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/ModelTheory/Definability.lean
|
FirstOrder.Language.DefinableSet.coe_inf
|
[] |
[
361,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.domLCongr_apply
|
[] |
[
778,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
776,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometryEquiv.coe_vaddConst
|
[] |
[
702,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
701,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.ball_eq_ball'
|
[
{
"state_after": "case h\nα : Type u\nβ : Type v\nX : Type ?u.28557\nι : Type ?u.28560\ninst✝ : PseudoMetricSpace α\nx✝¹ y z : α\nδ ε✝ ε₁ ε₂ : ℝ\ns : Set α\nε : ℝ\nx x✝ : α\n⊢ x✝ ∈ UniformSpace.ball x {p | dist p.fst p.snd < ε} ↔ x✝ ∈ ball x ε",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.28557\nι : Type ?u.28560\ninst✝ : PseudoMetricSpace α\nx✝ y z : α\nδ ε✝ ε₁ ε₂ : ℝ\ns : Set α\nε : ℝ\nx : α\n⊢ UniformSpace.ball x {p | dist p.fst p.snd < ε} = ball x ε",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\nβ : Type v\nX : Type ?u.28557\nι : Type ?u.28560\ninst✝ : PseudoMetricSpace α\nx✝¹ y z : α\nδ ε✝ ε₁ ε₂ : ℝ\ns : Set α\nε : ℝ\nx x✝ : α\n⊢ x✝ ∈ UniformSpace.ball x {p | dist p.fst p.snd < ε} ↔ x✝ ∈ ball x ε",
"tactic": "simp [dist_comm, UniformSpace.ball]"
}
] |
[
463,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
460,
1
] |
Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean
|
subset_balancedHull
|
[] |
[
144,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
UniformSpace.comap_inf
|
[] |
[
1285,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1283,
1
] |
Std/Data/Int/DivMod.lean
|
Int.emod_add_emod
|
[
{
"state_after": "m n k : Int\nthis : (m % n + k) % n = (m % n + k + n * (m / n)) % n\n⊢ (m % n + k) % n = (m + k) % n",
"state_before": "m n k : Int\n⊢ (m % n + k) % n = (m + k) % n",
"tactic": "have := (add_mul_emod_self_left (m % n + k) n (m / n)).symm"
},
{
"state_after": "no goals",
"state_before": "m n k : Int\nthis : (m % n + k) % n = (m % n + k + n * (m / n)) % n\n⊢ (m % n + k) % n = (m + k) % n",
"tactic": "rwa [Int.add_right_comm, emod_add_ediv] at this"
}
] |
[
422,
50
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
420,
9
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.ediam_eq_top_iff_unbounded
|
[] |
[
2657,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2656,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.norm_Lp_toLp_restrict_le
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.334371\nE : Type u_2\nF : Type ?u.334377\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type ?u.334386\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\ns : Set α\nf : { x // x ∈ Lp E p }\n⊢ snorm (↑↑(Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p))) p (Measure.restrict μ s) ≤ snorm (↑↑f) p μ",
"state_before": "α : Type u_1\nβ : Type ?u.334371\nE : Type u_2\nF : Type ?u.334377\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type ?u.334386\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\ns : Set α\nf : { x // x ∈ Lp E p }\n⊢ ‖Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p)‖ ≤ ‖f‖",
"tactic": "rw [Lp.norm_def, Lp.norm_def, ENNReal.toReal_le_toReal (Lp.snorm_ne_top _) (Lp.snorm_ne_top _)]"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.334371\nE : Type u_2\nF : Type ?u.334377\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type ?u.334386\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\ns : Set α\nf : { x // x ∈ Lp E p }\n⊢ snorm (↑↑(Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p))) p (Measure.restrict μ s) =\n snorm (↑↑f) p (Measure.restrict μ ?refine'_2)\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.334371\nE : Type u_2\nF : Type ?u.334377\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type ?u.334386\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\ns : Set α\nf : { x // x ∈ Lp E p }\n⊢ Set α",
"state_before": "α : Type u_1\nβ : Type ?u.334371\nE : Type u_2\nF : Type ?u.334377\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type ?u.334386\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\ns : Set α\nf : { x // x ∈ Lp E p }\n⊢ snorm (↑↑(Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p))) p (Measure.restrict μ s) ≤ snorm (↑↑f) p μ",
"tactic": "refine' (le_of_eq _).trans (snorm_mono_measure _ Measure.restrict_le_self)"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.334371\nE : Type u_2\nF : Type ?u.334377\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type ?u.334386\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\ns : Set α\nf : { x // x ∈ Lp E p }\n⊢ snorm (↑↑(Memℒp.toLp ↑↑f (_ : Memℒp (↑↑f) p))) p (Measure.restrict μ s) =\n snorm (↑↑f) p (Measure.restrict μ ?refine'_2)\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.334371\nE : Type u_2\nF : Type ?u.334377\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type ?u.334386\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : ℝ≥0∞\nμ : Measure α\ns : Set α\nf : { x // x ∈ Lp E p }\n⊢ Set α",
"tactic": "exact snorm_congr_ae (Memℒp.coeFn_toLp _)"
}
] |
[
906,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
902,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.append_assoc
|
[
{
"state_after": "case h\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\n⊢ append (append a b) c i = (append a (append b c) ∘ ↑(cast (_ : m + n + p = m + (n + p)))) i",
"state_before": "m n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni : Fin n\ny : α✝ (succ i)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\n⊢ append (append a b) c = append a (append b c) ∘ ↑(cast (_ : m + n + p = m + (n + p)))",
"tactic": "ext i"
},
{
"state_after": "case h\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\n⊢ append (append a b) c i = append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) i)",
"state_before": "case h\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\n⊢ append (append a b) c i = (append a (append b c) ∘ ↑(cast (_ : m + n + p = m + (n + p)))) i",
"tactic": "rw [Function.comp_apply]"
},
{
"state_after": "case h.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nl : Fin (m + n)\n⊢ append (append a b) c (↑(castAdd p) l) =\n append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(castAdd p) l))\n\ncase h.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nr : Fin p\n⊢ append (append a b) c (↑(natAdd (m + n)) r) =\n append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(natAdd (m + n)) r))",
"state_before": "case h\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\n⊢ append (append a b) c i = append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) i)",
"tactic": "refine' Fin.addCases (fun l => _) (fun r => _) i"
},
{
"state_after": "case h.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nl : Fin (m + n)\n⊢ append a b l = append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(castAdd p) l))",
"state_before": "case h.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nl : Fin (m + n)\n⊢ append (append a b) c (↑(castAdd p) l) =\n append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(castAdd p) l))",
"tactic": "rw [append_left]"
},
{
"state_after": "case h.refine'_1.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nl : Fin (m + n)\nll : Fin m\n⊢ append a b (↑(castAdd n) ll) =\n append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(castAdd p) (↑(castAdd n) ll)))\n\ncase h.refine'_1.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nl : Fin (m + n)\nlr : Fin n\n⊢ append a b (↑(natAdd m) lr) =\n append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(castAdd p) (↑(natAdd m) lr)))",
"state_before": "case h.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nl : Fin (m + n)\n⊢ append a b l = append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(castAdd p) l))",
"tactic": "refine' Fin.addCases (fun ll => _) (fun lr => _) l"
},
{
"state_after": "case h.refine'_1.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nl : Fin (m + n)\nll : Fin m\n⊢ a ll = append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(castAdd p) (↑(castAdd n) ll)))",
"state_before": "case h.refine'_1.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nl : Fin (m + n)\nll : Fin m\n⊢ append a b (↑(castAdd n) ll) =\n append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(castAdd p) (↑(castAdd n) ll)))",
"tactic": "rw [append_left]"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_1.refine'_1\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nl : Fin (m + n)\nll : Fin m\n⊢ a ll = append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(castAdd p) (↑(castAdd n) ll)))",
"tactic": "simp [castAdd_castAdd]"
},
{
"state_after": "case h.refine'_1.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nl : Fin (m + n)\nlr : Fin n\n⊢ b lr = append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(castAdd p) (↑(natAdd m) lr)))",
"state_before": "case h.refine'_1.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nl : Fin (m + n)\nlr : Fin n\n⊢ append a b (↑(natAdd m) lr) =\n append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(castAdd p) (↑(natAdd m) lr)))",
"tactic": "rw [append_right]"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_1.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nl : Fin (m + n)\nlr : Fin n\n⊢ b lr = append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(castAdd p) (↑(natAdd m) lr)))",
"tactic": "simp [castAdd_natAdd]"
},
{
"state_after": "case h.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nr : Fin p\n⊢ c r = append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(natAdd (m + n)) r))",
"state_before": "case h.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nr : Fin p\n⊢ append (append a b) c (↑(natAdd (m + n)) r) =\n append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(natAdd (m + n)) r))",
"tactic": "rw [append_right]"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_2\nm n : ℕ\nα✝ : Fin (n + 1) → Type u\nx : α✝ 0\nq : (i : Fin (n + 1)) → α✝ i\np✝ : (i : Fin n) → α✝ (succ i)\ni✝ : Fin n\ny : α✝ (succ i✝)\nz : α✝ 0\np : ℕ\nα : Type u_1\na : Fin m → α\nb : Fin n → α\nc : Fin p → α\ni : Fin (m + n + p)\nr : Fin p\n⊢ c r = append a (append b c) (↑(cast (_ : m + n + p = m + (n + p))) (↑(natAdd (m + n)) r))",
"tactic": "simp [← natAdd_natAdd]"
}
] |
[
349,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/Topology/Bases.lean
|
TopologicalSpace.isOpen_of_mem_countableBasis
|
[] |
[
651,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
649,
1
] |
Mathlib/Order/SuccPred/Limit.lean
|
Order.IsSuccLimit.succ_lt_iff
|
[] |
[
157,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Algebra/EuclideanDomain/Defs.lean
|
EuclideanDomain.gcd_zero_left
|
[
{
"state_after": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na : R\n⊢ (fun b =>\n if a0 : 0 = 0 then b\n else\n let_fun x := (_ : b % 0 ≺ 0);\n gcd (b % 0) 0)\n a =\n a",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na : R\n⊢ gcd 0 a = a",
"tactic": "rw [gcd]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na : R\n⊢ (fun b =>\n if a0 : 0 = 0 then b\n else\n let_fun x := (_ : b % 0 ≺ 0);\n gcd (b % 0) 0)\n a =\n a",
"tactic": "exact if_pos rfl"
}
] |
[
214,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Analysis/NormedSpace/Extr.lean
|
IsLocalMax.norm_add_sameRay
|
[] |
[
93,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Order/CompleteBooleanAlgebra.lean
|
PUnit.sSup_eq
|
[] |
[
401,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
1
] |
Mathlib/Algebra/Hom/Centroid.lean
|
CentroidHom.coe_toAddMonoidHom_injective
|
[] |
[
134,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Order/Circular.lean
|
sbtw_trans_right
|
[] |
[
244,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.coe_deleteEdges_eq
|
[
{
"state_after": "case Adj.h.mk.h.mk.a\nι : Sort ?u.246328\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : Subgraph G\ns✝ s : Set (Sym2 V)\nv : V\nhv : v ∈ (deleteEdges G' s).verts\nw : V\nhw : w ∈ (deleteEdges G' s).verts\n⊢ SimpleGraph.Adj (Subgraph.coe (deleteEdges G' s)) { val := v, property := hv } { val := w, property := hw } ↔\n SimpleGraph.Adj (SimpleGraph.deleteEdges (Subgraph.coe G') (Sym2.map Subtype.val ⁻¹' s))\n { val := v, property := hv } { val := w, property := hw }",
