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/Sym/Basic.lean
|
Sym.cons_swap
|
[] |
[
131,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
collinear_iff_not_affineIndependent
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type ?u.274898\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : Fin 3 → P\n⊢ Collinear k (Set.range p) ↔ ¬AffineIndependent k p",
"tactic": "rw [collinear_iff_finrank_le_one,\n finrank_vectorSpan_le_iff_not_affineIndependent k p (Fintype.card_fin 3)]"
}
] |
[
457,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
454,
1
] |
Mathlib/LinearAlgebra/SModEq.lean
|
SModEq.def
|
[] |
[
43,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
41,
11
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.prob_add_prob_compl
|
[] |
[
3221,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3220,
1
] |
Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean
|
CategoryTheory.FreeMonoidalCategory.discrete_functor_map_eq_id
|
[] |
[
234,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Data/List/Infix.lean
|
List.dropLast_subset
|
[] |
[
187,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.xz_sub
|
[
{
"state_after": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : n ≤ m\n⊢ xz a1 (m - n) = xz a1 m * xz a1 n + ↑(Pell.d a1) * yz a1 m * -yz a1 n",
"state_before": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : n ≤ m\n⊢ xz a1 (m - n) = xz a1 m * xz a1 n - ↑(Pell.d a1) * yz a1 m * yz a1 n",
"tactic": "rw [sub_eq_add_neg, ← mul_neg]"
},
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : n ≤ m\n⊢ xz a1 (m - n) = xz a1 m * xz a1 n + ↑(Pell.d a1) * yz a1 m * -yz a1 n",
"tactic": "exact congr_arg Zsqrtd.re (pellZd_sub a1 h)"
}
] |
[
389,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/Computability/Reduce.lean
|
ComputablePred.computable_of_manyOneReducible
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\nσ : Type ?u.29812\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\np : α → Prop\nq : β → Prop\nh₂ : ComputablePred q\nf : α → β\nc : Computable f\nhf : ∀ (a : α), p a ↔ q (f a)\n⊢ ComputablePred p",
"state_before": "α : Type u_1\nβ : Type u_2\nσ : Type ?u.29812\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\np : α → Prop\nq : β → Prop\nh₁ : p ≤₀ q\nh₂ : ComputablePred q\n⊢ ComputablePred p",
"tactic": "rcases h₁ with ⟨f, c, hf⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\nσ : Type ?u.29812\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\np : α → Prop\nq : β → Prop\nh₂ : ComputablePred q\nf : α → β\nc : Computable f\nhf : ∀ (a : α), p a ↔ q (f a)\n⊢ ComputablePred fun a => q (f a)",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\nσ : Type ?u.29812\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\np : α → Prop\nq : β → Prop\nh₂ : ComputablePred q\nf : α → β\nc : Computable f\nhf : ∀ (a : α), p a ↔ q (f a)\n⊢ ComputablePred p",
"tactic": "rw [show p = fun a => q (f a) from Set.ext hf]"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nσ : Type ?u.29812\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\np : α → Prop\nf : α → β\nc : Computable f\ng : β → Bool\nhg : Computable g\nh₂ : ComputablePred fun a => g a = true\nhf : ∀ (a : α), p a ↔ (fun a => g a = true) (f a)\n⊢ ComputablePred fun a => (fun a => g a = true) (f a)",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\nσ : Type ?u.29812\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\np : α → Prop\nq : β → Prop\nh₂ : ComputablePred q\nf : α → β\nc : Computable f\nhf : ∀ (a : α), p a ↔ q (f a)\n⊢ ComputablePred fun a => q (f a)",
"tactic": "rcases computable_iff.1 h₂ with ⟨g, hg, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nσ : Type ?u.29812\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\np : α → Prop\nf : α → β\nc : Computable f\ng : β → Bool\nhg : Computable g\nh₂ : ComputablePred fun a => g a = true\nhf : ∀ (a : α), p a ↔ (fun a => g a = true) (f a)\n⊢ ComputablePred fun a => (fun a => g a = true) (f a)",
"tactic": "exact ⟨by infer_instance, by simpa using hg.comp c⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nσ : Type ?u.29812\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\np : α → Prop\nf : α → β\nc : Computable f\ng : β → Bool\nhg : Computable g\nh₂ : ComputablePred fun a => g a = true\nhf : ∀ (a : α), p a ↔ (fun a => g a = true) (f a)\n⊢ DecidablePred fun a => (fun a => g a = true) (f a)",
"tactic": "infer_instance"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nσ : Type ?u.29812\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\np : α → Prop\nf : α → β\nc : Computable f\ng : β → Bool\nhg : Computable g\nh₂ : ComputablePred fun a => g a = true\nhf : ∀ (a : α), p a ↔ (fun a => g a = true) (f a)\n⊢ Computable fun a => decide ((fun a => (fun a => g a = true) (f a)) a)",
"tactic": "simpa using hg.comp c"
}
] |
[
142,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/CategoryTheory/Subobject/MonoOver.lean
|
CategoryTheory.MonoOver.mk'_arrow
|
[] |
[
97,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/ContinuousLinearMap.lean
|
measurable_smul_const
|
[] |
[
109,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/LinearAlgebra/Projection.lean
|
LinearMap.IsProj.eq_conj_prodMap
|
[
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\np : Submodule R E\nf : E →ₗ[R] E\nh : IsProj p f\n⊢ f =\n comp (comp (↑(Submodule.prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f)))) (prodMap id 0))\n ↑(LinearEquiv.symm (Submodule.prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f))))",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\np : Submodule R E\nf : E →ₗ[R] E\nh : IsProj p f\n⊢ f = ↑(LinearEquiv.conj (Submodule.prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f)))) (prodMap id 0)",
"tactic": "rw [LinearEquiv.conj_apply]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\np : Submodule R E\nf : E →ₗ[R] E\nh : IsProj p f\n⊢ f =\n comp (comp (↑(Submodule.prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f)))) (prodMap id 0))\n ↑(LinearEquiv.symm (Submodule.prodEquivOfIsCompl p (ker f) (_ : IsCompl p (ker f))))",
"tactic": "exact h.eq_conj_prod_map'"
}
] |
[
459,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
456,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_add_const_Ioc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => x + a) ⁻¹' Ioc b c = Ioc (b - a) (c - a)",
"tactic": "simp [← Ioi_inter_Iic]"
}
] |
[
119,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Data/Int/Log.lean
|
Int.log_of_left_le_one
|
[
{
"state_after": "case inl\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : b ≤ 1\nr : R\nh : 1 ≤ r\n⊢ log b r = 0\n\ncase inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : b ≤ 1\nr : R\nh : r ≤ 1\n⊢ log b r = 0",
"state_before": "R : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : b ≤ 1\nr : R\n⊢ log b r = 0",
"tactic": "cases' le_total 1 r with h h"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : b ≤ 1\nr : R\nh : 1 ≤ r\n⊢ log b r = 0",
"tactic": "rw [log_of_one_le_right _ h, Nat.log_of_left_le_one hb, Int.ofNat_zero]"
},
{
"state_after": "no goals",
"state_before": "case inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : b ≤ 1\nr : R\nh : r ≤ 1\n⊢ log b r = 0",
"tactic": "rw [log_of_right_le_one _ h, Nat.clog_of_left_le_one hb, Int.ofNat_zero, neg_zero]"
}
] |
[
87,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/MeasureTheory/Function/Floor.lean
|
Int.measurable_ceil
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.1579\nR : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : LinearOrderedRing R\ninst✝⁴ : FloorRing R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : OpensMeasurableSpace R\nx : R\n⊢ MeasurableSet (ceil ⁻¹' {⌈x⌉})",
"tactic": "simpa only [Int.preimage_ceil_singleton] using measurableSet_Ioc"
}
] |
[
41,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Algebra/GroupWithZero/Semiconj.lean
|
SemiconjBy.inv_right_iff₀
|
[] |
[
61,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.add_iSup
|
[
{
"state_after": "α : Type ?u.136001\nβ : Type ?u.136004\nγ : Type ?u.136007\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns✝ : Set ℝ≥0∞\nι : Sort u_1\ns : ι → ℝ≥0∞\ninst✝ : Nonempty ι\n⊢ (⨆ (b : ι), s b + a) = ⨆ (b : ι), a + s b",
"state_before": "α : Type ?u.136001\nβ : Type ?u.136004\nγ : Type ?u.136007\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns✝ : Set ℝ≥0∞\nι : Sort u_1\ns : ι → ℝ≥0∞\ninst✝ : Nonempty ι\n⊢ a + iSup s = ⨆ (b : ι), a + s b",
"tactic": "rw [add_comm, iSup_add]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.136001\nβ : Type ?u.136004\nγ : Type ?u.136007\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns✝ : Set ℝ≥0∞\nι : Sort u_1\ns : ι → ℝ≥0∞\ninst✝ : Nonempty ι\n⊢ (⨆ (b : ι), s b + a) = ⨆ (b : ι), a + s b",
"tactic": "simp [add_comm]"
}
] |
[
594,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
593,
1
] |
Mathlib/Data/List/Cycle.lean
|
List.prev_reverse_eq_next
|
[
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\n⊢ prev (reverse l) (nthLe l k hk) (_ : nthLe l k hk ∈ reverse l) = next l (nthLe l k hk) hx",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl : List α\nh : Nodup l\nx : α\nhx : x ∈ l\n⊢ prev (reverse l) x (_ : x ∈ reverse l) = next l x hx",
"tactic": "obtain ⟨k, hk, rfl⟩ := nthLe_of_mem hx"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\n⊢ prev (reverse l) (nthLe l k hk) (_ : nthLe l k hk ∈ reverse l) = next l (nthLe l k hk) hx",
"state_before": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\n⊢ prev (reverse l) (nthLe l k hk) (_ : nthLe l k hk ∈ reverse l) = next l (nthLe l k hk) hx",
"tactic": "have lpos : 0 < l.length := k.zero_le.trans_lt hk"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ prev (reverse l) (nthLe l k hk) (_ : nthLe l k hk ∈ reverse l) = next l (nthLe l k hk) hx",
"state_before": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\n⊢ prev (reverse l) (nthLe l k hk) (_ : nthLe l k hk ∈ reverse l) = next l (nthLe l k hk) hx",
"tactic": "have key : l.length - 1 - k < l.length :=\n (Nat.sub_le _ _).trans_lt (tsub_lt_self lpos Nat.succ_pos')"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ prev (reverse l) (nthLe l k hk) (_ : nthLe l k hk ∈ reverse l) =\n nthLe (pmap (next l) l (_ : ∀ (x : α), x ∈ l → x ∈ l)) k\n (_ : k < length (pmap (next l) l (_ : ∀ (x : α), x ∈ l → x ∈ l)))",
"state_before": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ prev (reverse l) (nthLe l k hk) (_ : nthLe l k hk ∈ reverse l) = next l (nthLe l k hk) hx",
"tactic": "rw [← nthLe_pmap l.next (fun _ h => h) (by simpa using hk)]"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ prev (reverse l) (nthLe (reverse l) (length l - 1 - k) (_ : length l - 1 - k < length (reverse l)))\n (_ : nthLe (reverse l) (length l - 1 - k) (_ : length l - 1 - k < length (reverse l)) ∈ reverse l) =\n nthLe (rotate l 1) k (_ : k < length (rotate l 1))",
"state_before": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ prev (reverse l) (nthLe l k hk) (_ : nthLe l k hk ∈ reverse l) =\n nthLe (pmap (next l) l (_ : ∀ (x : α), x ∈ l → x ∈ l)) k\n (_ : k < length (pmap (next l) l (_ : ∀ (x : α), x ∈ l → x ∈ l)))",
"tactic": "simp_rw [← nthLe_reverse l k (key.trans_le (by simp)), pmap_next_eq_rotate_one _ h]"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ nthLe (pmap (prev (reverse l)) (reverse l) (_ : ∀ (x : α), x ∈ reverse l → x ∈ reverse l)) (length l - 1 - k)\n ?intro.intro =\n nthLe (rotate l 1) k (_ : k < length (rotate l 1))\n\ncase intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ length l - 1 - k < length (pmap (prev (reverse l)) (reverse l) (_ : ∀ (x : α), x ∈ reverse l → x ∈ reverse l))",
"state_before": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ prev (reverse l) (nthLe (reverse l) (length l - 1 - k) (_ : length l - 1 - k < length (reverse l)))\n (_ : nthLe (reverse l) (length l - 1 - k) (_ : length l - 1 - k < length (reverse l)) ∈ reverse l) =\n nthLe (rotate l 1) k (_ : k < length (rotate l 1))",
"tactic": "rw [← nthLe_pmap l.reverse.prev fun _ h => h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ k < length (pmap (next l) l (_ : ∀ (x : α), x ∈ l → x ∈ l))",
"tactic": "simpa using hk"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ length l ≤ length (reverse l)",
"tactic": "simp"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ nthLe (reverse (rotate l (Nat.succ 0))) (length l - 1 - k)\n (_ : length l - 1 - k < length (reverse (rotate l (Nat.succ 0)))) =\n nthLe (rotate l 1) k (_ : k < length (rotate l 1))",
"state_before": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ nthLe (pmap (prev (reverse l)) (reverse l) (_ : ∀ (x : α), x ∈ reverse l → x ∈ reverse l)) (length l - 1 - k)\n ?intro.intro =\n nthLe (rotate l 1) k (_ : k < length (rotate l 1))",
"tactic": "simp_rw [pmap_prev_eq_rotate_length_sub_one _ (nodup_reverse.mpr h), rotate_reverse,\n length_reverse, Nat.mod_eq_of_lt (tsub_lt_self lpos Nat.succ_pos'),\n tsub_tsub_cancel_of_le (Nat.succ_le_of_lt lpos)]"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ nthLe (reverse (reverse (rotate l (Nat.succ 0)))) (length (reverse (rotate l (Nat.succ 0))) - 1 - (length l - 1 - k))\n ?intro.intro.h1 =\n nthLe (rotate l 1) k (_ : k < length (rotate l 1))\n\ncase intro.intro.h1\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ length (reverse (rotate l (Nat.succ 0))) - 1 - (length l - 1 - k) < length (reverse (reverse (rotate l (Nat.succ 0))))\n\ncase intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ length l - 1 - k < length (pmap (prev (reverse l)) (reverse l) (_ : ∀ (x : α), x ∈ reverse l → x ∈ reverse l))",
