url
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values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
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| start
stringlengths 6
10
| end
stringlengths 6
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stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
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|---|---|---|---|---|---|---|---|---|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MinkowskiCaratheodory.lean | convexHull_extremePoints | [20, 1] | [23, 53] | sorry | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
s B : Set E
hscomp : IsCompact s
hsconv : Convex ℝ s
⊢ (convexHull ℝ) (extremePoints ℝ s) = s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MinkowskiCaratheodory.lean | closed_convexHull_extremePoints_of_compact_of_convex | [32, 1] | [35, 96] | rw [closure_convexHull_extremePoints hscomp hsconv, convexHull_extremePoints hscomp hsconv] | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
s B : Set E
hscomp : IsCompact s
hsconv : Convex ℝ s
⊢ closure ((convexHull ℝ) (extremePoints ℝ s)) = (convexHull ℝ) (extremePoints ℝ s) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.totalDegree_f₁_add_totalDegree_f₂ | [48, 1] | [56, 50] | refine (add_le_add (totalDegree_finset_sum _ _) $ (totalDegree_finset_sum _ _).trans $
Finset.sup_mono_fun fun a _ ↦ totalDegree_smul_le _ _).trans_lt ?_ | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
⊢ (ZMod.f₁ s).totalDegree + (ZMod.f₂ s).totalDegree < 2 * p - 1 | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
⊢ ((s.toEnumFinset.attach.sup fun i => (X i ^ (p - 1)).totalDegree) +
s.toEnumFinset.attach.sup fun a => (X a ^ (p - 1)).totalDegree) <
2 * p - 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.totalDegree_f₁_add_totalDegree_f₂ | [48, 1] | [56, 50] | simp only [totalDegree_X_pow, ← two_mul] | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
⊢ ((s.toEnumFinset.attach.sup fun i => (X i ^ (p - 1)).totalDegree) +
s.toEnumFinset.attach.sup fun a => (X a ^ (p - 1)).totalDegree) <
2 * p - 1 | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
⊢ (2 * s.toEnumFinset.attach.sup fun i => p - 1) < 2 * p - 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.totalDegree_f₁_add_totalDegree_f₂ | [48, 1] | [56, 50] | refine (mul_le_mul_left' Finset.sup_const_le _).trans_lt ?_ | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
⊢ (2 * s.toEnumFinset.attach.sup fun i => p - 1) < 2 * p - 1 | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
⊢ 2 * (p - 1) < 2 * p - 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.totalDegree_f₁_add_totalDegree_f₂ | [48, 1] | [56, 50] | rw [mul_tsub, mul_one] | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
⊢ 2 * (p - 1) < 2 * p - 1 | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
⊢ 2 * p - 2 < 2 * p - 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.totalDegree_f₁_add_totalDegree_f₂ | [48, 1] | [56, 50] | exact tsub_lt_tsub_left_of_le ((Fact.out : p.Prime).two_le.trans $
le_mul_of_one_le_left' one_le_two) one_lt_two | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
⊢ 2 * p - 2 < 2 * p - 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | haveI : NeZero p := inferInstance | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | set N := Fintype.card {x // eval x (f₁ s) = 0 ∧ eval x (f₂ s) = 0} | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | let zero_sol : {x // eval x (f₁ s) = 0 ∧ eval x (f₂ s) = 0} :=