"state_before": "ι : Sort ?u.246328\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : Subgraph G\ns✝ s : Set (Sym2 V)\n⊢ Subgraph.coe (deleteEdges G' s) = SimpleGraph.deleteEdges (Subgraph.coe G') (Sym2.map Subtype.val ⁻¹' s)",
"tactic": "ext ⟨v, hv⟩ ⟨w, hw⟩"
},
{
"state_after": "no goals",
"state_before": "case Adj.h.mk.h.mk.a\nι : Sort ?u.246328\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : Subgraph G\ns✝ s : Set (Sym2 V)\nv : V\nhv : v ∈ (deleteEdges G' s).verts\nw : V\nhw : w ∈ (deleteEdges G' s).verts\n⊢ SimpleGraph.Adj (Subgraph.coe (deleteEdges G' s)) { val := v, property := hv } { val := w, property := hw } ↔\n SimpleGraph.Adj (SimpleGraph.deleteEdges (Subgraph.coe G') (Sym2.map Subtype.val ⁻¹' s))\n { val := v, property := hv } { val := w, property := hw }",
"tactic": "simp"
}
] |
[
1084,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1081,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.Red.cons_cons_iff
|
[
{
"state_after": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\np : α × Bool\nLL₁ : List (α × Bool)\neq₁ : p :: L₁ = LL₁\n⊢ Red LL₁ (p :: L₂) → Red L₁ L₂",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\np : α × Bool\n⊢ Red (p :: L₁) (p :: L₂) → Red L₁ L₂",
"tactic": "generalize eq₁ : (p :: L₁ : List _) = LL₁"
},
{
"state_after": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\np : α × Bool\nLL₁ : List (α × Bool)\neq₁ : p :: L₁ = LL₁\nLL₂ : List (α × Bool)\neq₂ : p :: L₂ = LL₂\n⊢ Red LL₁ LL₂ → Red L₁ L₂",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\np : α × Bool\nLL₁ : List (α × Bool)\neq₁ : p :: L₁ = LL₁\n⊢ Red LL₁ (p :: L₂) → Red L₁ L₂",
"tactic": "generalize eq₂ : (p :: L₂ : List _) = LL₂"
},
{
"state_after": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\np : α × Bool\nLL₁ : List (α × Bool)\neq₁ : p :: L₁ = LL₁\nLL₂ : List (α × Bool)\neq₂ : p :: L₂ = LL₂\nh : Red LL₁ LL₂\n⊢ Red L₁ L₂",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\np : α × Bool\nLL₁ : List (α × Bool)\neq₁ : p :: L₁ = LL₁\nLL₂ : List (α × Bool)\neq₂ : p :: L₂ = LL₂\n⊢ Red LL₁ LL₂ → Red L₁ L₂",
"tactic": "intro h"
},
{
"state_after": "case refl\nα : Type u\nL L₁✝ L₂✝ L₃ L₄ : List (α × Bool)\np : α × Bool\nLL₁ : List (α × Bool)\neq₁✝ : p :: L₁✝ = LL₁\nLL₂ : List (α × Bool)\neq₂✝ : p :: L₂✝ = LL₂\nL₁ L₂ : List (α × Bool)\neq₁ : p :: L₁ = LL₂\neq₂ : p :: L₂ = LL₂\n⊢ Red L₁ L₂\n\ncase head\nα : Type u\nL L₁✝¹ L₂✝¹ L₃ L₄ : List (α × Bool)\np : α × Bool\nLL₁ : List (α × Bool)\neq₁✝ : p :: L₁✝¹ = LL₁\nLL₂ : List (α × Bool)\neq₂✝ : p :: L₂✝¹ = LL₂\nL₁✝ L₂✝ : List (α × Bool)\nh₁₂ : Step L₁✝ L₂✝\nh : ReflTransGen Step L₂✝ LL₂\nih : ∀ {L₁ L₂ : List (α × Bool)}, p :: L₁ = L₂✝ → p :: L₂ = LL₂ → Red L₁ L₂\nL₁ L₂ : List (α × Bool)\neq₁ : p :: L₁ = L₁✝\neq₂ : p :: L₂ = LL₂\n⊢ Red L₁ L₂",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\np : α × Bool\nLL₁ : List (α × Bool)\neq₁ : p :: L₁ = LL₁\nLL₂ : List (α × Bool)\neq₂ : p :: L₂ = LL₂\nh : Red LL₁ LL₂\n⊢ Red L₁ L₂",
"tactic": "induction' h using Relation.ReflTransGen.head_induction_on\n with L₁ L₂ h₁₂ h ih\n generalizing L₁ L₂"
},
{
"state_after": "case refl\nα : Type u\nL L₁✝ L₂✝ L₃ L₄ : List (α × Bool)\np : α × Bool\nL₁ L₂ : List (α × Bool)\neq₁ : p :: L₁ = p :: L₂✝\neq₂ : p :: L₂ = p :: L₂✝\n⊢ Red L₁ L₂",
"state_before": "case refl\nα : Type u\nL L₁✝ L₂✝ L₃ L₄ : List (α × Bool)\np : α × Bool\nLL₁ : List (α × Bool)\neq₁✝ : p :: L₁✝ = LL₁\nLL₂ : List (α × Bool)\neq₂✝ : p :: L₂✝ = LL₂\nL₁ L₂ : List (α × Bool)\neq₁ : p :: L₁ = LL₂\neq₂ : p :: L₂ = LL₂\n⊢ Red L₁ L₂",
"tactic": "subst_vars"
},
{
"state_after": "case refl.refl\nα : Type u\nL L₁✝ L₂ L₃ L₄ : List (α × Bool)\np : α × Bool\nL₁ : List (α × Bool)\neq₁ : p :: L₁ = p :: L₂\n⊢ Red L₁ L₂",
"state_before": "case refl\nα : Type u\nL L₁✝ L₂✝ L₃ L₄ : List (α × Bool)\np : α × Bool\nL₁ L₂ : List (α × Bool)\neq₁ : p :: L₁ = p :: L₂✝\neq₂ : p :: L₂ = p :: L₂✝\n⊢ Red L₁ L₂",
"tactic": "cases eq₂"
},
{
"state_after": "case refl.refl.refl\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\np : α × Bool\n⊢ Red L₂ L₂",
"state_before": "case refl.refl\nα : Type u\nL L₁✝ L₂ L₃ L₄ : List (α × Bool)\np : α × Bool\nL₁ : List (α × Bool)\neq₁ : p :: L₁ = p :: L₂\n⊢ Red L₁ L₂",
"tactic": "cases eq₁"
},
{
"state_after": "no goals",
"state_before": "case refl.refl.refl\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\np : α × Bool\n⊢ Red L₂ L₂",
"tactic": "constructor"
},
{
"state_after": "case head\nα : Type u\nL L₁✝ L₂✝¹ L₃ L₄ : List (α × Bool)\np : α × Bool\nL₂✝ L₁ L₂ : List (α × Bool)\nh : ReflTransGen Step L₂✝ (p :: L₂✝¹)\nih : ∀ {L₁ L₂ : List (α × Bool)}, p :: L₁ = L₂✝ → p :: L₂ = p :: L₂✝¹ → Red L₁ L₂\neq₂ : p :: L₂ = p :: L₂✝¹\nh₁₂ : Step (p :: L₁) L₂✝\n⊢ Red L₁ L₂",
"state_before": "case head\nα : Type u\nL L₁✝¹ L₂✝¹ L₃ L₄ : List (α × Bool)\np : α × Bool\nLL₁ : List (α × Bool)\neq₁✝ : p :: L₁✝¹ = LL₁\nLL₂ : List (α × Bool)\neq₂✝ : p :: L₂✝¹ = LL₂\nL₁✝ L₂✝ : List (α × Bool)\nh₁₂ : Step L₁✝ L₂✝\nh : ReflTransGen Step L₂✝ LL₂\nih : ∀ {L₁ L₂ : List (α × Bool)}, p :: L₁ = L₂✝ → p :: L₂ = LL₂ → Red L₁ L₂\nL₁ L₂ : List (α × Bool)\neq₁ : p :: L₁ = L₁✝\neq₂ : p :: L₂ = LL₂\n⊢ Red L₁ L₂",
"tactic": "subst_vars"
},
{
"state_after": "case head.refl\nα : Type u\nL L₁✝ L₂✝ L₃ L₄ : List (α × Bool)\np : α × Bool\nL₂ L₁ : List (α × Bool)\nh : ReflTransGen Step L₂ (p :: L₂✝)\nih : ∀ {L₁ L₂_1 : List (α × Bool)}, p :: L₁ = L₂ → p :: L₂_1 = p :: L₂✝ → Red L₁ L₂_1\nh₁₂ : Step (p :: L₁) L₂\n⊢ Red L₁ L₂✝",
"state_before": "case head\nα : Type u\nL L₁✝ L₂✝¹ L₃ L₄ : List (α × Bool)\np : α × Bool\nL₂✝ L₁ L₂ : List (α × Bool)\nh : ReflTransGen Step L₂✝ (p :: L₂✝¹)\nih : ∀ {L₁ L₂ : List (α × Bool)}, p :: L₁ = L₂✝ → p :: L₂ = p :: L₂✝¹ → Red L₁ L₂\neq₂ : p :: L₂ = p :: L₂✝¹\nh₁₂ : Step (p :: L₁) L₂✝\n⊢ Red L₁ L₂",
"tactic": "cases eq₂"
},
{
"state_after": "case head.refl.mk\nα : Type u\nL L₁✝ L₂✝ L₃ L₄ L₂ L₁ : List (α × Bool)\na : α\nb : Bool\nh : ReflTransGen Step L₂ ((a, b) :: L₂✝)\nih : ∀ {L₁ L₂_1 : List (α × Bool)}, (a, b) :: L₁ = L₂ → (a, b) :: L₂_1 = (a, b) :: L₂✝ → Red L₁ L₂_1\nh₁₂ : Step ((a, b) :: L₁) L₂\n⊢ Red L₁ L₂✝",
"state_before": "case head.refl\nα : Type u\nL L₁✝ L₂✝ L₃ L₄ : List (α × Bool)\np : α × Bool\nL₂ L₁ : List (α × Bool)\nh : ReflTransGen Step L₂ (p :: L₂✝)\nih : ∀ {L₁ L₂_1 : List (α × Bool)}, p :: L₁ = L₂ → p :: L₂_1 = p :: L₂✝ → Red L₁ L₂_1\nh₁₂ : Step (p :: L₁) L₂\n⊢ Red L₁ L₂✝",
"tactic": "cases' p with a b"
},
{
"state_after": "case head.refl.mk\nα : Type u\nL L₁✝ L₂✝ L₃ L₄ L₂ L₁ : List (α × Bool)\na : α\nb : Bool\nh : ReflTransGen Step L₂ ((a, b) :: L₂✝)\nih : ∀ {L₁ L₂_1 : List (α × Bool)}, (a, b) :: L₁ = L₂ → (a, b) :: L₂_1 = (a, b) :: L₂✝ → Red L₁ L₂_1\nh₁₂ : (∃ L, Step L₁ L ∧ L₂ = (a, b) :: L) ∨ L₁ = (a, !b) :: L₂\n⊢ Red L₁ L₂✝",
"state_before": "case head.refl.mk\nα : Type u\nL L₁✝ L₂✝ L₃ L₄ L₂ L₁ : List (α × Bool)\na : α\nb : Bool\nh : ReflTransGen Step L₂ ((a, b) :: L₂✝)\nih : ∀ {L₁ L₂_1 : List (α × Bool)}, (a, b) :: L₁ = L₂ → (a, b) :: L₂_1 = (a, b) :: L₂✝ → Red L₁ L₂_1\nh₁₂ : Step ((a, b) :: L₁) L₂\n⊢ Red L₁ L₂✝",
"tactic": "rw [Step.cons_left_iff] at h₁₂"
},
{
"state_after": "case head.refl.mk.inl.intro.intro\nα : Type u\nL✝ L₁✝ L₂ L₃ L₄ L₁ : List (α × Bool)\na : α\nb : Bool\nL : List (α × Bool)\nh₁₂ : Step L₁ L\nh : ReflTransGen Step ((a, b) :: L) ((a, b) :: L₂)\nih : ∀ {L₁ L₂_1 : List (α × Bool)}, (a, b) :: L₁ = (a, b) :: L → (a, b) :: L₂_1 = (a, b) :: L₂ → Red L₁ L₂_1\n⊢ Red L₁ L₂\n\ncase head.refl.mk.inr\nα : Type u\nL L₁ L₂✝ L₃ L₄ L₂ : List (α × Bool)\na : α\nb : Bool\nh : ReflTransGen Step L₂ ((a, b) :: L₂✝)\nih : ∀ {L₁ L₂_1 : List (α × Bool)}, (a, b) :: L₁ = L₂ → (a, b) :: L₂_1 = (a, b) :: L₂✝ → Red L₁ L₂_1\n⊢ Red ((a, !b) :: L₂) L₂✝",
"state_before": "case head.refl.mk\nα : Type u\nL L₁✝ L₂✝ L₃ L₄ L₂ L₁ : List (α × Bool)\na : α\nb : Bool\nh : ReflTransGen Step L₂ ((a, b) :: L₂✝)\nih : ∀ {L₁ L₂_1 : List (α × Bool)}, (a, b) :: L₁ = L₂ → (a, b) :: L₂_1 = (a, b) :: L₂✝ → Red L₁ L₂_1\nh₁₂ : (∃ L, Step L₁ L ∧ L₂ = (a, b) :: L) ∨ L₁ = (a, !b) :: L₂\n⊢ Red L₁ L₂✝",
"tactic": "rcases h₁₂ with (⟨L, h₁₂, rfl⟩ | rfl)"
},
{
"state_after": "no goals",
"state_before": "case head.refl.mk.inl.intro.intro\nα : Type u\nL✝ L₁✝ L₂ L₃ L₄ L₁ : List (α × Bool)\na : α\nb : Bool\nL : List (α × Bool)\nh₁₂ : Step L₁ L\nh : ReflTransGen Step ((a, b) :: L) ((a, b) :: L₂)\nih : ∀ {L₁ L₂_1 : List (α × Bool)}, (a, b) :: L₁ = (a, b) :: L → (a, b) :: L₂_1 = (a, b) :: L₂ → Red L₁ L₂_1\n⊢ Red L₁ L₂",
"tactic": "exact (ih rfl rfl).head h₁₂"
},
{
"state_after": "no goals",
"state_before": "case head.refl.mk.inr\nα : Type u\nL L₁ L₂✝ L₃ L₄ L₂ : List (α × Bool)\na : α\nb : Bool\nh : ReflTransGen Step L₂ ((a, b) :: L₂✝)\nih : ∀ {L₁ L₂_1 : List (α × Bool)}, (a, b) :: L₁ = L₂ → (a, b) :: L₂_1 = (a, b) :: L₂✝ → Red L₁ L₂_1\n⊢ Red ((a, !b) :: L₂) L₂✝",
"tactic": "exact (cons_cons h).tail Step.cons_not_rev"
}
] |
[
275,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
255,
1
] |
Mathlib/Algebra/Symmetrized.lean
|
SymAlg.sym_injective
|
[] |
[
103,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/GroupTheory/Commutator.lean
|
Subgroup.commutator_le
|
[] |
[
94,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Algebra/Module/Projective.lean
|
Module.Projective.of_basis
|
[
{
"state_after": "R : Type u_2\ninst✝² : Ring R\nP : Type u_3\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nι : Type u_1\nb : Basis ι R P\n⊢ Function.LeftInverse ↑(Finsupp.total P P R id) ↑(↑(Basis.constr b ℕ) fun i => Finsupp.single (↑b i) 1)",
"state_before": "R : Type u_2\ninst✝² : Ring R\nP : Type u_3\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nι : Type u_1\nb : Basis ι R P\n⊢ Projective R P",
"tactic": "use b.constr ℕ fun i => Finsupp.single (b i) (1 : R)"
},
{
"state_after": "R : Type u_2\ninst✝² : Ring R\nP : Type u_3\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nι : Type u_1\nb : Basis ι R P\nm : P\n⊢ ↑(Finsupp.total P P R id) (↑(↑(Basis.constr b ℕ) fun i => Finsupp.single (↑b i) 1) m) = m",
"state_before": "R : Type u_2\ninst✝² : Ring R\nP : Type u_3\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nι : Type u_1\nb : Basis ι R P\n⊢ Function.LeftInverse ↑(Finsupp.total P P R id) ↑(↑(Basis.constr b ℕ) fun i => Finsupp.single (↑b i) 1)",
"tactic": "intro m"
},
{
"state_after": "R : Type u_2\ninst✝² : Ring R\nP : Type u_3\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nι : Type u_1\nb : Basis ι R P\nm : P\n⊢ (sum (↑b.repr m) fun i d => d • ↑b i) = m",
"state_before": "R : Type u_2\ninst✝² : Ring R\nP : Type u_3\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nι : Type u_1\nb : Basis ι R P\nm : P\n⊢ ↑(Finsupp.total P P R id) (↑(↑(Basis.constr b ℕ) fun i => Finsupp.single (↑b i) 1) m) = m",
"tactic": "simp only [b.constr_apply, mul_one, id.def, Finsupp.smul_single', Finsupp.total_single,\n LinearMap.map_finsupp_sum]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝² : Ring R\nP : Type u_3\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nι : Type u_1\nb : Basis ι R P\nm : P\n⊢ (sum (↑b.repr m) fun i d => d • ↑b i) = m",
"tactic": "exact b.total_repr m"
}
] |
[
165,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.colimit.pre_eq
|
[
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝⁶ : Category J\nK : Type u₂\ninst✝⁵ : Category K\nC : Type u\ninst✝⁴ : Category C\nF : J ⥤ C\ninst✝³ : HasColimit F\nE : K ⥤ J\ninst✝² : HasColimit (E ⋙ F)\nL : Type u₃\ninst✝¹ : Category L\nD : L ⥤ K\ninst✝ : HasColimit (D ⋙ E ⋙ F)\ns : ColimitCocone (E ⋙ F)\nt : ColimitCocone F\n⊢ pre F E = (isoColimitCocone s).hom ≫ IsColimit.desc s.isColimit (Cocone.whisker E t.cocone) ≫ (isoColimitCocone t).inv",
"tactic": "aesop_cat"
}
] |
[
1009,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1005,
1
] |
Mathlib/Algebra/Module/Torsion.lean
|
Submodule.torsionBy_le_torsionBy_of_dvd
|
[
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set R\na✝ a b : R\ndvd : a ∣ b\n⊢ torsionBySet R M ↑(span R {a}) ≤ torsionBySet R M {b}",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set R\na✝ a b : R\ndvd : a ∣ b\n⊢ torsionBy R M a ≤ torsionBy R M b",
"tactic": "rw [← torsionBySet_span_singleton_eq, ← torsionBySet_singleton_eq]"
},
{
"state_after": "case st\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set R\na✝ a b : R\ndvd : a ∣ b\n⊢ {b} ⊆ ↑(span R {a})",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set R\na✝ a b : R\ndvd : a ∣ b\n⊢ torsionBySet R M ↑(span R {a}) ≤ torsionBySet R M {b}",
"tactic": "apply torsionBySet_le_torsionBySet_of_subset"