"state_before": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ nthLe (reverse (rotate l (Nat.succ 0))) (length l - 1 - k)\n (_ : length l - 1 - k < length (reverse (rotate l (Nat.succ 0)))) =\n nthLe (rotate l 1) k (_ : k < length (rotate l 1))",
"tactic": "rw [← nthLe_reverse]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ length l - 1 - k < length (pmap (prev (reverse l)) (reverse l) (_ : ∀ (x : α), x ∈ reverse l → x ∈ reverse l))",
"tactic": ". simpa"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ nthLe (reverse (reverse (rotate l (Nat.succ 0)))) (length (reverse (rotate l (Nat.succ 0))) - 1 - (length l - 1 - k))\n ?intro.intro.h1 =\n nthLe (rotate l 1) k (_ : k < length (rotate l 1))",
"tactic": "simp [tsub_tsub_cancel_of_le (Nat.le_pred_of_lt hk)]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.h1\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ length (reverse (rotate l (Nat.succ 0))) - 1 - (length l - 1 - k) < length (reverse (reverse (rotate l (Nat.succ 0))))",
"tactic": "simpa using (Nat.sub_le _ _).trans_lt (tsub_lt_self lpos Nat.succ_pos')"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\nlpos : 0 < length l\nkey : length l - 1 - k < length l\n⊢ length l - 1 - k < length (pmap (prev (reverse l)) (reverse l) (_ : ∀ (x : α), x ∈ reverse l → x ∈ reverse l))",
"tactic": "simpa"
}
] |
[
417,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
402,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.zeroLocus_anti_mono_ideal
|
[] |
[
245,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/Data/List/Dedup.lean
|
List.sum_map_count_dedup_filter_eq_countp
|
[
{
"state_after": "case nil\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\n⊢ sum (map (fun x => count x []) (filter p (dedup []))) = countp p []\n\ncase cons\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\n⊢ sum (map (fun x => count x (a :: as)) (filter p (dedup (a :: as)))) = countp p (a :: as)",
"state_before": "α : Type u\ninst✝ : DecidableEq α\np : α → Bool\nl : List α\n⊢ sum (map (fun x => count x l) (filter p (dedup l))) = countp p l",
"tactic": "induction' l with a as h"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\n⊢ sum (map (fun x => count x []) (filter p (dedup []))) = countp p []",
"tactic": "simp"
},
{
"state_after": "case cons\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\n⊢ sum (map (fun i => count i as) (filter p (dedup (a :: as)))) +\n sum (map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))) =\n countp p as + if p a = true then 1 else 0",
"state_before": "case cons\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\n⊢ sum (map (fun x => count x (a :: as)) (filter p (dedup (a :: as)))) = countp p (a :: as)",
"tactic": "simp_rw [List.countp_cons, List.count_cons', List.sum_map_add]"
},
{
"state_after": "case cons.e_a\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\n⊢ sum (map (fun i => count i as) (filter p (dedup (a :: as)))) = countp p as\n\ncase cons.e_a\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\n⊢ sum (map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))) = if p a = true then 1 else 0",
"state_before": "case cons\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\n⊢ sum (map (fun i => count i as) (filter p (dedup (a :: as)))) +\n sum (map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))) =\n countp p as + if p a = true then 1 else 0",
"tactic": "congr 1"
},
{
"state_after": "case cons.e_a\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\n⊢ sum (map (fun i => count i as) (filter p (dedup (a :: as)))) = sum (map (fun x => count x as) (filter p (dedup as)))",
"state_before": "case cons.e_a\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\n⊢ sum (map (fun i => count i as) (filter p (dedup (a :: as)))) = countp p as",
"tactic": "refine' _root_.trans _ h"
},
{
"state_after": "case pos\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nha : a ∈ as\n⊢ sum (map (fun i => count i as) (filter p (dedup (a :: as)))) = sum (map (fun x => count x as) (filter p (dedup as)))\n\ncase neg\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nha : ¬a ∈ as\n⊢ sum (map (fun i => count i as) (filter p (dedup (a :: as)))) = sum (map (fun x => count x as) (filter p (dedup as)))",
"state_before": "case cons.e_a\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\n⊢ sum (map (fun i => count i as) (filter p (dedup (a :: as)))) = sum (map (fun x => count x as) (filter p (dedup as)))",
"tactic": "by_cases ha : a ∈ as"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nha : a ∈ as\n⊢ sum (map (fun i => count i as) (filter p (dedup (a :: as)))) = sum (map (fun x => count x as) (filter p (dedup as)))",
"tactic": "simp [dedup_cons_of_mem ha]"
},
{
"state_after": "case neg\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nha : ¬a ∈ as\n⊢ sum\n (map (fun i => count i as)\n (match p a with\n | true => a :: filter p (dedup as)\n | false => filter p (dedup as))) =\n sum (map (fun i => count i as) (filter p (dedup as)))",
"state_before": "case neg\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nha : ¬a ∈ as\n⊢ sum (map (fun i => count i as) (filter p (dedup (a :: as)))) = sum (map (fun x => count x as) (filter p (dedup as)))",
"tactic": "simp only [dedup_cons_of_not_mem ha, List.filter]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nha : ¬a ∈ as\n⊢ sum\n (map (fun i => count i as)\n (match p a with\n | true => a :: filter p (dedup as)\n | false => filter p (dedup as))) =\n sum (map (fun i => count i as) (filter p (dedup as)))",
"tactic": "match p a with\n| true => simp only [List.map_cons, List.sum_cons, List.count_eq_zero.2 ha, zero_add]\n| false => simp only"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nha : ¬a ∈ as\n⊢ sum\n (map (fun i => count i as)\n (match true with\n | true => a :: filter p (dedup as)\n | false => filter p (dedup as))) =\n sum (map (fun i => count i as) (filter p (dedup as)))",
"tactic": "simp only [List.map_cons, List.sum_cons, List.count_eq_zero.2 ha, zero_add]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nha : ¬a ∈ as\n⊢ sum\n (map (fun i => count i as)\n (match false with\n | true => a :: filter p (dedup as)\n | false => filter p (dedup as))) =\n sum (map (fun i => count i as) (filter p (dedup as)))",
"tactic": "simp only"
},
{
"state_after": "case pos\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : p a = true\n⊢ sum (map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))) = if p a = true then 1 else 0\n\ncase neg\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : ¬p a = true\n⊢ sum (map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))) = if p a = true then 1 else 0",
"state_before": "case cons.e_a\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\n⊢ sum (map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))) = if p a = true then 1 else 0",
"tactic": "by_cases hp : p a"
},
{
"state_after": "case pos\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : p a = true\n⊢ (count a (filter p (dedup (a :: as))) • if a = a then 1 else 0) = if p a = true then 1 else 0",
"state_before": "case pos\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : p a = true\n⊢ sum (map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))) = if p a = true then 1 else 0",
"tactic": "refine' _root_.trans (sum_map_eq_nsmul_single a _ fun _ h _ => by simp [h]) _"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : p a = true\n⊢ (count a (filter p (dedup (a :: as))) • if a = a then 1 else 0) = if p a = true then 1 else 0",
"tactic": "simp [hp, count_dedup]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh✝ : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : p a = true\nx✝¹ : α\nh : x✝¹ ≠ a\nx✝ : x✝¹ ∈ filter p (dedup (a :: as))\n⊢ (if x✝¹ = a then 1 else 0) = 0",
"tactic": "simp [h]"
},
{
"state_after": "case neg\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : ¬p a = true\nn : ℕ\nhn : n ∈ map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))\n⊢ n = 0",
"state_before": "case neg\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : ¬p a = true\n⊢ sum (map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))) = if p a = true then 1 else 0",
"tactic": "refine' _root_.trans (List.sum_eq_zero fun n hn => _) (by simp [hp])"
},
{
"state_after": "case neg.intro\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : ¬p a = true\nn : ℕ\nhn : n ∈ map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))\na' : α\nha' : a' ∈ filter p (dedup (a :: as)) ∧ (if a' = a then 1 else 0) = n\n⊢ n = 0",
"state_before": "case neg\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : ¬p a = true\nn : ℕ\nhn : n ∈ map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))\n⊢ n = 0",
"tactic": "obtain ⟨a', ha'⟩ := List.mem_map.1 hn"
},
{
"state_after": "case neg.intro.inl\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : ¬p a = true\nn : ℕ\nhn : n ∈ map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))\na' : α\nha : a' = a\nha' : a' ∈ filter p (dedup (a :: as)) ∧ 1 = n\n⊢ n = 0\n\ncase neg.intro.inr\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : ¬p a = true\nn : ℕ\nhn : n ∈ map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))\na' : α\nha : ¬a' = a\nha' : a' ∈ filter p (dedup (a :: as)) ∧ 0 = n\n⊢ n = 0",
"state_before": "case neg.intro\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : ¬p a = true\nn : ℕ\nhn : n ∈ map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))\na' : α\nha' : a' ∈ filter p (dedup (a :: as)) ∧ (if a' = a then 1 else 0) = n\n⊢ n = 0",
"tactic": "split_ifs at ha' with ha"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : ¬p a = true\n⊢ 0 = if p a = true then 1 else 0",
"tactic": "simp [hp]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.inl\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : ¬p a = true\nn : ℕ\nhn : n ∈ map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))\na' : α\nha : a' = a\nha' : a' ∈ filter p (dedup (a :: as)) ∧ 1 = n\n⊢ n = 0",
"tactic": "simp only [ha, mem_filter, mem_dedup, find?, mem_cons, true_or, hp,\n and_false, false_and] at ha'"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.inr\nα : Type u\ninst✝ : DecidableEq α\np : α → Bool\na : α\nas : List α\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : ¬p a = true\nn : ℕ\nhn : n ∈ map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))\na' : α\nha : ¬a' = a\nha' : a' ∈ filter p (dedup (a :: as)) ∧ 0 = n\n⊢ n = 0",
"tactic": "exact ha'.2.symm"
}
] |
[
135,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.tendsto_translationNumber_of_dist_bounded_aux
|
[
{
"state_after": "case refine'_1\nf g : CircleDeg1Lift\nx : ℕ → ℝ\nC : ℝ\nH : ∀ (n : ℕ), dist (↑(f ^ n) 0) (x n) ≤ C\n⊢ ℕ → ℝ\n\ncase refine'_2\nf g : CircleDeg1Lift\nx : ℕ → ℝ\nC : ℝ\nH : ∀ (n : ℕ), dist (↑(f ^ n) 0) (x n) ≤ C\n⊢ ∀ (t : ℕ), dist (transnumAuxSeq f t) (x (2 ^ t) / 2 ^ t) ≤ ?refine'_1 t\n\ncase refine'_3\nf g : CircleDeg1Lift\nx : ℕ → ℝ\nC : ℝ\nH : ∀ (n : ℕ), dist (↑(f ^ n) 0) (x n) ≤ C\n⊢ Tendsto ?refine'_1 atTop (𝓝 0)",
"state_before": "f g : CircleDeg1Lift\nx : ℕ → ℝ\nC : ℝ\nH : ∀ (n : ℕ), dist (↑(f ^ n) 0) (x n) ≤ C\n⊢ Tendsto (fun n => x (2 ^ n) / 2 ^ n) atTop (𝓝 (τ f))",
"tactic": "refine' f.tendsto_translationNumber_aux.congr_dist (squeeze_zero (fun _ => dist_nonneg) _ _)"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nf g : CircleDeg1Lift\nx : ℕ → ℝ\nC : ℝ\nH : ∀ (n : ℕ), dist (↑(f ^ n) 0) (x n) ≤ C\n⊢ ℕ → ℝ",
"tactic": "exact fun n => C / 2 ^ n"
},
{
"state_after": "case refine'_2\nf g : CircleDeg1Lift\nx : ℕ → ℝ\nC : ℝ\nH : ∀ (n : ℕ), dist (↑(f ^ n) 0) (x n) ≤ C\nn : ℕ\n⊢ dist (transnumAuxSeq f n) (x (2 ^ n) / 2 ^ n) ≤ C / 2 ^ n",
"state_before": "case refine'_2\nf g : CircleDeg1Lift\nx : ℕ → ℝ\nC : ℝ\nH : ∀ (n : ℕ), dist (↑(f ^ n) 0) (x n) ≤ C\n⊢ ∀ (t : ℕ), dist (transnumAuxSeq f t) (x (2 ^ t) / 2 ^ t) ≤ C / 2 ^ t",
"tactic": "intro n"
},
{
"state_after": "case refine'_2\nf g : CircleDeg1Lift\nx : ℕ → ℝ\nC : ℝ\nH : ∀ (n : ℕ), dist (↑(f ^ n) 0) (x n) ≤ C\nn : ℕ\nthis : 0 < 2 ^ n\n⊢ dist (transnumAuxSeq f n) (x (2 ^ n) / 2 ^ n) ≤ C / 2 ^ n",
"state_before": "case refine'_2\nf g : CircleDeg1Lift\nx : ℕ → ℝ\nC : ℝ\nH : ∀ (n : ℕ), dist (↑(f ^ n) 0) (x n) ≤ C\nn : ℕ\n⊢ dist (transnumAuxSeq f n) (x (2 ^ n) / 2 ^ n) ≤ C / 2 ^ n",
"tactic": "have : 0 < (2 ^ n : ℝ) := pow_pos zero_lt_two _"
},
{
"state_after": "case h.e'_3\nf g : CircleDeg1Lift\nx : ℕ → ℝ\nC : ℝ\nH : ∀ (n : ℕ), dist (↑(f ^ n) 0) (x n) ≤ C\nn : ℕ\nthis : 0 < 2 ^ n\n⊢ dist (transnumAuxSeq f n) (x (2 ^ n) / 2 ^ n) = dist (↑(f ^ 2 ^ n) 0) (x (2 ^ n)) / 2 ^ n",
"state_before": "case refine'_2\nf g : CircleDeg1Lift\nx : ℕ → ℝ\nC : ℝ\nH : ∀ (n : ℕ), dist (↑(f ^ n) 0) (x n) ≤ C\nn : ℕ\nthis : 0 < 2 ^ n\n⊢ dist (transnumAuxSeq f n) (x (2 ^ n) / 2 ^ n) ≤ C / 2 ^ n",
