},
{
"state_after": "case mk.mk.cons\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\nh : Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝) ∈ Walk.edges (Walk.append q (Walk.reverse p))\n⊢ Walk.cons ph p = q",
"state_before": "case mk.mk.cons\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\nh : Adj G u✝ v✝ ∧ ∀ (p : Walk G u✝ v✝), Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝) ∈ Walk.edges p\n⊢ Walk.cons ph p = q",
"tactic": "replace h := h.2 (q.append p.reverse)"
},
{
"state_after": "case mk.mk.cons\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\nh : Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝) ∈ Walk.edges q ∨ Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝) ∈ Walk.edges p\n⊢ Walk.cons ph p = q",
"state_before": "case mk.mk.cons\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\nh : Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝) ∈ Walk.edges (Walk.append q (Walk.reverse p))\n⊢ Walk.cons ph p = q",
"tactic": "simp only [Walk.edges_append, Walk.edges_reverse, List.mem_append, List.mem_reverse'] at h"
},
{
"state_after": "case mk.mk.cons.inl\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\nh : Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝) ∈ Walk.edges q\n⊢ Walk.cons ph p = q\n\ncase mk.mk.cons.inr\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\nh : Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝) ∈ Walk.edges p\n⊢ Walk.cons ph p = q",
"state_before": "case mk.mk.cons\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\nh : Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝) ∈ Walk.edges q ∨ Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝) ∈ Walk.edges p\n⊢ Walk.cons ph p = q",
"tactic": "cases' h with h h"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.cons.inl.nil\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ u✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ u✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nhq : Walk.IsPath Walk.nil\nh : Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝) ∈ Walk.edges Walk.nil\n⊢ Walk.cons ph p = Walk.nil",
"tactic": "simp [Walk.isPath_def] at hp"
},
{
"state_after": "case mk.mk.cons.inl.cons\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝¹ w✝ : V\nph : Adj G u✝ v✝¹\np : Walk G v✝¹ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝¹ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath p ∧ ¬u✝ ∈ Walk.support p\nv✝ : V\nh✝ : Adj G u✝ v✝\nq : Walk G v✝ w✝\nhq : Walk.IsPath q ∧ ¬u✝ ∈ Walk.support q\nh : Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝¹) ∈ Walk.edges (Walk.cons h✝ q)\n⊢ Walk.cons ph p = Walk.cons h✝ q",
"state_before": "case mk.mk.cons.inl.cons\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝¹ w✝ : V\nph : Adj G u✝ v✝¹\np : Walk G v✝¹ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝¹ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nv✝ : V\nh✝ : Adj G u✝ v✝\nq : Walk G v✝ w✝\nhq : Walk.IsPath (Walk.cons h✝ q)\nh : Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝¹) ∈ Walk.edges (Walk.cons h✝ q)\n⊢ Walk.cons ph p = Walk.cons h✝ q",
"tactic": "rw [Walk.cons_isPath_iff] at hp hq"
},
{
"state_after": "case mk.mk.cons.inl.cons\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝¹ w✝ : V\nph : Adj G u✝ v✝¹\np : Walk G v✝¹ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝¹ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath p ∧ ¬u✝ ∈ Walk.support p\nv✝ : V\nh✝ : Adj G u✝ v✝\nq : Walk G v✝ w✝\nhq : Walk.IsPath q ∧ ¬u✝ ∈ Walk.support q\nh : (v✝¹ = v✝ ∨ u✝ = v✝ ∧ v✝¹ = u✝) ∨ Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝¹) ∈ Walk.edges q\n⊢ Walk.cons ph p = Walk.cons h✝ q",
"state_before": "case mk.mk.cons.inl.cons\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝¹ w✝ : V\nph : Adj G u✝ v✝¹\np : Walk G v✝¹ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝¹ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath p ∧ ¬u✝ ∈ Walk.support p\nv✝ : V\nh✝ : Adj G u✝ v✝\nq : Walk G v✝ w✝\nhq : Walk.IsPath q ∧ ¬u✝ ∈ Walk.support q\nh : Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝¹) ∈ Walk.edges (Walk.cons h✝ q)\n⊢ Walk.cons ph p = Walk.cons h✝ q",
"tactic": "simp only [Walk.edges_cons, List.mem_cons, Sym2.eq_iff, true_and] at h"
},
{
"state_after": "case mk.mk.cons.inl.cons.inl.inl.refl\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath p ∧ ¬u✝ ∈ Walk.support p\nh✝ : Adj G u✝ v✝\nq : Walk G v✝ w✝\nhq : Walk.IsPath q ∧ ¬u✝ ∈ Walk.support q\n⊢ Walk.cons ph p = Walk.cons h✝ q\n\ncase mk.mk.cons.inl.cons.inl.inr.intro\nV : Type u\nG✝ G : SimpleGraph V\nv w v✝ w✝ : V\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nph : Adj G v✝ v✝\nhp : Walk.IsPath p ∧ ¬v✝ ∈ Walk.support p\nh✝ : Adj G v✝ v✝\nq : Walk G v✝ w✝\nhq : Walk.IsPath q ∧ ¬v✝ ∈ Walk.support q\n⊢ Walk.cons ph p = Walk.cons h✝ q\n\ncase mk.mk.cons.inl.cons.inr\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝¹ w✝ : V\nph : Adj G u✝ v✝¹\np : Walk G v✝¹ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝¹ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath p ∧ ¬u✝ ∈ Walk.support p\nv✝ : V\nh✝ : Adj G u✝ v✝\nq : Walk G v✝ w✝\nhq : Walk.IsPath q ∧ ¬u✝ ∈ Walk.support q\nh : Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝¹) ∈ Walk.edges q\n⊢ Walk.cons ph p = Walk.cons h✝ q",
"state_before": "case mk.mk.cons.inl.cons\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝¹ w✝ : V\nph : Adj G u✝ v✝¹\np : Walk G v✝¹ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝¹ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath p ∧ ¬u✝ ∈ Walk.support p\nv✝ : V\nh✝ : Adj G u✝ v✝\nq : Walk G v✝ w✝\nhq : Walk.IsPath q ∧ ¬u✝ ∈ Walk.support q\nh : (v✝¹ = v✝ ∨ u✝ = v✝ ∧ v✝¹ = u✝) ∨ Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝¹) ∈ Walk.edges q\n⊢ Walk.cons ph p = Walk.cons h✝ q",
"tactic": "rcases h with (⟨h, rfl⟩ | ⟨rfl, rfl⟩) | h"
},
{
"state_after": "case mk.mk.cons.inl.cons.inl.inl.refl.refl\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath p ∧ ¬u✝ ∈ Walk.support p\nh✝ : Adj G u✝ v✝\nhq : Walk.IsPath p ∧ ¬u✝ ∈ Walk.support p\n⊢ Walk.cons ph p = Walk.cons h✝ p",
"state_before": "case mk.mk.cons.inl.cons.inl.inl.refl\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath p ∧ ¬u✝ ∈ Walk.support p\nh✝ : Adj G u✝ v✝\nq : Walk G v✝ w✝\nhq : Walk.IsPath q ∧ ¬u✝ ∈ Walk.support q\n⊢ Walk.cons ph p = Walk.cons h✝ q",
"tactic": "cases ih hp.1 q hq.1"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.cons.inl.cons.inl.inl.refl.refl\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath p ∧ ¬u✝ ∈ Walk.support p\nh✝ : Adj G u✝ v✝\nhq : Walk.IsPath p ∧ ¬u✝ ∈ Walk.support p\n⊢ Walk.cons ph p = Walk.cons h✝ p",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.cons.inl.cons.inl.inr.intro\nV : Type u\nG✝ G : SimpleGraph V\nv w v✝ w✝ : V\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nph : Adj G v✝ v✝\nhp : Walk.IsPath p ∧ ¬v✝ ∈ Walk.support p\nh✝ : Adj G v✝ v✝\nq : Walk G v✝ w✝\nhq : Walk.IsPath q ∧ ¬v✝ ∈ Walk.support q\n⊢ Walk.cons ph p = Walk.cons h✝ q",
"tactic": "simp at hq"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.cons.inl.cons.inr\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝¹ w✝ : V\nph : Adj G u✝ v✝¹\np : Walk G v✝¹ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝¹ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath p ∧ ¬u✝ ∈ Walk.support p\nv✝ : V\nh✝ : Adj G u✝ v✝\nq : Walk G v✝ w✝\nhq : Walk.IsPath q ∧ ¬u✝ ∈ Walk.support q\nh : Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝¹) ∈ Walk.edges q\n⊢ Walk.cons ph p = Walk.cons h✝ q",
"tactic": "exact absurd (Walk.fst_mem_support_of_mem_edges _ h) hq.2"
},
{
"state_after": "case mk.mk.cons.inr\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath p ∧ ¬u✝ ∈ Walk.support p\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\nh : Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝) ∈ Walk.edges p\n⊢ Walk.cons ph p = q",
"state_before": "case mk.mk.cons.inr\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath (Walk.cons ph p)\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\nh : Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝) ∈ Walk.edges p\n⊢ Walk.cons ph p = q",