⟨0, by simp [f₁, f₂, map_sum, (Fact.out : p.Prime).one_lt, tsub_eq_zero_iff_le]⟩ | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | have hN₀ : 0 < N := @Fintype.card_pos _ _ ⟨zero_sol⟩ | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | have hs' : 2 * p - 1 = Fintype.card s.toEnumFinset := by simp [hs] | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | have hpN : p ∣ N := char_dvd_card_solutions_of_add_lt p
(totalDegree_f₁_add_totalDegree_f₂.trans_eq hs') | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | obtain ⟨x, hx⟩ := Fintype.exists_ne_of_one_lt_card ((Fact.out : p.Prime).one_lt.trans_le $
Nat.le_of_dvd hN₀ hpN) zero_sol | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 | case intro
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | refine ⟨(s.toEnumFinset.attach.filter $ fun a ↦ x.1 a ≠ 0).1.map
(Prod.fst ∘ ((↑) : s.toEnumFinset → ZMod p × ℕ)), le_iff_count.2 $ fun a ↦ ?_, ?_, ?_⟩ | case intro
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 | case intro.refine_1
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
a : ZMod p
⊢ count a (Multiset.map (Prod.fst ∘ Subtype.val) (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach).val) ≤
count a s
case intro.refine_2
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ Multiset.card (Multiset.map (Prod.fst ∘ Subtype.val) (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach).val) =
p
case intro.refine_3
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ (Multiset.map (Prod.fst ∘ Subtype.val) (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach).val).sum = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | simp [f₁, f₂, map_sum, (Fact.out : p.Prime).one_lt, tsub_eq_zero_iff_le] | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
⊢ (eval 0) (ZMod.f₁ s) = 0 ∧ (eval 0) (ZMod.f₂ s) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | simp [hs] | n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
⊢ 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset } | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | simp only [← Finset.filter_val, Finset.card_val, Function.comp_apply, count_map] | case intro.refine_1
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
a : ZMod p
⊢ count a (Multiset.map (Prod.fst ∘ Subtype.val) (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach).val) ≤
count a s | case intro.refine_1
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
a : ZMod p
⊢ (Finset.filter (fun a_1 => a = (↑a_1).1) (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach)).card ≤ count a s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | refine (Finset.card_le_card $ Finset.filter_subset_filter _ $
Finset.filter_subset _ _).trans_eq ?_ | case intro.refine_1
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
a : ZMod p
⊢ (Finset.filter (fun a_1 => a = (↑a_1).1) (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach)).card ≤ count a s | case intro.refine_1
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
a : ZMod p
⊢ (Finset.filter (fun a_1 => a = (↑a_1).1) s.toEnumFinset.attach).card = count a s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | refine (Finset.card_filter_attach (fun c : ZMod p × ℕ ↦ a = c.1) _).trans ?_ | case intro.refine_1
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
a : ZMod p
⊢ (Finset.filter (fun a_1 => a = (↑a_1).1) s.toEnumFinset.attach).card = count a s | case intro.refine_1
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
a : ZMod p
⊢ (Finset.filter (fun c => a = c.1) s.toEnumFinset).card = count a s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | simp [toEnumFinset_filter_eq] | case intro.refine_1
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
a : ZMod p