"tactic": "convert (div_le_div_right this).2 (H (2 ^ n)) using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nf g : CircleDeg1Lift\nx : ℕ → ℝ\nC : ℝ\nH : ∀ (n : ℕ), dist (↑(f ^ n) 0) (x n) ≤ C\nn : ℕ\nthis : 0 < 2 ^ n\n⊢ dist (transnumAuxSeq f n) (x (2 ^ n) / 2 ^ n) = dist (↑(f ^ 2 ^ n) 0) (x (2 ^ n)) / 2 ^ n",
"tactic": "rw [transnumAuxSeq, Real.dist_eq, ← sub_div, abs_div, abs_of_pos this, Real.dist_eq]"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nf g : CircleDeg1Lift\nx : ℕ → ℝ\nC : ℝ\nH : ∀ (n : ℕ), dist (↑(f ^ n) 0) (x n) ≤ C\n⊢ Tendsto (fun n => C / 2 ^ n) atTop (𝓝 0)",
"tactic": "exact mul_zero C ▸ tendsto_const_nhds.mul <| tendsto_inv_atTop_zero.comp <|\n tendsto_pow_atTop_atTop_of_one_lt one_lt_two"
}
] |
[
702,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
692,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.strictMono_unique
|
[] |
[
606,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
603,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.option_some_iff
|
[] |
[
289,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Products.lean
|
CategoryTheory.Limits.Sigma.ι_reindex_hom
|
[
{
"state_after": "β : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\n⊢ ι (f ∘ ↑ε) b ≫\n (HasColimit.isoOfEquivalence (Discrete.equivalence ε)\n (Discrete.natIso fun x =>\n Iso.refl ((Discrete.functor f).obj ((Discrete.functor (Discrete.mk ∘ ↑ε)).obj x)))).hom =\n ι f (↑ε b)",
"state_before": "β : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\n⊢ ι (f ∘ ↑ε) b ≫ (reindex ε f).hom = ι f (↑ε b)",
"tactic": "dsimp [Sigma.reindex]"
},
{
"state_after": "β : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\n⊢ (Discrete.functor (f ∘ ↑ε)).map ((Equivalence.unit (Discrete.equivalence ε)).app { as := b }) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) } =\n ι f (↑ε b)",
"state_before": "β : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\n⊢ ι (f ∘ ↑ε) b ≫\n (HasColimit.isoOfEquivalence (Discrete.equivalence ε)\n (Discrete.natIso fun x =>\n Iso.refl ((Discrete.functor f).obj ((Discrete.functor (Discrete.mk ∘ ↑ε)).obj x)))).hom =\n ι f (↑ε b)",
"tactic": "simp only [HasColimit.isoOfEquivalence_hom_π, Functor.id_obj, Discrete.functor_obj,\n Function.comp_apply, Discrete.equivalence_functor, Discrete.equivalence_inverse,\n Functor.comp_obj, Discrete.natIso_inv_app, Iso.refl_inv, Category.id_comp]"
},
{
"state_after": "β : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : ↑ε (↑ε.symm (↑ε b)) = ↑ε b)) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε b } =\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) }\n⊢ (Discrete.functor (f ∘ ↑ε)).map ((Equivalence.unit (Discrete.equivalence ε)).app { as := b }) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) } =\n ι f (↑ε b)",
"state_before": "β : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\n⊢ (Discrete.functor (f ∘ ↑ε)).map ((Equivalence.unit (Discrete.equivalence ε)).app { as := b }) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) } =\n ι f (↑ε b)",
"tactic": "have h := colimit.w (Discrete.functor f) (Discrete.eqToHom' (ε.apply_symm_apply (ε b)))"
},
{
"state_after": "β : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : ↑ε (↑ε.symm (↑ε b)) = ↑ε b)) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε b } =\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) }\n⊢ (Discrete.functor (f ∘ ↑ε)).map ((Equivalence.unit (Discrete.equivalence ε)).app { as := b }) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) } =\n ι f (↑ε b)",
"state_before": "β : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : ↑ε (↑ε.symm (↑ε b)) = ↑ε b)) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε b } =\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) }\n⊢ (Discrete.functor (f ∘ ↑ε)).map ((Equivalence.unit (Discrete.equivalence ε)).app { as := b }) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) } =\n ι f (↑ε b)",
"tactic": "simp only [Discrete.functor_obj] at h"
},
{
"state_after": "β : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : ↑ε (↑ε.symm (↑ε b)) = ↑ε b)) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε b } =\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) }\n⊢ eqToHom (_ : f (↑ε b) = f (↑ε b)) ≫ colimit.ι (Discrete.functor f) { as := ↑ε b } = ι f (↑ε b)\n\ncase p\nβ : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : ↑ε (↑ε.symm (↑ε b)) = ↑ε b)) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε b } =\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) }\n⊢ { as := b } = { as := ↑ε.symm (↑ε b) }\n\ncase p\nβ : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : ↑ε (↑ε.symm (↑ε b)) = ↑ε b)) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε b } =\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) }\n⊢ { as := b } = { as := ↑ε.symm (↑ε b) }\n\ncase p\nβ : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : ↑ε (↑ε.symm (↑ε b)) = ↑ε b)) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε b } =\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) }\n⊢ { as := b } = { as := ↑ε.symm (↑ε b) }",
"state_before": "β : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : ↑ε (↑ε.symm (↑ε b)) = ↑ε b)) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε b } =\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) }\n⊢ (Discrete.functor (f ∘ ↑ε)).map ((Equivalence.unit (Discrete.equivalence ε)).app { as := b }) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) } =\n ι f (↑ε b)",
"tactic": "erw [← h, eqToHom_map, eqToHom_map, eqToHom_trans_assoc]"
},
{
"state_after": "no goals",
"state_before": "β : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : ↑ε (↑ε.symm (↑ε b)) = ↑ε b)) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε b } =\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) }\n⊢ eqToHom (_ : f (↑ε b) = f (↑ε b)) ≫ colimit.ι (Discrete.functor f) { as := ↑ε b } = ι f (↑ε b)\n\ncase p\nβ : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : ↑ε (↑ε.symm (↑ε b)) = ↑ε b)) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε b } =\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) }\n⊢ { as := b } = { as := ↑ε.symm (↑ε b) }\n\ncase p\nβ : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : ↑ε (↑ε.symm (↑ε b)) = ↑ε b)) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε b } =\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) }\n⊢ { as := b } = { as := ↑ε.symm (↑ε b) }\n\ncase p\nβ : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : ↑ε (↑ε.symm (↑ε b)) = ↑ε b)) ≫\n colimit.ι (Discrete.functor f) { as := ↑ε b } =\n colimit.ι (Discrete.functor f) { as := ↑ε (↑ε.symm (↑ε b)) }\n⊢ { as := b } = { as := ↑ε.symm (↑ε b) }",
"tactic": "all_goals { simp }"
}
] |
[
440,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
431,
1
] |
Mathlib/Analysis/Complex/UnitDisc/Basic.lean
|
Complex.UnitDisc.coe_conj
|
[] |
[
218,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Algebra/Order/Archimedean.lean
|
exists_rat_gt
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : Archimedean α\nx✝ y ε x : α\nn : ℕ\nh : x < ↑n\n⊢ x < ↑↑n",
"tactic": "rwa [Rat.cast_coe_nat]"
}
] |
[
255,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.StronglyMeasurable.smul
|
[] |
[
461,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
11
] |
Mathlib/Analysis/Convex/Complex.lean
|
convex_halfspace_im_lt
|
[] |
[
39,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Fintype.choose_subtype_eq
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.144755\nβ : Type ?u.144758\nγ : Type ?u.144761\ninst✝³ : Fintype α✝\np✝ : α✝ → Prop\ninst✝² : DecidablePred p✝\nα : Type ?u.144781\np : α → Prop\ninst✝¹ : Fintype { a // p a }\ninst✝ : DecidableEq α\nx y : { a // p a }\nhy : (fun a => ↑a = ↑x) y\n⊢ y = x",
"tactic": "simpa [Subtype.ext_iff] using hy"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.144755\nβ : Type ?u.144758\nγ : Type ?u.144761\ninst✝³ : Fintype α✝\np✝ : α✝ → Prop\ninst✝² : DecidablePred p✝\nα : Type u_1\np : α → Prop\ninst✝¹ : Fintype { a // p a }\ninst✝ : DecidableEq α\nx : { a // p a }\nh : optParam (∃! a, ↑a = ↑x) (_ : ∃ x_1, (fun a => ↑a = ↑x) x_1 ∧ ∀ (y : { a // p a }), (fun a => ↑a = ↑x) y → y = x_1)\n⊢ choose (fun y => ↑y = ↑x) h = x",
"tactic": "rw [Subtype.ext_iff, Fintype.choose_spec (fun y : { a : α // p a } => (y : α) = x) _]"
}
] |
[
1124,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1119,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
SeminormFamily.basisSets_smul_right
|
[
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\n⊢ ∀ᶠ (x : 𝕜) in 𝓝 0, x • v ∈ U",
"state_before": "𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU : U ∈ basisSets p\n⊢ ∀ᶠ (x : 𝕜) in 𝓝 0, x • v ∈ U",
"tactic": "rcases p.basisSets_iff.mp hU with ⟨s, r, hr, hU⟩"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\n⊢ {x | x • v ∈ ball (Finset.sup s p) 0 r} ∈ 𝓝 0",
"state_before": "case intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\n⊢ ∀ᶠ (x : 𝕜) in 𝓝 0, x • v ∈ U",
"tactic": "rw [hU, Filter.eventually_iff]"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\n⊢ {x | ‖x‖ * ↑(Finset.sup s p) v < r} ∈ 𝓝 0",
"state_before": "case intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\n⊢ {x | x • v ∈ ball (Finset.sup s p) 0 r} ∈ 𝓝 0",
"tactic": "simp_rw [(s.sup p).mem_ball_zero, map_smul_eq_mul]"
},
{
"state_after": "case pos\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\nh : 0 < ↑(Finset.sup s p) v\n⊢ {x | ‖x‖ * ↑(Finset.sup s p) v < r} ∈ 𝓝 0\n\ncase neg\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\nh : ¬0 < ↑(Finset.sup s p) v\n⊢ {x | ‖x‖ * ↑(Finset.sup s p) v < r} ∈ 𝓝 0",
"state_before": "case intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\n⊢ {x | ‖x‖ * ↑(Finset.sup s p) v < r} ∈ 𝓝 0",
"tactic": "by_cases h : 0 < (s.sup p) v"
},
{
"state_after": "case neg\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\nh : ¬0 < ↑(Finset.sup s p) v\n⊢ {x | True} ∈ 𝓝 0",
"state_before": "case neg\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\nh : ¬0 < ↑(Finset.sup s p) v\n⊢ {x | ‖x‖ * ↑(Finset.sup s p) v < r} ∈ 𝓝 0",
"tactic": "simp_rw [le_antisymm (not_lt.mp h) (map_nonneg _ v), MulZeroClass.mul_zero, hr]"
},
{
"state_after": "no goals",
"state_before": "case neg\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\nh : ¬0 < ↑(Finset.sup s p) v\n⊢ {x | True} ∈ 𝓝 0",
"tactic": "exact IsOpen.mem_nhds isOpen_univ (mem_univ 0)"
},
{
"state_after": "case pos\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\nh : 0 < ↑(Finset.sup s p) v\n⊢ {x | ‖x‖ < r / ↑(Finset.sup s p) v} ∈ 𝓝 0",
"state_before": "case pos\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\nh : 0 < ↑(Finset.sup s p) v\n⊢ {x | ‖x‖ * ↑(Finset.sup s p) v < r} ∈ 𝓝 0",
"tactic": "simp_rw [(lt_div_iff h).symm]"
},
{
"state_after": "case pos\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\nh : 0 < ↑(Finset.sup s p) v\n⊢ Metric.ball 0 (r / ↑(Finset.sup s p) v) ∈ 𝓝 0",
"state_before": "case pos\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\nh : 0 < ↑(Finset.sup s p) v\n⊢ {x | ‖x‖ < r / ↑(Finset.sup s p) v} ∈ 𝓝 0",
"tactic": "rw [← _root_.ball_zero_eq]"
},
{
"state_after": "no goals",
"state_before": "case pos\n𝕜 : Type u_2\n𝕜₂ : Type ?u.53503\n𝕝 : Type ?u.53506\n𝕝₂ : Type ?u.53509\nE : Type u_1\nF : Type ?u.53515\nG : Type ?u.53518\nι : Type u_3\nι' : Type ?u.53524\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nv : E\nU : Set E\nhU✝ : U ∈ basisSets p\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = ball (Finset.sup s p) 0 r\nh : 0 < ↑(Finset.sup s p) v\n⊢ Metric.ball 0 (r / ↑(Finset.sup s p) v) ∈ 𝓝 0",
"tactic": "exact Metric.ball_mem_nhds 0 (div_pos hr h)"
}
] |
[
156,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Data/Finset/Slice.lean
|
Finset.mem_slice
|
[] |
[
143,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/CategoryTheory/Subobject/Basic.lean
|
CategoryTheory.Subobject.underlyingIso_hom_comp_eq_mk
|
[] |
[
245,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/Algebra/Module/Submodule/Lattice.lean
|
Submodule.mem_sup_right
|
[
{
"state_after": "R : Type u_1\nS✝ : Type ?u.151745\nM : Type u_2\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S✝ M\ninst✝¹ : SMul S✝ R\ninst✝ : IsScalarTower S✝ R M\np q S T : Submodule R M\nthis : T ≤ S ⊔ T\n⊢ ∀ {x : M}, x ∈ T → x ∈ S ⊔ T",
"state_before": "R : Type u_1\nS✝ : Type ?u.151745\nM : Type u_2\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S✝ M\ninst✝¹ : SMul S✝ R\ninst✝ : IsScalarTower S✝ R M\np q S T : Submodule R M\n⊢ ∀ {x : M}, x ∈ T → x ∈ S ⊔ T",
"tactic": "have : T ≤ S ⊔ T := le_sup_right"
},
{
"state_after": "R : Type u_1\nS✝ : Type ?u.151745\nM : Type u_2\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S✝ M\ninst✝¹ : SMul S✝ R\ninst✝ : IsScalarTower S✝ R M\np q S T : Submodule R M\nthis : Preorder.toLE.1 T (S ⊔ T)\n⊢ ∀ {x : M}, x ∈ T → x ∈ S ⊔ T",
"state_before": "R : Type u_1\nS✝ : Type ?u.151745\nM : Type u_2\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S✝ M\ninst✝¹ : SMul S✝ R\ninst✝ : IsScalarTower S✝ R M\np q S T : Submodule R M\nthis : T ≤ S ⊔ T\n⊢ ∀ {x : M}, x ∈ T → x ∈ S ⊔ T",
"tactic": "rw [LE.le] at this"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS✝ : Type ?u.151745\nM : Type u_2\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S✝ M\ninst✝¹ : SMul S✝ R\ninst✝ : IsScalarTower S✝ R M\np q S T : Submodule R M\nthis : Preorder.toLE.1 T (S ⊔ T)\n⊢ ∀ {x : M}, x ∈ T → x ∈ S ⊔ T",