},
{
"state_after": "case st\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set R\na✝ a c : R\ndvd : a ∣ c\n⊢ c ∈ ↑(span R {a})",
"state_before": "case st\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set R\na✝ a b : R\ndvd : a ∣ b\n⊢ {b} ⊆ ↑(span R {a})",
"tactic": "rintro c (rfl : c = b)"
},
{
"state_after": "no goals",
"state_before": "case st\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set R\na✝ a c : R\ndvd : a ∣ c\n⊢ c ∈ ↑(span R {a})",
"tactic": "exact Ideal.mem_span_singleton.mpr dvd"
}
] |
[
300,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
UniformContinuous.comp_tendstoUniformly
|
[] |
[
272,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/Analysis/Normed/Group/HomCompletion.lean
|
NormedAddGroupHom.completion_coe
|
[] |
[
83,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.comap_neBot_iff_compl_range
|
[] |
[
2352,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2351,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.conj_comp
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u\nβ : Sort v\nγ : Sort w\ne : α ≃ β\nf₁ f₂ : α → α\n⊢ ↑(conj e) (f₁ ∘ f₂) = ↑(conj e) f₁ ∘ ↑(conj e) f₂",
"tactic": "apply arrowCongr_comp"
}
] |
[
586,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
585,
1
] |
src/lean/Init/SizeOfLemmas.lean
|
Fin.sizeOf
|
[
{
"state_after": "case mk\nn val✝ : Nat\nisLt✝ : val✝ < n\n⊢ SizeOf.sizeOf { val := val✝, isLt := isLt✝ } = { val := val✝, isLt := isLt✝ }.val + 1",
"state_before": "n : Nat\na : Fin n\n⊢ SizeOf.sizeOf a = a.val + 1",
"tactic": "cases a"
},
{
"state_after": "no goals",
"state_before": "case mk\nn val✝ : Nat\nisLt✝ : val✝ < n\n⊢ SizeOf.sizeOf { val := val✝, isLt := isLt✝ } = { val := val✝, isLt := isLt✝ }.val + 1",
"tactic": "simp_arith"
}
] |
[
12,
22
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
11,
9
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
TendstoUniformly.comp
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng✝ : ι → α\nh : TendstoUniformlyOnFilter F f p ⊤\ng : γ → α\n⊢ TendstoUniformlyOnFilter (fun n => F n ∘ g) (f ∘ g) p ⊤",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng✝ : ι → α\nh : TendstoUniformly F f p\ng : γ → α\n⊢ TendstoUniformly (fun n => F n ∘ g) (f ∘ g) p",
"tactic": "rw [tendstoUniformly_iff_tendstoUniformlyOnFilter] at h⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng✝ : ι → α\nh : TendstoUniformlyOnFilter F f p ⊤\ng : γ → α\n⊢ TendstoUniformlyOnFilter (fun n => F n ∘ g) (f ∘ g) p ⊤",
"tactic": "simpa [principal_univ, comap_principal] using h.comp g"
}
] |
[
252,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.IsCycle.ne_one
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.306885\nα : Type u_1\nβ : Type ?u.306891\nf g : Perm α\nx y : α\nh : IsCycle f\nhf : f = 1\n⊢ False",
"tactic": "simp [hf, IsCycle] at h"
}
] |
[
291,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
291,
1
] |
Mathlib/CategoryTheory/MorphismProperty.lean
|
CategoryTheory.MorphismProperty.monomorphisms.iff
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type ?u.59403\ninst✝ : Category D\nX Y : C\nf : X ⟶ Y\n⊢ monomorphisms C f ↔ Mono f",
"tactic": "rfl"
}
] |
[
408,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
408,
1
] |
Mathlib/RingTheory/OreLocalization/Basic.lean
|
OreLocalization.div_eq_one'
|
[
{
"state_after": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\nhr : r ∈ S\n⊢ ∃ u v, 1 * ↑u = r * v ∧ ↑1 * ↑u = ↑{ val := r, property := hr } * v",
"state_before": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\nhr : r ∈ S\n⊢ r /ₒ { val := r, property := hr } = 1",
"tactic": "rw [OreLocalization.one_def, oreDiv_eq_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\nhr : r ∈ S\n⊢ ∃ u v, 1 * ↑u = r * v ∧ ↑1 * ↑u = ↑{ val := r, property := hr } * v",
"tactic": "exact ⟨⟨r, hr⟩, 1, by simp, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\nhr : r ∈ S\n⊢ 1 * ↑{ val := r, property := hr } = r * 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\nhr : r ∈ S\n⊢ ↑1 * ↑{ val := r, property := hr } = ↑{ val := r, property := hr } * 1",
"tactic": "simp"
}
] |
[
287,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
11
] |
Mathlib/Analysis/Normed/Field/Basic.lean
|
Filter.tendsto_mul_left_cobounded
|
[] |
[
611,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
608,
1
] |
Mathlib/Data/Polynomial/Inductions.lean
|
Polynomial.divX_C
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na✝ b : R\nn✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\na : R\nn : ℕ\n⊢ coeff (divX (↑C a)) n = coeff 0 n",
"tactic": "simp [coeff_divX, coeff_C, Finsupp.single_eq_of_ne _]"
}
] |
[
59,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Order/SymmDiff.lean
|
inf_symmDiff_symmDiff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.34523\nα : Type u_1\nβ : Type ?u.34529\nπ : ι → Type ?u.34534\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\n⊢ (a ⊓ b) ∆ (a ∆ b) = a ⊔ b",
"tactic": "rw [symmDiff_comm, symmDiff_symmDiff_inf]"
}
] |
[
214,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Data/List/Pairwise.lean
|
List.pwFilter_map
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\nh' : ∀ (b : β), b ∈ pwFilter (fun x y => R (f x) (f y)) xs → R (f x) (f b)\n⊢ pwFilter R (map f (x :: xs)) = map f (pwFilter (fun x y => R (f x) (f y)) (x :: xs))",
"state_before": "α : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\n⊢ pwFilter R (map f (x :: xs)) = map f (pwFilter (fun x y => R (f x) (f y)) (x :: xs))",
"tactic": "have h' : ∀ b : β, b ∈ pwFilter (fun x y : β => R (f x) (f y)) xs → R (f x) (f b) :=\n fun b hb => h _ (by rw [pwFilter_map f xs]; apply mem_map_of_mem _ hb)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\nh' : ∀ (b : β), b ∈ pwFilter (fun x y => R (f x) (f y)) xs → R (f x) (f b)\n⊢ pwFilter R (map f (x :: xs)) = map f (pwFilter (fun x y => R (f x) (f y)) (x :: xs))",
"tactic": "rw [map, pwFilter_cons_of_pos h, pwFilter_cons_of_pos h', pwFilter_map f xs, map]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\nb : β\nhb : b ∈ pwFilter (fun x y => R (f x) (f y)) xs\n⊢ f b ∈ map f (pwFilter (fun x y => R (f x) (f y)) xs)",
"state_before": "α : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\nb : β\nhb : b ∈ pwFilter (fun x y => R (f x) (f y)) xs\n⊢ f b ∈ pwFilter R (map f xs)",
"tactic": "rw [pwFilter_map f xs]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\nb : β\nhb : b ∈ pwFilter (fun x y => R (f x) (f y)) xs\n⊢ f b ∈ map f (pwFilter (fun x y => R (f x) (f y)) xs)",
"tactic": "apply mem_map_of_mem _ hb"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ¬∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\nh' : ¬∀ (b : β), b ∈ pwFilter (fun x y => R (f x) (f y)) xs → R (f x) (f b)\n⊢ pwFilter R (map f (x :: xs)) = map f (pwFilter (fun x y => R (f x) (f y)) (x :: xs))",
"state_before": "α : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ¬∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\n⊢ pwFilter R (map f (x :: xs)) = map f (pwFilter (fun x y => R (f x) (f y)) (x :: xs))",
"tactic": "have h' : ¬∀ b : β, b ∈ pwFilter (fun x y : β => R (f x) (f y)) xs → R (f x) (f b) :=\n fun hh =>\n h fun a ha => by\n rw [pwFilter_map f xs, mem_map] at ha\n rcases ha with ⟨b, hb₀, hb₁⟩\n subst a\n exact hh _ hb₀"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ¬∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\nh' : ¬∀ (b : β), b ∈ pwFilter (fun x y => R (f x) (f y)) xs → R (f x) (f b)\n⊢ pwFilter R (map f (x :: xs)) = map f (pwFilter (fun x y => R (f x) (f y)) (x :: xs))",
"tactic": "rw [map, pwFilter_cons_of_neg h, pwFilter_cons_of_neg h', pwFilter_map f xs]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na✝ : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ¬∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\nhh : ∀ (b : β), b ∈ pwFilter (fun x y => R (f x) (f y)) xs → R (f x) (f b)\na : α\nha : ∃ a_1, a_1 ∈ pwFilter (fun x y => R (f x) (f y)) xs ∧ f a_1 = a\n⊢ R (f x) a",
"state_before": "α : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na✝ : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ¬∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\nhh : ∀ (b : β), b ∈ pwFilter (fun x y => R (f x) (f y)) xs → R (f x) (f b)\na : α\nha : a ∈ pwFilter R (map f xs)\n⊢ R (f x) a",
"tactic": "rw [pwFilter_map f xs, mem_map] at ha"
},
{
"state_after": "case intro.intro\nα : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na✝ : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ¬∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\nhh : ∀ (b : β), b ∈ pwFilter (fun x y => R (f x) (f y)) xs → R (f x) (f b)\na : α\nb : β\nhb₀ : b ∈ pwFilter (fun x y => R (f x) (f y)) xs\nhb₁ : f b = a\n⊢ R (f x) a",
"state_before": "α : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na✝ : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ¬∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\nhh : ∀ (b : β), b ∈ pwFilter (fun x y => R (f x) (f y)) xs → R (f x) (f b)\na : α\nha : ∃ a_1, a_1 ∈ pwFilter (fun x y => R (f x) (f y)) xs ∧ f a_1 = a\n⊢ R (f x) a",
"tactic": "rcases ha with ⟨b, hb₀, hb₁⟩"
},
{
"state_after": "case intro.intro\nα : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ¬∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\nhh : ∀ (b : β), b ∈ pwFilter (fun x y => R (f x) (f y)) xs → R (f x) (f b)\nb : β\nhb₀ : b ∈ pwFilter (fun x y => R (f x) (f y)) xs\n⊢ R (f x) (f b)",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na✝ : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ¬∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\nhh : ∀ (b : β), b ∈ pwFilter (fun x y => R (f x) (f y)) xs → R (f x) (f b)\na : α\nb : β\nhb₀ : b ∈ pwFilter (fun x y => R (f x) (f y)) xs\nhb₁ : f b = a\n⊢ R (f x) a",
"tactic": "subst a"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type u_1\nR S T : α → α → Prop\na : α\nl : List α\ninst✝ : DecidableRel R\nf : β → α\nx : β\nxs : List β\nh : ¬∀ (b : α), b ∈ pwFilter R (map f xs) → R (f x) b\nhh : ∀ (b : β), b ∈ pwFilter (fun x y => R (f x) (f y)) xs → R (f x) (f b)\nb : β\nhb₀ : b ∈ pwFilter (fun x y => R (f x) (f y)) xs\n⊢ R (f x) (f b)",
"tactic": "exact hh _ hb₀"
}
] |
[
388,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
1
] |
Mathlib/Data/Real/Sqrt.lean
|
Real.sqrt_inj
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx : 0 ≤ x\nhy : 0 ≤ y\n⊢ sqrt x = sqrt y ↔ x = y",
"tactic": "simp [le_antisymm_iff, hx, hy]"
}
] |
[
333,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/Topology/Maps.lean
|
IsOpenMap.preimage_frontier_eq_frontier_preimage
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.196791\nδ : Type ?u.196794\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α → β\nhf : IsOpenMap f\nhfc : Continuous f\ns : Set β\n⊢ f ⁻¹' frontier s = frontier (f ⁻¹' s)",
"tactic": "simp only [frontier_eq_closure_inter_closure, preimage_inter, preimage_compl,\n hf.preimage_closure_eq_closure_preimage hfc]"
}
] |
[
440,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/Tactic/NormNum/Core.lean
|
Mathlib.Meta.NormNum.IsNat.of_raw
|
[] |
[
54,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
BddBelow.finite
|
[] |
[
464,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/Order/Height.lean
|
Set.le_chainHeight_TFAE
|
[
{
"state_after": "case tfae_1_to_2\nα : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\n⊢ ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n\n\nα : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\ntfae_1_to_2 : ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n\n⊢ TFAE [↑n ≤ chainHeight s, ∃ l, l ∈ subchain s ∧ length l = n, ∃ l, l ∈ subchain s ∧ n ≤ length l]",
"state_before": "α : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\n⊢ TFAE [↑n ≤ chainHeight s, ∃ l, l ∈ subchain s ∧ length l = n, ∃ l, l ∈ subchain s ∧ n ≤ length l]",
"tactic": "tfae_have 1 → 2"
},
{
"state_after": "case tfae_2_to_3\nα : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\ntfae_1_to_2 : ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n\n⊢ (∃ l, l ∈ subchain s ∧ length l = n) → ∃ l, l ∈ subchain s ∧ n ≤ length l\n\nα : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\ntfae_1_to_2 : ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n\ntfae_2_to_3 : (∃ l, l ∈ subchain s ∧ length l = n) → ∃ l, l ∈ subchain s ∧ n ≤ length l\n⊢ TFAE [↑n ≤ chainHeight s, ∃ l, l ∈ subchain s ∧ length l = n, ∃ l, l ∈ subchain s ∧ n ≤ length l]",