⊢ (Finset.filter (fun c => a = c.1) s.toEnumFinset).card = count a s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | simp only [card_map, ← Finset.filter_val, Finset.card_val, Function.comp_apply,
count_map, ← Finset.map_val] | case intro.refine_2
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ Multiset.card (Multiset.map (Prod.fst ∘ Subtype.val) (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach).val) =
p | case intro.refine_2
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach).card = p |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | refine Nat.eq_of_dvd_of_lt_two_mul (Finset.card_pos.2 ?_).ne' ?_ $
(Finset.card_filter_le _ _).trans_lt ?_ | case intro.refine_2
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach).card = p | case intro.refine_2.refine_1
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach).Nonempty
case intro.refine_2.refine_2
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ p ∣ (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach).card
case intro.refine_2.refine_3
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ s.toEnumFinset.attach.card < 2 * p |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | rw [← Subtype.coe_ne_coe, Function.ne_iff] at hx | case intro.refine_2.refine_1
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach).Nonempty | case intro.refine_2.refine_1
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : ∃ a, ↑x a ≠ ↑zero_sol a
⊢ (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach).Nonempty |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | exact hx.imp (fun a ha ↦ mem_filter.2 ⟨Finset.mem_attach _ _, ha⟩) | case intro.refine_2.refine_1
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : ∃ a, ↑x a ≠ ↑zero_sol a
⊢ (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach).Nonempty | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | rw [← CharP.cast_eq_zero_iff (ZMod p), ← Finset.sum_boole] | case intro.refine_2.refine_2
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ p ∣ (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach).card | case intro.refine_2.refine_2
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ (∑ x_1 ∈ s.toEnumFinset.attach, if ↑x x_1 ≠ 0 then 1 else 0) = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | simpa only [f₁, map_sum, ZMod.pow_card_sub_one, map_pow, eval_X] using x.2.1 | case intro.refine_2.refine_2
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ (∑ x_1 ∈ s.toEnumFinset.attach, if ↑x x_1 ≠ 0 then 1 else 0) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | rw [Finset.card_attach, card_toEnumFinset, hs] | case intro.refine_2.refine_3
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ s.toEnumFinset.attach.card < 2 * p | case intro.refine_2.refine_3
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ 2 * p - 1 < 2 * p |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | exact tsub_lt_self (mul_pos zero_lt_two (Fact.out : p.Prime).pos) zero_lt_one | case intro.refine_2.refine_3
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ 2 * p - 1 < 2 * p | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.aux | [58, 1] | [101, 74] | simpa only [f₂, map_sum, ZMod.pow_card_sub_one, Finset.sum_map_val, Finset.sum_filter,
smul_eval, map_pow, eval_X, mul_ite, mul_zero, mul_one] using x.2.2 | case intro.refine_3
n p : ℕ
inst✝ : Fact p.Prime
s : Multiset (ZMod p)
hs : Multiset.card s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
zero_sol : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype.card { x // x ∈ s.toEnumFinset }
hpN : p ∣ N
x : { x // (eval x) (ZMod.f₁ s) = 0 ∧ (eval x) (ZMod.f₂ s) = 0 }