"tactic": "rw [Walk.cons_isPath_iff] at hp"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.cons.inr\nV : Type u\nG✝ G : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : Adj G u✝ v✝\np : Walk G v✝ w✝\nih : Walk.IsPath p → ∀ (q : Walk G v✝ w✝), Walk.IsPath q → p = q\nhp : Walk.IsPath p ∧ ¬u✝ ∈ Walk.support p\nq : Walk G u✝ w✝\nhq : Walk.IsPath q\nh : Quotient.mk (Sym2.Rel.setoid V) (u✝, v✝) ∈ Walk.edges p\n⊢ Walk.cons ph p = q",
"tactic": "exact absurd (Walk.fst_mem_support_of_mem_edges _ h) hp.2"
}
] |
[
113,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Algebra/Order/Module.lean
|
smul_lt_smul_of_neg
|
[
{
"state_after": "k : Type u_2\nM : Type u_1\nN : Type ?u.7657\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nh : a < b\nhc : c < 0\n⊢ -c • a < -c • b",
"state_before": "k : Type u_2\nM : Type u_1\nN : Type ?u.7657\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nh : a < b\nhc : c < 0\n⊢ c • b < c • a",
"tactic": "rw [← neg_neg c, neg_smul, neg_smul (-c), neg_lt_neg_iff]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_2\nM : Type u_1\nN : Type ?u.7657\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nh : a < b\nhc : c < 0\n⊢ -c • a < -c • b",
"tactic": "exact smul_lt_smul_of_pos h (neg_pos_of_neg hc)"
}
] |
[
58,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
RingHom.range_eq_map
|
[
{
"state_after": "case h\nR : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ng : S →+* T\nf✝ f : R →+* S\nx✝ : S\n⊢ x✝ ∈ range f ↔ x✝ ∈ Subring.map f ⊤",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ng : S →+* T\nf✝ f : R →+* S\n⊢ range f = Subring.map f ⊤",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ng : S →+* T\nf✝ f : R →+* S\nx✝ : S\n⊢ x✝ ∈ range f ↔ x✝ ∈ Subring.map f ⊤",
"tactic": "simp"
}
] |
[
664,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
662,
1
] |
Mathlib/CategoryTheory/Iso.lean
|
CategoryTheory.IsIso.comp_inv_eq
|
[] |
[
434,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_X_pow_le
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\n⊢ degree (X ^ n) ≤ ↑n",
"tactic": "simpa only [C_1, one_mul] using degree_C_mul_X_pow_le n (1 : R)"
}
] |
[
472,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.absorbent_closedBall_zero
|
[] |
[
991,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
990,
11
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
lt_inv_iff_mul_lt_one'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c : α\n⊢ b * a < b * b⁻¹ ↔ b * a < 1",
"tactic": "rw [mul_inv_self]"
}
] |
[
194,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.concat_cons
|
[] |
[
289,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.trim_smul
|
[] |
[
1758,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1756,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.mk_add_moveRight_inr
|
[] |
[
1548,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1545,
1
] |
Mathlib/Computability/Ackermann.lean
|
lt_ack_right
|
[] |
[
200,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/Data/Int/ModEq.lean
|
Int.ModEq.refl
|
[] |
[
50,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
11
] |
Mathlib/Topology/ExtremallyDisconnected.lean
|
StoneCech.projective
|
[
{
"state_after": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\n⊢ ∃ h, Continuous h ∧ g ∘ h = f",
"state_before": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\n⊢ CompactT2.Projective (StoneCech X)",
"tactic": "intro Y Z _tsY _tsZ _csY _t2Y _csZ _csZ f g hf hg g_sur"
},
{
"state_after": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\ns : Z → Y := fun z => choose (_ : ∃ a, g a = z)\n⊢ ∃ h, Continuous h ∧ g ∘ h = f",
"state_before": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\n⊢ ∃ h, Continuous h ∧ g ∘ h = f",
"tactic": "let s : Z → Y := fun z => Classical.choose <| g_sur z"
},
{
"state_after": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\ns : Z → Y := fun z => choose (_ : ∃ a, g a = z)\nhs : g ∘ s = id\n⊢ ∃ h, Continuous h ∧ g ∘ h = f",
"state_before": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\ns : Z → Y := fun z => choose (_ : ∃ a, g a = z)\n⊢ ∃ h, Continuous h ∧ g ∘ h = f",
"tactic": "have hs : g ∘ s = id := funext fun z => Classical.choose_spec (g_sur z)"
},
{
"state_after": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\ns : Z → Y := fun z => choose (_ : ∃ a, g a = z)\nhs : g ∘ s = id\nt : X → Y := s ∘ f ∘ stoneCechUnit\n⊢ ∃ h, Continuous h ∧ g ∘ h = f",
"state_before": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\ns : Z → Y := fun z => choose (_ : ∃ a, g a = z)\nhs : g ∘ s = id\n⊢ ∃ h, Continuous h ∧ g ∘ h = f",
"tactic": "let t := s ∘ f ∘ stoneCechUnit"
},
{
"state_after": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\ns : Z → Y := fun z => choose (_ : ∃ a, g a = z)\nhs : g ∘ s = id\nt : X → Y := s ∘ f ∘ stoneCechUnit\nht : Continuous t\n⊢ ∃ h, Continuous h ∧ g ∘ h = f",
"state_before": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\ns : Z → Y := fun z => choose (_ : ∃ a, g a = z)\nhs : g ∘ s = id\nt : X → Y := s ∘ f ∘ stoneCechUnit\n⊢ ∃ h, Continuous h ∧ g ∘ h = f",
"tactic": "have ht : Continuous t := continuous_of_discreteTopology"
},
{
"state_after": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\ns : Z → Y := fun z => choose (_ : ∃ a, g a = z)\nhs : g ∘ s = id\nt : X → Y := s ∘ f ∘ stoneCechUnit\nht : Continuous t\nh : StoneCech X → Y := stoneCechExtend ht\n⊢ ∃ h, Continuous h ∧ g ∘ h = f",
"state_before": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\ns : Z → Y := fun z => choose (_ : ∃ a, g a = z)\nhs : g ∘ s = id\nt : X → Y := s ∘ f ∘ stoneCechUnit\nht : Continuous t\n⊢ ∃ h, Continuous h ∧ g ∘ h = f",
"tactic": "let h : StoneCech X → Y := stoneCechExtend ht"
},
{
"state_after": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\ns : Z → Y := fun z => choose (_ : ∃ a, g a = z)\nhs : g ∘ s = id\nt : X → Y := s ∘ f ∘ stoneCechUnit\nht : Continuous t\nh : StoneCech X → Y := stoneCechExtend ht\nhh : Continuous h\n⊢ ∃ h, Continuous h ∧ g ∘ h = f",
"state_before": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\ns : Z → Y := fun z => choose (_ : ∃ a, g a = z)\nhs : g ∘ s = id\nt : X → Y := s ∘ f ∘ stoneCechUnit\nht : Continuous t\nh : StoneCech X → Y := stoneCechExtend ht\n⊢ ∃ h, Continuous h ∧ g ∘ h = f",
"tactic": "have hh : Continuous h := continuous_stoneCechExtend ht"
},
{
"state_after": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\ns : Z → Y := fun z => choose (_ : ∃ a, g a = z)\nhs : g ∘ s = id\nt : X → Y := s ∘ f ∘ stoneCechUnit\nht : Continuous t\nh : StoneCech X → Y := stoneCechExtend ht\nhh : Continuous h\n⊢ (g ∘ h) ∘ stoneCechUnit = f ∘ stoneCechUnit",
"state_before": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\ns : Z → Y := fun z => choose (_ : ∃ a, g a = z)\nhs : g ∘ s = id\nt : X → Y := s ∘ f ∘ stoneCechUnit\nht : Continuous t\nh : StoneCech X → Y := stoneCechExtend ht\nhh : Continuous h\n⊢ ∃ h, Continuous h ∧ g ∘ h = f",
"tactic": "refine' ⟨h, hh, denseRange_stoneCechUnit.equalizer (hg.comp hh) hf _⟩"
},
{
"state_after": "no goals",
"state_before": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nY Z : Type u\n_tsY : TopologicalSpace Y\n_tsZ : TopologicalSpace Z\n_csY : CompactSpace Y\n_t2Y : T2Space Y\n_csZ✝ : CompactSpace Z\n_csZ : T2Space Z\nf : StoneCech X → Z\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\ng_sur : Surjective g\ns : Z → Y := fun z => choose (_ : ∃ a, g a = z)\nhs : g ∘ s = id\nt : X → Y := s ∘ f ∘ stoneCechUnit\nht : Continuous t\nh : StoneCech X → Y := stoneCechExtend ht\nhh : Continuous h\n⊢ (g ∘ h) ∘ stoneCechUnit = f ∘ stoneCechUnit",
"tactic": "rw [comp.assoc, stoneCechExtend_extends ht, ← comp.assoc, hs, comp.left_id]"
}
] |
[
83,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.integral_zero
|
[
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.871391\n𝕜 : Type ?u.871394\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : SMulCommClass ℝ 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\n⊢ (if hf : Integrable fun x => 0 then ↑L1.integralCLM (Integrable.toL1 (fun x => 0) hf) else 0) = 0",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.871391\n𝕜 : Type ?u.871394\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : SMulCommClass ℝ 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\n⊢ (∫ (x : α), 0 ∂μ) = 0",