"state_before": "α : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\ntfae_1_to_2 : ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n\n⊢ TFAE [↑n ≤ chainHeight s, ∃ l, l ∈ subchain s ∧ length l = n, ∃ l, l ∈ subchain s ∧ n ≤ length l]",
"tactic": "tfae_have 2 → 3"
},
{
"state_after": "case tfae_3_to_1\nα : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\ntfae_1_to_2 : ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n\ntfae_2_to_3 : (∃ l, l ∈ subchain s ∧ length l = n) → ∃ l, l ∈ subchain s ∧ n ≤ length l\n⊢ (∃ l, l ∈ subchain s ∧ n ≤ length l) → ↑n ≤ chainHeight s\n\nα : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\ntfae_1_to_2 : ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n\ntfae_2_to_3 : (∃ l, l ∈ subchain s ∧ length l = n) → ∃ l, l ∈ subchain s ∧ n ≤ length l\ntfae_3_to_1 : (∃ l, l ∈ subchain s ∧ n ≤ length l) → ↑n ≤ chainHeight s\n⊢ TFAE [↑n ≤ chainHeight s, ∃ l, l ∈ subchain s ∧ length l = n, ∃ l, l ∈ subchain s ∧ n ≤ length l]",
"state_before": "α : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\ntfae_1_to_2 : ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n\ntfae_2_to_3 : (∃ l, l ∈ subchain s ∧ length l = n) → ∃ l, l ∈ subchain s ∧ n ≤ length l\n⊢ TFAE [↑n ≤ chainHeight s, ∃ l, l ∈ subchain s ∧ length l = n, ∃ l, l ∈ subchain s ∧ n ≤ length l]",
"tactic": "tfae_have 3 → 1"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\ntfae_1_to_2 : ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n\ntfae_2_to_3 : (∃ l, l ∈ subchain s ∧ length l = n) → ∃ l, l ∈ subchain s ∧ n ≤ length l\ntfae_3_to_1 : (∃ l, l ∈ subchain s ∧ n ≤ length l) → ↑n ≤ chainHeight s\n⊢ TFAE [↑n ≤ chainHeight s, ∃ l, l ∈ subchain s ∧ length l = n, ∃ l, l ∈ subchain s ∧ n ≤ length l]",
"tactic": "tfae_finish"
},
{
"state_after": "no goals",
"state_before": "case tfae_1_to_2\nα : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\n⊢ ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n",
"tactic": "exact s.exists_chain_of_le_chainHeight"
},
{
"state_after": "case tfae_2_to_3.intro.intro\nα : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl✝ : List α\na : α\nn : ℕ\ntfae_1_to_2 : ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n\nl : List α\nhls : l ∈ subchain s\nhe : length l = n\n⊢ ∃ l, l ∈ subchain s ∧ n ≤ length l",
"state_before": "case tfae_2_to_3\nα : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\ntfae_1_to_2 : ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n\n⊢ (∃ l, l ∈ subchain s ∧ length l = n) → ∃ l, l ∈ subchain s ∧ n ≤ length l",
"tactic": "rintro ⟨l, hls, he⟩"
},
{
"state_after": "no goals",
"state_before": "case tfae_2_to_3.intro.intro\nα : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl✝ : List α\na : α\nn : ℕ\ntfae_1_to_2 : ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n\nl : List α\nhls : l ∈ subchain s\nhe : length l = n\n⊢ ∃ l, l ∈ subchain s ∧ n ≤ length l",
"tactic": "exact ⟨l, hls, he.ge⟩"
},
{
"state_after": "case tfae_3_to_1.intro.intro\nα : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl✝ : List α\na : α\nn : ℕ\ntfae_1_to_2 : ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n\ntfae_2_to_3 : (∃ l, l ∈ subchain s ∧ length l = n) → ∃ l, l ∈ subchain s ∧ n ≤ length l\nl : List α\nhs : l ∈ subchain s\nhn : n ≤ length l\n⊢ ↑n ≤ chainHeight s",
"state_before": "case tfae_3_to_1\nα : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\ntfae_1_to_2 : ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n\ntfae_2_to_3 : (∃ l, l ∈ subchain s ∧ length l = n) → ∃ l, l ∈ subchain s ∧ n ≤ length l\n⊢ (∃ l, l ∈ subchain s ∧ n ≤ length l) → ↑n ≤ chainHeight s",
"tactic": "rintro ⟨l, hs, hn⟩"
},
{
"state_after": "no goals",
"state_before": "case tfae_3_to_1.intro.intro\nα : Type u_1\nβ : Type ?u.5660\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl✝ : List α\na : α\nn : ℕ\ntfae_1_to_2 : ↑n ≤ chainHeight s → ∃ l, l ∈ subchain s ∧ length l = n\ntfae_2_to_3 : (∃ l, l ∈ subchain s ∧ length l = n) → ∃ l, l ∈ subchain s ∧ n ≤ length l\nl : List α\nhs : l ∈ subchain s\nhn : n ≤ length l\n⊢ ↑n ≤ chainHeight s",
"tactic": "exact le_iSup₂_of_le l hs (WithTop.coe_le_coe.2 hn)"
}
] |
[
117,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/CategoryTheory/Bicategory/Basic.lean
|
CategoryTheory.Bicategory.pentagon_hom_hom_inv_inv_hom
|
[
{
"state_after": "no goals",
"state_before": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng : b ⟶ c\nh : c ⟶ d\ni : d ⟶ e\n⊢ inv ((α_ f (g ≫ h) i).hom ≫ f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv) =\n inv ((α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom)",
"tactic": "simp"
}
] |
[
304,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.sum_curry_index
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\n⊢ (sum (sum f fun p c => single p.fst (single p.snd c)) fun a f => sum f (g a)) = sum f fun p c => g p.fst p.snd c",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\n⊢ (sum (Finsupp.curry f) fun a f => sum f (g a)) = sum f fun p c => g p.fst p.snd c",
"tactic": "rw [Finsupp.curry]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\n⊢ (sum (sum f fun p c => single p.fst (single p.snd c)) fun a f => sum f (g a)) = ?m.540167\n\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\n⊢ ?m.540167 = sum f fun p c => g p.fst p.snd c\n\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\n⊢ N",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\n⊢ (sum (sum f fun p c => single p.fst (single p.snd c)) fun a f => sum f (g a)) = sum f fun p c => g p.fst p.snd c",
"tactic": "trans"
},
{
"state_after": "case e_g\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\n⊢ (fun a b => sum (single a.fst (single a.snd b)) fun a f => sum f (g a)) = fun p c => g p.fst p.snd c",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\n⊢ (sum f fun a b => sum (single a.fst (single a.snd b)) fun a f => sum f (g a)) = sum f fun p c => g p.fst p.snd c",
"tactic": "congr"
},
{
"state_after": "case e_g.h.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\np : α × β\nc : M\n⊢ (sum (single p.fst (single p.snd c)) fun a f => sum f (g a)) = g p.fst p.snd c",
"state_before": "case e_g\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\n⊢ (fun a b => sum (single a.fst (single a.snd b)) fun a f => sum f (g a)) = fun p c => g p.fst p.snd c",
"tactic": "funext p c"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\np : α × β\nc : M\n⊢ (sum (single p.fst (single p.snd c)) fun a f => sum f (g a)) = ?m.541818\n\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\np : α × β\nc : M\n⊢ ?m.541818 = g p.fst p.snd c\n\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\np : α × β\nc : M\n⊢ N",
"state_before": "case e_g.h.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\np : α × β\nc : M\n⊢ (sum (single p.fst (single p.snd c)) fun a f => sum f (g a)) = g p.fst p.snd c",
"tactic": "trans"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\np : α × β\nc : M\n⊢ sum (single p.snd c) (g p.fst) = g p.fst p.snd c",
"tactic": "exact sum_single_index (hg₀ _ _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\n⊢ (sum (sum f fun p c => single p.fst (single p.snd c)) fun a f => sum f (g a)) = ?m.540167",
"tactic": "exact\n sum_sum_index (fun a => sum_zero_index) fun a b₀ b₁ =>\n sum_add_index' (fun a => hg₀ _ _) fun c d₀ d₁ => hg₁ _ _ _ _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.536703\nι : Type ?u.536706\nM : Type u_3\nM' : Type ?u.536712\nN : Type u_4\nP : Type ?u.536718\nG : Type ?u.536721\nH : Type ?u.536724\nR : Type ?u.536727\nS : Type ?u.536730\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\nhg₀ : ∀ (a : α) (b : β), g a b 0 = 0\nhg₁ : ∀ (a : α) (b : β) (c₀ c₁ : M), g a b (c₀ + c₁) = g a b c₀ + g a b c₁\np : α × β\nc : M\n⊢ (sum (single p.fst (single p.snd c)) fun a f => sum f (g a)) = ?m.541818",
"tactic": "exact sum_single_index sum_zero_index"
}
] |
[
1240,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1228,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.constantCoeff_invOfUnit
|
[
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : Ring R\nφ : MvPowerSeries σ R\nu : Rˣ\n⊢ ↑(constantCoeff σ R) (invOfUnit φ u) = ↑u⁻¹",
"tactic": "rw [← coeff_zero_eq_constantCoeff_apply, coeff_invOfUnit, if_pos rfl]"
}
] |
[
856,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
854,
1
] |
Mathlib/Data/Finset/Sigma.lean
|
Finset.not_mem_sigmaLift_of_ne_right
|
[
{
"state_after": "ι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\nγ : ι → Type u_1\ninst✝ : DecidableEq ι\nf : ⦃i : ι⦄ → α i → β i → Finset (γ i)\na : Sigma α\nb : Sigma β\nx : Sigma γ\nh : b.fst ≠ x.fst\n⊢ ¬∃ ha hb, x.snd ∈ f (ha ▸ a.snd) (hb ▸ b.snd)",
"state_before": "ι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\nγ : ι → Type u_1\ninst✝ : DecidableEq ι\nf : ⦃i : ι⦄ → α i → β i → Finset (γ i)\na : Sigma α\nb : Sigma β\nx : Sigma γ\nh : b.fst ≠ x.fst\n⊢ ¬x ∈ sigmaLift f a b",
"tactic": "rw [mem_sigmaLift]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\nγ : ι → Type u_1\ninst✝ : DecidableEq ι\nf : ⦃i : ι⦄ → α i → β i → Finset (γ i)\na : Sigma α\nb : Sigma β\nx : Sigma γ\nh : b.fst ≠ x.fst\n⊢ ¬∃ ha hb, x.snd ∈ f (ha ▸ a.snd) (hb ▸ b.snd)",
"tactic": "exact fun H => h H.snd.fst"
}
] |
[
164,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/AlgebraicTopology/SplitSimplicialObject.lean
|
SimplicialObject.Splitting.IndexSet.eqId_iff_len_le
|
[
{
"state_after": "C : Type ?u.4924\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ len A.fst.unop = len Δ.unop ↔ len Δ.unop ≤ len A.fst.unop",
"state_before": "C : Type ?u.4924\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ EqId A ↔ len Δ.unop ≤ len A.fst.unop",
"tactic": "rw [eqId_iff_len_eq]"
},
{
"state_after": "case mp\nC : Type ?u.4924\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ len A.fst.unop = len Δ.unop → len Δ.unop ≤ len A.fst.unop\n\ncase mpr\nC : Type ?u.4924\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ len Δ.unop ≤ len A.fst.unop → len A.fst.unop = len Δ.unop",
"state_before": "C : Type ?u.4924\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ len A.fst.unop = len Δ.unop ↔ len Δ.unop ≤ len A.fst.unop",
"tactic": "constructor"
},
{
"state_after": "case mp\nC : Type ?u.4924\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : len A.fst.unop = len Δ.unop\n⊢ len Δ.unop ≤ len A.fst.unop",
"state_before": "case mp\nC : Type ?u.4924\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ len A.fst.unop = len Δ.unop → len Δ.unop ≤ len A.fst.unop",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mp\nC : Type ?u.4924\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : len A.fst.unop = len Δ.unop\n⊢ len Δ.unop ≤ len A.fst.unop",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nC : Type ?u.4924\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ len Δ.unop ≤ len A.fst.unop → len A.fst.unop = len Δ.unop",
"tactic": "exact le_antisymm (len_le_of_epi (inferInstance : Epi A.e))"
}
] |
[
163,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.eq_of_le
|
[] |
[
91,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pullbackComparison_comp_snd
|
[] |
[
1457,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1454,
1
] |
Mathlib/Topology/Order/Basic.lean
|
exists_Icc_mem_subset_of_mem_nhds
|
[
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhs : s ∈ 𝓝 a\nb : α\nhba : b ≤ a\nhb_nhds : Icc b a ∈ 𝓝[Iic a] a\nhbs : Icc b a ⊆ s\n⊢ ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhs : s ∈ 𝓝 a\n⊢ ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s",
"tactic": "rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Iic (nhdsWithin_le_nhds hs) with\n ⟨b, hba, hb_nhds, hbs⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhs : s ∈ 𝓝 a\nb : α\nhba : b ≤ a\nhb_nhds : Icc b a ∈ 𝓝[Iic a] a\nhbs : Icc b a ⊆ s\nc : α\nhac : a ≤ c\nhc_nhds : Icc a c ∈ 𝓝[Ici a] a\nhcs : Icc a c ⊆ s\n⊢ ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s",
"state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhs : s ∈ 𝓝 a\nb : α\nhba : b ≤ a\nhb_nhds : Icc b a ∈ 𝓝[Iic a] a\nhbs : Icc b a ⊆ s\n⊢ ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s",