hx : x ≠ zero_sol
⊢ (Multiset.map (Prod.fst ∘ Subtype.val) (Finset.filter (fun a => ↑x a ≠ 0) s.toEnumFinset.attach).val).sum = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | induction n using Nat.prime_composite_induction | n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
s : Multiset (ZMod n)
hs : 2 * n - 1 ≤ Multiset.card s
⊢ ∃ t ≤ s, Multiset.card t = n ∧ t.sum = 0 | case zero
n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
s : Multiset (ZMod 0)
hs : 2 * 0 - 1 ≤ Multiset.card s
⊢ ∃ t ≤ s, Multiset.card t = 0 ∧ t.sum = 0
case one
n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
s : Multiset (ZMod 1)
hs : 2 * 1 - 1 ≤ Multiset.card s
⊢ ∃ t ≤ s, Multiset.card t = 1 ∧ t.sum = 0
case prime
n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
p✝ : ℕ
a✝ : p✝.Prime
s : Multiset (ZMod p✝)
hs : 2 * p✝ - 1 ≤ Multiset.card s
⊢ ∃ t ≤ s, Multiset.card t = p✝ ∧ t.sum = 0
case composite
n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a✝⁴ : ℕ
a✝³ : 2 ≤ a✝⁴
a✝² : ∀ {s : Multiset (ZMod a✝⁴)}, 2 * a✝⁴ - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a✝⁴ ∧ t.sum = 0
b✝ : ℕ
a✝¹ : 2 ≤ b✝
a✝ : ∀ {s : Multiset (ZMod b✝)}, 2 * b✝ - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b✝ ∧ t.sum = 0
s : Multiset (ZMod (a✝⁴ * b✝))
hs : 2 * (a✝⁴ * b✝) - 1 ≤ Multiset.card s
⊢ ∃ t ≤ s, Multiset.card t = a✝⁴ * b✝ ∧ t.sum = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | case zero => exact ⟨0, s.zero_le, card_zero, sum_zero⟩ | n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
s : Multiset (ZMod 0)
hs : 2 * 0 - 1 ≤ Multiset.card s
⊢ ∃ t ≤ s, Multiset.card t = 0 ∧ t.sum = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | case one => obtain ⟨t, ht, hn⟩ := exists_le_card_eq hs; exact ⟨t, ht, hn, Subsingleton.elim _ _⟩ | n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
s : Multiset (ZMod 1)
hs : 2 * 1 - 1 ≤ Multiset.card s
⊢ ∃ t ≤ s, Multiset.card t = 1 ∧ t.sum = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | case prime p hp =>
haveI := Fact.mk hp
obtain ⟨t, hts, ht⟩ := exists_le_card_eq hs
obtain ⟨u, hut, hu⟩ := aux ht
exact ⟨u, hut.trans hts, hu⟩ | n p✝ : ℕ
inst✝ : Fact p✝.Prime
s✝ : Multiset (ZMod p✝)
p : ℕ
hp : p.Prime
s : Multiset (ZMod p)
hs : 2 * p - 1 ≤ Multiset.card s
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | exact ⟨0, s.zero_le, card_zero, sum_zero⟩ | n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
s : Multiset (ZMod 0)
hs : 2 * 0 - 1 ≤ Multiset.card s
⊢ ∃ t ≤ s, Multiset.card t = 0 ∧ t.sum = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | obtain ⟨t, ht, hn⟩ := exists_le_card_eq hs | n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
s : Multiset (ZMod 1)
hs : 2 * 1 - 1 ≤ Multiset.card s
⊢ ∃ t ≤ s, Multiset.card t = 1 ∧ t.sum = 0 | case intro.intro
n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
s : Multiset (ZMod 1)
hs : 2 * 1 - 1 ≤ Multiset.card s
t : Multiset (ZMod 1)
ht : t ≤ s
hn : Multiset.card t = 2 * 1 - 1
⊢ ∃ t ≤ s, Multiset.card t = 1 ∧ t.sum = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | exact ⟨t, ht, hn, Subsingleton.elim _ _⟩ | case intro.intro
n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
s : Multiset (ZMod 1)
hs : 2 * 1 - 1 ≤ Multiset.card s
t : Multiset (ZMod 1)
ht : t ≤ s
hn : Multiset.card t = 2 * 1 - 1
⊢ ∃ t ≤ s, Multiset.card t = 1 ∧ t.sum = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | haveI := Fact.mk hp | n p✝ : ℕ
inst✝ : Fact p✝.Prime
s✝ : Multiset (ZMod p✝)
p : ℕ
hp : p.Prime
s : Multiset (ZMod p)
hs : 2 * p - 1 ≤ Multiset.card s
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 | n p✝ : ℕ