"tactic": "simp only [integral, L1.integral]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.871391\n𝕜 : Type ?u.871394\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : SMulCommClass ℝ 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\n⊢ (if hf : Integrable fun x => 0 then ↑L1.integralCLM (Integrable.toL1 (fun x => 0) hf) else 0) = 0",
"tactic": "exact setToFun_zero (dominatedFinMeasAdditive_weightedSMul μ)"
}
] |
[
847,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
845,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.toList_eq_singleton_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.31600\nγ : Type ?u.31603\na : α\nm : Multiset α\n⊢ toList m = [a] ↔ m = {a}",
"tactic": "rw [← perm_singleton, ← coe_eq_coe, coe_toList, coe_singleton]"
}
] |
[
483,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
482,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.surjOn_iUnion₂
|
[] |
[
1598,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1596,
1
] |
Mathlib/Data/Set/Countable.lean
|
Set.countable_iff_exists_injOn
|
[] |
[
61,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Data/MvPolynomial/CommRing.lean
|
MvPolynomial.support_neg
|
[] |
[
88,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.map_cast_heq
|
[
{
"state_after": "α✝ : Type ?u.17749\nβ : Type ?u.17752\nγ : Type ?u.17755\nf : α✝ ↪ β\ns✝ : Finset α✝\nα : Type u_1\ns : Finset α\n⊢ HEq (map (Equiv.toEmbedding (Equiv.cast (_ : α = α))) s) s",
"state_before": "α✝ : Type ?u.17749\nβ✝ : Type ?u.17752\nγ : Type ?u.17755\nf : α✝ ↪ β✝\ns✝ : Finset α✝\nα β : Type u_1\nh : α = β\ns : Finset α\n⊢ HEq (map (Equiv.toEmbedding (Equiv.cast h)) s) s",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.17749\nβ : Type ?u.17752\nγ : Type ?u.17755\nf : α✝ ↪ β\ns✝ : Finset α✝\nα : Type u_1\ns : Finset α\n⊢ HEq (map (Equiv.toEmbedding (Equiv.cast (_ : α = α))) s) s",
"tactic": "simp"
}
] |
[
134,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.mem_filter_of_mem
|
[] |
[
1991,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1990,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.inter_diff_distrib_left
|
[] |
[
1613,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1612,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean
|
Nat.descFactorial_eq_zero_iff_lt
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ descFactorial n 0 = 0 ↔ n < 0",
"tactic": "simp only [descFactorial_zero, Nat.one_ne_zero, Nat.not_lt_zero]"
},
{
"state_after": "n k : ℕ\n⊢ n ≤ k → n ≠ k → n ≤ k",
"state_before": "n k : ℕ\n⊢ descFactorial n (succ k) = 0 ↔ n < succ k",
"tactic": "rw [descFactorial_succ, mul_eq_zero, descFactorial_eq_zero_iff_lt, lt_succ_iff,\n tsub_eq_zero_iff_le, lt_iff_le_and_ne, or_iff_left_iff_imp, and_imp]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\n⊢ n ≤ k → n ≠ k → n ≤ k",
"tactic": "exact fun h _ => h"
}
] |
[
402,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
397,
1
] |
Mathlib/Data/List/Indexes.lean
|
List.foldlIdxSpec_cons
|
[] |
[
262,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Std/Data/String/Lemmas.lean
|
String.revFind_of_valid
|
[
{
"state_after": "no goals",
"state_before": "p : Char → Bool\ns : String\n⊢ revFind s p =\n Option.map (fun x => { byteIdx := utf8Len x }) (List.tail? (List.dropWhile (fun x => !p x) (List.reverse s.data)))",
"tactic": "simpa using revFindAux_of_valid p s.1.reverse []"
}
] |
[
353,
51
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
351,
1
] |
Mathlib/Topology/UnitInterval.lean
|
unitInterval.one_minus_le_one
|
[
{
"state_after": "no goals",
"state_before": "x : ↑I\n⊢ 1 - ↑x ≤ 1",
"tactic": "simpa using x.2.1"
}
] |
[
152,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Data/Set/Countable.lean
|
Set.Countable.image
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ns : Set α\nhs : Set.Countable s\nf : α → β\n⊢ Set.Countable (range fun x => f ↑x)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ns : Set α\nhs : Set.Countable s\nf : α → β\n⊢ Set.Countable (f '' s)",
"tactic": "rw [image_eq_range]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ns : Set α\nhs : Set.Countable s\nf : α → β\nthis : Countable ↑s\n⊢ Set.Countable (range fun x => f ↑x)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ns : Set α\nhs : Set.Countable s\nf : α → β\n⊢ Set.Countable (range fun x => f ↑x)",
"tactic": "haveI := hs.to_subtype"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ns : Set α\nhs : Set.Countable s\nf : α → β\nthis : Countable ↑s\n⊢ Set.Countable (range fun x => f ↑x)",
"tactic": "apply countable_range"
}
] |
[
137,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/LinearAlgebra/Dimension.lean
|
Basis.finite_index_of_rank_lt_aleph0
|
[] |
[
856,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
854,
1
] |
Std/Data/Array/Lemmas.lean
|
Array.get_set_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.14725\na : Array α\ni : Fin (size a)\nv : α\n⊢ i.val < size (set a i v)",
"tactic": "simp [i.2]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\na : Array α\ni : Fin (size a)\nv : α\n⊢ (set a i v)[i.val] = v",
"tactic": "simp only [set, getElem_eq_data_get, List.get_set_eq]"
}
] |
[
89,
56
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
87,
9
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.arccos_inj
|
[] |
[
379,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/Init/Data/Bool/Lemmas.lean
|
Bool.coe_true
|
[
{
"state_after": "no goals",
"state_before": "⊢ (true = true) = True",
"tactic": "simp"
}
] |
[
115,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Combinatorics/SimpleGraph/Prod.lean
|
SimpleGraph.Preconnected.ofBoxProdLeft
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.82005\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty β\nh : Preconnected (G □ H)\n⊢ Preconnected G",
"tactic": "classical\nrintro a₁ a₂\nobtain ⟨w⟩ := h (a₁, Classical.arbitrary _) (a₂, Classical.arbitrary _)\nexact ⟨w.ofBoxProdLeft⟩"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.82005\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty β\nh : Preconnected (G □ H)\na₁ a₂ : α\n⊢ Reachable G a₁ a₂",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.82005\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty β\nh : Preconnected (G □ H)\n⊢ Preconnected G",
"tactic": "rintro a₁ a₂"
},
{
"state_after": "case intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.82005\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty β\nh : Preconnected (G □ H)\na₁ a₂ : α\nw : Walk (G □ H) (a₁, Classical.arbitrary β) (a₂, Classical.arbitrary β)\n⊢ Reachable G a₁ a₂",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.82005\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty β\nh : Preconnected (G □ H)\na₁ a₂ : α\n⊢ Reachable G a₁ a₂",
"tactic": "obtain ⟨w⟩ := h (a₁, Classical.arbitrary _) (a₂, Classical.arbitrary _)"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.82005\nG : SimpleGraph α\nH : SimpleGraph β\ninst✝ : Nonempty β\nh : Preconnected (G □ H)\na₁ a₂ : α\nw : Walk (G □ H) (a₁, Classical.arbitrary β) (a₂, Classical.arbitrary β)\n⊢ Reachable G a₁ a₂",
"tactic": "exact ⟨w.ofBoxProdLeft⟩"
}
] |
[
186,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
11
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieSubmodule.mem_coeSubmodule
|
[] |
[
106,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Analysis/Convex/Join.lean
|
convexJoin_empty_right
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.18687\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns✝ t s₁ s₂ t₁ t₂ u : Set E\nx y : E\ns : Set E\n⊢ convexJoin 𝕜 s ∅ = ∅",
"tactic": "simp [convexJoin]"
}
] |
[
66,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Data/List/Permutation.lean
|
List.permutationsAux2_fst
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nt : α\nts : List α\nr : List β\ny : α\nys : List α\nf : List α → β\n⊢ (permutationsAux2 t ts r (y :: ys) f).fst = y :: ys ++ ts",
"tactic": "simp [permutationsAux2, permutationsAux2_fst t _ _ ys]"
}
] |
[
60,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean
|
Geometry.SimplicialComplex.faces_bot
|
[] |
[
255,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
255,
1
] |
Mathlib/Algebra/Polynomial/BigOperators.lean
|
Polynomial.natDegree_list_prod_le
|
[
{