"tactic": "rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici (nhdsWithin_le_nhds hs) with\n ⟨c, hac, hc_nhds, hcs⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhs : s ∈ 𝓝 a\nb : α\nhba : b ≤ a\nhb_nhds : Icc b a ∈ 𝓝[Iic a] a\nhbs : Icc b a ⊆ s\nc : α\nhac : a ≤ c\nhc_nhds : Icc a c ∈ 𝓝[Ici a] a\nhcs : Icc a c ⊆ s\n⊢ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhs : s ∈ 𝓝 a\nb : α\nhba : b ≤ a\nhb_nhds : Icc b a ∈ 𝓝[Iic a] a\nhbs : Icc b a ⊆ s\nc : α\nhac : a ≤ c\nhc_nhds : Icc a c ∈ 𝓝[Ici a] a\nhcs : Icc a c ⊆ s\n⊢ ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s",
"tactic": "refine' ⟨b, c, ⟨hba, hac⟩, _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhs : s ∈ 𝓝 a\nb : α\nhba : b ≤ a\nhb_nhds : Icc b a ∈ 𝓝[Iic a] a\nhbs : Icc b a ⊆ s\nc : α\nhac : a ≤ c\nhc_nhds : Icc a c ∈ 𝓝[Ici a] a\nhcs : Icc a c ⊆ s\n⊢ Icc b a ∪ Icc a c ∈ 𝓝[Iic a] a ⊔ 𝓝[Ici a] a ∧ Icc b a ∪ Icc a c ⊆ s",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhs : s ∈ 𝓝 a\nb : α\nhba : b ≤ a\nhb_nhds : Icc b a ∈ 𝓝[Iic a] a\nhbs : Icc b a ⊆ s\nc : α\nhac : a ≤ c\nhc_nhds : Icc a c ∈ 𝓝[Ici a] a\nhcs : Icc a c ⊆ s\n⊢ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s",
"tactic": "rw [← Icc_union_Icc_eq_Icc hba hac, ← nhds_left_sup_nhds_right]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhs : s ∈ 𝓝 a\nb : α\nhba : b ≤ a\nhb_nhds : Icc b a ∈ 𝓝[Iic a] a\nhbs : Icc b a ⊆ s\nc : α\nhac : a ≤ c\nhc_nhds : Icc a c ∈ 𝓝[Ici a] a\nhcs : Icc a c ⊆ s\n⊢ Icc b a ∪ Icc a c ∈ 𝓝[Iic a] a ⊔ 𝓝[Ici a] a ∧ Icc b a ∪ Icc a c ⊆ s",
"tactic": "exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩"
}
] |
[
1275,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1267,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.one_left
|
[] |
[
190,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Tactic/NormNum/Basic.lean
|
Mathlib.Meta.NormNum.isRat_pow
|
[
{
"state_after": "case mk.mk\nα : Type u_1\ninst✝ : Ring α\nan : ℤ\nad b : ℕ\ninv✝ : Invertible ↑ad\n⊢ IsRat ((↑an * ⅟↑ad) ^ b) (an ^ b) (ad ^ b)",
"state_before": "α : Type u_1\ninst✝ : Ring α\nf : α → ℕ → α\na : α\nan cn : ℤ\nad b b' cd : ℕ\n⊢ f = HPow.hPow → IsRat a an ad → IsNat b b' → Int.pow an b' = cn → Nat.pow ad b' = cd → IsRat (f a b) cn cd",
"tactic": "rintro rfl ⟨_, rfl⟩ ⟨rfl⟩ (rfl : an ^ b = _) (rfl : ad ^ b = _)"
},
{
"state_after": "case mk.mk\nα : Type u_1\ninst✝ : Ring α\nan : ℤ\nad b : ℕ\ninv✝ : Invertible ↑ad\nthis : Invertible (↑ad ^ b)\n⊢ IsRat ((↑an * ⅟↑ad) ^ b) (an ^ b) (ad ^ b)",
"state_before": "case mk.mk\nα : Type u_1\ninst✝ : Ring α\nan : ℤ\nad b : ℕ\ninv✝ : Invertible ↑ad\n⊢ IsRat ((↑an * ⅟↑ad) ^ b) (an ^ b) (ad ^ b)",
"tactic": "have := invertiblePow (ad:α) b"
},
{
"state_after": "case mk.mk\nα : Type u_1\ninst✝ : Ring α\nan : ℤ\nad b : ℕ\ninv✝ : Invertible ↑ad\nthis : Invertible ↑(ad ^ b)\n⊢ IsRat ((↑an * ⅟↑ad) ^ b) (an ^ b) (ad ^ b)",
"state_before": "case mk.mk\nα : Type u_1\ninst✝ : Ring α\nan : ℤ\nad b : ℕ\ninv✝ : Invertible ↑ad\nthis : Invertible (↑ad ^ b)\n⊢ IsRat ((↑an * ⅟↑ad) ^ b) (an ^ b) (ad ^ b)",
"tactic": "rw [← Nat.cast_pow] at this"
},
{
"state_after": "case mk.mk\nα : Type u_1\ninst✝ : Ring α\nan : ℤ\nad b : ℕ\ninv✝ : Invertible ↑ad\nthis : Invertible ↑(ad ^ b)\n⊢ (↑an * ⅟↑ad) ^ b = ↑(an ^ b) * ⅟↑(ad ^ b)",
"state_before": "case mk.mk\nα : Type u_1\ninst✝ : Ring α\nan : ℤ\nad b : ℕ\ninv✝ : Invertible ↑ad\nthis : Invertible ↑(ad ^ b)\n⊢ IsRat ((↑an * ⅟↑ad) ^ b) (an ^ b) (ad ^ b)",
"tactic": "use this"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nα : Type u_1\ninst✝ : Ring α\nan : ℤ\nad b : ℕ\ninv✝ : Invertible ↑ad\nthis : Invertible ↑(ad ^ b)\n⊢ (↑an * ⅟↑ad) ^ b = ↑(an ^ b) * ⅟↑(ad ^ b)",
"tactic": "simp [invOf_pow, Commute.mul_pow]"
}
] |
[
442,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
435,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
MonoidHom.comp_mul
|
[
{
"state_after": "case h\nα : Type ?u.215866\nβ : Type ?u.215869\nM : Type u_1\nN : Type u_2\nP : Type u_3\nG : Type ?u.215881\nH : Type ?u.215884\nF : Type ?u.215887\ninst✝⁴ : Group G\ninst✝³ : CommGroup H\ninst✝² : MulOneClass M\ninst✝¹ : CommMonoid N\ninst✝ : CommMonoid P\ng : N →* P\nf₁ f₂ : M →* N\nx✝ : M\n⊢ ↑(comp g (f₁ * f₂)) x✝ = ↑(comp g f₁ * comp g f₂) x✝",
"state_before": "α : Type ?u.215866\nβ : Type ?u.215869\nM : Type u_1\nN : Type u_2\nP : Type u_3\nG : Type ?u.215881\nH : Type ?u.215884\nF : Type ?u.215887\ninst✝⁴ : Group G\ninst✝³ : CommGroup H\ninst✝² : MulOneClass M\ninst✝¹ : CommMonoid N\ninst✝ : CommMonoid P\ng : N →* P\nf₁ f₂ : M →* N\n⊢ comp g (f₁ * f₂) = comp g f₁ * comp g f₂",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type ?u.215866\nβ : Type ?u.215869\nM : Type u_1\nN : Type u_2\nP : Type u_3\nG : Type ?u.215881\nH : Type ?u.215884\nF : Type ?u.215887\ninst✝⁴ : Group G\ninst✝³ : CommGroup H\ninst✝² : MulOneClass M\ninst✝¹ : CommMonoid N\ninst✝ : CommMonoid P\ng : N →* P\nf₁ f₂ : M →* N\nx✝ : M\n⊢ ↑(comp g (f₁ * f₂)) x✝ = ↑(comp g f₁ * comp g f₂) x✝",
"tactic": "simp only [mul_apply, Function.comp_apply, map_mul, coe_comp]"
}
] |
[
1531,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1528,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearEquiv.coe_curry_symm
|
[] |
[
1992,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1991,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
OrthogonalFamily.comp
|
[] |
[
2065,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2063,
1
] |
Mathlib/Algebra/GroupWithZero/Power.lean
|
Commute.zpow_right₀
|
[] |
[
124,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Analysis/NormedSpace/PiLp.lean
|
PiLp.dist_eq_iSup
|
[
{
"state_after": "p : ℝ≥0∞\n𝕜 : Type ?u.22498\n𝕜' : Type ?u.22501\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.22514\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → Dist (α i)\nf g : PiLp ⊤ α\n⊢ (if ⊤ = 0 then ↑(Finset.card (Finite.toFinset (_ : Set.Finite {i | f i ≠ g i})))\n else if ⊤ = ⊤ then ⨆ (i : ι), dist (f i) (g i) else (∑ i : ι, dist (f i) (g i) ^ 0) ^ (1 / 0)) =\n ⨆ (i : ι), dist (f i) (g i)",
"state_before": "p : ℝ≥0∞\n𝕜 : Type ?u.22498\n𝕜' : Type ?u.22501\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.22514\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → Dist (α i)\nf g : PiLp ⊤ α\n⊢ dist f g = ⨆ (i : ι), dist (f i) (g i)",
"tactic": "dsimp [dist]"
},
{
"state_after": "no goals",
"state_before": "p : ℝ≥0∞\n𝕜 : Type ?u.22498\n𝕜' : Type ?u.22501\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.22514\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → Dist (α i)\nf g : PiLp ⊤ α\n⊢ (if ⊤ = 0 then ↑(Finset.card (Finite.toFinset (_ : Set.Finite {i | f i ≠ g i})))\n else if ⊤ = ⊤ then ⨆ (i : ι), dist (f i) (g i) else (∑ i : ι, dist (f i) (g i) ^ 0) ^ (1 / 0)) =\n ⨆ (i : ι), dist (f i) (g i)",
"tactic": "exact if_neg ENNReal.top_ne_zero"
}
] |
[
221,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Analysis/Normed/Group/AddCircle.lean
|
AddCircle.norm_coe_mul
|
[
{
"state_after": "p x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\n⊢ ‖↑(t * x)‖ = abs t * ‖↑x‖",
"state_before": "p x t : ℝ\n⊢ ‖↑(t * x)‖ = abs t * ‖↑x‖",
"tactic": "have aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b) := fun {a b c} h => by\n simp only [mem_zmultiples_iff] at h⊢\n obtain ⟨n, rfl⟩ := h\n exact ⟨n, (mul_smul_comm n c b).symm⟩"
},
{
"state_after": "case inl\np x : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\n⊢ ‖↑(0 * x)‖ = abs 0 * ‖↑x‖\n\ncase inr\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\n⊢ ‖↑(t * x)‖ = abs t * ‖↑x‖",
"state_before": "p x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\n⊢ ‖↑(t * x)‖ = abs t * ‖↑x‖",
"tactic": "rcases eq_or_ne t 0 with (rfl | ht)"
},
{
"state_after": "case inr\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\n⊢ ‖↑(t * x)‖ = abs t * ‖↑x‖",
"state_before": "case inr\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\n⊢ ‖↑(t * x)‖ = abs t * ‖↑x‖",
"tactic": "have ht' : |t| ≠ 0 := (not_congr abs_eq_zero).mpr ht"
},
{
"state_after": "case inr\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\n⊢ sInf ((fun a => abs a) '' {m | ↑m = ↑(t * x)}) = abs t * sInf ((fun a => abs a) '' {m | ↑m = ↑x})",
"state_before": "case inr\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\n⊢ ‖↑(t * x)‖ = abs t * ‖↑x‖",
"tactic": "simp only [quotient_norm_eq, Real.norm_eq_abs]"
},
{
"state_after": "case inr\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\n⊢ sInf ((fun a => abs a) '' {m | ↑m = ↑(t * x)}) = sInf (abs t • (fun a => abs a) '' {m | ↑m = ↑x})",
"state_before": "case inr\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\n⊢ sInf ((fun a => abs a) '' {m | ↑m = ↑(t * x)}) = abs t * sInf ((fun a => abs a) '' {m | ↑m = ↑x})",
"tactic": "conv_rhs => rw [← smul_eq_mul, ← Real.sInf_smul_of_nonneg (abs_nonneg t)]"
},
{
"state_after": "case inr\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\n⊢ sInf ((fun a => abs a) '' {m | m - t * x ∈ zmultiples (t * p)}) =\n sInf (abs t • (fun a => abs a) '' {m | m - x ∈ zmultiples p})",
"state_before": "case inr\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\n⊢ sInf ((fun a => abs a) '' {m | ↑m = ↑(t * x)}) = sInf (abs t • (fun a => abs a) '' {m | ↑m = ↑x})",
"tactic": "simp only [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_iff_sub_mem]"
},
{
"state_after": "case inr.e_a\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\n⊢ (fun a => abs a) '' {m | m - t * x ∈ zmultiples (t * p)} = abs t • (fun a => abs a) '' {m | m - x ∈ zmultiples p}",
"state_before": "case inr\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\n⊢ sInf ((fun a => abs a) '' {m | m - t * x ∈ zmultiples (t * p)}) =\n sInf (abs t • (fun a => abs a) '' {m | m - x ∈ zmultiples p})",
"tactic": "congr 1"
},
{
"state_after": "case inr.e_a.h\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz : ℝ\n⊢ z ∈ (fun a => abs a) '' {m | m - t * x ∈ zmultiples (t * p)} ↔\n z ∈ abs t • (fun a => abs a) '' {m | m - x ∈ zmultiples p}",
"state_before": "case inr.e_a\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\n⊢ (fun a => abs a) '' {m | m - t * x ∈ zmultiples (t * p)} = abs t • (fun a => abs a) '' {m | m - x ∈ zmultiples p}",
"tactic": "ext z"
},
{
"state_after": "case inr.e_a.h\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz : ℝ\n⊢ z ∈ (fun a => abs a) '' {m | m - t * x ∈ zmultiples (t * p)} ↔\n (abs t)⁻¹ • z ∈ (fun a => abs a) '' {m | m - x ∈ zmultiples p}",
"state_before": "case inr.e_a.h\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz : ℝ\n⊢ z ∈ (fun a => abs a) '' {m | m - t * x ∈ zmultiples (t * p)} ↔\n z ∈ abs t • (fun a => abs a) '' {m | m - x ∈ zmultiples p}",
"tactic": "rw [mem_smul_set_iff_inv_smul_mem₀ ht']"
},
{
"state_after": "case inr.e_a.h\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz : ℝ\n⊢ (∃ y, y - t * x ∈ zmultiples (t * p) ∧ abs y = z) ↔ ∃ w, w - x ∈ zmultiples p ∧ abs w = (abs t)⁻¹ * z",
"state_before": "case inr.e_a.h\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz : ℝ\n⊢ z ∈ (fun a => abs a) '' {m | m - t * x ∈ zmultiples (t * p)} ↔\n (abs t)⁻¹ • z ∈ (fun a => abs a) '' {m | m - x ∈ zmultiples p}",
"tactic": "show\n (∃ y, y - t * x ∈ zmultiples (t * p) ∧ |y| = z) ↔ ∃ w, w - x ∈ zmultiples p ∧ |w| = (|t|)⁻¹ * z"
},
{
"state_after": "case inr.e_a.h.mp\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz : ℝ\n⊢ (∃ y, y - t * x ∈ zmultiples (t * p) ∧ abs y = z) → ∃ w, w - x ∈ zmultiples p ∧ abs w = (abs t)⁻¹ * z\n\ncase inr.e_a.h.mpr\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz : ℝ\n⊢ (∃ w, w - x ∈ zmultiples p ∧ abs w = (abs t)⁻¹ * z) → ∃ y, y - t * x ∈ zmultiples (t * p) ∧ abs y = z",
"state_before": "case inr.e_a.h\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz : ℝ\n⊢ (∃ y, y - t * x ∈ zmultiples (t * p) ∧ abs y = z) ↔ ∃ w, w - x ∈ zmultiples p ∧ abs w = (abs t)⁻¹ * z",
"tactic": "constructor"
},
{
"state_after": "p x t a b c : ℝ\nh : ∃ k, k • b = a\n⊢ ∃ k, k • (c * b) = c * a",
"state_before": "p x t a b c : ℝ\nh : a ∈ zmultiples b\n⊢ c * a ∈ zmultiples (c * b)",
"tactic": "simp only [mem_zmultiples_iff] at h⊢"
},
{
"state_after": "case intro\np x t b c : ℝ\nn : ℤ\n⊢ ∃ k, k • (c * b) = c * n • b",
"state_before": "p x t a b c : ℝ\nh : ∃ k, k • b = a\n⊢ ∃ k, k • (c * b) = c * a",
"tactic": "obtain ⟨n, rfl⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case intro\np x t b c : ℝ\nn : ℤ\n⊢ ∃ k, k • (c * b) = c * n • b",
"tactic": "exact ⟨n, (mul_smul_comm n c b).symm⟩"
},
{
"state_after": "no goals",