inst✝ : Fact p✝.Prime
s✝ : Multiset (ZMod p✝)
p : ℕ
hp : p.Prime
s : Multiset (ZMod p)
hs : 2 * p - 1 ≤ Multiset.card s
this : Fact p.Prime
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | obtain ⟨t, hts, ht⟩ := exists_le_card_eq hs | n p✝ : ℕ
inst✝ : Fact p✝.Prime
s✝ : Multiset (ZMod p✝)
p : ℕ
hp : p.Prime
s : Multiset (ZMod p)
hs : 2 * p - 1 ≤ Multiset.card s
this : Fact p.Prime
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 | case intro.intro
n p✝ : ℕ
inst✝ : Fact p✝.Prime
s✝ : Multiset (ZMod p✝)
p : ℕ
hp : p.Prime
s : Multiset (ZMod p)
hs : 2 * p - 1 ≤ Multiset.card s
this : Fact p.Prime
t : Multiset (ZMod p)
hts : t ≤ s
ht : Multiset.card t = 2 * p - 1
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | obtain ⟨u, hut, hu⟩ := aux ht | case intro.intro
n p✝ : ℕ
inst✝ : Fact p✝.Prime
s✝ : Multiset (ZMod p✝)
p : ℕ
hp : p.Prime
s : Multiset (ZMod p)
hs : 2 * p - 1 ≤ Multiset.card s
this : Fact p.Prime
t : Multiset (ZMod p)
hts : t ≤ s
ht : Multiset.card t = 2 * p - 1
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 | case intro.intro.intro.intro
n p✝ : ℕ
inst✝ : Fact p✝.Prime
s✝ : Multiset (ZMod p✝)
p : ℕ
hp : p.Prime
s : Multiset (ZMod p)
hs : 2 * p - 1 ≤ Multiset.card s
this : Fact p.Prime
t : Multiset (ZMod p)
hts : t ≤ s
ht : Multiset.card t = 2 * p - 1
u : Multiset (ZMod p)
hut : u ≤ t
hu : Multiset.card u = p ∧ u.sum = 0
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | exact ⟨u, hut.trans hts, hu⟩ | case intro.intro.intro.intro
n p✝ : ℕ
inst✝ : Fact p✝.Prime
s✝ : Multiset (ZMod p✝)
p : ℕ
hp : p.Prime
s : Multiset (ZMod p)
hs : 2 * p - 1 ≤ Multiset.card s
this : Fact p.Prime
t : Multiset (ZMod p)
hts : t ≤ s
ht : Multiset.card t = 2 * p - 1
u : Multiset (ZMod p)
hut : u ≤ t
hu : Multiset.card u = p ∧ u.sum = 0
⊢ ∃ t ≤ s, Multiset.card t = p ∧ t.sum = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | suffices ∀ n ≤ 2 * b - 1, ∃ m : Multiset (Multiset $ ZMod $ a * b), Multiset.card m = n ∧
m.Pairwise _root_.Disjoint ∧ ∀ ⦃u : Multiset $ ZMod $ a * b⦄, u ∈ m →
Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples (a : ZMod $ a * b) by
obtain ⟨m, hm⟩ := this _ le_rfl
sorry | n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
⊢ ∃ t ≤ s, Multiset.card t = a * b ∧ t.sum = 0 | n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
⊢ ∀ n ≤ 2 * b - 1,
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | rintro n hn | n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
⊢ ∀ n ≤ 2 * b - 1,
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a | n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
hn : n ≤ 2 * b - 1
⊢ ∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | induction' n with n ih | n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
hn : n ≤ 2 * b - 1
⊢ ∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a | case zero
n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
hn : 0 ≤ 2 * b - 1
⊢ ∃ m,
Multiset.card m = 0 ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
case succ
n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
⊢ ∃ m,
Multiset.card m = n + 1 ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | obtain ⟨m, hm⟩ := ih (Nat.le_of_succ_le hn) | case succ
n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
⊢ ∃ m,
Multiset.card m = n + 1 ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a | case succ.intro
n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
⊢ ∃ m,
Multiset.card m = n + 1 ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | have : 2 * a - 1 ≤ Multiset.card ((s - m.sum).map $ castHom (dvd_mul_right _ _) $ ZMod a) := by