"tactic": "exact this"
}
] |
[
287,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/FieldTheory/Finiteness.lean
|
IsNoetherian.range_finsetBasis
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : IsNoetherian K V\n⊢ Set.range ↑(finsetBasis K V) = Basis.ofVectorSpaceIndex K V",
"tactic": "rw [finsetBasis, Basis.range_reindex, Basis.range_ofVectorSpace]"
}
] |
[
106,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.factorization_inj
|
[
{
"state_after": "no goals",
"state_before": "a : ℕ\nha : a ∈ {x | x ≠ 0}\nb : ℕ\nhb : b ∈ {x | x ≠ 0}\nh : factorization a = factorization b\np : ℕ\n⊢ ↑(factorization a) p = ↑(factorization b) p",
"tactic": "simp [h]"
}
] |
[
119,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/RingTheory/OreLocalization/Basic.lean
|
OreLocalization.add_assoc
|
[
{
"state_after": "case c\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\ny z : OreLocalization R S\nr₁ : R\ns₁ : { x // x ∈ S }\n⊢ r₁ /ₒ s₁ + y + z = r₁ /ₒ s₁ + (y + z)",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nx y z : OreLocalization R S\n⊢ x + y + z = x + (y + z)",
"tactic": "induction' x using OreLocalization.ind with r₁ s₁"
},
{
"state_after": "case c.c\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nz : OreLocalization R S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\n⊢ r₁ /ₒ s₁ + r₂ /ₒ s₂ + z = r₁ /ₒ s₁ + (r₂ /ₒ s₂ + z)",
"state_before": "case c\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\ny z : OreLocalization R S\nr₁ : R\ns₁ : { x // x ∈ S }\n⊢ r₁ /ₒ s₁ + y + z = r₁ /ₒ s₁ + (y + z)",
"tactic": "induction' y using OreLocalization.ind with r₂ s₂"
},
{
"state_after": "case c.c.c\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\n⊢ r₁ /ₒ s₁ + r₂ /ₒ s₂ + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ /ₒ s₂ + r₃ /ₒ s₃)",
"state_before": "case c.c\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nz : OreLocalization R S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\n⊢ r₁ /ₒ s₁ + r₂ /ₒ s₂ + z = r₁ /ₒ s₁ + (r₂ /ₒ s₂ + z)",
"tactic": "induction' z using OreLocalization.ind with r₃ s₃"
},
{
"state_after": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nha' : r₁ /ₒ s₁ + r₂ /ₒ s₂ = (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa)\n⊢ r₁ /ₒ s₁ + r₂ /ₒ s₂ + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ /ₒ s₂ + r₃ /ₒ s₃)",
"state_before": "case c.c.c\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\n⊢ r₁ /ₒ s₁ + r₂ /ₒ s₂ + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ /ₒ s₂ + r₃ /ₒ s₃)",
"tactic": "rcases oreDivAddChar' r₁ r₂ s₁ s₂ with ⟨ra, sa, ha, ha'⟩"
},
{
"state_after": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nha' : r₁ /ₒ s₁ + r₂ /ₒ s₂ = (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa)\n⊢ (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ /ₒ s₂ + r₃ /ₒ s₃)",
"state_before": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nha' : r₁ /ₒ s₁ + r₂ /ₒ s₂ = (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa)\n⊢ r₁ /ₒ s₁ + r₂ /ₒ s₂ + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ /ₒ s₂ + r₃ /ₒ s₃)",
"tactic": "rw [ha']"
},
{
"state_after": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\n⊢ (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ /ₒ s₂ + r₃ /ₒ s₃)",
"state_before": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nha' : r₁ /ₒ s₁ + r₂ /ₒ s₂ = (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa)\n⊢ (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ /ₒ s₂ + r₃ /ₒ s₃)",
"tactic": "clear ha'"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nhb' : r₂ /ₒ s₂ + r₃ /ₒ s₃ = (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)\n⊢ (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ /ₒ s₂ + r₃ /ₒ s₃)",
"state_before": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\n⊢ (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ /ₒ s₂ + r₃ /ₒ s₃)",
"tactic": "rcases oreDivAddChar' r₂ r₃ s₂ s₃ with ⟨rb, sb, hb, hb'⟩"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nhb' : r₂ /ₒ s₂ + r₃ /ₒ s₃ = (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)\n⊢ (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nhb' : r₂ /ₒ s₂ + r₃ /ₒ s₃ = (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)\n⊢ (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ /ₒ s₂ + r₃ /ₒ s₃)",
"tactic": "rw [hb']"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\n⊢ (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nhb' : r₂ /ₒ s₂ + r₃ /ₒ s₃ = (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)\n⊢ (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)",
"tactic": "clear hb'"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nq : (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = ((r₁ * ↑sa + r₂ * ra) * ↑sc + r₃ * rc) /ₒ (s₁ * sa * sc)\n⊢ (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\n⊢ (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)",
"tactic": "rcases oreDivAddChar' (r₁ * sa + r₂ * ra) r₃ (s₁ * sa) s₃ with ⟨rc, sc, hc, q⟩"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nq : (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = ((r₁ * ↑sa + r₂ * ra) * ↑sc + r₃ * rc) /ₒ (s₁ * sa * sc)\n⊢ ((r₁ * ↑sa + r₂ * ra) * ↑sc + r₃ * rc) /ₒ (s₁ * sa * sc) = r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nq : (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = ((r₁ * ↑sa + r₂ * ra) * ↑sc + r₃ * rc) /ₒ (s₁ * sa * sc)\n⊢ (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)",
"tactic": "rw [q]"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\n⊢ ((r₁ * ↑sa + r₂ * ra) * ↑sc + r₃ * rc) /ₒ (s₁ * sa * sc) = r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nq : (r₁ * ↑sa + r₂ * ra) /ₒ (s₁ * sa) + r₃ /ₒ s₃ = ((r₁ * ↑sa + r₂ * ra) * ↑sc + r₃ * rc) /ₒ (s₁ * sa * sc)\n⊢ ((r₁ * ↑sa + r₂ * ra) * ↑sc + r₃ * rc) /ₒ (s₁ * sa * sc) = r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)",
"tactic": "clear q"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\nq : r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb) = (r₁ * ↑sd + (r₂ * ↑sb + r₃ * rb) * rd) /ₒ (s₁ * sd)\n⊢ ((r₁ * ↑sa + r₂ * ra) * ↑sc + r₃ * rc) /ₒ (s₁ * sa * sc) = r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\n⊢ ((r₁ * ↑sa + r₂ * ra) * ↑sc + r₃ * rc) /ₒ (s₁ * sa * sc) = r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)",
"tactic": "rcases oreDivAddChar' r₁ (r₂ * sb + r₃ * rb) s₁ (s₂ * sb) with ⟨rd, sd, hd, q⟩"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\nq : r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb) = (r₁ * ↑sd + (r₂ * ↑sb + r₃ * rb) * rd) /ₒ (s₁ * sd)\n⊢ ((r₁ * ↑sa + r₂ * ra) * ↑sc + r₃ * rc) /ₒ (s₁ * sa * sc) = (r₁ * ↑sd + (r₂ * ↑sb + r₃ * rb) * rd) /ₒ (s₁ * sd)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\nq : r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb) = (r₁ * ↑sd + (r₂ * ↑sb + r₃ * rb) * rd) /ₒ (s₁ * sd)\n⊢ ((r₁ * ↑sa + r₂ * ra) * ↑sc + r₃ * rc) /ₒ (s₁ * sa * sc) = r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb)",
"tactic": "rw [q]"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\n⊢ ((r₁ * ↑sa + r₂ * ra) * ↑sc + r₃ * rc) /ₒ (s₁ * sa * sc) = (r₁ * ↑sd + (r₂ * ↑sb + r₃ * rb) * rd) /ₒ (s₁ * sd)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\nq : r₁ /ₒ s₁ + (r₂ * ↑sb + r₃ * rb) /ₒ (s₂ * sb) = (r₁ * ↑sd + (r₂ * ↑sb + r₃ * rb) * rd) /ₒ (s₁ * sd)\n⊢ ((r₁ * ↑sa + r₂ * ra) * ↑sc + r₃ * rc) /ₒ (s₁ * sa * sc) = (r₁ * ↑sd + (r₂ * ↑sb + r₃ * rb) * rd) /ₒ (s₁ * sd)",
"tactic": "clear q"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\n⊢ (r₁ * (↑sa * ↑sc) + (r₂ * (ra * ↑sc) + r₃ * rc)) /ₒ (s₁ * (sa * sc)) =\n (r₁ * ↑sd + (r₂ * (↑sb * rd) + r₃ * (rb * rd))) /ₒ (s₁ * sd)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\n⊢ ((r₁ * ↑sa + r₂ * ra) * ↑sc + r₃ * rc) /ₒ (s₁ * sa * sc) = (r₁ * ↑sd + (r₂ * ↑sb + r₃ * rb) * rd) /ₒ (s₁ * sd)",
"tactic": "simp only [right_distrib, mul_assoc, add_assoc]"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\n⊢ r₁ * (↑sa * ↑sc) /ₒ (s₁ * (sa * sc)) + (r₂ * (ra * ↑sc) /ₒ (s₁ * (sa * sc)) + r₃ * rc /ₒ (s₁ * (sa * sc))) =\n r₁ * ↑sd /ₒ (s₁ * sd) + (r₂ * (↑sb * rd) /ₒ (s₁ * sd) + r₃ * (rb * rd) /ₒ (s₁ * sd))",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\n⊢ (r₁ * (↑sa * ↑sc) + (r₂ * (ra * ↑sc) + r₃ * rc)) /ₒ (s₁ * (sa * sc)) =\n (r₁ * ↑sd + (r₂ * (↑sb * rd) + r₃ * (rb * rd))) /ₒ (s₁ * sd)",
"tactic": "simp only [← add_oreDiv]"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\n⊢ r₁ * (↑sa * ↑sc) /ₒ (s₁ * (sa * sc)) = r₁ * ↑sd /ₒ (s₁ * sd)\n\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\n⊢ r₂ * (ra * ↑sc) /ₒ (s₁ * (sa * sc)) + r₃ * rc /ₒ (s₁ * (sa * sc)) =\n r₂ * (↑sb * rd) /ₒ (s₁ * sd) + r₃ * (rb * rd) /ₒ (s₁ * sd)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\n⊢ r₁ * (↑sa * ↑sc) /ₒ (s₁ * (sa * sc)) + (r₂ * (ra * ↑sc) /ₒ (s₁ * (sa * sc)) + r₃ * rc /ₒ (s₁ * (sa * sc))) =\n r₁ * ↑sd /ₒ (s₁ * sd) + (r₂ * (↑sb * rd) /ₒ (s₁ * sd) + r₃ * (rb * rd) /ₒ (s₁ * sd))",
"tactic": "congr 1"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\n⊢ r₂ * (ra * ↑sc) /ₒ (s₁ * (sa * sc)) = r₂ * (↑sb * rd) /ₒ (s₁ * sd)\n\ncase c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\n⊢ r₃ * rc /ₒ (s₁ * (sa * sc)) = r₃ * (rb * rd) /ₒ (s₁ * sd)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\n⊢ r₂ * (ra * ↑sc) /ₒ (s₁ * (sa * sc)) + r₃ * rc /ₒ (s₁ * (sa * sc)) =\n r₂ * (↑sb * rd) /ₒ (s₁ * sd) + r₃ * (rb * rd) /ₒ (s₁ * sd)",
"tactic": "congr 1"
},
{
"state_after": "no goals",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\n⊢ r₁ * (↑sa * ↑sc) /ₒ (s₁ * (sa * sc)) = r₁ * ↑sd /ₒ (s₁ * sd)",
"tactic": "rw [← OreLocalization.expand', ← mul_assoc, ← mul_assoc, ← OreLocalization.expand', ←\n OreLocalization.expand']"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑(s₁ * sa) = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑(s₁ * sd) = ↑(s₂ * sb) * rd\n⊢ r₂ * (ra * ↑sc) /ₒ (s₁ * (sa * sc)) = r₂ * (↑sb * rd) /ₒ (s₁ * sd)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\n⊢ r₂ * (ra * ↑sc) /ₒ (s₁ * (sa * sc)) = r₂ * (↑sb * rd) /ₒ (s₁ * sd)",
"tactic": "simp_rw [← Submonoid.coe_mul] at ha hd"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑(s₁ * sa) = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑(s₁ * sd) = ↑(s₂ * sb) * rd\n⊢ r₂ /ₒ s₂ = r₂ * ↑sb /ₒ (s₂ * sb)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑(s₁ * sa) = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑(s₁ * sd) = ↑(s₂ * sb) * rd\n⊢ r₂ * (ra * ↑sc) /ₒ (s₁ * (sa * sc)) = r₂ * (↑sb * rd) /ₒ (s₁ * sd)",
"tactic": "rw [Subtype.coe_eq_of_eq_mk hd, ← mul_assoc, ← mul_assoc, ← mul_assoc, ← OreLocalization.expand,\n ← OreLocalization.expand', Subtype.coe_eq_of_eq_mk ha, ← OreLocalization.expand]"
},
{
"state_after": "no goals",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑(s₁ * sa) = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑(s₁ * sd) = ↑(s₂ * sb) * rd\n⊢ r₂ /ₒ s₂ = r₂ * ↑sb /ₒ (s₂ * sb)",
"tactic": "apply OreLocalization.expand'"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a.mk.mk\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\nre : R\nfst✝ : { x // x ∈ S }\nsnd✝ : ↑sd * ↑fst✝ = ↑(sa * sc) * re\n⊢ r₃ * rc /ₒ (s₁ * (sa * sc)) = r₃ * (rb * rd) /ₒ (s₁ * sd)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\n⊢ r₃ * rc /ₒ (s₁ * (sa * sc)) = r₃ * (rb * rd) /ₒ (s₁ * sd)",
"tactic": "rcases oreCondition (sd : R) (sa * sc) with ⟨re, _, _⟩"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a.mk.mk\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑(s₂ * sb) = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa * sc) = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑(s₁ * sd) = ↑(s₂ * sb) * rd\nre : R\nfst✝ : { x // x ∈ S }\nsnd✝ : ↑sd * ↑fst✝ = ↑(sa * sc) * re\n⊢ r₃ * rc /ₒ (s₁ * (sa * sc)) = r₃ * (rb * rd) /ₒ (s₁ * sd)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a.mk.mk\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑s₂ * ↑sb = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa) * ↑sc = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑s₁ * ↑sd = ↑(s₂ * sb) * rd\nre : R\nfst✝ : { x // x ∈ S }\nsnd✝ : ↑sd * ↑fst✝ = ↑(sa * sc) * re\n⊢ r₃ * rc /ₒ (s₁ * (sa * sc)) = r₃ * (rb * rd) /ₒ (s₁ * sd)",
"tactic": "simp_rw [← Submonoid.coe_mul] at hb hc hd"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a.mk.mk\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑(s₂ * sb) = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa * sc) = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑(s₁ * sd) = ↑(s₂ * sb) * rd\nre : R\nfst✝ : { x // x ∈ S }\nsnd✝ : ↑sd * ↑fst✝ = ↑(sa * sc) * re\n⊢ r₃ * rc /ₒ { val := ↑s₃ * rc, property := (_ : ↑s₃ * rc ∈ ↑S) } = r₃ * (rb * rd) /ₒ (s₁ * sd)",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a.mk.mk\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑(s₂ * sb) = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa * sc) = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑(s₁ * sd) = ↑(s₂ * sb) * rd\nre : R\nfst✝ : { x // x ∈ S }\nsnd✝ : ↑sd * ↑fst✝ = ↑(sa * sc) * re\n⊢ r₃ * rc /ₒ (s₁ * (sa * sc)) = r₃ * (rb * rd) /ₒ (s₁ * sd)",
"tactic": "rw [← mul_assoc, Subtype.coe_eq_of_eq_mk hc]"
},
{