"state_after": "case nil\nR : Type u\nι : Type w\ns : Finset ι\nS : Type u_1\ninst✝ : Semiring S\n⊢ natDegree (List.prod []) ≤ List.sum (List.map natDegree [])\n\ncase cons\nR : Type u\nι : Type w\ns : Finset ι\nS : Type u_1\ninst✝ : Semiring S\nhd : S[X]\ntl : List S[X]\nIH : natDegree (List.prod tl) ≤ List.sum (List.map natDegree tl)\n⊢ natDegree (List.prod (hd :: tl)) ≤ List.sum (List.map natDegree (hd :: tl))",
"state_before": "R : Type u\nι : Type w\ns : Finset ι\nS : Type u_1\ninst✝ : Semiring S\nl : List S[X]\n⊢ natDegree (List.prod l) ≤ List.sum (List.map natDegree l)",
"tactic": "induction' l with hd tl IH"
},
{
"state_after": "no goals",
"state_before": "case nil\nR : Type u\nι : Type w\ns : Finset ι\nS : Type u_1\ninst✝ : Semiring S\n⊢ natDegree (List.prod []) ≤ List.sum (List.map natDegree [])",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nR : Type u\nι : Type w\ns : Finset ι\nS : Type u_1\ninst✝ : Semiring S\nhd : S[X]\ntl : List S[X]\nIH : natDegree (List.prod tl) ≤ List.sum (List.map natDegree tl)\n⊢ natDegree (List.prod (hd :: tl)) ≤ List.sum (List.map natDegree (hd :: tl))",
"tactic": "simpa using natDegree_mul_le.trans (add_le_add_left IH _)"
}
] |
[
82,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.image₂_right
|
[
{
"state_after": "α : Type u_2\nα' : Type ?u.69695\nβ : Type u_1\nβ' : Type ?u.69701\nγ : Type ?u.69704\nγ' : Type ?u.69707\nδ : Type ?u.69710\nδ' : Type ?u.69713\nε : Type ?u.69716\nε' : Type ?u.69719\nζ : Type ?u.69722\nζ' : Type ?u.69725\nν : Type ?u.69728\ninst✝⁸ : DecidableEq α'\ninst✝⁷ : DecidableEq β'\ninst✝⁶ : DecidableEq γ\ninst✝⁵ : DecidableEq γ'\ninst✝⁴ : DecidableEq δ\ninst✝³ : DecidableEq δ'\ninst✝² : DecidableEq ε\ninst✝¹ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝ : DecidableEq β\nh : Finset.Nonempty s\n⊢ image2 (fun x y => y) ↑s ↑t = ↑t",
"state_before": "α : Type u_2\nα' : Type ?u.69695\nβ : Type u_1\nβ' : Type ?u.69701\nγ : Type ?u.69704\nγ' : Type ?u.69707\nδ : Type ?u.69710\nδ' : Type ?u.69713\nε : Type ?u.69716\nε' : Type ?u.69719\nζ : Type ?u.69722\nζ' : Type ?u.69725\nν : Type ?u.69728\ninst✝⁸ : DecidableEq α'\ninst✝⁷ : DecidableEq β'\ninst✝⁶ : DecidableEq γ\ninst✝⁵ : DecidableEq γ'\ninst✝⁴ : DecidableEq δ\ninst✝³ : DecidableEq δ'\ninst✝² : DecidableEq ε\ninst✝¹ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝ : DecidableEq β\nh : Finset.Nonempty s\n⊢ ↑(image₂ (fun x y => y) s t) = ↑t",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nα' : Type ?u.69695\nβ : Type u_1\nβ' : Type ?u.69701\nγ : Type ?u.69704\nγ' : Type ?u.69707\nδ : Type ?u.69710\nδ' : Type ?u.69713\nε : Type ?u.69716\nε' : Type ?u.69719\nζ : Type ?u.69722\nζ' : Type ?u.69725\nν : Type ?u.69728\ninst✝⁸ : DecidableEq α'\ninst✝⁷ : DecidableEq β'\ninst✝⁶ : DecidableEq γ\ninst✝⁵ : DecidableEq γ'\ninst✝⁴ : DecidableEq δ\ninst✝³ : DecidableEq δ'\ninst✝² : DecidableEq ε\ninst✝¹ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝ : DecidableEq β\nh : Finset.Nonempty s\n⊢ image2 (fun x y => y) ↑s ↑t = ↑t",
"tactic": "exact image2_right h"
}
] |
[
368,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/Algebra/GradedMonoid.lean
|
SetLike.coe_gMul
|
[] |
[
544,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
541,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isBigO_of_le
|
[] |
[
558,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
557,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean
|
hasStrictFDerivAt_euclidean
|
[
{
"state_after": "𝕜 : Type u_1\nι : Type u_3\nH : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup H\ninst✝¹ : NormedSpace 𝕜 H\ninst✝ : Fintype ι\nf : H → EuclideanSpace 𝕜 ι\nf' : H →L[𝕜] EuclideanSpace 𝕜 ι\nt : Set H\ny : H\n⊢ (∀ (i : ι),\n HasStrictFDerivAt (fun x => (↑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i)\n (comp (proj i) (comp (↑(EuclideanSpace.equiv ι 𝕜)) f')) y) ↔\n ∀ (i : ι), HasStrictFDerivAt (fun x => f x i) (comp (EuclideanSpace.proj i) f') y",
"state_before": "𝕜 : Type u_1\nι : Type u_3\nH : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup H\ninst✝¹ : NormedSpace 𝕜 H\ninst✝ : Fintype ι\nf : H → EuclideanSpace 𝕜 ι\nf' : H →L[𝕜] EuclideanSpace 𝕜 ι\nt : Set H\ny : H\n⊢ HasStrictFDerivAt f f' y ↔ ∀ (i : ι), HasStrictFDerivAt (fun x => f x i) (comp (EuclideanSpace.proj i) f') y",
"tactic": "rw [← (EuclideanSpace.equiv ι 𝕜).comp_hasStrictFDerivAt_iff, hasStrictFDerivAt_pi']"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nι : Type u_3\nH : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup H\ninst✝¹ : NormedSpace 𝕜 H\ninst✝ : Fintype ι\nf : H → EuclideanSpace 𝕜 ι\nf' : H →L[𝕜] EuclideanSpace 𝕜 ι\nt : Set H\ny : H\n⊢ (∀ (i : ι),\n HasStrictFDerivAt (fun x => (↑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i)\n (comp (proj i) (comp (↑(EuclideanSpace.equiv ι 𝕜)) f')) y) ↔\n ∀ (i : ι), HasStrictFDerivAt (fun x => f x i) (comp (EuclideanSpace.proj i) f') y",
"tactic": "rfl"
}
] |
[
326,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
norm_toNNReal'
|
[] |
[
900,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
899,
1
] |
Mathlib/Tactic/NormNum/Basic.lean
|
Mathlib.Meta.NormNum.isInt_eq_true
|
[] |
[
692,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
691,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
Finset.prod_lt_prod_of_nonempty'
|
[
{
"state_after": "case Hle\nι : Type u_1\nα : Type ?u.128549\nβ : Type ?u.128552\nM : Type u_2\nN : Type ?u.128558\nG : Type ?u.128561\nk : Type ?u.128564\nR : Type ?u.128567\ninst✝ : OrderedCancelCommMonoid M\nf g : ι → M\ns t : Finset ι\nhs : Finset.Nonempty s\nHlt : ∀ (i : ι), i ∈ s → f i < g i\n⊢ ∀ (i : ι), i ∈ s → f i ≤ g i\n\ncase Hlt\nι : Type u_1\nα : Type ?u.128549\nβ : Type ?u.128552\nM : Type u_2\nN : Type ?u.128558\nG : Type ?u.128561\nk : Type ?u.128564\nR : Type ?u.128567\ninst✝ : OrderedCancelCommMonoid M\nf g : ι → M\ns t : Finset ι\nhs : Finset.Nonempty s\nHlt : ∀ (i : ι), i ∈ s → f i < g i\n⊢ ∃ i, i ∈ s ∧ f i < g i",
"state_before": "ι : Type u_1\nα : Type ?u.128549\nβ : Type ?u.128552\nM : Type u_2\nN : Type ?u.128558\nG : Type ?u.128561\nk : Type ?u.128564\nR : Type ?u.128567\ninst✝ : OrderedCancelCommMonoid M\nf g : ι → M\ns t : Finset ι\nhs : Finset.Nonempty s\nHlt : ∀ (i : ι), i ∈ s → f i < g i\n⊢ ∏ i in s, f i < ∏ i in s, g i",
"tactic": "apply prod_lt_prod'"
},
{
"state_after": "case Hlt.intro\nι : Type u_1\nα : Type ?u.128549\nβ : Type ?u.128552\nM : Type u_2\nN : Type ?u.128558\nG : Type ?u.128561\nk : Type ?u.128564\nR : Type ?u.128567\ninst✝ : OrderedCancelCommMonoid M\nf g : ι → M\ns t : Finset ι\nHlt : ∀ (i : ι), i ∈ s → f i < g i\ni : ι\nhi : i ∈ s\n⊢ ∃ i, i ∈ s ∧ f i < g i",
"state_before": "case Hlt\nι : Type u_1\nα : Type ?u.128549\nβ : Type ?u.128552\nM : Type u_2\nN : Type ?u.128558\nG : Type ?u.128561\nk : Type ?u.128564\nR : Type ?u.128567\ninst✝ : OrderedCancelCommMonoid M\nf g : ι → M\ns t : Finset ι\nhs : Finset.Nonempty s\nHlt : ∀ (i : ι), i ∈ s → f i < g i\n⊢ ∃ i, i ∈ s ∧ f i < g i",
"tactic": "cases' hs with i hi"
},
{
"state_after": "no goals",
"state_before": "case Hlt.intro\nι : Type u_1\nα : Type ?u.128549\nβ : Type ?u.128552\nM : Type u_2\nN : Type ?u.128558\nG : Type ?u.128561\nk : Type ?u.128564\nR : Type ?u.128567\ninst✝ : OrderedCancelCommMonoid M\nf g : ι → M\ns t : Finset ι\nHlt : ∀ (i : ι), i ∈ s → f i < g i\ni : ι\nhi : i ∈ s\n⊢ ∃ i, i ∈ s ∧ f i < g i",
"tactic": "exact ⟨i, hi, Hlt i hi⟩"
},
{
"state_after": "case Hle\nι : Type u_1\nα : Type ?u.128549\nβ : Type ?u.128552\nM : Type u_2\nN : Type ?u.128558\nG : Type ?u.128561\nk : Type ?u.128564\nR : Type ?u.128567\ninst✝ : OrderedCancelCommMonoid M\nf g : ι → M\ns t : Finset ι\nhs : Finset.Nonempty s\nHlt : ∀ (i : ι), i ∈ s → f i < g i\ni : ι\nhi : i ∈ s\n⊢ f i ≤ g i",
"state_before": "case Hle\nι : Type u_1\nα : Type ?u.128549\nβ : Type ?u.128552\nM : Type u_2\nN : Type ?u.128558\nG : Type ?u.128561\nk : Type ?u.128564\nR : Type ?u.128567\ninst✝ : OrderedCancelCommMonoid M\nf g : ι → M\ns t : Finset ι\nhs : Finset.Nonempty s\nHlt : ∀ (i : ι), i ∈ s → f i < g i\n⊢ ∀ (i : ι), i ∈ s → f i ≤ g i",
"tactic": "intro i hi"
},
{
"state_after": "no goals",
"state_before": "case Hle\nι : Type u_1\nα : Type ?u.128549\nβ : Type ?u.128552\nM : Type u_2\nN : Type ?u.128558\nG : Type ?u.128561\nk : Type ?u.128564\nR : Type ?u.128567\ninst✝ : OrderedCancelCommMonoid M\nf g : ι → M\ns t : Finset ι\nhs : Finset.Nonempty s\nHlt : ∀ (i : ι), i ∈ s → f i < g i\ni : ι\nhi : i ∈ s\n⊢ f i ≤ g i",