rw [card_map]
refine (le_tsub_of_add_le_left $ le_trans ?_ hs).trans le_card_sub
have : m.map Multiset.card = replicate (2 * a - 1) n := sorry
rw [map_multiset_sum, this, sum_replicate, ← le_tsub_iff_right, tsub_tsub_tsub_cancel_right,
← mul_tsub, ← mul_tsub_one]
sorry
sorry
sorry | case succ.intro
n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
⊢ ∃ m,
Multiset.card m = n + 1 ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a | case succ.intro
n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : 2 * a - 1 ≤ Multiset.card (Multiset.map (⇑(castHom ⋯ (ZMod a))) (s - m.sum))
⊢ ∃ m,
Multiset.card m = n + 1 ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | sorry | case succ.intro
n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : 2 * a - 1 ≤ Multiset.card (Multiset.map (⇑(castHom ⋯ (ZMod a))) (s - m.sum))
⊢ ∃ m,
Multiset.card m = n + 1 ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | obtain ⟨m, hm⟩ := this _ le_rfl | n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
this :
∀ n ≤ 2 * b - 1,
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
⊢ ∃ t ≤ s, Multiset.card t = a * b ∧ t.sum = 0 | case intro
n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
this :
∀ n ≤ 2 * b - 1,
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = 2 * b - 1 ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
⊢ ∃ t ≤ s, Multiset.card t = a * b ∧ t.sum = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | sorry | case intro
n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
this :
∀ n ≤ 2 * b - 1,
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = 2 * b - 1 ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
⊢ ∃ t ≤ s, Multiset.card t = a * b ∧ t.sum = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | exact ⟨0, by simp⟩ | case zero
n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
hn : 0 ≤ 2 * b - 1
⊢ ∃ m,
Multiset.card m = 0 ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | simp | n p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
hn : 0 ≤ 2 * b - 1
⊢ Multiset.card 0 = 0 ∧
Multiset.Pairwise _root_.Disjoint 0 ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ 0 → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | rw [card_map] | n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
⊢ 2 * a - 1 ≤ Multiset.card (Multiset.map (⇑(castHom ⋯ (ZMod a))) (s - m.sum)) | n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
⊢ 2 * a - 1 ≤ Multiset.card (s - m.sum) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | refine (le_tsub_of_add_le_left $ le_trans ?_ hs).trans le_card_sub | n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
⊢ 2 * a - 1 ≤ Multiset.card (s - m.sum) | n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
⊢ Multiset.card m.sum + (2 * a - 1) ≤ 2 * (a * b) - 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | have : m.map Multiset.card = replicate (2 * a - 1) n := sorry | n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
⊢ Multiset.card m.sum + (2 * a - 1) ≤ 2 * (a * b) - 1 | n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : Multiset.map (⇑Multiset.card) m = replicate (2 * a - 1) n
⊢ Multiset.card m.sum + (2 * a - 1) ≤ 2 * (a * b) - 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | rw [map_multiset_sum, this, sum_replicate, ← le_tsub_iff_right, tsub_tsub_tsub_cancel_right,
← mul_tsub, ← mul_tsub_one] | n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : Multiset.map (⇑Multiset.card) m = replicate (2 * a - 1) n
⊢ Multiset.card m.sum + (2 * a - 1) ≤ 2 * (a * b) - 1 | n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : Multiset.map (⇑Multiset.card) m = replicate (2 * a - 1) n