"state_after": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a.mk.mk\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑(s₂ * sb) = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa * sc) = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑(s₁ * sd) = ↑(s₂ * sb) * rd\nre : R\nfst✝ : { x // x ∈ S }\nsnd✝ : ↑sd * ↑fst✝ = ↑(sa * sc) * re\n⊢ r₃ /ₒ s₃ = r₃ * rb /ₒ { val := ↑s₃ * rb, property := (_ : ↑s₃ * rb ∈ ↑S) }",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a.mk.mk\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑(s₂ * sb) = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa * sc) = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑(s₁ * sd) = ↑(s₂ * sb) * rd\nre : R\nfst✝ : { x // x ∈ S }\nsnd✝ : ↑sd * ↑fst✝ = ↑(sa * sc) * re\n⊢ r₃ * rc /ₒ { val := ↑s₃ * rc, property := (_ : ↑s₃ * rc ∈ ↑S) } = r₃ * (rb * rd) /ₒ (s₁ * sd)",
"tactic": "rw [← OreLocalization.expand, Subtype.coe_eq_of_eq_mk hd, ← mul_assoc, ←\n OreLocalization.expand, Subtype.coe_eq_of_eq_mk hb]"
},
{
"state_after": "no goals",
"state_before": "case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro.mk.mk.intro.e_a.e_a.mk.mk\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ : R\ns₁ : { x // x ∈ S }\nr₂ : R\ns₂ : { x // x ∈ S }\nr₃ : R\ns₃ : { x // x ∈ S }\nra : R\nsa : { x // x ∈ S }\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrb : R\nsb : { x // x ∈ S }\nhb : ↑(s₂ * sb) = ↑s₃ * rb\nrc : R\nsc : { x // x ∈ S }\nhc : ↑(s₁ * sa * sc) = ↑s₃ * rc\nrd : R\nsd : { x // x ∈ S }\nhd : ↑(s₁ * sd) = ↑(s₂ * sb) * rd\nre : R\nfst✝ : { x // x ∈ S }\nsnd✝ : ↑sd * ↑fst✝ = ↑(sa * sc) * re\n⊢ r₃ /ₒ s₃ = r₃ * rb /ₒ { val := ↑s₃ * rb, property := (_ : ↑s₃ * rb ∈ ↑S) }",
"tactic": "apply OreLocalization.expand"
}
] |
[
655,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
630,
11
] |
Mathlib/Data/Matrix/PEquiv.lean
|
PEquiv.mul_toPEquiv_toMatrix
|
[
{
"state_after": "no goals",
"state_before": "k : Type ?u.16360\nl : Type ?u.16363\nm✝ : Type ?u.16366\nn✝ : Type ?u.16369\nα✝ : Type v\nm : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : Semiring α\nf : n ≃ n\nM : Matrix m n α\ni : m\nj : n\n⊢ (M ⬝ toMatrix (Equiv.toPEquiv f)) i j = submatrix M id (↑f.symm) i j",
"tactic": "rw [PEquiv.matrix_mul_apply, ← Equiv.toPEquiv_symm, Equiv.toPEquiv_apply,\n Matrix.submatrix_apply, id.def]"
}
] |
[
115,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Data/Real/Basic.lean
|
Real.mk_mul
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\nf g : CauSeq ℚ abs\n⊢ mk (f * g) = mk f * mk g",
"tactic": "simp [mk, ← ofCauchy_mul]"
}
] |
[
337,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/CategoryTheory/Preadditive/Generator.lean
|
CategoryTheory.Preadditive.isSeparating_iff
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Preadditive C\n𝒢 : Set C\nh𝒢 : IsSeparating 𝒢\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ X), h ≫ f = 0\n⊢ ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ X), h ≫ f = h ≫ 0",
"tactic": "simpa only [Limits.comp_zero] using hf"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Preadditive C\n𝒢 : Set C\nh𝒢 : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ X), h ≫ f = 0) → f = 0\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ X), h ≫ f = h ≫ g\n⊢ ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ X), h ≫ (f - g) = 0",
"tactic": "simpa only [Preadditive.comp_sub, sub_eq_zero] using hfg"
}
] |
[
34,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
31,
1
] |
Mathlib/Algebra/Divisibility/Units.lean
|
isUnit_of_dvd_unit
|
[] |
[
136,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.length_darts
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\n⊢ List.length (darts p) = length p",
"tactic": "induction p <;> simp [*]"
}
] |
[
777,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
776,
1
] |
Mathlib/Algebra/Invertible.lean
|
commute_invOf
|
[] |
[
342,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.sInter_eq_iInter
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.167787\nγ : Type ?u.167790\nι : Sort ?u.167793\nι' : Sort ?u.167796\nι₂ : Sort ?u.167799\nκ : ι → Sort ?u.167804\nκ₁ : ι → Sort ?u.167809\nκ₂ : ι → Sort ?u.167814\nκ' : ι' → Sort ?u.167819\ns : Set (Set α)\n⊢ ⋂₀ s = ⋂ (i : ↑s), ↑i",
"tactic": "simp only [← sInter_range, Subtype.range_coe]"
}
] |
[
1326,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1325,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.L1.SimpleFunc.setToL1S_smul_real
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.680254\nG : Type ?u.680257\n𝕜 : Type ?u.680260\np : ℝ≥0∞\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : NormedSpace ℝ F'\ninst✝² : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nT : Set α → E →L[ℝ] F\nh_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0\nh_add : FinMeasAdditive μ T\nc : ℝ\nf : { x // x ∈ simpleFunc E 1 μ }\n⊢ SimpleFunc.setToSimpleFunc T (toSimpleFunc (c • f)) = c • SimpleFunc.setToSimpleFunc T (toSimpleFunc f)",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.680254\nG : Type ?u.680257\n𝕜 : Type ?u.680260\np : ℝ≥0∞\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : NormedSpace ℝ F'\ninst✝² : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nT : Set α → E →L[ℝ] F\nh_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0\nh_add : FinMeasAdditive μ T\nc : ℝ\nf : { x // x ∈ simpleFunc E 1 μ }\n⊢ setToL1S T (c • f) = c • setToL1S T f",
"tactic": "simp_rw [setToL1S]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.680254\nG : Type ?u.680257\n𝕜 : Type ?u.680260\np : ℝ≥0∞\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : NormedSpace ℝ F'\ninst✝² : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nT : Set α → E →L[ℝ] F\nh_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0\nh_add : FinMeasAdditive μ T\nc : ℝ\nf : { x // x ∈ simpleFunc E 1 μ }\n⊢ SimpleFunc.setToSimpleFunc T (toSimpleFunc (c • f)) = SimpleFunc.setToSimpleFunc T (c • toSimpleFunc f)",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.680254\nG : Type ?u.680257\n𝕜 : Type ?u.680260\np : ℝ≥0∞\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : NormedSpace ℝ F'\ninst✝² : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nT : Set α → E →L[ℝ] F\nh_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0\nh_add : FinMeasAdditive μ T\nc : ℝ\nf : { x // x ∈ simpleFunc E 1 μ }\n⊢ SimpleFunc.setToSimpleFunc T (toSimpleFunc (c • f)) = c • SimpleFunc.setToSimpleFunc T (toSimpleFunc f)",
"tactic": "rw [← SimpleFunc.setToSimpleFunc_smul_real T h_add c (SimpleFunc.integrable f)]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.680254\nG : Type ?u.680257\n𝕜 : Type ?u.680260\np : ℝ≥0∞\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : NormedSpace ℝ F'\ninst✝² : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nT : Set α → E →L[ℝ] F\nh_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0\nh_add : FinMeasAdditive μ T\nc : ℝ\nf : { x // x ∈ simpleFunc E 1 μ }\n⊢ ↑(toSimpleFunc (c • f)) =ᵐ[μ] ↑(c • toSimpleFunc f)",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.680254\nG : Type ?u.680257\n𝕜 : Type ?u.680260\np : ℝ≥0∞\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : NormedSpace ℝ F'\ninst✝² : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nT : Set α → E →L[ℝ] F\nh_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0\nh_add : FinMeasAdditive μ T\nc : ℝ\nf : { x // x ∈ simpleFunc E 1 μ }\n⊢ SimpleFunc.setToSimpleFunc T (toSimpleFunc (c • f)) = SimpleFunc.setToSimpleFunc T (c • toSimpleFunc f)",
"tactic": "refine' SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable _) _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.680254\nG : Type ?u.680257\n𝕜 : Type ?u.680260\np : ℝ≥0∞\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : NormedSpace ℝ F'\ninst✝² : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nT : Set α → E →L[ℝ] F\nh_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s = 0 → T s = 0\nh_add : FinMeasAdditive μ T\nc : ℝ\nf : { x // x ∈ simpleFunc E 1 μ }\n⊢ ↑(toSimpleFunc (c • f)) =ᵐ[μ] ↑(c • toSimpleFunc f)",
"tactic": "exact smul_toSimpleFunc c f"
}
] |
[
789,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
783,
1
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
TopologicalSpace.Opens.openEmbedding
|
[] |
[
138,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Topology/Bases.lean
|
Embedding.secondCountableTopology
|
[] |
[
894,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
892,
11
] |
Mathlib/Analysis/Convex/Between.lean
|
sbtw_vadd_const_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.121133\nP : Type u_3\nP' : Type ?u.121139\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y z : V\np : P\n⊢ Sbtw R (x +ᵥ p) (y +ᵥ p) (z +ᵥ p) ↔ Sbtw R x y z",
"tactic": "rw [Sbtw, Sbtw, wbtw_vadd_const_iff, (vadd_right_injective p).ne_iff,\n (vadd_right_injective p).ne_iff]"
}
] |
[
221,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/Analysis/Convex/Combination.lean
|
mem_Icc_of_mem_stdSimplex
|
[] |
[
479,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
478,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
HasFPowerSeriesAt.apply_eq_zero
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nh : HasFPowerSeriesAt 0 p x\nn k : ℕ\nhk : ∀ (m : ℕ), m < k → ∀ (y : E), (↑(p m) fun x => y) = 0\n⊢ ∀ (y : E), (↑(p k) fun x => y) = 0",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nh : HasFPowerSeriesAt 0 p x\nn : ℕ\n⊢ ∀ (y : E), (↑(p n) fun x => y) = 0",
"tactic": "refine' Nat.strong_induction_on n fun k hk => _"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nn k : ℕ\nhk : ∀ (m : ℕ), m < k → ∀ (y : E), (↑(p m) fun x => y) = 0\npsum_eq : FormalMultilinearSeries.partialSum p (k + 1) = fun y => ↑(p k) fun x => y\nh :\n (fun y => OfNat.ofNat 0 (x + y) - FormalMultilinearSeries.partialSum p (Nat.succ k) y) =O[𝓝 0] fun y =>\n ‖y‖ ^ Nat.succ k\n⊢ ∀ (y : E), (↑(p k) fun x => y) = 0",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nh : HasFPowerSeriesAt 0 p x\nn k : ℕ\nhk : ∀ (m : ℕ), m < k → ∀ (y : E), (↑(p m) fun x => y) = 0\npsum_eq : FormalMultilinearSeries.partialSum p (k + 1) = fun y => ↑(p k) fun x => y\n⊢ ∀ (y : E), (↑(p k) fun x => y) = 0",
"tactic": "replace h := h.isBigO_sub_partialSum_pow k.succ"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nn k : ℕ\nhk : ∀ (m : ℕ), m < k → ∀ (y : E), (↑(p m) fun x => y) = 0\npsum_eq : FormalMultilinearSeries.partialSum p (k + 1) = fun y => ↑(p k) fun x => y\nh : (fun x => ↑(p k) fun x_1 => x) =O[𝓝 0] fun y => ‖y‖ ^ Nat.succ k\n⊢ ∀ (y : E), (↑(p k) fun x => y) = 0",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nn k : ℕ\nhk : ∀ (m : ℕ), m < k → ∀ (y : E), (↑(p m) fun x => y) = 0\npsum_eq : FormalMultilinearSeries.partialSum p (k + 1) = fun y => ↑(p k) fun x => y\nh :\n (fun y => OfNat.ofNat 0 (x + y) - FormalMultilinearSeries.partialSum p (Nat.succ k) y) =O[𝓝 0] fun y =>\n ‖y‖ ^ Nat.succ k\n⊢ ∀ (y : E), (↑(p k) fun x => y) = 0",
"tactic": "simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nn k : ℕ\nhk : ∀ (m : ℕ), m < k → ∀ (y : E), (↑(p m) fun x => y) = 0\npsum_eq : FormalMultilinearSeries.partialSum p (k + 1) = fun y => ↑(p k) fun x => y\nh : (fun x => ↑(p k) fun x_1 => x) =O[𝓝 0] fun y => ‖y‖ ^ Nat.succ k\n⊢ ∀ (y : E), (↑(p k) fun x => y) = 0",
"tactic": "exact h.continuousMultilinearMap_apply_eq_zero"
},
{
"state_after": "case h\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nh : HasFPowerSeriesAt 0 p x\nn k : ℕ\nhk : ∀ (m : ℕ), m < k → ∀ (y : E), (↑(p m) fun x => y) = 0\nz : E\n⊢ FormalMultilinearSeries.partialSum p (k + 1) z = ↑(p k) fun x => z",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nh : HasFPowerSeriesAt 0 p x\nn k : ℕ\nhk : ∀ (m : ℕ), m < k → ∀ (y : E), (↑(p m) fun x => y) = 0\n⊢ FormalMultilinearSeries.partialSum p (k + 1) = fun y => ↑(p k) fun x => y",
"tactic": "funext z"
},
{
"state_after": "case h.refine'_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nh : HasFPowerSeriesAt 0 p x\nn k : ℕ\nhk : ∀ (m : ℕ), m < k → ∀ (y : E), (↑(p m) fun x => y) = 0\nz : E\nb : ℕ\nhb : b ∈ Finset.range (k + 1)\nhnb : b ≠ k\n⊢ (↑(p b) fun x => z) = 0\n\ncase h.refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nh : HasFPowerSeriesAt 0 p x\nn k : ℕ\nhk : ∀ (m : ℕ), m < k → ∀ (y : E), (↑(p m) fun x => y) = 0\nz : E\nhn : ¬k ∈ Finset.range (k + 1)\n⊢ (↑(p k) fun x => z) = 0",
"state_before": "case h\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nh : HasFPowerSeriesAt 0 p x\nn k : ℕ\nhk : ∀ (m : ℕ), m < k → ∀ (y : E), (↑(p m) fun x => y) = 0\nz : E\n⊢ FormalMultilinearSeries.partialSum p (k + 1) z = ↑(p k) fun x => z",
"tactic": "refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _"
},
{
"state_after": "case h.refine'_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nh : HasFPowerSeriesAt 0 p x\nn k : ℕ\nhk : ∀ (m : ℕ), m < k → ∀ (y : E), (↑(p m) fun x => y) = 0\nz : E\nb : ℕ\nhb : b ∈ Finset.range (k + 1)\nhnb : b ≠ k\nthis : b ≤ k\n⊢ (↑(p b) fun x => z) = 0",
"state_before": "case h.refine'_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nh : HasFPowerSeriesAt 0 p x\nn k : ℕ\nhk : ∀ (m : ℕ), m < k → ∀ (y : E), (↑(p m) fun x => y) = 0\nz : E\nb : ℕ\nhb : b ∈ Finset.range (k + 1)\nhnb : b ≠ k\n⊢ (↑(p b) fun x => z) = 0",