"state_before": "case inl\np x : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\n⊢ ‖↑(0 * x)‖ = abs 0 * ‖↑x‖",
"tactic": "simp"
},
{
"state_after": "case inr.e_a.h.mp.intro.intro\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\ny : ℝ\nhy : y - t * x ∈ zmultiples (t * p)\n⊢ ∃ w, w - x ∈ zmultiples p ∧ abs w = (abs t)⁻¹ * abs y",
"state_before": "case inr.e_a.h.mp\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz : ℝ\n⊢ (∃ y, y - t * x ∈ zmultiples (t * p) ∧ abs y = z) → ∃ w, w - x ∈ zmultiples p ∧ abs w = (abs t)⁻¹ * z",
"tactic": "rintro ⟨y, hy, rfl⟩"
},
{
"state_after": "case inr.e_a.h.mp.intro.intro\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\ny : ℝ\nhy : y - t * x ∈ zmultiples (t * p)\n⊢ t⁻¹ * y - x ∈ zmultiples p",
"state_before": "case inr.e_a.h.mp.intro.intro\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\ny : ℝ\nhy : y - t * x ∈ zmultiples (t * p)\n⊢ ∃ w, w - x ∈ zmultiples p ∧ abs w = (abs t)⁻¹ * abs y",
"tactic": "refine' ⟨t⁻¹ * y, _, by rw [abs_mul, abs_inv]⟩"
},
{
"state_after": "case inr.e_a.h.mp.intro.intro\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\ny : ℝ\nhy : y - t * x ∈ zmultiples (t * p)\n⊢ t⁻¹ * (y - t * x) ∈ zmultiples (t⁻¹ * (t * p))",
"state_before": "case inr.e_a.h.mp.intro.intro\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\ny : ℝ\nhy : y - t * x ∈ zmultiples (t * p)\n⊢ t⁻¹ * y - x ∈ zmultiples p",
"tactic": "rw [← inv_mul_cancel_left₀ ht x, ← inv_mul_cancel_left₀ ht p, ← mul_sub]"
},
{
"state_after": "no goals",
"state_before": "case inr.e_a.h.mp.intro.intro\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\ny : ℝ\nhy : y - t * x ∈ zmultiples (t * p)\n⊢ t⁻¹ * (y - t * x) ∈ zmultiples (t⁻¹ * (t * p))",
"tactic": "exact aux hy"
},
{
"state_after": "no goals",
"state_before": "p x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\ny : ℝ\nhy : y - t * x ∈ zmultiples (t * p)\n⊢ abs (t⁻¹ * y) = (abs t)⁻¹ * abs y",
"tactic": "rw [abs_mul, abs_inv]"
},
{
"state_after": "case inr.e_a.h.mpr.intro.intro\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz w : ℝ\nhw : w - x ∈ zmultiples p\nhw' : abs w = (abs t)⁻¹ * z\n⊢ ∃ y, y - t * x ∈ zmultiples (t * p) ∧ abs y = z",
"state_before": "case inr.e_a.h.mpr\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz : ℝ\n⊢ (∃ w, w - x ∈ zmultiples p ∧ abs w = (abs t)⁻¹ * z) → ∃ y, y - t * x ∈ zmultiples (t * p) ∧ abs y = z",
"tactic": "rintro ⟨w, hw, hw'⟩"
},
{
"state_after": "case inr.e_a.h.mpr.intro.intro\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz w : ℝ\nhw : w - x ∈ zmultiples p\nhw' : abs w = (abs t)⁻¹ * z\n⊢ t * w - t * x ∈ zmultiples (t * p)",
"state_before": "case inr.e_a.h.mpr.intro.intro\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz w : ℝ\nhw : w - x ∈ zmultiples p\nhw' : abs w = (abs t)⁻¹ * z\n⊢ ∃ y, y - t * x ∈ zmultiples (t * p) ∧ abs y = z",
"tactic": "refine' ⟨t * w, _, by rw [← (eq_inv_mul_iff_mul_eq₀ ht').mp hw', abs_mul]⟩"
},
{
"state_after": "case inr.e_a.h.mpr.intro.intro\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz w : ℝ\nhw : w - x ∈ zmultiples p\nhw' : abs w = (abs t)⁻¹ * z\n⊢ t * (w - x) ∈ zmultiples (t * p)",
"state_before": "case inr.e_a.h.mpr.intro.intro\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz w : ℝ\nhw : w - x ∈ zmultiples p\nhw' : abs w = (abs t)⁻¹ * z\n⊢ t * w - t * x ∈ zmultiples (t * p)",
"tactic": "rw [← mul_sub]"
},
{
"state_after": "no goals",
"state_before": "case inr.e_a.h.mpr.intro.intro\np x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz w : ℝ\nhw : w - x ∈ zmultiples p\nhw' : abs w = (abs t)⁻¹ * z\n⊢ t * (w - x) ∈ zmultiples (t * p)",
"tactic": "exact aux hw"
},
{
"state_after": "no goals",
"state_before": "p x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)\nht : t ≠ 0\nht' : abs t ≠ 0\nz w : ℝ\nhw : w - x ∈ zmultiples p\nhw' : abs w = (abs t)⁻¹ * z\n⊢ abs (t * w) = z",
"tactic": "rw [← (eq_inv_mul_iff_mul_eq₀ ht').mp hw', abs_mul]"
}
] |
[
71,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.mk_of_mem
|
[] |
[
588,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
587,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.TM1to1.supportsStmt_write
|
[
{
"state_after": "no goals",
"state_before": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nS : Finset Λ'\nl : List Bool\nq : Stmt Bool Λ' σ\n⊢ SupportsStmt S (write l q) = SupportsStmt S q",
"tactic": "induction' l with _ l IH <;> simp only [write, SupportsStmt, *]"
}
] |
[
1718,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1716,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.extend_mono
|
[
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nm : (s : Set α) → MeasurableSet s → ℝ≥0∞\nm0 : m ∅ (_ : MeasurableSet ∅) = 0\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),\n Pairwise (Disjoint on f) →\n m (⋃ (i : ℕ), f i) (_ : MeasurableSet (⋃ (b : ℕ), f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i))\ns₁ s₂ : Set α\nh₁ : MeasurableSet s₁\nhs : s₁ ⊆ s₂\n⊢ ∀ (i : (fun s => MeasurableSet s) s₂), extend m s₁ ≤ m s₂ i",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm : (s : Set α) → MeasurableSet s → ℝ≥0∞\nm0 : m ∅ (_ : MeasurableSet ∅) = 0\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),\n Pairwise (Disjoint on f) →\n m (⋃ (i : ℕ), f i) (_ : MeasurableSet (⋃ (b : ℕ), f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i))\ns₁ s₂ : Set α\nh₁ : MeasurableSet s₁\nhs : s₁ ⊆ s₂\n⊢ extend m s₁ ≤ extend m s₂",
"tactic": "refine' le_iInf _"
},
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nm : (s : Set α) → MeasurableSet s → ℝ≥0∞\nm0 : m ∅ (_ : MeasurableSet ∅) = 0\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),\n Pairwise (Disjoint on f) →\n m (⋃ (i : ℕ), f i) (_ : MeasurableSet (⋃ (b : ℕ), f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i))\ns₁ s₂ : Set α\nh₁ : MeasurableSet s₁\nhs : s₁ ⊆ s₂\nh₂ : (fun s => MeasurableSet s) s₂\n⊢ extend m s₁ ≤ m s₂ h₂",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm : (s : Set α) → MeasurableSet s → ℝ≥0∞\nm0 : m ∅ (_ : MeasurableSet ∅) = 0\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),\n Pairwise (Disjoint on f) →\n m (⋃ (i : ℕ), f i) (_ : MeasurableSet (⋃ (b : ℕ), f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i))\ns₁ s₂ : Set α\nh₁ : MeasurableSet s₁\nhs : s₁ ⊆ s₂\n⊢ ∀ (i : (fun s => MeasurableSet s) s₂), extend m s₁ ≤ m s₂ i",
"tactic": "intro h₂"
},
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nm : (s : Set α) → MeasurableSet s → ℝ≥0∞\nm0 : m ∅ (_ : MeasurableSet ∅) = 0\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),\n Pairwise (Disjoint on f) →\n m (⋃ (i : ℕ), f i) (_ : MeasurableSet (⋃ (b : ℕ), f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i))\ns₁ s₂ : Set α\nh₁ : MeasurableSet s₁\nhs : s₁ ⊆ s₂\nh₂ : (fun s => MeasurableSet s) s₂\nthis : extend m (s₁ ∪ s₂ \\ s₁) = extend m s₁ + extend m (s₂ \\ s₁)\n⊢ extend m s₁ ≤ m s₂ h₂",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm : (s : Set α) → MeasurableSet s → ℝ≥0∞\nm0 : m ∅ (_ : MeasurableSet ∅) = 0\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),\n Pairwise (Disjoint on f) →\n m (⋃ (i : ℕ), f i) (_ : MeasurableSet (⋃ (b : ℕ), f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i))\ns₁ s₂ : Set α\nh₁ : MeasurableSet s₁\nhs : s₁ ⊆ s₂\nh₂ : (fun s => MeasurableSet s) s₂\n⊢ extend m s₁ ≤ m s₂ h₂",
"tactic": "have :=\n extend_union MeasurableSet.empty m0 MeasurableSet.iUnion mU disjoint_sdiff_self_right h₁\n (h₂.diff h₁)"
},
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nm : (s : Set α) → MeasurableSet s → ℝ≥0∞\nm0 : m ∅ (_ : MeasurableSet ∅) = 0\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),\n Pairwise (Disjoint on f) →\n m (⋃ (i : ℕ), f i) (_ : MeasurableSet (⋃ (b : ℕ), f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i))\ns₁ s₂ : Set α\nh₁ : MeasurableSet s₁\nhs : s₁ ⊆ s₂\nh₂ : (fun s => MeasurableSet s) s₂\nthis : extend m s₂ = extend m s₁ + extend m (s₂ \\ s₁)\n⊢ extend m s₁ ≤ m s₂ h₂",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm : (s : Set α) → MeasurableSet s → ℝ≥0∞\nm0 : m ∅ (_ : MeasurableSet ∅) = 0\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),\n Pairwise (Disjoint on f) →\n m (⋃ (i : ℕ), f i) (_ : MeasurableSet (⋃ (b : ℕ), f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i))\ns₁ s₂ : Set α\nh₁ : MeasurableSet s₁\nhs : s₁ ⊆ s₂\nh₂ : (fun s => MeasurableSet s) s₂\nthis : extend m (s₁ ∪ s₂ \\ s₁) = extend m s₁ + extend m (s₂ \\ s₁)\n⊢ extend m s₁ ≤ m s₂ h₂",
"tactic": "rw [union_diff_cancel hs] at this"
},
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nm : (s : Set α) → MeasurableSet s → ℝ≥0∞\nm0 : m ∅ (_ : MeasurableSet ∅) = 0\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),\n Pairwise (Disjoint on f) →\n m (⋃ (i : ℕ), f i) (_ : MeasurableSet (⋃ (b : ℕ), f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i))\ns₁ s₂ : Set α\nh₁ : MeasurableSet s₁\nhs : s₁ ⊆ s₂\nh₂ : (fun s => MeasurableSet s) s₂\nthis : extend m s₂ = extend m s₁ + extend m (s₂ \\ s₁)\n⊢ extend m s₁ ≤ extend m s₂",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm : (s : Set α) → MeasurableSet s → ℝ≥0∞\nm0 : m ∅ (_ : MeasurableSet ∅) = 0\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),\n Pairwise (Disjoint on f) →\n m (⋃ (i : ℕ), f i) (_ : MeasurableSet (⋃ (b : ℕ), f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i))\ns₁ s₂ : Set α\nh₁ : MeasurableSet s₁\nhs : s₁ ⊆ s₂\nh₂ : (fun s => MeasurableSet s) s₂\nthis : extend m s₂ = extend m s₁ + extend m (s₂ \\ s₁)\n⊢ extend m s₁ ≤ m s₂ h₂",
"tactic": "rw [← extend_eq m]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm : (s : Set α) → MeasurableSet s → ℝ≥0∞\nm0 : m ∅ (_ : MeasurableSet ∅) = 0\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),\n Pairwise (Disjoint on f) →\n m (⋃ (i : ℕ), f i) (_ : MeasurableSet (⋃ (b : ℕ), f b)) = ∑' (i : ℕ), m (f i) (_ : MeasurableSet (f i))\ns₁ s₂ : Set α\nh₁ : MeasurableSet s₁\nhs : s₁ ⊆ s₂\nh₂ : (fun s => MeasurableSet s) s₂\nthis : extend m s₂ = extend m s₁ + extend m (s₂ \\ s₁)\n⊢ extend m s₁ ≤ extend m s₂",
"tactic": "exact le_iff_exists_add.2 ⟨_, this⟩"
}
] |
[
1571,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1563,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.prod_inter_prod
|
[
{
"state_after": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.13537\nδ : Type ?u.13540\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\n⊢ (x, y) ∈ s₁ ×ˢ t₁ ∩ s₂ ×ˢ t₂ ↔ (x, y) ∈ (s₁ ∩ s₂) ×ˢ (t₁ ∩ t₂)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.13537\nδ : Type ?u.13540\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\n⊢ s₁ ×ˢ t₁ ∩ s₂ ×ˢ t₂ = (s₁ ∩ s₂) ×ˢ (t₁ ∩ t₂)",
"tactic": "ext ⟨x, y⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.13537\nδ : Type ?u.13540\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nx : α\ny : β\n⊢ (x, y) ∈ s₁ ×ˢ t₁ ∩ s₂ ×ˢ t₂ ↔ (x, y) ∈ (s₁ ∩ s₂) ×ˢ (t₁ ∩ t₂)",
"tactic": "simp [and_assoc, and_left_comm]"
}
] |
[
173,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Order/Hom/Bounded.lean
|
BotHom.coe_comp
|
[] |
[
452,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
451,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.int_cast_zmod_eq_zero_iff_dvd
|
[
{
"state_after": "no goals",
"state_before": "a : ℤ\nb : ℕ\n⊢ ↑a = 0 ↔ ↑b ∣ a",
"tactic": "rw [← Int.cast_zero, ZMod.int_cast_eq_int_cast_iff, Int.modEq_zero_iff_dvd]"
}
] |
[
475,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
474,
1
] |
Mathlib/CategoryTheory/Abelian/NonPreadditive.lean
|
CategoryTheory.NonPreadditiveAbelian.sub_add
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b c : X ⟶ Y\n⊢ a - b + c = a - (b - c)",
"tactic": "rw [add_def, neg_def, sub_sub_sub, sub_zero]"
}
] |
[
419,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
418,
1
] |
Mathlib/Topology/Basic.lean
|
AccPt.mono
|
[] |
[
1197,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1196,
1
] |
Mathlib/Data/Option/Basic.lean
|
Option.map_comp_some
|
[] |
[
76,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Order/BooleanAlgebra.lean
|
inf_sdiff_eq_bot_iff
|
[
{
"state_after": "α : Type u\nβ : Type ?u.14724\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhz : z ≤ y\nhx : x ≤ y\n⊢ Disjoint z (y \\ x) ↔ z ≤ x",
"state_before": "α : Type u\nβ : Type ?u.14724\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhz : z ≤ y\nhx : x ≤ y\n⊢ z ⊓ y \\ x = ⊥ ↔ z ≤ x",