⊢ (2 * a - 1) • n ≤ 2 * (a * (b - 1))
n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : Multiset.map (⇑Multiset.card) m = replicate (2 * a - 1) n
⊢ 1 ≤ 2 * a
n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : Multiset.map (⇑Multiset.card) m = replicate (2 * a - 1) n
⊢ 2 * a - 1 ≤ 2 * (a * b) - 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | sorry | n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : Multiset.map (⇑Multiset.card) m = replicate (2 * a - 1) n
⊢ (2 * a - 1) • n ≤ 2 * (a * (b - 1))
n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : Multiset.map (⇑Multiset.card) m = replicate (2 * a - 1) n
⊢ 1 ≤ 2 * a
n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : Multiset.map (⇑Multiset.card) m = replicate (2 * a - 1) n
⊢ 2 * a - 1 ≤ 2 * (a * b) - 1 | n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : Multiset.map (⇑Multiset.card) m = replicate (2 * a - 1) n
⊢ 1 ≤ 2 * a
n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : Multiset.map (⇑Multiset.card) m = replicate (2 * a - 1) n
⊢ 2 * a - 1 ≤ 2 * (a * b) - 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | sorry | n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : Multiset.map (⇑Multiset.card) m = replicate (2 * a - 1) n
⊢ 1 ≤ 2 * a
n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : Multiset.map (⇑Multiset.card) m = replicate (2 * a - 1) n
⊢ 2 * a - 1 ≤ 2 * (a * b) - 1 | n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : Multiset.map (⇑Multiset.card) m = replicate (2 * a - 1) n
⊢ 2 * a - 1 ≤ 2 * (a * b) - 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/ErdosGinzburgZiv.lean | ZMod.exists_submultiset_eq_zero | [105, 1] | [136, 10] | sorry | n✝ p : ℕ
inst✝ : Fact p.Prime
s✝ : Multiset (ZMod p)
a : ℕ
ha : 2 ≤ a
iha : ∀ {s : Multiset (ZMod a)}, 2 * a - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = a ∧ t.sum = 0
b : ℕ
hb : 2 ≤ b
ihb : ∀ {s : Multiset (ZMod b)}, 2 * b - 1 ≤ Multiset.card s → ∃ t ≤ s, Multiset.card t = b ∧ t.sum = 0
s : Multiset (ZMod (a * b))
hs : 2 * (a * b) - 1 ≤ Multiset.card s
n : ℕ
ih :
n ≤ 2 * b - 1 →
∃ m,
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
hn : n + 1 ≤ 2 * b - 1
m : Multiset (Multiset (ZMod (a * b)))
hm :
Multiset.card m = n ∧
Multiset.Pairwise _root_.Disjoint m ∧
∀ ⦃u : Multiset (ZMod (a * b))⦄, u ∈ m → Multiset.card u = 2 * a + 1 ∧ u.sum ∈ AddSubgroup.zmultiples ↑a
this : Multiset.map (⇑Multiset.card) m = replicate (2 * a - 1) n
⊢ 2 * a - 1 ≤ 2 * (a * b) - 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Algebra/Order/BigOperators/LocallyFinite.lean | Finset.mul_prod_Ico | [12, 1] | [14, 36] | rw [Icc_eq_cons_Ico h, prod_cons] | α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : CommMonoid β
f : α → β
a b : α
inst✝ : LocallyFiniteOrder α
h : a ≤ b
⊢ f b * ∏ x ∈ Ico a b, f x = ∏ x ∈ Icc a b, f x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Algebra/Order/BigOperators/LocallyFinite.lean | Finset.mul_prod_Ioc | [16, 1] | [18, 36] | rw [Icc_eq_cons_Ioc h, prod_cons] | α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : CommMonoid β
f : α → β
a b : α
inst✝ : LocallyFiniteOrder α
h : a ≤ b
⊢ f a * ∏ x ∈ Ioc a b, f x = ∏ x ∈ Icc a b, f x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Algebra/Order/BigOperators/LocallyFinite.lean | Finset.mul_prod_Ioi | [25, 1] | [27, 34] | rw [Ici_eq_cons_Ioi, prod_cons] | α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : CommMonoid β
f : α → β
a✝ b : α
inst✝ : LocallyFiniteOrderTop α
a : α