"tactic": "apply le_of_lt (Hlt i hi)"
}
] |
[
456,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
450,
1
] |
Mathlib/Data/Polynomial/Degree/Lemmas.lean
|
Polynomial.coeff_pow_of_natDegree_le
|
[
{
"state_after": "case zero\nR : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q r : R[X]\npn : natDegree p ≤ n\n⊢ coeff (p ^ Nat.zero) (n * Nat.zero) = coeff p n ^ Nat.zero\n\ncase succ\nR : Type u\nS : Type v\nι : Type w\na b : R\nm✝ n : ℕ\ninst✝ : Semiring R\np q r : R[X]\npn : natDegree p ≤ n\nm : ℕ\nhm : coeff (p ^ m) (n * m) = coeff p n ^ m\n⊢ coeff (p ^ Nat.succ m) (n * Nat.succ m) = coeff p n ^ Nat.succ m",
"state_before": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q r : R[X]\npn : natDegree p ≤ n\n⊢ coeff (p ^ m) (n * m) = coeff p n ^ m",
"tactic": "induction' m with m hm"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q r : R[X]\npn : natDegree p ≤ n\n⊢ coeff (p ^ Nat.zero) (n * Nat.zero) = coeff p n ^ Nat.zero",
"tactic": "simp"
},
{
"state_after": "R : Type u\nS : Type v\nι : Type w\na b : R\nm✝ n : ℕ\ninst✝ : Semiring R\np q r : R[X]\npn : natDegree p ≤ n\nm : ℕ\nhm : coeff (p ^ m) (n * m) = coeff p n ^ m\n⊢ natDegree (p ^ m) ≤ n * m",
"state_before": "case succ\nR : Type u\nS : Type v\nι : Type w\na b : R\nm✝ n : ℕ\ninst✝ : Semiring R\np q r : R[X]\npn : natDegree p ≤ n\nm : ℕ\nhm : coeff (p ^ m) (n * m) = coeff p n ^ m\n⊢ coeff (p ^ Nat.succ m) (n * Nat.succ m) = coeff p n ^ Nat.succ m",
"tactic": "rw [pow_succ', pow_succ', ← hm, Nat.mul_succ, coeff_mul_of_natDegree_le _ pn]"
},
{
"state_after": "R : Type u\nS : Type v\nι : Type w\na b : R\nm✝ n : ℕ\ninst✝ : Semiring R\np q r : R[X]\npn : natDegree p ≤ n\nm : ℕ\nhm : coeff (p ^ m) (n * m) = coeff p n ^ m\n⊢ m * natDegree p ≤ m * n",
"state_before": "R : Type u\nS : Type v\nι : Type w\na b : R\nm✝ n : ℕ\ninst✝ : Semiring R\np q r : R[X]\npn : natDegree p ≤ n\nm : ℕ\nhm : coeff (p ^ m) (n * m) = coeff p n ^ m\n⊢ natDegree (p ^ m) ≤ n * m",
"tactic": "refine' natDegree_pow_le.trans (le_trans _ (mul_comm _ _).le)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nι : Type w\na b : R\nm✝ n : ℕ\ninst✝ : Semiring R\np q r : R[X]\npn : natDegree p ≤ n\nm : ℕ\nhm : coeff (p ^ m) (n * m) = coeff p n ^ m\n⊢ m * natDegree p ≤ m * n",
"tactic": "exact mul_le_mul_of_nonneg_left pn m.zero_le"
}
] |
[
184,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearEquiv.ext
|
[] |
[
1815,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1814,
1
] |
Mathlib/Analysis/Normed/Group/Quotient.lean
|
NormedAddGroupHom.isQuotientQuotient
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type ?u.464490\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nm : M\n⊢ ‖↑(AddSubgroup.normedMk S) m‖ = sInf ((fun m_1 => ‖m + m_1‖) '' ↑(ker (AddSubgroup.normedMk S)))",
"tactic": "simpa [S.ker_normedMk] using quotient_norm_mk_eq _ m"
}
] |
[
378,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.ae_eq_trim_iff
|
[] |
[
1807,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1804,
1
] |
Mathlib/CategoryTheory/Sites/Canonical.lean
|
CategoryTheory.Sheaf.isSheaf_of_representable
|
[] |
[
228,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
Measurable.isGLB
|
[
{
"state_after": "α : Type u_3\nβ : Type ?u.1210770\nγ : Type ?u.1210773\nγ₂ : Type ?u.1210776\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)\n⊢ Measurable g",
"state_before": "α : Type u_3\nβ : Type ?u.1210770\nγ : Type ?u.1210773\nγ₂ : Type ?u.1210776\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsGLB {a | ∃ i, f i b = a} (g b)\n⊢ Measurable g",
"tactic": "change ∀ b, IsGLB (range fun i => f i b) (g b) at hg"
},
{
"state_after": "α : Type u_3\nβ : Type ?u.1210770\nγ : Type ?u.1210773\nγ₂ : Type ?u.1210776\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)\n⊢ Measurable g",
"state_before": "α : Type u_3\nβ : Type ?u.1210770\nγ : Type ?u.1210773\nγ₂ : Type ?u.1210776\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)\n⊢ Measurable g",
"tactic": "rw [‹BorelSpace α›.measurable_eq, borel_eq_generateFrom_Iio α]"
},
{
"state_after": "case h\nα : Type u_3\nβ : Type ?u.1210770\nγ : Type ?u.1210773\nγ₂ : Type ?u.1210776\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)\n⊢ ∀ (t : Set α), t ∈ range Iio → MeasurableSet (g ⁻¹' t)",
"state_before": "α : Type u_3\nβ : Type ?u.1210770\nγ : Type ?u.1210773\nγ₂ : Type ?u.1210776\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)\n⊢ Measurable g",
"tactic": "apply measurable_generateFrom"
},
{
"state_after": "case h.intro\nα : Type u_3\nβ : Type ?u.1210770\nγ : Type ?u.1210773\nγ₂ : Type ?u.1210776\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)\na : α\n⊢ MeasurableSet (g ⁻¹' Iio a)",
"state_before": "case h\nα : Type u_3\nβ : Type ?u.1210770\nγ : Type ?u.1210773\nγ₂ : Type ?u.1210776\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)\n⊢ ∀ (t : Set α), t ∈ range Iio → MeasurableSet (g ⁻¹' t)",
"tactic": "rintro _ ⟨a, rfl⟩"
},
{
"state_after": "case h.intro\nα : Type u_3\nβ : Type ?u.1210770\nγ : Type ?u.1210773\nγ₂ : Type ?u.1210776\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)\na : α\n⊢ MeasurableSet (⋃ (i : ι), {x | f i x < a})",
"state_before": "case h.intro\nα : Type u_3\nβ : Type ?u.1210770\nγ : Type ?u.1210773\nγ₂ : Type ?u.1210776\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)\na : α\n⊢ MeasurableSet (g ⁻¹' Iio a)",
"tactic": "simp_rw [Set.preimage, mem_Iio, isGLB_lt_iff (hg _), exists_range_iff, setOf_exists]"
},
{
"state_after": "no goals",
"state_before": "case h.intro\nα : Type u_3\nβ : Type ?u.1210770\nγ : Type ?u.1210773\nγ₂ : Type ?u.1210776\nδ : Type u_2\nι✝ : Sort y\ns t u : Set α\ninst✝¹³ : TopologicalSpace α\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : BorelSpace α\ninst✝¹⁰ : TopologicalSpace β\ninst✝⁹ : MeasurableSpace β\ninst✝⁸ : BorelSpace β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : MeasurableSpace γ\ninst✝⁵ : BorelSpace γ\ninst✝⁴ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_1\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)\na : α\n⊢ MeasurableSet (⋃ (i : ι), {x | f i x < a})",
"tactic": "exact MeasurableSet.iUnion fun i => hf i (isOpen_gt' _).measurableSet"
}
] |
[
1138,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1131,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.choose_mem
|
[] |
[
3060,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3059,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.subtypeDomain_add
|
[] |
[
510,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
508,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
OrderTop.upperBounds_univ
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝³ : Preorder α\ninst✝² : Preorder β\ns t : Set α\na b : α\ninst✝¹ : PartialOrder γ\ninst✝ : OrderTop γ\n⊢ upperBounds univ = {⊤}",
"tactic": "rw [isGreatest_univ.upperBounds_eq, Ici_top]"
}
] |
[
812,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
811,
1
] |
Mathlib/GroupTheory/Subgroup/ZPowers.lean
|
Subgroup.zpowers_one_eq_bot
|
[] |
[
225,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/Data/Dfinsupp/Interval.lean
|
Dfinsupp.card_Ioc
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : ι → Type u_2\ninst✝⁴ : DecidableEq ι\ninst✝³ : (i : ι) → DecidableEq (α i)\ninst✝² : (i : ι) → PartialOrder (α i)\ninst✝¹ : (i : ι) → Zero (α i)\ninst✝ : (i : ι) → LocallyFiniteOrder (α i)\nf g : Π₀ (i : ι), α i\n⊢ card (Ioc f g) = ∏ i in support f ∪ support g, card (Icc (↑f i) (↑g i)) - 1",
"tactic": "rw [card_Ioc_eq_card_Icc_sub_one, card_Icc]"
}
] |
[
190,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.NeBot.of_smul_filter
|
[] |
[
1221,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1220,
1
] |
Mathlib/Topology/FiberBundle/Constructions.lean
|
Bundle.Trivial.trivialization_source
|
[] |
[
88,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.set_integral_restrict
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_2\nι : Type ?u.1446410\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\ns t✝ : Set β\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : β → E\na : α\nhs : MeasurableSet s\nt : Set β\n⊢ (∫ (x : β) in t, f x ∂↑(kernel.restrict κ hs) a) = ∫ (x : β) in t ∩ s, f x ∂↑κ a",
"tactic": "rw [restrict_apply, Measure.restrict_restrict' hs]"
}
] |
[