"tactic": "have := Finset.mem_range_succ_iff.mp hb"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nh : HasFPowerSeriesAt 0 p x\nn k : ℕ\nhk : ∀ (m : ℕ), m < k → ∀ (y : E), (↑(p m) fun x => y) = 0\nz : E\nb : ℕ\nhb : b ∈ Finset.range (k + 1)\nhnb : b ≠ k\nthis : b ≤ k\n⊢ (↑(p b) fun x => z) = 0",
"tactic": "simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1119376\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nh : HasFPowerSeriesAt 0 p x\nn k : ℕ\nhk : ∀ (m : ℕ), m < k → ∀ (y : E), (↑(p m) fun x => y) = 0\nz : E\nhn : ¬k ∈ Finset.range (k + 1)\n⊢ (↑(p k) fun x => z) = 0",
"tactic": "exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))"
}
] |
[
999,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
988,
1
] |
Mathlib/NumberTheory/Padics/RingHoms.lean
|
PadicInt.lift_self
|
[
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type ?u.654027\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nz : ℤ_[p]\n⊢ ↑(lift\n (_ :\n ∀ (m n : ℕ) (h : m ≤ n),\n RingHom.comp (ZMod.castHom (_ : p ^ m ∣ p ^ n) (ZMod (p ^ m))) (toZModPow n) = toZModPow m))\n z =\n ↑(RingHom.id ℤ_[p]) z",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type ?u.654027\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nz : ℤ_[p]\n⊢ ↑(lift\n (_ :\n ∀ (m n : ℕ) (h : m ≤ n),\n RingHom.comp (ZMod.castHom (_ : p ^ m ∣ p ^ n) (ZMod (p ^ m))) (toZModPow n) = toZModPow m))\n z =\n z",
"tactic": "show _ = RingHom.id _ z"
},
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type ?u.654027\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nz : ℤ_[p]\n⊢ ∀ (n : ℕ), RingHom.comp (toZModPow n) (RingHom.id ℤ_[p]) = toZModPow n",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type ?u.654027\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nz : ℤ_[p]\n⊢ ↑(lift\n (_ :\n ∀ (m n : ℕ) (h : m ≤ n),\n RingHom.comp (ZMod.castHom (_ : p ^ m ∣ p ^ n) (ZMod (p ^ m))) (toZModPow n) = toZModPow m))\n z =\n ↑(RingHom.id ℤ_[p]) z",
"tactic": "rw [@lift_unique p _ ℤ_[p] _ _ zmod_cast_comp_toZModPow (RingHom.id ℤ_[p])]"
},
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type ?u.654027\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nz : ℤ_[p]\nn✝ : ℕ\n⊢ RingHom.comp (toZModPow n✝) (RingHom.id ℤ_[p]) = toZModPow n✝",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type ?u.654027\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nz : ℤ_[p]\n⊢ ∀ (n : ℕ), RingHom.comp (toZModPow n) (RingHom.id ℤ_[p]) = toZModPow n",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type ?u.654027\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nz : ℤ_[p]\nn✝ : ℕ\n⊢ RingHom.comp (toZModPow n✝) (RingHom.id ℤ_[p]) = toZModPow n✝",
"tactic": "rw [RingHom.comp_id]"
}
] |
[
666,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
663,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.map_filter
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.30514\nf✝ : α ↪ β\ns : Finset α\nf : α ≃ β\np : α → Prop\ninst✝ : DecidablePred p\n⊢ map (Equiv.toEmbedding f) (filter p s) = filter (p ∘ ↑f.symm) (map (Equiv.toEmbedding f) s)",
"tactic": "simp only [filter_map, Function.comp, Equiv.toEmbedding_apply, Equiv.symm_apply_apply]"
}
] |
[
189,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Data/Analysis/Topology.lean
|
Ctop.Realizer.isClosed_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.8619\nσ : Type ?u.8622\nτ : Type ?u.8625\ninst✝ : TopologicalSpace α\nF : Realizer α\ns : Set α\na : α\nthis : (a : Prop) → Decidable a\n⊢ (¬a ∈ s → ∃ b, a ∈ f F.F b ∧ ∀ (z : α), z ∈ f F.F b → ¬z ∈ s) ↔\n (∀ (b : F.σ), a ∈ f F.F b → ∃ z, z ∈ f F.F b ∩ s) → a ∈ s",
"state_before": "α : Type u_1\nβ : Type ?u.8619\nσ : Type ?u.8622\nτ : Type ?u.8625\ninst✝ : TopologicalSpace α\nF : Realizer α\ns : Set α\na : α\n⊢ (¬a ∈ s → ∃ b, a ∈ f F.F b ∧ ∀ (z : α), z ∈ f F.F b → ¬z ∈ s) ↔\n (∀ (b : F.σ), a ∈ f F.F b → ∃ z, z ∈ f F.F b ∩ s) → a ∈ s",
"tactic": "haveI := Classical.propDecidable"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.8619\nσ : Type ?u.8622\nτ : Type ?u.8625\ninst✝ : TopologicalSpace α\nF : Realizer α\ns : Set α\na : α\nthis : (a : Prop) → Decidable a\n⊢ (¬∃ b, a ∈ f F.F b ∧ ∀ (z : α), z ∈ f F.F b → ¬z ∈ s) → a ∈ s ↔\n (∀ (b : F.σ), a ∈ f F.F b → ∃ z, z ∈ f F.F b ∩ s) → a ∈ s",
"state_before": "α : Type u_1\nβ : Type ?u.8619\nσ : Type ?u.8622\nτ : Type ?u.8625\ninst✝ : TopologicalSpace α\nF : Realizer α\ns : Set α\na : α\nthis : (a : Prop) → Decidable a\n⊢ (¬a ∈ s → ∃ b, a ∈ f F.F b ∧ ∀ (z : α), z ∈ f F.F b → ¬z ∈ s) ↔\n (∀ (b : F.σ), a ∈ f F.F b → ∃ z, z ∈ f F.F b ∩ s) → a ∈ s",
"tactic": "rw [not_imp_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.8619\nσ : Type ?u.8622\nτ : Type ?u.8625\ninst✝ : TopologicalSpace α\nF : Realizer α\ns : Set α\na : α\nthis : (a : Prop) → Decidable a\n⊢ (¬∃ b, a ∈ f F.F b ∧ ∀ (z : α), z ∈ f F.F b → ¬z ∈ s) → a ∈ s ↔\n (∀ (b : F.σ), a ∈ f F.F b → ∃ z, z ∈ f F.F b ∩ s) → a ∈ s",
"tactic": "simp [not_exists, not_and, not_forall, and_comm]"
}
] |
[
152,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Topology/Instances/NNReal.lean
|
NNReal.tsum_mul_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\na : ℝ≥0\nf : α → ℝ≥0\n⊢ ↑(∑' (x : α), a * f x) = ↑(a * ∑' (x : α), f x)",
"tactic": "simp only [coe_tsum, NNReal.coe_mul, tsum_mul_left]"
}
] |
[
195,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
8
] |
Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean
|
Finset.card_mul_mul_card_le_card_mul_mul_card_mul
|
[
{
"state_after": "case inl\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B C✝ A C : Finset α\n⊢ card (A * C) * card ∅ ≤ card (A * ∅) * card (∅ * C)\n\ncase inr\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\n⊢ card (A * C) * card B ≤ card (A * B) * card (B * C)",
"state_before": "α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ card (A * C) * card B ≤ card (A * B) * card (B * C)",
"tactic": "obtain rfl | hB := B.eq_empty_or_nonempty"
},
{
"state_after": "case inr\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\n⊢ card (A * C) * card B ≤ card (A * B) * card (B * C)",
"state_before": "case inr\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\n⊢ card (A * C) * card B ≤ card (A * B) * card (B * C)",
"tactic": "have hB' : B ∈ B.powerset.erase ∅ := mem_erase_of_ne_of_mem hB.ne_empty (mem_powerset_self _)"
},
{
"state_after": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : U ∈ erase (powerset B) ∅\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ card (A * C) * card B ≤ card (A * B) * card (B * C)",
"state_before": "case inr\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\n⊢ card (A * C) * card B ≤ card (A * B) * card (B * C)",
"tactic": "obtain ⟨U, hU, hUA⟩ :=\n exists_min_image (B.powerset.erase ∅) (fun U ↦ (U * A).card / U.card : _ → ℚ≥0) ⟨B, hB'⟩"
},
{
"state_after": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ card (A * C) * card B ≤ card (A * B) * card (B * C)",
"state_before": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : U ∈ erase (powerset B) ∅\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ card (A * C) * card B ≤ card (A * B) * card (B * C)",
"tactic": "rw [mem_erase, mem_powerset, ← nonempty_iff_ne_empty] at hU"
},
{
"state_after": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (A * C) * card B) ≤ ↑(card (A * B) * card (B * C))",
"state_before": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ card (A * C) * card B ≤ card (A * B) * card (B * C)",
"tactic": "refine' cast_le.1 (_ : (_ : ℚ≥0) ≤ _)"
},
{
"state_after": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (A * C)) * ↑(card B) ≤ ↑(card (A * B)) * ↑(card (B * C))",
"state_before": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (A * C) * card B) ≤ ↑(card (A * B) * card (B * C))",
"tactic": "push_cast"
},
{
"state_after": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (A * C)) ≤ ↑(card (A * B)) * ↑(card (B * C)) / ↑(card B)",
"state_before": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (A * C)) * ↑(card B) ≤ ↑(card (A * B)) * ↑(card (B * C))",
"tactic": "refine' (le_div_iff <| cast_pos.2 hB.card_pos).1 _"
},
{
"state_after": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (A * C)) ≤ ↑(card (B * A)) / ↑(card B) * ↑(card (B * C))",
"state_before": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (A * C)) ≤ ↑(card (A * B)) * ↑(card (B * C)) / ↑(card B)",
"tactic": "rw [mul_div_right_comm, mul_comm _ B]"
},
{
"state_after": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (U * (A * C))) ≤ ↑(card (B * A)) / ↑(card B) * ↑(card (B * C))",
"state_before": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (A * C)) ≤ ↑(card (B * A)) / ↑(card B) * ↑(card (B * C))",
"tactic": "refine' (cast_le.2 <| card_le_card_mul_left _ hU.1).trans _"
},
{
"state_after": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (U * (A * C))) ≤ ↑(card (U * A)) / ↑(card U) * ↑(card (U * C))",
"state_before": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (U * (A * C))) ≤ ↑(card (B * A)) / ↑(card B) * ↑(card (B * C))",
"tactic": "refine' le_trans _\n (mul_le_mul (hUA _ hB') (cast_le.2 <| card_le_of_subset <| mul_subset_mul_right hU.2)\n (zero_le _) (zero_le _))"
},
{
"state_after": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (U * A * C)) ≤ ↑(card (U * A)) * ↑(card (U * C)) / ↑(card U)",
"state_before": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (U * (A * C))) ≤ ↑(card (U * A)) / ↑(card U) * ↑(card (U * C))",
"tactic": "rw [← mul_div_right_comm, ← mul_assoc]"
},
{
"state_after": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (U * A * C)) * ↑(card U) ≤ ↑(card (U * A)) * ↑(card (U * C))",
"state_before": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (U * A * C)) ≤ ↑(card (U * A)) * ↑(card (U * C)) / ↑(card U)",
"tactic": "refine' (le_div_iff <| cast_pos.2 hU.1.card_pos).2 _"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\nhB : Finset.Nonempty B\nhB' : B ∈ erase (powerset B) ∅\nU : Finset α\nhU : Finset.Nonempty U ∧ U ⊆ B\nhUA : ∀ (x' : Finset α), x' ∈ erase (powerset B) ∅ → ↑(card (U * A)) / ↑(card U) ≤ ↑(card (x' * A)) / ↑(card x')\n⊢ ↑(card (U * A * C)) * ↑(card U) ≤ ↑(card (U * A)) * ↑(card (U * C))",
"tactic": "exact_mod_cast mul_pluennecke_petridis C (mul_aux hU.1 hU.2 hUA)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B C✝ A C : Finset α\n⊢ card (A * C) * card ∅ ≤ card (A * ∅) * card (∅ * C)",
"tactic": "simp"
}
] |
[
160,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_eraseMin
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nsize✝ sz : ℕ\nl' : Ordnode α\ny : α\nr' : Ordnode α\nx : α\nr : Ordnode α\n⊢ dual (eraseMin (node size✝ (node sz l' y r') x r)) = eraseMax (dual (node size✝ (node sz l' y r') x r))",
"tactic": "rw [eraseMin, dual_balanceR, dual_eraseMin (node sz l' y r'), dual, dual, dual, eraseMax]"
}
] |
[
602,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
598,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
Submonoid.comap_le_comap_iff_of_surjective
|
[] |
[
495,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
494,
1
] |
Mathlib/Computability/Ackermann.lean
|
ack_succ_zero
|
[
{
"state_after": "no goals",
"state_before": "m : ℕ\n⊢ ack (m + 1) 0 = ack m 1",
"tactic": "rw [ack]"
}
] |
[
76,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Data/PEquiv.lean
|
PEquiv.trans_eq_none
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\ng : β ≃. γ\na : α\n⊢ (∀ (a_1 : γ), ¬∃ b, b ∈ ↑f a ∧ a_1 ∈ ↑g b) ↔ ∀ (b : β) (c : γ), b ∈ ↑f a → ¬c ∈ ↑g b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\ng : β ≃. γ\na : α\n⊢ ↑(PEquiv.trans f g) a = none ↔ ∀ (b : β) (c : γ), ¬b ∈ ↑f a ∨ ¬c ∈ ↑g b",
"tactic": "simp only [eq_none_iff_forall_not_mem, mem_trans, imp_iff_not_or.symm]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\ng : β ≃. γ\na : α\n⊢ (∀ (a_1 : γ) (b : β), b ∈ ↑f a → ¬a_1 ∈ ↑g b) ↔ ∀ (b : β) (c : γ), b ∈ ↑f a → ¬c ∈ ↑g b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\ng : β ≃. γ\na : α\n⊢ (∀ (a_1 : γ), ¬∃ b, b ∈ ↑f a ∧ a_1 ∈ ↑g b) ↔ ∀ (b : β) (c : γ), b ∈ ↑f a → ¬c ∈ ↑g b",
"tactic": "push_neg"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\ng : β ≃. γ\na : α\n⊢ (∀ (a_1 : γ) (b : β), b ∈ ↑f a → ¬a_1 ∈ ↑g b) ↔ ∀ (b : β) (c : γ), b ∈ ↑f a → ¬c ∈ ↑g b",
"tactic": "exact forall_swap"
}
] |
[
164,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.le_min'_iff
|
[] |
[
1342,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1341,
1
] |
Mathlib/Analysis/Convex/Gauge.lean
|
gauge_of_subset_zero
|
[
{
"state_after": "case inl\n𝕜 : Type ?u.34254\nE : Type u_1\nF : Type ?u.34260\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nt : Set E\na : ℝ\nh : ∅ ⊆ 0\n⊢ gauge ∅ = 0\n\ncase inr\n𝕜 : Type ?u.34254\nE : Type u_1\nF : Type ?u.34260\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nt : Set E\na : ℝ\nh : {0} ⊆ 0\n⊢ gauge {0} = 0",
"state_before": "𝕜 : Type ?u.34254\nE : Type u_1\nF : Type ?u.34260\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nh : s ⊆ 0\n⊢ gauge s = 0",
"tactic": "obtain rfl | rfl := subset_singleton_iff_eq.1 h"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type ?u.34254\nE : Type u_1\nF : Type ?u.34260\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nt : Set E\na : ℝ\nh : ∅ ⊆ 0\n⊢ gauge ∅ = 0\n\ncase inr\n𝕜 : Type ?u.34254\nE : Type u_1\nF : Type ?u.34260\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nt : Set E\na : ℝ\nh : {0} ⊆ 0\n⊢ gauge {0} = 0",
"tactic": "exacts [gauge_empty, gauge_zero']"