"tactic": "rw [← disjoint_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.14724\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhz : z ≤ y\nhx : x ≤ y\n⊢ Disjoint z (y \\ x) ↔ z ≤ x",
"tactic": "exact disjoint_sdiff_iff_le hz hx"
}
] |
[
265,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Data/Fintype/Card.lean
|
Finite.of_surjective
|
[] |
[
444,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Mathlib/Analysis/Calculus/Deriv/Star.lean
|
deriv.star
|
[] |
[
63,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
11
] |
Mathlib/MeasureTheory/Group/Integration.lean
|
MeasureTheory.Integrable.comp_inv
|
[] |
[
42,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/Data/Finset/Pi.lean
|
Finset.pi_empty
|
[] |
[
91,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.integrable_smul_const
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.1111558\nγ : Type ?u.1111561\nδ : Type ?u.1111564\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁶ : MeasurableSpace δ\ninst✝⁵ : NormedAddCommGroup β\ninst✝⁴ : NormedAddCommGroup γ\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : CompleteSpace 𝕜\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : α → 𝕜\nc : E\nhc : c ≠ 0\n⊢ AEStronglyMeasurable f μ → ((∫⁻ (a : α), ↑‖f a‖₊ * ↑‖c‖₊ ∂μ) < ⊤ ↔ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) < ⊤)",
"state_before": "α : Type u_2\nβ : Type ?u.1111558\nγ : Type ?u.1111561\nδ : Type ?u.1111564\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁶ : MeasurableSpace δ\ninst✝⁵ : NormedAddCommGroup β\ninst✝⁴ : NormedAddCommGroup γ\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : CompleteSpace 𝕜\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : α → 𝕜\nc : E\nhc : c ≠ 0\n⊢ (Integrable fun x => f x • c) ↔ Integrable f",
"tactic": "simp_rw [Integrable, aestronglyMeasurable_smul_const_iff (f := f) hc, and_congr_right_iff,\n HasFiniteIntegral, nnnorm_smul, ENNReal.coe_mul]"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.1111558\nγ : Type ?u.1111561\nδ : Type ?u.1111564\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁶ : MeasurableSpace δ\ninst✝⁵ : NormedAddCommGroup β\ninst✝⁴ : NormedAddCommGroup γ\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : CompleteSpace 𝕜\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : α → 𝕜\nc : E\nhc : c ≠ 0\na✝ : AEStronglyMeasurable f μ\n⊢ (∫⁻ (a : α), ↑‖f a‖₊ * ↑‖c‖₊ ∂μ) < ⊤ ↔ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) < ⊤",
"state_before": "α : Type u_2\nβ : Type ?u.1111558\nγ : Type ?u.1111561\nδ : Type ?u.1111564\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁶ : MeasurableSpace δ\ninst✝⁵ : NormedAddCommGroup β\ninst✝⁴ : NormedAddCommGroup γ\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : CompleteSpace 𝕜\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : α → 𝕜\nc : E\nhc : c ≠ 0\n⊢ AEStronglyMeasurable f μ → ((∫⁻ (a : α), ↑‖f a‖₊ * ↑‖c‖₊ ∂μ) < ⊤ ↔ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) < ⊤)",
"tactic": "intro _"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.1111558\nγ : Type ?u.1111561\nδ : Type ?u.1111564\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁶ : MeasurableSpace δ\ninst✝⁵ : NormedAddCommGroup β\ninst✝⁴ : NormedAddCommGroup γ\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : CompleteSpace 𝕜\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : α → 𝕜\nc : E\nhc : c ≠ 0\na✝ : AEStronglyMeasurable f μ\n⊢ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) < ⊤ ∧ ↑‖c‖₊ < ⊤ ∨ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) = 0 ∨ ↑‖c‖₊ = 0 ↔ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) < ⊤",
"state_before": "α : Type u_2\nβ : Type ?u.1111558\nγ : Type ?u.1111561\nδ : Type ?u.1111564\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁶ : MeasurableSpace δ\ninst✝⁵ : NormedAddCommGroup β\ninst✝⁴ : NormedAddCommGroup γ\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : CompleteSpace 𝕜\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : α → 𝕜\nc : E\nhc : c ≠ 0\na✝ : AEStronglyMeasurable f μ\n⊢ (∫⁻ (a : α), ↑‖f a‖₊ * ↑‖c‖₊ ∂μ) < ⊤ ↔ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) < ⊤",
"tactic": "rw [lintegral_mul_const' _ _ ENNReal.coe_ne_top, ENNReal.mul_lt_top_iff]"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.1111558\nγ : Type ?u.1111561\nδ : Type ?u.1111564\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁶ : MeasurableSpace δ\ninst✝⁵ : NormedAddCommGroup β\ninst✝⁴ : NormedAddCommGroup γ\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : CompleteSpace 𝕜\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : α → 𝕜\nc : E\nhc : c ≠ 0\na✝ : AEStronglyMeasurable f μ\nthis : ∀ (x : ℝ≥0∞), x = 0 → x < ⊤\n⊢ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) < ⊤ ∧ ↑‖c‖₊ < ⊤ ∨ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) = 0 ∨ ↑‖c‖₊ = 0 ↔ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) < ⊤",
"state_before": "α : Type u_2\nβ : Type ?u.1111558\nγ : Type ?u.1111561\nδ : Type ?u.1111564\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁶ : MeasurableSpace δ\ninst✝⁵ : NormedAddCommGroup β\ninst✝⁴ : NormedAddCommGroup γ\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : CompleteSpace 𝕜\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : α → 𝕜\nc : E\nhc : c ≠ 0\na✝ : AEStronglyMeasurable f μ\n⊢ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) < ⊤ ∧ ↑‖c‖₊ < ⊤ ∨ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) = 0 ∨ ↑‖c‖₊ = 0 ↔ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) < ⊤",
"tactic": "have : ∀ x : ℝ≥0∞, x = 0 → x < ∞ := by simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.1111558\nγ : Type ?u.1111561\nδ : Type ?u.1111564\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁶ : MeasurableSpace δ\ninst✝⁵ : NormedAddCommGroup β\ninst✝⁴ : NormedAddCommGroup γ\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : CompleteSpace 𝕜\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : α → 𝕜\nc : E\nhc : c ≠ 0\na✝ : AEStronglyMeasurable f μ\nthis : ∀ (x : ℝ≥0∞), x = 0 → x < ⊤\n⊢ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) < ⊤ ∧ ↑‖c‖₊ < ⊤ ∨ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) = 0 ∨ ↑‖c‖₊ = 0 ↔ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) < ⊤",
"tactic": "simp [hc, or_iff_left_of_imp (this _)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.1111558\nγ : Type ?u.1111561\nδ : Type ?u.1111564\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁶ : MeasurableSpace δ\ninst✝⁵ : NormedAddCommGroup β\ninst✝⁴ : NormedAddCommGroup γ\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : CompleteSpace 𝕜\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : α → 𝕜\nc : E\nhc : c ≠ 0\na✝ : AEStronglyMeasurable f μ\n⊢ ∀ (x : ℝ≥0∞), x = 0 → x < ⊤",
"tactic": "simp"
}
] |
[
1068,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1062,
1
] |
Mathlib/Order/GaloisConnection.lean
|
sSup_image2_eq_sSup_sInf
|
[] |
[
389,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/Analysis/Normed/Group/HomCompletion.lean
|
NormedAddGroupHom.extension_coe
|
[] |
[
222,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.sin_add
|
[
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ sin (x + y) = sin x * cos y + cos x * sin y",
"tactic": "rw [← mul_left_inj' I_ne_zero, ← sinh_mul_I, add_mul, add_mul, mul_right_comm, ← sinh_mul_I,\n mul_assoc, ← sinh_mul_I, ← cosh_mul_I, ← cosh_mul_I, sinh_add]"
}
] |
[
849,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
847,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.isCauSeq_abs_exp
|
[
{
"state_after": "no goals",
"state_before": "z : ℂ\nn : ℕ\nhn : ↑abs z < ↑n\nhn0 : 0 < ↑n\n⊢ ↑abs z / ↑n < 1",
"tactic": "rwa [div_lt_iff hn0, one_mul]"
},
{
"state_after": "z : ℂ\nn : ℕ\nhn : ↑abs z < ↑n\nhn0 : 0 < ↑n\nm : ℕ\nhm : n ≤ m\n⊢ ↑abs z / ↑(Nat.succ m) * ↑abs (z ^ m / ↑(Nat.factorial m)) ≤ ↑abs z / ↑n * ↑abs (z ^ m / ↑(Nat.factorial m))",
"state_before": "z : ℂ\nn : ℕ\nhn : ↑abs z < ↑n\nhn0 : 0 < ↑n\nm : ℕ\nhm : n ≤ m\n⊢ abs' (↑abs (z ^ Nat.succ m / ↑(Nat.factorial (Nat.succ m)))) ≤ ↑abs z / ↑n * abs' (↑abs (z ^ m / ↑(Nat.factorial m)))",
"tactic": "rw [abs_abs, abs_abs, Nat.factorial_succ, pow_succ, mul_comm m.succ, Nat.cast_mul, ← div_div,\n mul_div_assoc, mul_div_right_comm, map_mul, map_div₀, abs_cast_nat]"
},
{
"state_after": "case h.h.h\nz : ℂ\nn : ℕ\nhn : ↑abs z < ↑n\nhn0 : 0 < ↑n\nm : ℕ\nhm : n ≤ m\n⊢ n ≤ Nat.succ m",
"state_before": "z : ℂ\nn : ℕ\nhn : ↑abs z < ↑n\nhn0 : 0 < ↑n\nm : ℕ\nhm : n ≤ m\n⊢ ↑abs z / ↑(Nat.succ m) * ↑abs (z ^ m / ↑(Nat.factorial m)) ≤ ↑abs z / ↑n * ↑abs (z ^ m / ↑(Nat.factorial m))",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case h.h.h\nz : ℂ\nn : ℕ\nhn : ↑abs z < ↑n\nhn0 : 0 < ↑n\nm : ℕ\nhm : n ≤ m\n⊢ n ≤ Nat.succ m",
"tactic": "exact le_trans hm (Nat.le_succ _)"
}
] |
[
363,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/Topology/Homeomorph.lean
|
Homeomorph.trans_apply
|
[] |
[
124,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Data/List/Rotate.lean
|
List.rotate_append_length_eq
|
[
{
"state_after": "α : Type u\nl l' : List α\n⊢ rotate' (l ++ l') (length l) = l' ++ l",
"state_before": "α : Type u\nl l' : List α\n⊢ rotate (l ++ l') (length l) = l' ++ l",
"tactic": "rw [rotate_eq_rotate']"
},
{
"state_after": "case nil\nα : Type u\nl' : List α\n⊢ rotate' ([] ++ l') (length []) = l' ++ []\n\ncase cons\nα : Type u\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : ∀ (l' : List α), rotate' (tail✝ ++ l') (length tail✝) = l' ++ tail✝\nl' : List α\n⊢ rotate' (head✝ :: tail✝ ++ l') (length (head✝ :: tail✝)) = l' ++ head✝ :: tail✝",
"state_before": "α : Type u\nl l' : List α\n⊢ rotate' (l ++ l') (length l) = l' ++ l",
"tactic": "induction l generalizing l'"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nl' : List α\n⊢ rotate' ([] ++ l') (length []) = l' ++ []",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : ∀ (l' : List α), rotate' (tail✝ ++ l') (length tail✝) = l' ++ tail✝\nl' : List α\n⊢ rotate' (head✝ :: tail✝ ++ l') (length (head✝ :: tail✝)) = l' ++ head✝ :: tail✝",
"tactic": "simp_all [rotate']"
}
] |
[
159,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Algebra/Star/Pointwise.lean
|
Set.mem_star
|
[] |
[
62,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Data/List/Card.lean
|
List.card_le_card_cons
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Sort ?u.22986\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\na : α\nas : List α\n⊢ card as ≤ card (a :: as)",
"tactic": "cases Decidable.em (a ∈ as) with\n| inl h => simp [h, Nat.le_refl]\n| inr h => simp [h, Nat.le_succ]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Sort ?u.22986\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\na : α\nas : List α\nh : a ∈ as\n⊢ card as ≤ card (a :: as)",
"tactic": "simp [h, Nat.le_refl]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Sort ?u.22986\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\na : α\nas : List α\nh : ¬a ∈ as\n⊢ card as ≤ card (a :: as)",
"tactic": "simp [h, Nat.le_succ]"
}
] |
[
90,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Analysis/Convex/Jensen.lean
|
ConcaveOn.exists_le_of_centerMass
|
[] |
[
125,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Analysis/Complex/Arg.lean
|
Complex.abs_sub_eq_iff
|
[] |
[
57,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
LinearMap.lTensor_id
|
[] |
[
1096,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1095,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
QuotientGroup.isOpenMap_coe
|
[
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ns : Set G\ns_op : IsOpen s\n⊢ IsOpen (mk '' s)",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\n⊢ IsOpenMap mk",
"tactic": "intro s s_op"
},
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ns : Set G\ns_op : IsOpen s\n⊢ IsOpen (mk ⁻¹' (mk '' s))",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ns : Set G\ns_op : IsOpen s\n⊢ IsOpen (mk '' s)",
"tactic": "change IsOpen (((↑) : G → G ⧸ N) ⁻¹' ((↑) '' s))"
},
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ns : Set G\ns_op : IsOpen s\n⊢ IsOpen (⋃ (x : { x // x ∈ N }), (fun y => y * ↑x) ⁻¹' s)",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ns : Set G\ns_op : IsOpen s\n⊢ IsOpen (mk ⁻¹' (mk '' s))",
"tactic": "rw [QuotientGroup.preimage_image_mk N s]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ns : Set G\ns_op : IsOpen s\n⊢ IsOpen (⋃ (x : { x // x ∈ N }), (fun y => y * ↑x) ⁻¹' s)",