524,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
521,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.lf_of_lf_of_equiv
|
[] |
[
831,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
830,
1
] |
Mathlib/Data/Nat/Factors.lean
|
Nat.mem_factors_mul_left
|
[
{
"state_after": "case inl\np b : ℕ\nhb : b ≠ 0\nhpa : p ∈ factors 0\n⊢ p ∈ factors (0 * b)\n\ncase inr\np a b : ℕ\nhpa : p ∈ factors a\nhb : b ≠ 0\nha : a ≠ 0\n⊢ p ∈ factors (a * b)",
"state_before": "p a b : ℕ\nhpa : p ∈ factors a\nhb : b ≠ 0\n⊢ p ∈ factors (a * b)",
"tactic": "rcases eq_or_ne a 0 with (rfl | ha)"
},
{
"state_after": "no goals",
"state_before": "case inr\np a b : ℕ\nhpa : p ∈ factors a\nhb : b ≠ 0\nha : a ≠ 0\n⊢ p ∈ factors (a * b)",
"tactic": "apply (mem_factors_mul ha hb).2 (Or.inl hpa)"
},
{
"state_after": "no goals",
"state_before": "case inl\np b : ℕ\nhb : b ≠ 0\nhpa : p ∈ factors 0\n⊢ p ∈ factors (0 * b)",
"tactic": "simp at hpa"
}
] |
[
289,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.eventually_lift'_iff
|
[] |
[
270,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/Analysis/Complex/PhragmenLindelof.lean
|
PhragmenLindelof.eq_zero_on_quadrant_I
|
[] |
[
438,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
431,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.pow_subset_pow
|
[
{
"state_after": "F : Type ?u.52486\nα : Type u_1\nβ : Type ?u.52492\nγ : Type ?u.52495\ninst✝ : Monoid α\ns t : Set α\na : α\nm n : ℕ\nhst : s ⊆ t\n⊢ 1 ⊆ t ^ 0",
"state_before": "F : Type ?u.52486\nα : Type u_1\nβ : Type ?u.52492\nγ : Type ?u.52495\ninst✝ : Monoid α\ns t : Set α\na : α\nm n : ℕ\nhst : s ⊆ t\n⊢ s ^ 0 ⊆ t ^ 0",
"tactic": "rw [pow_zero]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.52486\nα : Type u_1\nβ : Type ?u.52492\nγ : Type ?u.52495\ninst✝ : Monoid α\ns t : Set α\na : α\nm n : ℕ\nhst : s ⊆ t\n⊢ 1 ⊆ t ^ 0",
"tactic": "exact Subset.rfl"
},
{
"state_after": "F : Type ?u.52486\nα : Type u_1\nβ : Type ?u.52492\nγ : Type ?u.52495\ninst✝ : Monoid α\ns t : Set α\na : α\nm n✝ : ℕ\nhst : s ⊆ t\nn : ℕ\n⊢ s * s ^ n ⊆ t ^ (n + 1)",
"state_before": "F : Type ?u.52486\nα : Type u_1\nβ : Type ?u.52492\nγ : Type ?u.52495\ninst✝ : Monoid α\ns t : Set α\na : α\nm n✝ : ℕ\nhst : s ⊆ t\nn : ℕ\n⊢ s ^ (n + 1) ⊆ t ^ (n + 1)",
"tactic": "rw [pow_succ]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.52486\nα : Type u_1\nβ : Type ?u.52492\nγ : Type ?u.52495\ninst✝ : Monoid α\ns t : Set α\na : α\nm n✝ : ℕ\nhst : s ⊆ t\nn : ℕ\n⊢ s * s ^ n ⊆ t ^ (n + 1)",
"tactic": "exact mul_subset_mul hst (pow_subset_pow hst _)"
}
] |
[
960,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
954,
1
] |
Mathlib/Analysis/LocallyConvex/AbsConvex.lean
|
AbsConvexOpenSets.coe_balanced
|
[] |
[
119,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.hasPullback_assoc_symm
|
[] |
[
2381,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2380,
1
] |
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean
|
Polynomial.nextCoeffUp_C_eq_zero
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\nc : R\n⊢ (if natTrailingDegree (↑C c) = 0 then 0 else coeff (↑C c) (natTrailingDegree (↑C c) + 1)) = 0",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\nc : R\n⊢ nextCoeffUp (↑C c) = 0",
"tactic": "rw [nextCoeffUp]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\nc : R\n⊢ (if natTrailingDegree (↑C c) = 0 then 0 else coeff (↑C c) (natTrailingDegree (↑C c) + 1)) = 0",
"tactic": "simp"
}
] |
[
490,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
488,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.disjoint_iff_right
|
[] |
[
735,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
733,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.isLocalization_of_base_ringEquiv
|
[
{
"state_after": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n⊢ IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S",
"state_before": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\n⊢ IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S",
"tactic": "letI : Algebra P S := ((algebraMap R S).comp h.symm.toRingHom).toAlgebra"
},
{
"state_after": "case map_units'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n⊢ ∀ (y : { x // x ∈ Submonoid.map (RingEquiv.toMonoidHom h) M }), IsUnit (↑(algebraMap P S) ↑y)\n\ncase surj'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n⊢ ∀ (z : S), ∃ x, z * ↑(algebraMap P S) ↑x.snd = ↑(algebraMap P S) x.fst\n\ncase eq_iff_exists'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n⊢ ∀ {x y : P}, ↑(algebraMap P S) x = ↑(algebraMap P S) y ↔ ∃ c, ↑c * x = ↑c * y",
"state_before": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n⊢ IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S",
"tactic": "constructor"
},
{
"state_after": "case map_units'.mk.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : R\nhy : y ∈ ↑M\n⊢ IsUnit\n (↑(algebraMap P S)\n ↑{ val := ↑(RingEquiv.toMonoidHom h) y,\n property := (_ : ∃ a, a ∈ ↑M ∧ ↑(RingEquiv.toMonoidHom h) a = ↑(RingEquiv.toMonoidHom h) y) })",
"state_before": "case map_units'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n⊢ ∀ (y : { x // x ∈ Submonoid.map (RingEquiv.toMonoidHom h) M }), IsUnit (↑(algebraMap P S) ↑y)",
"tactic": "rintro ⟨_, ⟨y, hy, rfl⟩⟩"
},
{
"state_after": "case h.e'_3\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : R\nhy : y ∈ ↑M\n⊢ ↑(algebraMap P S)\n ↑{ val := ↑(RingEquiv.toMonoidHom h) y,\n property := (_ : ∃ a, a ∈ ↑M ∧ ↑(RingEquiv.toMonoidHom h) a = ↑(RingEquiv.toMonoidHom h) y) } =\n ↑(algebraMap R S) ↑{ val := y, property := hy }",
"state_before": "case map_units'.mk.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : R\nhy : y ∈ ↑M\n⊢ IsUnit\n (↑(algebraMap P S)\n ↑{ val := ↑(RingEquiv.toMonoidHom h) y,\n property := (_ : ∃ a, a ∈ ↑M ∧ ↑(RingEquiv.toMonoidHom h) a = ↑(RingEquiv.toMonoidHom h) y) })",
"tactic": "convert IsLocalization.map_units S ⟨y, hy⟩"
},
{
"state_after": "case h.e'_3\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : R\nhy : y ∈ ↑M\n⊢ ↑(algebraMap R S) (↑(RingEquiv.toRingHom (RingEquiv.symm h)) (↑(RingEquiv.toMonoidHom h) y)) = ↑(algebraMap R S) y",
"state_before": "case h.e'_3\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : R\nhy : y ∈ ↑M\n⊢ ↑(algebraMap P S)\n ↑{ val := ↑(RingEquiv.toMonoidHom h) y,\n property := (_ : ∃ a, a ∈ ↑M ∧ ↑(RingEquiv.toMonoidHom h) a = ↑(RingEquiv.toMonoidHom h) y) } =\n ↑(algebraMap R S) ↑{ val := y, property := hy }",
"tactic": "dsimp only [RingHom.algebraMap_toAlgebra, RingHom.comp_apply]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : R\nhy : y ∈ ↑M\n⊢ ↑(algebraMap R S) (↑(RingEquiv.toRingHom (RingEquiv.symm h)) (↑(RingEquiv.toMonoidHom h) y)) = ↑(algebraMap R S) y",
"tactic": "exact congr_arg _ (h.symm_apply_apply _)"
},
{
"state_after": "case surj'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\n⊢ ∃ x, y * ↑(algebraMap P S) ↑x.snd = ↑(algebraMap P S) x.fst",
"state_before": "case surj'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n⊢ ∀ (z : S), ∃ x, z * ↑(algebraMap P S) ↑x.snd = ↑(algebraMap P S) x.fst",
"tactic": "intro y"
},
{
"state_after": "case surj'.intro.mk\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x ∈ M }\ne : y * ↑(algebraMap R S) ↑(x, s).snd = ↑(algebraMap R S) (x, s).fst\n⊢ ∃ x, y * ↑(algebraMap P S) ↑x.snd = ↑(algebraMap P S) x.fst",
"state_before": "case surj'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\n⊢ ∃ x, y * ↑(algebraMap P S) ↑x.snd = ↑(algebraMap P S) x.fst",
"tactic": "obtain ⟨⟨x, s⟩, e⟩ := IsLocalization.surj M y"
},
{
"state_after": "case surj'.intro.mk\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x ∈ M }\ne : y * ↑(algebraMap R S) ↑(x, s).snd = ↑(algebraMap R S) (x, s).fst\n⊢ y *\n ↑(algebraMap P S)\n ↑(↑h x,\n { val := ↑(RingEquiv.toMonoidHom h) ↑s,\n property := (_ : ∃ a, a ∈ ↑M ∧ ↑(RingEquiv.toMonoidHom h) a = ↑(RingEquiv.toMonoidHom h) ↑s) }).snd =\n ↑(algebraMap P S)\n (↑h x,\n { val := ↑(RingEquiv.toMonoidHom h) ↑s,\n property := (_ : ∃ a, a ∈ ↑M ∧ ↑(RingEquiv.toMonoidHom h) a = ↑(RingEquiv.toMonoidHom h) ↑s) }).fst",
"state_before": "case surj'.intro.mk\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x ∈ M }\ne : y * ↑(algebraMap R S) ↑(x, s).snd = ↑(algebraMap R S) (x, s).fst\n⊢ ∃ x, y * ↑(algebraMap P S) ↑x.snd = ↑(algebraMap P S) x.fst",
"tactic": "refine' ⟨⟨h x, _, _, s.prop, rfl⟩, _⟩"