}
] |
[
128,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/RingTheory/TensorProduct.lean
|
TensorProduct.AlgebraTensorModule.lift_tmul
|
[] |
[
155,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean
|
FreeAbelianGroup.lift.map_hom
|
[
{
"state_after": "α✝ : Type u\nβ✝ : Type v\ninst✝² : AddCommGroup β✝\nf✝ : α✝ → β✝\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : AddCommGroup β\ninst✝ : AddCommGroup γ\na : FreeAbelianGroup α\nf : α → β\ng : β →+ γ\n⊢ ↑(AddMonoidHom.comp g (↑lift f)) a = ↑(↑lift (↑g ∘ f)) a",
"state_before": "α✝ : Type u\nβ✝ : Type v\ninst✝² : AddCommGroup β✝\nf✝ : α✝ → β✝\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : AddCommGroup β\ninst✝ : AddCommGroup γ\na : FreeAbelianGroup α\nf : α → β\ng : β →+ γ\n⊢ ↑g (↑(↑lift f) a) = ↑(↑lift (↑g ∘ f)) a",
"tactic": "show (g.comp (lift f)) a = lift (g ∘ f) a"
},
{
"state_after": "case hg\nα✝ : Type u\nβ✝ : Type v\ninst✝² : AddCommGroup β✝\nf✝ : α✝ → β✝\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : AddCommGroup β\ninst✝ : AddCommGroup γ\na : FreeAbelianGroup α\nf : α → β\ng : β →+ γ\n⊢ ∀ (x : α), ↑(AddMonoidHom.comp g (↑lift f)) (of x) = (↑g ∘ f) x",
"state_before": "α✝ : Type u\nβ✝ : Type v\ninst✝² : AddCommGroup β✝\nf✝ : α✝ → β✝\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : AddCommGroup β\ninst✝ : AddCommGroup γ\na : FreeAbelianGroup α\nf : α → β\ng : β →+ γ\n⊢ ↑(AddMonoidHom.comp g (↑lift f)) a = ↑(↑lift (↑g ∘ f)) a",
"tactic": "apply lift.unique"
},
{
"state_after": "case hg\nα✝ : Type u\nβ✝ : Type v\ninst✝² : AddCommGroup β✝\nf✝ : α✝ → β✝\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : AddCommGroup β\ninst✝ : AddCommGroup γ\na✝ : FreeAbelianGroup α\nf : α → β\ng : β →+ γ\na : α\n⊢ ↑(AddMonoidHom.comp g (↑lift f)) (of a) = (↑g ∘ f) a",
"state_before": "case hg\nα✝ : Type u\nβ✝ : Type v\ninst✝² : AddCommGroup β✝\nf✝ : α✝ → β✝\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : AddCommGroup β\ninst✝ : AddCommGroup γ\na : FreeAbelianGroup α\nf : α → β\ng : β →+ γ\n⊢ ∀ (x : α), ↑(AddMonoidHom.comp g (↑lift f)) (of x) = (↑g ∘ f) x",
"tactic": "intro a"
},
{
"state_after": "case hg\nα✝ : Type u\nβ✝ : Type v\ninst✝² : AddCommGroup β✝\nf✝ : α✝ → β✝\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : AddCommGroup β\ninst✝ : AddCommGroup γ\na✝ : FreeAbelianGroup α\nf : α → β\ng : β →+ γ\na : α\n⊢ ↑g (↑(↑lift f) (of a)) = ↑g (f a)",
"state_before": "case hg\nα✝ : Type u\nβ✝ : Type v\ninst✝² : AddCommGroup β✝\nf✝ : α✝ → β✝\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : AddCommGroup β\ninst✝ : AddCommGroup γ\na✝ : FreeAbelianGroup α\nf : α → β\ng : β →+ γ\na : α\n⊢ ↑(AddMonoidHom.comp g (↑lift f)) (of a) = (↑g ∘ f) a",
"tactic": "show g ((lift f) (of a)) = g (f a)"
}
] |
[
136,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
WithBot.succ_unbot
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.61733\ninst✝² : Preorder α\ninst✝¹ : OrderBot α\ninst✝ : SuccOrder α\na : WithBot α\nha : a ≠ ⊥\n⊢ succ a ≠ ⊥",
"tactic": "induction a using WithBot.recBotCoe <;> simp"
}
] |
[
1224,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1220,
1
] |
Mathlib/Topology/MetricSpace/Holder.lean
|
HolderWith.nndist_le_of_le
|
[
{
"state_after": "X : Type u_1\nY : Type u_2\nZ : Type ?u.903365\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ nndist (f x) (f y) ≤ C * d ^ ↑r",
"state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.903365\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ ↑(nndist (f x) (f y)) ≤ ↑C * ↑d ^ ↑r",
"tactic": "norm_cast"
},
{
"state_after": "X : Type u_1\nY : Type u_2\nZ : Type ?u.903365\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ edist (f x) (f y) ≤ ↑C * ↑d ^ ↑r",
"state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.903365\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ nndist (f x) (f y) ≤ C * d ^ ↑r",
"tactic": "rw [← ENNReal.coe_le_coe, ← edist_nndist, ENNReal.coe_mul, ←\n ENNReal.coe_rpow_of_nonneg _ r.coe_nonneg]"
},
{
"state_after": "X : Type u_1\nY : Type u_2\nZ : Type ?u.903365\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ edist x y ≤ ↑d",
"state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.903365\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ edist (f x) (f y) ≤ ↑C * ↑d ^ ↑r",
"tactic": "apply hf.edist_le_of_le"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.903365\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ edist x y ≤ ↑d",
"tactic": "rwa [edist_nndist, ENNReal.coe_le_coe]"
}
] |
[
246,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/GroupTheory/Perm/Subgroup.lean
|
Equiv.Perm.sigmaCongrRightHom.card_range
|
[] |
[
55,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Std/Data/Array/Init/Lemmas.lean
|
Array.foldl_data_eq_bind
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nl : List α\nacc : Array β\nF : Array β → α → Array β\nG : α → List β\nH : ∀ (acc : Array β) (a : α), (F acc a).data = acc.data ++ G a\n⊢ (List.foldl F acc l).data = acc.data ++ List.bind l G",
"tactic": "induction l generalizing acc <;> simp [*, List.bind]"
}
] |
[
210,
55
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
206,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.trans_apply
|
[] |
[
280,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
9
] |
Mathlib/Topology/Sets/Compacts.lean
|
TopologicalSpace.PositiveCompacts.isCompact
|
[] |
[
333,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
11
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.SimpleFunc.integral_piecewise_zero
|
[
{
"state_after": "case refine'_1\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\n⊢ filter (fun x => x ≠ 0) (SimpleFunc.range (piecewise s hs f 0)) ⊆ filter (fun x => x ≠ 0) (SimpleFunc.range f)\n\ncase refine'_2\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\ny : F\nhy : y ∈ filter (fun x => x ≠ 0) (SimpleFunc.range f)\n⊢ ENNReal.toReal (↑↑μ (↑(piecewise s hs f 0) ⁻¹' {y})) • y = ENNReal.toReal (↑↑(Measure.restrict μ s) (↑f ⁻¹' {y})) • y",
"state_before": "α : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\n⊢ integral μ (piecewise s hs f 0) = integral (Measure.restrict μ s) f",
"tactic": "refine' (integral_eq_sum_of_subset _).trans\n ((sum_congr rfl fun y hy => _).trans (integral_eq_sum_filter _ _).symm)"
},
{
"state_after": "case refine'_1\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\ny : F\nhy : y ∈ filter (fun x => x ≠ 0) (SimpleFunc.range (piecewise s hs f 0))\n⊢ y ∈ filter (fun x => x ≠ 0) (SimpleFunc.range f)",
"state_before": "case refine'_1\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\n⊢ filter (fun x => x ≠ 0) (SimpleFunc.range (piecewise s hs f 0)) ⊆ filter (fun x => x ≠ 0) (SimpleFunc.range f)",
"tactic": "intro y hy"
},
{
"state_after": "case refine'_1\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\ny : F\nhy : (y = 0 ∧ s ≠ Set.univ ∨ y ∈ ↑f '' s) ∧ y ≠ 0\n⊢ y ∈ Set.range ↑f ∧ y ≠ 0",
"state_before": "case refine'_1\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\ny : F\nhy : y ∈ filter (fun x => x ≠ 0) (SimpleFunc.range (piecewise s hs f 0))\n⊢ y ∈ filter (fun x => x ≠ 0) (SimpleFunc.range f)",
"tactic": "simp only [mem_filter, mem_range, coe_piecewise, coe_zero, piecewise_eq_indicator,\n mem_range_indicator] at *"
},
{
"state_after": "case refine'_1.intro.inl.intro\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nh₀ : 0 ≠ 0\n⊢ 0 ∈ Set.range ↑f ∧ 0 ≠ 0\n\ncase refine'_1.intro.inr.intro.intro\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nx : α\nh₀ : ↑f x ≠ 0\n⊢ ↑f x ∈ Set.range ↑f ∧ ↑f x ≠ 0",
"state_before": "case refine'_1\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\ny : F\nhy : (y = 0 ∧ s ≠ Set.univ ∨ y ∈ ↑f '' s) ∧ y ≠ 0\n⊢ y ∈ Set.range ↑f ∧ y ≠ 0",
"tactic": "rcases hy with ⟨⟨rfl, -⟩ | ⟨x, -, rfl⟩, h₀⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.inl.intro\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nh₀ : 0 ≠ 0\n⊢ 0 ∈ Set.range ↑f ∧ 0 ≠ 0\n\ncase refine'_1.intro.inr.intro.intro\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nx : α\nh₀ : ↑f x ≠ 0\n⊢ ↑f x ∈ Set.range ↑f ∧ ↑f x ≠ 0",
"tactic": "exacts [(h₀ rfl).elim, ⟨Set.mem_range_self _, h₀⟩]"
},
{
"state_after": "case refine'_2\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\ny : F\nhy : y ∈ filter (fun x => x ≠ 0) (SimpleFunc.range f)\n⊢ ENNReal.toReal (↑↑μ (Set.piecewise s (↑f) 0 ⁻¹' {y})) • y = ENNReal.toReal (↑↑(Measure.restrict μ s) (↑f ⁻¹' {y})) • y",
"state_before": "case refine'_2\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\ny : F\nhy : y ∈ filter (fun x => x ≠ 0) (SimpleFunc.range f)\n⊢ ENNReal.toReal (↑↑μ (↑(piecewise s hs f 0) ⁻¹' {y})) • y = ENNReal.toReal (↑↑(Measure.restrict μ s) (↑f ⁻¹' {y})) • y",
"tactic": "dsimp"
},
{
"state_after": "case refine'_2.ht\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\ny : F\nhy : y ∈ filter (fun x => x ≠ 0) (SimpleFunc.range f)\n⊢ ¬0 ∈ {y}",
"state_before": "case refine'_2\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\ny : F\nhy : y ∈ filter (fun x => x ≠ 0) (SimpleFunc.range f)\n⊢ ENNReal.toReal (↑↑μ (Set.piecewise s (↑f) 0 ⁻¹' {y})) • y = ENNReal.toReal (↑↑(Measure.restrict μ s) (↑f ⁻¹' {y})) • y",
"tactic": "rw [Set.piecewise_eq_indicator, indicator_preimage_of_not_mem,\n Measure.restrict_apply (f.measurableSet_preimage _)]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.ht\nα : Type u_1\nE : Type ?u.125245\nF : Type u_2\n𝕜 : Type ?u.125251\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.125353\nF' : Type ?u.125356\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\ny : F\nhy : y ∈ filter (fun x => x ≠ 0) (SimpleFunc.range f)\n⊢ ¬0 ∈ {y}",
"tactic": "exact fun h₀ => (mem_filter.1 hy).2 (Eq.symm h₀)"
}
] |
[
370,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingIso.coe_orderIso_refl
|
[] |
[
452,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
451,
1
] |
Mathlib/Topology/ContinuousFunction/Compact.lean
|
ContinuousMap.continuous_eval_const
|
[] |
[
174,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.uniformInducing_iff
|
[
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.25437\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nf : α → β\n⊢ (∀ (i' : ℝ≥0∞),\n 0 < i' →\n ∃ i,\n 0 < i ∧\n ∀ (a b : α),\n (a, b) ∈ Prod.map f f ⁻¹' {p | edist p.fst p.snd < i} → (a, b) ∈ {p | edist p.fst p.snd < i'}) ↔\n ∀ (δ : ℝ≥0∞), δ > 0 → ∃ ε, ε > 0 ∧ ∀ {a b : α}, edist (f a) (f b) < ε → edist a b < δ",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.25437\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nf : α → β\n⊢ (∀ (i' : ℝ≥0∞), 0 < i' → ∃ i, 0 < i ∧ Prod.map f f ⁻¹' {p | edist p.fst p.snd < i} ⊆ {p | edist p.fst p.snd < i'}) ↔\n ∀ (δ : ℝ≥0∞), δ > 0 → ∃ ε, ε > 0 ∧ ∀ {a b : α}, edist (f a) (f b) < ε → edist a b < δ",
"tactic": "simp only [subset_def, Prod.forall]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.25437\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nf : α → β\n⊢ (∀ (i' : ℝ≥0∞),\n 0 < i' →\n ∃ i,\n 0 < i ∧\n ∀ (a b : α),\n (a, b) ∈ Prod.map f f ⁻¹' {p | edist p.fst p.snd < i} → (a, b) ∈ {p | edist p.fst p.snd < i'}) ↔\n ∀ (δ : ℝ≥0∞), δ > 0 → ∃ ε, ε > 0 ∧ ∀ {a b : α}, edist (f a) (f b) < ε → edist a b < δ",
"tactic": "rfl"
}
] |
[
296,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
291,
1
] |
Mathlib/LinearAlgebra/Ray.lean
|
SameRay.sameRay_map_iff
|
[] |
[
182,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/SetTheory/Ordinal/Principal.lean
|
Ordinal.unbounded_principal
|
[] |
[
127,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Init/Data/Bool/Lemmas.lean
|
Bool.eq_true_of_not_eq_false
|
[] |
[
83,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.bijOn_iInter
|
[] |
[
1632,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1628,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.div_eq_inv_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.213054\nβ : Type ?u.213057\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ a / b = b⁻¹ * a",
"tactic": "rw [div_eq_mul_inv, mul_comm]"
}
] |
[
1330,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1330,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.singleton_smul_singleton
|
[] |
[
165,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.log_of_ne_zero
|
[] |
[
49,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
1
] |
Mathlib/Order/Max.lean
|
IsBot.mono
|
[] |
[
301,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Data/Nat/Fib.lean
|
Nat.fast_fib_aux_bit_tt
|
[
{
"state_after": "n : ℕ\n⊢ (if true = true then\n ((binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).snd ^\n 2 +\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).fst ^\n 2,\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).snd *\n (2 *\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).fst +\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).snd))\n else\n ((binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).fst *\n (2 *\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).snd -\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).fst),\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).snd ^\n 2 +\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).fst ^\n 2)) =\n let p :=\n binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n;\n (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n\ncase h\nn : ℕ\n⊢ (if false = true then\n ((fib 0, fib 1).snd ^ 2 + (fib 0, fib 1).fst ^ 2,\n (fib 0, fib 1).snd * (2 * (fib 0, fib 1).fst + (fib 0, fib 1).snd))\n else\n ((fib 0, fib 1).fst * (2 * (fib 0, fib 1).snd - (fib 0, fib 1).fst),\n (fib 0, fib 1).snd ^ 2 + (fib 0, fib 1).fst ^ 2)) =\n (fib 0, fib 1)",