"tactic": "exact isOpen_iUnion fun n => (continuous_mul_right _).isOpen_preimage s s_op"
}
] |
[
980,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
976,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
zmultiplesAddHom_symm_apply
|
[] |
[
1016,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1014,
1
] |
Mathlib/Init/Algebra/Order.lean
|
lt_or_eq_of_le
|
[] |
[
233,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/Topology/Order.lean
|
TopologicalSpace.gc_generateFrom
|
[] |
[
191,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Data/MvPolynomial/Supported.lean
|
MvPolynomial.supported_le_supported_iff
|
[
{
"state_after": "case mp\nσ : Type u_1\nτ : Type ?u.129141\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\ninst✝ : Nontrivial R\n⊢ supported R s ≤ supported R t → s ⊆ t\n\ncase mpr\nσ : Type u_1\nτ : Type ?u.129141\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\ninst✝ : Nontrivial R\n⊢ s ⊆ t → supported R s ≤ supported R t",
"state_before": "σ : Type u_1\nτ : Type ?u.129141\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\ninst✝ : Nontrivial R\n⊢ supported R s ≤ supported R t ↔ s ⊆ t",
"tactic": "constructor"
},
{
"state_after": "case mp\nσ : Type u_1\nτ : Type ?u.129141\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\ninst✝ : Nontrivial R\nh : supported R s ≤ supported R t\ni : σ\n⊢ i ∈ s → i ∈ t",
"state_before": "case mp\nσ : Type u_1\nτ : Type ?u.129141\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\ninst✝ : Nontrivial R\n⊢ supported R s ≤ supported R t → s ⊆ t",
"tactic": "intro h i"
},
{
"state_after": "no goals",
"state_before": "case mp\nσ : Type u_1\nτ : Type ?u.129141\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\ninst✝ : Nontrivial R\nh : supported R s ≤ supported R t\ni : σ\n⊢ i ∈ s → i ∈ t",
"tactic": "simpa using @h (X i)"
},
{
"state_after": "no goals",
"state_before": "case mpr\nσ : Type u_1\nτ : Type ?u.129141\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\ninst✝ : Nontrivial R\n⊢ s ⊆ t → supported R s ≤ supported R t",
"tactic": "exact supported_mono"
}
] |
[
130,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
div_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.36695\nG : Type ?u.36698\ninst✝ : DivisionCommMonoid α\na b c d : α\n⊢ a / b * c = a / (b / c)",
"tactic": "simp"
}
] |
[
545,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
545,
1
] |
Mathlib/Order/PartialSups.lean
|
partialSups_mono
|
[
{
"state_after": "α : Type u_1\ninst✝ : SemilatticeSup α\nf g : ℕ → α\nh : f ≤ g\nn : ℕ\n⊢ ↑(partialSups f) n ≤ ↑(partialSups g) n",
"state_before": "α : Type u_1\ninst✝ : SemilatticeSup α\n⊢ Monotone partialSups",
"tactic": "rintro f g h n"
},
{
"state_after": "case zero\nα : Type u_1\ninst✝ : SemilatticeSup α\nf g : ℕ → α\nh : f ≤ g\n⊢ ↑(partialSups f) Nat.zero ≤ ↑(partialSups g) Nat.zero\n\ncase succ\nα : Type u_1\ninst✝ : SemilatticeSup α\nf g : ℕ → α\nh : f ≤ g\nn : ℕ\nih : ↑(partialSups f) n ≤ ↑(partialSups g) n\n⊢ ↑(partialSups f) (Nat.succ n) ≤ ↑(partialSups g) (Nat.succ n)",
"state_before": "α : Type u_1\ninst✝ : SemilatticeSup α\nf g : ℕ → α\nh : f ≤ g\nn : ℕ\n⊢ ↑(partialSups f) n ≤ ↑(partialSups g) n",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_1\ninst✝ : SemilatticeSup α\nf g : ℕ → α\nh : f ≤ g\n⊢ ↑(partialSups f) Nat.zero ≤ ↑(partialSups g) Nat.zero",
"tactic": "exact h 0"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u_1\ninst✝ : SemilatticeSup α\nf g : ℕ → α\nh : f ≤ g\nn : ℕ\nih : ↑(partialSups f) n ≤ ↑(partialSups g) n\n⊢ ↑(partialSups f) (Nat.succ n) ≤ ↑(partialSups g) (Nat.succ n)",
"tactic": "exact sup_le_sup ih (h _)"
}
] |
[
106,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Std/Data/Option/Lemmas.lean
|
Option.join_map_eq_map_join
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β\nx : Option (Option α)\n⊢ join (Option.map (Option.map f) x) = Option.map f (join x)",
"tactic": "cases x <;> simp"
}
] |
[
160,
70
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
159,
1
] |
Mathlib/Data/PNat/Prime.lean
|
PNat.coprime_one
|
[] |
[
255,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/CategoryTheory/Iso.lean
|
CategoryTheory.Iso.eq_inv_comp
|
[] |
[
226,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.mul_imJ
|
[] |
[
337,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.option_orElse
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.139430\nγ : Type ?u.139433\nδ : Type ?u.139436\nσ : Type ?u.139439\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nx✝ : Option α × Option α\no₁ o₂ : Option α\n⊢ (Option.casesOn (o₁, o₂).fst (o₁, o₂).snd fun b => ((o₁, o₂), b).fst.fst) =\n (fun x x_1 => HOrElse.hOrElse x fun x => x_1) (o₁, o₂).fst (o₁, o₂).snd",
"tactic": "cases o₁ <;> cases o₂ <;> rfl"
}
] |
[
765,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
764,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.repr_sum_self
|
[
{
"state_after": "case h\nι : Type u_1\nι' : Type ?u.673229\nR : Type u_2\nR₂ : Type ?u.673235\nK : Type ?u.673238\nM : Type u_3\nM' : Type ?u.673244\nM'' : Type ?u.673247\nV : Type u\nV' : Type ?u.673252\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nc : ι → R\nj : ι\n⊢ ↑(↑b.repr (∑ i : ι, c i • ↑b i)) j = c j",
"state_before": "ι : Type u_1\nι' : Type ?u.673229\nR : Type u_2\nR₂ : Type ?u.673235\nK : Type ?u.673238\nM : Type u_3\nM' : Type ?u.673244\nM'' : Type ?u.673247\nV : Type u\nV' : Type ?u.673252\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nc : ι → R\n⊢ ↑(↑b.repr (∑ i : ι, c i • ↑b i)) = c",
"tactic": "ext j"
},
{
"state_after": "case h\nι : Type u_1\nι' : Type ?u.673229\nR : Type u_2\nR₂ : Type ?u.673235\nK : Type ?u.673238\nM : Type u_3\nM' : Type ?u.673244\nM'' : Type ?u.673247\nV : Type u\nV' : Type ?u.673252\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nc : ι → R\nj : ι\n⊢ ∑ x : ι, ↑(Finsupp.single x (c x)) j = c j",
"state_before": "case h\nι : Type u_1\nι' : Type ?u.673229\nR : Type u_2\nR₂ : Type ?u.673235\nK : Type ?u.673238\nM : Type u_3\nM' : Type ?u.673244\nM'' : Type ?u.673247\nV : Type u\nV' : Type ?u.673252\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nc : ι → R\nj : ι\n⊢ ↑(↑b.repr (∑ i : ι, c i • ↑b i)) j = c j",
"tactic": "simp only [map_sum, LinearEquiv.map_smul, repr_self, Finsupp.smul_single, smul_eq_mul, mul_one,\n Finset.sum_apply']"
},
{
"state_after": "case h.h₀\nι : Type u_1\nι' : Type ?u.673229\nR : Type u_2\nR₂ : Type ?u.673235\nK : Type ?u.673238\nM : Type u_3\nM' : Type ?u.673244\nM'' : Type ?u.673247\nV : Type u\nV' : Type ?u.673252\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nc : ι → R\nj : ι\n⊢ ∀ (b : ι), b ∈ Finset.univ → b ≠ j → ↑(Finsupp.single b (c b)) j = 0\n\ncase h.h₁\nι : Type u_1\nι' : Type ?u.673229\nR : Type u_2\nR₂ : Type ?u.673235\nK : Type ?u.673238\nM : Type u_3\nM' : Type ?u.673244\nM'' : Type ?u.673247\nV : Type u\nV' : Type ?u.673252\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nc : ι → R\nj : ι\n⊢ ¬j ∈ Finset.univ → ↑(Finsupp.single j (c j)) j = 0",
"state_before": "case h\nι : Type u_1\nι' : Type ?u.673229\nR : Type u_2\nR₂ : Type ?u.673235\nK : Type ?u.673238\nM : Type u_3\nM' : Type ?u.673244\nM'' : Type ?u.673247\nV : Type u\nV' : Type ?u.673252\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nc : ι → R\nj : ι\n⊢ ∑ x : ι, ↑(Finsupp.single x (c x)) j = c j",
"tactic": "rw [Finset.sum_eq_single j, Finsupp.single_eq_same]"
},
{
"state_after": "case h.h₀\nι : Type u_1\nι' : Type ?u.673229\nR : Type u_2\nR₂ : Type ?u.673235\nK : Type ?u.673238\nM : Type u_3\nM' : Type ?u.673244\nM'' : Type ?u.673247\nV : Type u\nV' : Type ?u.673252\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nc : ι → R\nj i : ι\nhi : i ≠ j\n⊢ ↑(Finsupp.single i (c i)) j = 0",
"state_before": "case h.h₀\nι : Type u_1\nι' : Type ?u.673229\nR : Type u_2\nR₂ : Type ?u.673235\nK : Type ?u.673238\nM : Type u_3\nM' : Type ?u.673244\nM'' : Type ?u.673247\nV : Type u\nV' : Type ?u.673252\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nc : ι → R\nj : ι\n⊢ ∀ (b : ι), b ∈ Finset.univ → b ≠ j → ↑(Finsupp.single b (c b)) j = 0",
"tactic": "rintro i - hi"
},
{
"state_after": "no goals",
"state_before": "case h.h₀\nι : Type u_1\nι' : Type ?u.673229\nR : Type u_2\nR₂ : Type ?u.673235\nK : Type ?u.673238\nM : Type u_3\nM' : Type ?u.673244\nM'' : Type ?u.673247\nV : Type u\nV' : Type ?u.673252\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nc : ι → R\nj i : ι\nhi : i ≠ j\n⊢ ↑(Finsupp.single i (c i)) j = 0",
"tactic": "exact Finsupp.single_eq_of_ne hi"
},
{
"state_after": "case h.h₁\nι : Type u_1\nι' : Type ?u.673229\nR : Type u_2\nR₂ : Type ?u.673235\nK : Type ?u.673238\nM : Type u_3\nM' : Type ?u.673244\nM'' : Type ?u.673247\nV : Type u\nV' : Type ?u.673252\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nc : ι → R\nj : ι\na✝ : ¬j ∈ Finset.univ\n⊢ ↑(Finsupp.single j (c j)) j = 0",
"state_before": "case h.h₁\nι : Type u_1\nι' : Type ?u.673229\nR : Type u_2\nR₂ : Type ?u.673235\nK : Type ?u.673238\nM : Type u_3\nM' : Type ?u.673244\nM'' : Type ?u.673247\nV : Type u\nV' : Type ?u.673252\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nc : ι → R\nj : ι\n⊢ ¬j ∈ Finset.univ → ↑(Finsupp.single j (c j)) j = 0",
"tactic": "intros"
},
{
"state_after": "case h.h₁\nι : Type u_1\nι' : Type ?u.673229\nR : Type u_2\nR₂ : Type ?u.673235\nK : Type ?u.673238\nM : Type u_3\nM' : Type ?u.673244\nM'' : Type ?u.673247\nV : Type u\nV' : Type ?u.673252\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nc : ι → R\nj : ι\na✝ : ¬j ∈ Finset.univ\nthis : j ∈ Finset.univ\n⊢ ↑(Finsupp.single j (c j)) j = 0",
"state_before": "case h.h₁\nι : Type u_1\nι' : Type ?u.673229\nR : Type u_2\nR₂ : Type ?u.673235\nK : Type ?u.673238\nM : Type u_3\nM' : Type ?u.673244\nM'' : Type ?u.673247\nV : Type u\nV' : Type ?u.673252\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nc : ι → R\nj : ι\na✝ : ¬j ∈ Finset.univ\n⊢ ↑(Finsupp.single j (c j)) j = 0",
"tactic": "have := Finset.mem_univ j"
},
{
"state_after": "no goals",
"state_before": "case h.h₁\nι : Type u_1\nι' : Type ?u.673229\nR : Type u_2\nR₂ : Type ?u.673235\nK : Type ?u.673238\nM : Type u_3\nM' : Type ?u.673244\nM'' : Type ?u.673247\nV : Type u\nV' : Type ?u.673252\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Fintype ι\nb : Basis ι R M\nc : ι → R\nj : ι\na✝ : ¬j ∈ Finset.univ\nthis : j ∈ Finset.univ\n⊢ ↑(Finsupp.single j (c j)) j = 0",
"tactic": "contradiction"
}
] |
[
950,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
941,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biproduct.components_matrix
|
[
{
"state_after": "case w.w\nJ : Type\ninst✝⁴ : Fintype J\nK : Type\ninst✝³ : Fintype K\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasFiniteBiproducts C\nf : J → C\ng : K → C\nm : ⨁ f ⟶ ⨁ g\nj✝¹ : J\nj✝ : K\n⊢ (ι (fun j => f j) j✝¹ ≫ matrix fun j k => components m j k) ≫ π (fun k => g k) j✝ =\n (ι (fun j => f j) j✝¹ ≫ m) ≫ π (fun k => g k) j✝",
"state_before": "J : Type\ninst✝⁴ : Fintype J\nK : Type\ninst✝³ : Fintype K\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasFiniteBiproducts C\nf : J → C\ng : K → C\nm : ⨁ f ⟶ ⨁ g\n⊢ (matrix fun j k => components m j k) = m",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case w.w\nJ : Type\ninst✝⁴ : Fintype J\nK : Type\ninst✝³ : Fintype K\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasFiniteBiproducts C\nf : J → C\ng : K → C\nm : ⨁ f ⟶ ⨁ g\nj✝¹ : J\nj✝ : K\n⊢ (ι (fun j => f j) j✝¹ ≫ matrix fun j k => components m j k) ≫ π (fun k => g k) j✝ =\n (ι (fun j => f j) j✝¹ ≫ m) ≫ π (fun k => g k) j✝",
"tactic": "simp [biproduct.components]"
}
] |
[
858,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
855,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
measurable_of_subsingleton_codomain
|
[] |
[
254,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
Subgroup.smul_inf
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nG : Type u_1\nA : Type ?u.151816\nS✝ : Type ?u.151819\ninst✝³ : Group G\ninst✝² : AddGroup A\ns : Set G\ninst✝¹ : Group α\ninst✝ : MulDistribMulAction α G\na : α\nS T : Subgroup G\n⊢ a • (S ⊓ T) = a • S ⊓ a • T",
"tactic": "simp [SetLike.ext_iff, mem_pointwise_smul_iff_inv_smul_mem]"
}
] |
[
367,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
366,
1
] |
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