},
{
"state_after": "case surj'.intro.mk\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x ∈ M }\ne : y * ↑(algebraMap R S) ↑s = ↑(algebraMap R S) x\n⊢ y * ↑(algebraMap R S) (↑(RingEquiv.toRingHom (RingEquiv.symm h)) (↑(RingEquiv.toMonoidHom h) ↑s)) =\n ↑(algebraMap R S) (↑(RingEquiv.toRingHom (RingEquiv.symm h)) (↑h x))",
"state_before": "case surj'.intro.mk\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x ∈ M }\ne : y * ↑(algebraMap R S) ↑(x, s).snd = ↑(algebraMap R S) (x, s).fst\n⊢ y *\n ↑(algebraMap P S)\n ↑(↑h x,\n { val := ↑(RingEquiv.toMonoidHom h) ↑s,\n property := (_ : ∃ a, a ∈ ↑M ∧ ↑(RingEquiv.toMonoidHom h) a = ↑(RingEquiv.toMonoidHom h) ↑s) }).snd =\n ↑(algebraMap P S)\n (↑h x,\n { val := ↑(RingEquiv.toMonoidHom h) ↑s,\n property := (_ : ∃ a, a ∈ ↑M ∧ ↑(RingEquiv.toMonoidHom h) a = ↑(RingEquiv.toMonoidHom h) ↑s) }).fst",
"tactic": "dsimp only [RingHom.algebraMap_toAlgebra, RingHom.comp_apply] at e⊢"
},
{
"state_after": "no goals",
"state_before": "case surj'.intro.mk\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x ∈ M }\ne : y * ↑(algebraMap R S) ↑s = ↑(algebraMap R S) x\n⊢ y * ↑(algebraMap R S) (↑(RingEquiv.toRingHom (RingEquiv.symm h)) (↑(RingEquiv.toMonoidHom h) ↑s)) =\n ↑(algebraMap R S) (↑(RingEquiv.toRingHom (RingEquiv.symm h)) (↑h x))",
"tactic": "convert e <;> exact h.symm_apply_apply _"
},
{
"state_after": "case eq_iff_exists'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n⊢ ↑(algebraMap P S) x = ↑(algebraMap P S) y ↔ ∃ c, ↑c * x = ↑c * y",
"state_before": "case eq_iff_exists'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n⊢ ∀ {x y : P}, ↑(algebraMap P S) x = ↑(algebraMap P S) y ↔ ∃ c, ↑c * x = ↑c * y",
"tactic": "intro x y"
},
{
"state_after": "case eq_iff_exists'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n⊢ (∃ c, ↑c * ↑(RingEquiv.toRingHom (RingEquiv.symm h)) x = ↑c * ↑(RingEquiv.toRingHom (RingEquiv.symm h)) y) ↔\n ∃ c, ↑c * x = ↑c * y",
"state_before": "case eq_iff_exists'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n⊢ ↑(algebraMap P S) x = ↑(algebraMap P S) y ↔ ∃ c, ↑c * x = ↑c * y",
"tactic": "rw [RingHom.algebraMap_toAlgebra, RingHom.comp_apply, RingHom.comp_apply,\n IsLocalization.eq_iff_exists M S]"
},
{
"state_after": "case eq_iff_exists'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n⊢ (∃ c,\n ↑h.toEquiv (↑c * ↑(RingEquiv.toRingHom (RingEquiv.symm h)) x) =\n ↑h.toEquiv (↑c * ↑(RingEquiv.toRingHom (RingEquiv.symm h)) y)) ↔\n ∃ c, ↑c * x = ↑c * y",
"state_before": "case eq_iff_exists'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n⊢ (∃ c, ↑c * ↑(RingEquiv.toRingHom (RingEquiv.symm h)) x = ↑c * ↑(RingEquiv.toRingHom (RingEquiv.symm h)) y) ↔\n ∃ c, ↑c * x = ↑c * y",
"tactic": "simp_rw [← h.toEquiv.apply_eq_iff_eq]"
},
{
"state_after": "case eq_iff_exists'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n⊢ (∃ c, ↑h (↑c * ↑(RingEquiv.symm h) x) = ↑h (↑c * ↑(RingEquiv.symm h) y)) ↔ ∃ c, ↑c * x = ↑c * y",
"state_before": "case eq_iff_exists'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n⊢ (∃ c,\n ↑h.toEquiv (↑c * ↑(RingEquiv.toRingHom (RingEquiv.symm h)) x) =\n ↑h.toEquiv (↑c * ↑(RingEquiv.toRingHom (RingEquiv.symm h)) y)) ↔\n ∃ c, ↑c * x = ↑c * y",
"tactic": "change (∃ c : M, h (c * h.symm x) = h (c * h.symm y)) ↔ _"
},
{
"state_after": "case eq_iff_exists'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n⊢ (∃ c, ↑h ↑c * x = ↑h ↑c * y) ↔ ∃ c, ↑c * x = ↑c * y",
"state_before": "case eq_iff_exists'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n⊢ (∃ c, ↑h (↑c * ↑(RingEquiv.symm h) x) = ↑h (↑c * ↑(RingEquiv.symm h) y)) ↔ ∃ c, ↑c * x = ↑c * y",
"tactic": "simp only [RingEquiv.apply_symm_apply, RingEquiv.map_mul]"
},
{
"state_after": "no goals",
"state_before": "case eq_iff_exists'\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n⊢ (∃ c, ↑h ↑c * x = ↑h ↑c * y) ↔ ∃ c, ↑c * x = ↑c * y",
"tactic": "exact\n ⟨fun ⟨c, e⟩ => ⟨⟨_, _, c.prop, rfl⟩, e⟩, fun ⟨⟨_, c, h, e₁⟩, e₂⟩ => ⟨⟨_, h⟩, e₁.symm ▸ e₂⟩⟩"
}
] |
[
801,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
780,
1
] |
Std/Data/List/Lemmas.lean
|
List.drop_suffix
|
[] |
[
1719,
28
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1718,
1
] |
Mathlib/RingTheory/Ideal/LocalRing.lean
|
LocalRing.maximalIdeal_eq_bot
|
[] |
[
534,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
533,
1
] |
Mathlib/CategoryTheory/Adjunction/Opposites.lean
|
CategoryTheory.Adjunction.leftAdjointUniq_trans_app
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' F'' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nadj3 : F'' ⊣ G\nx : C\n⊢ (leftAdjointUniq adj1 adj2).hom.app x ≫ (leftAdjointUniq adj2 adj3).hom.app x =\n ((leftAdjointUniq adj1 adj2).hom ≫ (leftAdjointUniq adj2 adj3).hom).app x",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' F'' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nadj3 : F'' ⊣ G\nx : C\n⊢ (leftAdjointUniq adj1 adj2).hom.app x ≫ (leftAdjointUniq adj2 adj3).hom.app x = (leftAdjointUniq adj1 adj3).hom.app x",
"tactic": "rw [← leftAdjointUniq_trans adj1 adj2 adj3]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF F' F'' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nadj3 : F'' ⊣ G\nx : C\n⊢ (leftAdjointUniq adj1 adj2).hom.app x ≫ (leftAdjointUniq adj2 adj3).hom.app x =\n ((leftAdjointUniq adj1 adj2).hom ≫ (leftAdjointUniq adj2 adj3).hom).app x",
"tactic": "rfl"
}
] |
[
202,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
LinearMap.toBilinAux_eq
|
[] |
[
504,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.natAdd_natAdd
|
[] |
[
1448,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1446,
1
] |
Mathlib/Algebra/Star/Pointwise.lean
|
Set.star_univ
|
[] |
[
49,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/GroupTheory/Submonoid/Basic.lean
|
Submonoid.dense_induction
|
[
{
"state_after": "M : Type u_1\nN : Type ?u.25831\nA : Type ?u.25834\ninst✝¹ : MulOneClass M\ns✝ : Set M\ninst✝ : AddZeroClass A\nt : Set A\nS : Submonoid M\np : M → Prop\nx : M\ns : Set M\nhs : closure s = ⊤\nHs : ∀ (x : M), x ∈ s → p x\nH1 : p 1\nHmul : ∀ (x y : M), p x → p y → p (x * y)\nthis : ∀ (x : M), x ∈ closure s → p x\n⊢ p x",
"state_before": "M : Type u_1\nN : Type ?u.25831\nA : Type ?u.25834\ninst✝¹ : MulOneClass M\ns✝ : Set M\ninst✝ : AddZeroClass A\nt : Set A\nS : Submonoid M\np : M → Prop\nx : M\ns : Set M\nhs : closure s = ⊤\nHs : ∀ (x : M), x ∈ s → p x\nH1 : p 1\nHmul : ∀ (x y : M), p x → p y → p (x * y)\n⊢ p x",
"tactic": "have : ∀ x ∈ closure s, p x := fun x hx => closure_induction hx Hs H1 Hmul"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type ?u.25831\nA : Type ?u.25834\ninst✝¹ : MulOneClass M\ns✝ : Set M\ninst✝ : AddZeroClass A\nt : Set A\nS : Submonoid M\np : M → Prop\nx : M\ns : Set M\nhs : closure s = ⊤\nHs : ∀ (x : M), x ∈ s → p x\nH1 : p 1\nHmul : ∀ (x y : M), p x → p y → p (x * y)\nthis : ∀ (x : M), x ∈ closure s → p x\n⊢ p x",
"tactic": "simpa [hs] using this x"
}
] |
[
491,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
488,
1
] |
Mathlib/Data/Fin/VecNotation.lean
|
Matrix.cons_eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nm n o : ℕ\nm' : Type ?u.78559\nn' : Type ?u.78562\no' : Type ?u.78565\ninst✝ : Zero α\nv : Fin n → α\nx : α\nh : vecCons x v = 0\n⊢ v = 0",
"tactic": "convert congr_arg vecTail h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nm n o : ℕ\nm' : Type ?u.78559\nn' : Type ?u.78562\no' : Type ?u.78565\ninst✝ : Zero α\nv : Fin n → α\nx : α\nx✝ : x = 0 ∧ v = 0\nhx : x = 0\nhv : v = 0\n⊢ vecCons x v = 0",
"tactic": "simp [hx, hv]"
}
] |
[
559,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
555,
1
] |
Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean
|
TopCat.Sheaf.existsUnique_gluing'
|
[
{
"state_after": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\n⊢ ∃! s, ∀ (i : ι), (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = sf i",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\n⊢ ∃! s, ∀ (i : ι), (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = sf i",
"tactic": "have V_eq_supr_U : V = iSup U := le_antisymm hcover (iSup_le fun i => (iUV i).le)"
},
{
"state_after": "case intro.intro\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\n⊢ ∃! s, ∀ (i : ι), (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = sf i",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\n⊢ ∃! s, ∀ (i : ι), (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = sf i",