"state_before": "n : ℕ\n⊢ fastFibAux (bit true n) =\n let p := fastFibAux n;\n (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))",
"tactic": "rw [fastFibAux, binaryRec_eq]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ (if true = true then\n ((binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).snd ^\n 2 +\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).fst ^\n 2,\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).snd *\n (2 *\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).fst +\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).snd))\n else\n ((binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).fst *\n (2 *\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).snd -\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).fst),\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).snd ^\n 2 +\n (binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n).fst ^\n 2)) =\n let p :=\n binaryRec (fib 0, fib 1)\n (fun b x p =>\n if b = true then (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))\n else (p.fst * (2 * p.snd - p.fst), p.snd ^ 2 + p.fst ^ 2))\n n;\n (p.snd ^ 2 + p.fst ^ 2, p.snd * (2 * p.fst + p.snd))",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case h\nn : ℕ\n⊢ (if false = true then\n ((fib 0, fib 1).snd ^ 2 + (fib 0, fib 1).fst ^ 2,\n (fib 0, fib 1).snd * (2 * (fib 0, fib 1).fst + (fib 0, fib 1).snd))\n else\n ((fib 0, fib 1).fst * (2 * (fib 0, fib 1).snd - (fib 0, fib 1).fst),\n (fib 0, fib 1).snd ^ 2 + (fib 0, fib 1).fst ^ 2)) =\n (fib 0, fib 1)",
"tactic": "simp"
}
] |
[
233,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Order/Lattice.lean
|
lt_sup_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : LinearOrder α\na b c d : α\n⊢ a < b ⊔ c ↔ a < b ∨ a < c",
"tactic": "exact ⟨fun h =>\n (le_total c b).imp\n (fun bc => by rwa [sup_eq_left.2 bc] at h)\n (fun bc => by rwa [sup_eq_right.2 bc] at h),\n fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : LinearOrder α\na b c d : α\nh : a < b ⊔ c\nbc : c ≤ b\n⊢ a < b",
"tactic": "rwa [sup_eq_left.2 bc] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : LinearOrder α\na b c d : α\nh : a < b ⊔ c\nbc : b ≤ c\n⊢ a < c",
"tactic": "rwa [sup_eq_right.2 bc] at h"
}
] |
[
858,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
853,
1
] |
Mathlib/Topology/NhdsSet.lean
|
nhdsSet_union
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.13421\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t✝ s₁ s₂ t₁ t₂ : Set α\nx : α\ns t : Set α\n⊢ 𝓝ˢ (s ∪ t) = 𝓝ˢ s ⊔ 𝓝ˢ t",
"tactic": "simp only [nhdsSet, image_union, sSup_union]"
}
] |
[
135,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Data/Int/Log.lean
|
Int.log_natCast
|
[
{
"state_after": "case zero\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\n⊢ log b ↑Nat.zero = ↑(Nat.log b Nat.zero)\n\ncase succ\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb n✝ : ℕ\n⊢ log b ↑(Nat.succ n✝) = ↑(Nat.log b (Nat.succ n✝))",
"state_before": "R : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb n : ℕ\n⊢ log b ↑n = ↑(Nat.log b n)",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\n⊢ log b ↑Nat.zero = ↑(Nat.log b Nat.zero)",
"tactic": "simp [log_of_right_le_one]"
},
{
"state_after": "case succ.hr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb n✝ : ℕ\n⊢ 1 ≤ ↑(Nat.succ n✝)",
"state_before": "case succ\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb n✝ : ℕ\n⊢ log b ↑(Nat.succ n✝) = ↑(Nat.log b (Nat.succ n✝))",
"tactic": "rw [log_of_one_le_right, Nat.floor_coe]"
},
{
"state_after": "no goals",
"state_before": "case succ.hr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb n✝ : ℕ\n⊢ 1 ≤ ↑(Nat.succ n✝)",
"tactic": "simp"
}
] |
[
81,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
IsBoundedLinearMap.contDiff
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.95197\ninst✝¹ : NormedAddCommGroup X\ninst✝ : 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\nhf : IsBoundedLinearMap 𝕜 f\nh : ContDiff 𝕜 ⊤ f\n⊢ ContDiff 𝕜 n f\n\ncase h\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.95197\ninst✝¹ : NormedAddCommGroup X\ninst✝ : 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\nhf : IsBoundedLinearMap 𝕜 f\n⊢ ContDiff 𝕜 ⊤ f",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.95197\ninst✝¹ : NormedAddCommGroup X\ninst✝ : 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\nhf : IsBoundedLinearMap 𝕜 f\n⊢ ContDiff 𝕜 n f",
"tactic": "suffices h : ContDiff 𝕜 ∞ f"
},
{
"state_after": "case h\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.95197\ninst✝¹ : NormedAddCommGroup X\ninst✝ : 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\nhf : IsBoundedLinearMap 𝕜 f\n⊢ Differentiable 𝕜 f ∧ ContDiff 𝕜 ⊤ fun y => _root_.fderiv 𝕜 f y",
"state_before": "case h\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.95197\ninst✝¹ : NormedAddCommGroup X\ninst✝ : 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\nhf : IsBoundedLinearMap 𝕜 f\n⊢ ContDiff 𝕜 ⊤ f",
"tactic": "rw [contDiff_top_iff_fderiv]"
},
{
"state_after": "case h\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.95197\ninst✝¹ : NormedAddCommGroup X\ninst✝ : 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\nhf : IsBoundedLinearMap 𝕜 f\n⊢ ContDiff 𝕜 ⊤ fun y => _root_.fderiv 𝕜 f y",
"state_before": "case h\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.95197\ninst✝¹ : NormedAddCommGroup X\ninst✝ : 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\nhf : IsBoundedLinearMap 𝕜 f\n⊢ Differentiable 𝕜 f ∧ ContDiff 𝕜 ⊤ fun y => _root_.fderiv 𝕜 f y",
"tactic": "refine' ⟨hf.differentiable, _⟩"
},
{
"state_after": "case h\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.95197\ninst✝¹ : NormedAddCommGroup X\ninst✝ : 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\nhf : IsBoundedLinearMap 𝕜 f\n⊢ ContDiff 𝕜 ⊤ fun y => toContinuousLinearMap hf",
"state_before": "case h\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.95197\ninst✝¹ : NormedAddCommGroup X\ninst✝ : 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\nhf : IsBoundedLinearMap 𝕜 f\n⊢ ContDiff 𝕜 ⊤ fun y => _root_.fderiv 𝕜 f y",
"tactic": "simp_rw [hf.fderiv]"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.95197\ninst✝¹ : NormedAddCommGroup X\ninst✝ : 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\nhf : IsBoundedLinearMap 𝕜 f\n⊢ ContDiff 𝕜 ⊤ fun y => toContinuousLinearMap hf",
"tactic": "exact contDiff_const"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.95197\ninst✝¹ : NormedAddCommGroup X\ninst✝ : 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\nhf : IsBoundedLinearMap 𝕜 f\nh : ContDiff 𝕜 ⊤ f\n⊢ ContDiff 𝕜 n f",
"tactic": "exact h.of_le le_top"
}
] |
[
152,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Combinatorics/SimpleGraph/Matching.lean
|
SimpleGraph.Subgraph.IsMatching.toEdge.surjective
|
[
{
"state_after": "case mk\nV : Type u\nG : SimpleGraph V\nM✝ M : Subgraph G\nh : IsMatching M\ne : Sym2 V\nhe : e ∈ edgeSet M\n⊢ ∃ a, toEdge h a = { val := e, property := he }",
"state_before": "V : Type u\nG : SimpleGraph V\nM✝ M : Subgraph G\nh : IsMatching M\n⊢ Function.Surjective (toEdge h)",
"tactic": "rintro ⟨e, he⟩"
},
{
"state_after": "case mk\nV : Type u\nG : SimpleGraph V\nM✝ M : Subgraph G\nh : IsMatching M\ne : Sym2 V\nhe✝ : e ∈ edgeSet M\nx y : V\nhe : Quotient.mk (Sym2.Rel.setoid V) (x, y) ∈ edgeSet M\n⊢ ∃ a, toEdge h a = { val := Quotient.mk (Sym2.Rel.setoid V) (x, y), property := he }",
"state_before": "case mk\nV : Type u\nG : SimpleGraph V\nM✝ M : Subgraph G\nh : IsMatching M\ne : Sym2 V\nhe : e ∈ edgeSet M\n⊢ ∃ a, toEdge h a = { val := e, property := he }",
"tactic": "refine Sym2.ind (fun x y he => ?_) e he"
},
{
"state_after": "no goals",
"state_before": "case mk\nV : Type u\nG : SimpleGraph V\nM✝ M : Subgraph G\nh : IsMatching M\ne : Sym2 V\nhe✝ : e ∈ edgeSet M\nx y : V\nhe : Quotient.mk (Sym2.Rel.setoid V) (x, y) ∈ edgeSet M\n⊢ ∃ a, toEdge h a = { val := Quotient.mk (Sym2.Rel.setoid V) (x, y), property := he }",
"tactic": "exact ⟨⟨x, M.edge_vert he⟩, h.toEdge_eq_of_adj _ he⟩"
}
] |
[
77,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
summable_iff_cauchySeq_finset
|
[] |
[
1093,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1091,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.hasPullback_of_right_iso
|
[] |
[
1724,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1723,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.subset_congr_left
|
[] |
[
103,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Data/Sym/Sym2.lean
|
Sym2.coe_lift₂_symm_apply
|
[] |
[
235,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Order/BooleanAlgebra.lean
|
eq_compl_iff_isCompl
|
[
{
"state_after": "α : Type u\nβ : Type ?u.53845\nw x y z : α\ninst✝ : BooleanAlgebra α\nh : x = yᶜ\n⊢ IsCompl (yᶜ) y",
"state_before": "α : Type u\nβ : Type ?u.53845\nw x y z : α\ninst✝ : BooleanAlgebra α\nh : x = yᶜ\n⊢ IsCompl x y",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.53845\nw x y z : α\ninst✝ : BooleanAlgebra α\nh : x = yᶜ\n⊢ IsCompl (yᶜ) y",
"tactic": "exact isCompl_compl.symm"
}
] |
[
618,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
615,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
differentiableWithinAt_snd
|
[] |
[
298,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/ModelTheory/FinitelyGenerated.lean
|
FirstOrder.Language.Substructure.cg_def
|
[] |
[
111,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
abs_sub
|
[
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na✝ b✝ c d a b : α\n⊢ abs (a + -b) ≤ abs a + abs (-b)",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na✝ b✝ c d a b : α\n⊢ abs (a - b) ≤ abs a + abs b",
"tactic": "rw [sub_eq_add_neg, ← abs_neg b]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na✝ b✝ c d a b : α\n⊢ abs (a + -b) ≤ abs a + abs (-b)",
"tactic": "exact abs_add a _"
}
] |
[
281,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.integrable_congr
|
[] |
[
490,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/Topology/Algebra/Constructions.lean
|
MulOpposite.comap_unop_nhds
|
[] |
[
85,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.minDegree_le_degree
|
[
{
"state_after": "case intro\nι : Sort ?u.290851\n𝕜 : Type ?u.290854\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nv : V\nt : ℕ\nht : Finset.min (image (fun v => degree G v) univ) = ↑t\n⊢ minDegree G ≤ degree G v",
"state_before": "ι : Sort ?u.290851\n𝕜 : Type ?u.290854\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nv : V\n⊢ minDegree G ≤ degree G v",
"tactic": "obtain ⟨t, ht⟩ := Finset.min_of_mem (mem_image_of_mem (fun v => G.degree v) (mem_univ v))"
},
{
"state_after": "case intro\nι : Sort ?u.290851\n𝕜 : Type ?u.290854\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nv : V\nt : ℕ\nht : Finset.min (image (fun v => degree G v) univ) = ↑t\nthis : t ≤ degree G v\n⊢ minDegree G ≤ degree G v",
"state_before": "case intro\nι : Sort ?u.290851\n𝕜 : Type ?u.290854\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nv : V\nt : ℕ\nht : Finset.min (image (fun v => degree G v) univ) = ↑t\n⊢ minDegree G ≤ degree G v",
"tactic": "have := Finset.min_le_of_eq (mem_image_of_mem _ (mem_univ v)) ht"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Sort ?u.290851\n𝕜 : Type ?u.290854\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nv : V\nt : ℕ\nht : Finset.min (image (fun v => degree G v) univ) = ↑t\nthis : t ≤ degree G v\n⊢ minDegree G ≤ degree G v",
"tactic": "rwa [minDegree, ht]"
}
] |
[
1525,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1522,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.swap_eq_update
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u_1\ninst✝ : DecidableEq α\ni j x : α\n⊢ ↑(swap i j) x = update (update id j i) i j x",
"tactic": "rw [update_apply _ i j, update_apply _ j i, Equiv.swap_apply_def, id.def]"
}
] |
[
1599,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1598,
1
] |
Mathlib/CategoryTheory/Iso.lean
|
CategoryTheory.IsIso.Iso.inv_hom
|
[
{
"state_after": "case hom_inv_id\nC : Type u\ninst✝ : Category C\nX Y Z : C\nf✝ g : X ⟶ Y\nh : Y ⟶ Z\nf : X ≅ Y\n⊢ f.hom ≫ f.inv = 𝟙 X",
"state_before": "C : Type u\ninst✝ : Category C\nX Y Z : C\nf✝ g : X ⟶ Y\nh : Y ⟶ Z\nf : X ≅ Y\n⊢ inv f.hom = f.inv",
"tactic": "apply inv_eq_of_hom_inv_id"
},
{
"state_after": "no goals",
"state_before": "case hom_inv_id\nC : Type u\ninst✝ : Category C\nX Y Z : C\nf✝ g : X ⟶ Y\nh : Y ⟶ Z\nf : X ≅ Y\n⊢ f.hom ≫ f.inv = 𝟙 X",
"tactic": "simp"
}
] |
[
419,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
417,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPolynomial.coe_pow
|
[] |
[
1171,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1169,
1
] |
Std/Classes/LawfulMonad.lean
|
SatisfiesM.map_post
|
[] |
[
104,
30
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
102,
1
] |
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