"tactic": "obtain ⟨gl, gl_spec, gl_uniq⟩ := F.existsUnique_gluing U sf h"
},
{
"state_after": "case intro.intro.refine'_1\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\n⊢ (fun s => ∀ (i : ι), (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = sf i)\n ((CategoryTheory.forget C).map (F.val.map (eqToHom V_eq_supr_U).op) gl)\n\ncase intro.intro.refine'_2\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\n⊢ ∀ (y : (CategoryTheory.forget C).obj (F.val.obj V.op)),\n (fun s => ∀ (i : ι), (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = sf i) y →\n y = (CategoryTheory.forget C).map (F.val.map (eqToHom V_eq_supr_U).op) gl",
"state_before": "case intro.intro\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\n⊢ ∃! s, ∀ (i : ι), (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = sf i",
"tactic": "refine' ⟨F.1.map (eqToHom V_eq_supr_U).op gl, _, _⟩"
},
{
"state_after": "case intro.intro.refine'_1\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\ni : ι\n⊢ (CategoryTheory.forget C).map (F.val.map (iUV i).op)\n ((CategoryTheory.forget C).map (F.val.map (eqToHom V_eq_supr_U).op) gl) =\n sf i",
"state_before": "case intro.intro.refine'_1\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\n⊢ (fun s => ∀ (i : ι), (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = sf i)\n ((CategoryTheory.forget C).map (F.val.map (eqToHom V_eq_supr_U).op) gl)",
"tactic": "intro i"
},
{
"state_after": "case intro.intro.refine'_1\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\ni : ι\n⊢ (CategoryTheory.forget C).map (F.val.map ((eqToHom V_eq_supr_U).op ≫ (iUV i).op)) gl = sf i",
"state_before": "case intro.intro.refine'_1\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\ni : ι\n⊢ (CategoryTheory.forget C).map (F.val.map (iUV i).op)\n ((CategoryTheory.forget C).map (F.val.map (eqToHom V_eq_supr_U).op) gl) =\n sf i",
"tactic": "rw [← comp_apply, ← F.1.map_comp]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_1\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\ni : ι\n⊢ (CategoryTheory.forget C).map (F.val.map ((eqToHom V_eq_supr_U).op ≫ (iUV i).op)) gl = sf i",
"tactic": "exact gl_spec i"
},
{
"state_after": "case intro.intro.refine'_2\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\ngl' : (CategoryTheory.forget C).obj (F.val.obj V.op)\ngl'_spec : ∀ (i : ι), (CategoryTheory.forget C).map (F.val.map (iUV i).op) gl' = sf i\n⊢ gl' = (CategoryTheory.forget C).map (F.val.map (eqToHom V_eq_supr_U).op) gl",
"state_before": "case intro.intro.refine'_2\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\n⊢ ∀ (y : (CategoryTheory.forget C).obj (F.val.obj V.op)),\n (fun s => ∀ (i : ι), (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = sf i) y →\n y = (CategoryTheory.forget C).map (F.val.map (eqToHom V_eq_supr_U).op) gl",
"tactic": "intro gl' gl'_spec"
},
{
"state_after": "case h.e'_2\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\ngl' : (CategoryTheory.forget C).obj (F.val.obj V.op)\ngl'_spec : ∀ (i : ι), (CategoryTheory.forget C).map (F.val.map (iUV i).op) gl' = sf i\n⊢ gl' = (CategoryTheory.forget C).map (F.val.map ((eqToHom (_ : iSup U = V)).op ≫ (eqToHom V_eq_supr_U).op)) gl'\n\ncase intro.intro.refine'_2.convert_3\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\ngl' : (CategoryTheory.forget C).obj (F.val.obj V.op)\ngl'_spec : ∀ (i : ι), (CategoryTheory.forget C).map (F.val.map (iUV i).op) gl' = sf i\ni : ι\n⊢ (CategoryTheory.forget C).map (F.val.map ((eqToHom (_ : iSup U = V)).op ≫ (leSupr U i).op)) gl' = sf i",
"state_before": "case intro.intro.refine'_2\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\ngl' : (CategoryTheory.forget C).obj (F.val.obj V.op)\ngl'_spec : ∀ (i : ι), (CategoryTheory.forget C).map (F.val.map (iUV i).op) gl' = sf i\n⊢ gl' = (CategoryTheory.forget C).map (F.val.map (eqToHom V_eq_supr_U).op) gl",
"tactic": "convert congr_arg _ (gl_uniq (F.1.map (eqToHom V_eq_supr_U.symm).op gl') fun i => _) <;>\n rw [← comp_apply, ← F.1.map_comp]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\ngl' : (CategoryTheory.forget C).obj (F.val.obj V.op)\ngl'_spec : ∀ (i : ι), (CategoryTheory.forget C).map (F.val.map (iUV i).op) gl' = sf i\n⊢ gl' = (CategoryTheory.forget C).map (F.val.map ((eqToHom (_ : iSup U = V)).op ≫ (eqToHom V_eq_supr_U).op)) gl'",
"tactic": "rw [eqToHom_op, eqToHom_op, eqToHom_trans, eqToHom_refl, F.1.map_id, id_apply]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_2.convert_3\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : ConcreteCategory C\ninst✝² : HasLimits C\ninst✝¹ : ReflectsIsomorphisms ConcreteCategory.forget\ninst✝ : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\nι : Type v\nU : ι → Opens ↑X\nV : Opens ↑X\niUV : (i : ι) → U i ⟶ V\nhcover : V ≤ iSup U\nsf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (U i).op)\nh : IsCompatible F.val U sf\nV_eq_supr_U : V = iSup U\ngl : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)\ngl_spec : IsGluing F.val U sf gl\ngl_uniq : ∀ (y : (CategoryTheory.forget C).obj (F.val.obj (iSup U).op)), (fun s => IsGluing F.val U sf s) y → y = gl\ngl' : (CategoryTheory.forget C).obj (F.val.obj V.op)\ngl'_spec : ∀ (i : ι), (CategoryTheory.forget C).map (F.val.map (iUV i).op) gl' = sf i\ni : ι\n⊢ (CategoryTheory.forget C).map (F.val.map ((eqToHom (_ : iSup U = V)).op ≫ (leSupr U i).op)) gl' = sf i",
"tactic": "convert gl'_spec i"
}
] |
[
270,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Data/List/Basic.lean
|
List.append_concat
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.25691\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁✝ l₂✝ : List α\na : α\nl₁ l₂ : List α\n⊢ l₁ ++ concat l₂ a = concat (l₁ ++ l₂) a",
"tactic": "simp"
}
] |
[
579,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
579,
1
] |
Mathlib/Analysis/InnerProductSpace/Orientation.lean
|
OrthonormalBasis.det_eq_neg_det_of_opposite_orientation
|
[
{
"state_after": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\nh : Basis.orientation (OrthonormalBasis.toBasis e) ≠ Basis.orientation (OrthonormalBasis.toBasis f)\n⊢ ↑(Basis.det (OrthonormalBasis.toBasis e)) ↑(OrthonormalBasis.toBasis f) • Basis.det (OrthonormalBasis.toBasis f) =\n -Basis.det (OrthonormalBasis.toBasis f)",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\nh : Basis.orientation (OrthonormalBasis.toBasis e) ≠ Basis.orientation (OrthonormalBasis.toBasis f)\n⊢ Basis.det (OrthonormalBasis.toBasis e) = -Basis.det (OrthonormalBasis.toBasis f)",
"tactic": "rw [e.toBasis.det.eq_smul_basis_det f.toBasis]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\nh : Basis.orientation (OrthonormalBasis.toBasis e) ≠ Basis.orientation (OrthonormalBasis.toBasis f)\n⊢ ↑(Basis.det (OrthonormalBasis.toBasis e)) ↑(OrthonormalBasis.toBasis f) • Basis.det (OrthonormalBasis.toBasis f) =\n -Basis.det (OrthonormalBasis.toBasis f)",
"tactic": "simp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h,\n neg_one_smul ℝ (M := AlternatingMap ℝ E ℝ ι)]"
}
] |
[
99,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Data/Int/ModEq.lean
|
Int.exists_unique_equiv
|
[
{
"state_after": "m n a✝ b✝ c d a b : ℤ\nhb : 0 < b\nthis : a % b < abs b\n⊢ a % b < b",
"state_before": "m n a✝ b✝ c d a b : ℤ\nhb : 0 < b\n⊢ a % b < b",
"tactic": "have : a % b < |b| := emod_lt _ (ne_of_gt hb)"
},
{
"state_after": "no goals",
"state_before": "m n a✝ b✝ c d a b : ℤ\nhb : 0 < b\nthis : a % b < abs b\n⊢ a % b < b",
"tactic": "rwa [abs_of_pos hb] at this"
},
{
"state_after": "no goals",
"state_before": "m n a✝ b✝ c d a b : ℤ\nhb : 0 < b\n⊢ a % b ≡ a [ZMOD b]",
"tactic": "simp [ModEq]"
}
] |
[
311,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/Data/Real/Cardinality.lean
|
Cardinal.not_countable_real
|
[
{
"state_after": "c : ℝ\nf g : ℕ → Bool\nn : ℕ\n⊢ ℵ₀ < 𝔠",
"state_before": "c : ℝ\nf g : ℕ → Bool\nn : ℕ\n⊢ ¬Set.Countable Set.univ",
"tactic": "rw [← le_aleph0_iff_set_countable, not_le, mk_univ_real]"
},
{
"state_after": "no goals",
"state_before": "c : ℝ\nf g : ℕ → Bool\nn : ℕ\n⊢ ℵ₀ < 𝔠",
"tactic": "apply cantor"
}
] |
[
223,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
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