⊢ f a * ∏ x ∈ Ioi a, f x = ∏ x ∈ Ici a, f x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Algebra/Order/BigOperators/LocallyFinite.lean | Finset.mul_prod_Iio | [34, 1] | [36, 34] | rw [Iic_eq_cons_Iio, prod_cons] | α : Type u_1
β : Type u_2
inst✝² : PartialOrder α
inst✝¹ : CommMonoid β
f : α → β
a✝ b : α
inst✝ : LocallyFiniteOrderBot α
a : α
⊢ f a * ∏ x ∈ Iio a, f x = ∏ x ∈ Iic a, f x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Multipartite.lean | Finpartition.multipartiteGraph_adj_of_mem_parts | [18, 1] | [24, 72] | refine' ⟨_, fun hst u hu hau hbu => hst _⟩ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
P : Finpartition univ
s t : Finset α
a b : α
hs : s ∈ P.parts
ht : t ∈ P.parts
ha : a ∈ s
hb : b ∈ t
⊢ P.multipartiteGraph.Adj a b ↔ s ≠ t | case refine'_1
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
P : Finpartition univ
s t : Finset α
a b : α
hs : s ∈ P.parts
ht : t ∈ P.parts
ha : a ∈ s
hb : b ∈ t
⊢ P.multipartiteGraph.Adj a b → s ≠ t
case refine'_2
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
P : Finpartition univ
s t : Finset α
a b : α
hs : s ∈ P.parts
ht : t ∈ P.parts
ha : a ∈ s
hb : b ∈ t
hst : s ≠ t
u : Finset α
hu : u ∈ P.parts
hau : a ∈ u
hbu : b ∈ u
⊢ s = t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Multipartite.lean | Finpartition.multipartiteGraph_adj_of_mem_parts | [18, 1] | [24, 72] | rintro hab rfl | case refine'_1
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
P : Finpartition univ
s t : Finset α
a b : α
hs : s ∈ P.parts
ht : t ∈ P.parts
ha : a ∈ s
hb : b ∈ t
⊢ P.multipartiteGraph.Adj a b → s ≠ t | case refine'_1
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
P : Finpartition univ
s : Finset α
a b : α
hs : s ∈ P.parts
ha : a ∈ s
hab : P.multipartiteGraph.Adj a b
ht : s ∈ P.parts
hb : b ∈ s
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Multipartite.lean | Finpartition.multipartiteGraph_adj_of_mem_parts | [18, 1] | [24, 72] | exact hab hs ha hb | case refine'_1
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
P : Finpartition univ
s : Finset α
a b : α
hs : s ∈ P.parts
ha : a ∈ s
hab : P.multipartiteGraph.Adj a b
ht : s ∈ P.parts
hb : b ∈ s
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Multipartite.lean | Finpartition.multipartiteGraph_adj_of_mem_parts | [18, 1] | [24, 72] | rw [P.eq_of_mem_parts hs hu ha hau, P.eq_of_mem_parts ht hu hb hbu] | case refine'_2
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
P : Finpartition univ
s t : Finset α
a b : α
hs : s ∈ P.parts
ht : t ∈ P.parts
ha : a ∈ s
hb : b ∈ t
hst : s ≠ t
u : Finset α
hu : u ∈ P.parts
hau : a ∈ u
hbu : b ∈ u
⊢ s = t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.symm | [27, 1] | [28, 95] | simpa [dist_comm, add_comm] using d | V : Type u_1
inst✝ : MetricSpace V
u v w : V
huv : u ≠ v
huw : u ≠ w
hvw : v ≠ w
d : Dist.dist u v + Dist.dist v w = Dist.dist u w
⊢ Dist.dist w v + Dist.dist v u = Dist.dist w u | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_iff_of_ne | [32, 1] | [34, 48] | simp [MetricSpace.sbtw_iff, h12, h13, h23] | V : Type u_1
inst✝ : MetricSpace V
u v w : V
h12 : u ≠ v
h13 : u ≠ w
h23 : v ≠ w
⊢ sbtw u v w ↔ dist u v + dist v w = dist u w | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_mk | [36, 1] | [42, 14] | refine ⟨h12, ?_, h23, h.antisymm (dist_triangle _ _ _)⟩ | V : Type u_1
inst✝ : MetricSpace V
u v w : V
h12 : u ≠ v
h23 : v ≠ w
h : dist u v + dist v w ≤ dist u w
⊢ sbtw u v w | V : Type u_1
inst✝ : MetricSpace V
u v w : V
h12 : u ≠ v
h23 : v ≠ w
h : dist u v + dist v w ≤ dist u w
⊢ u ≠ w |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_mk | [36, 1] | [42, 14] | rintro rfl | V : Type u_1
inst✝ : MetricSpace V
u v w : V
h12 : u ≠ v
h23 : v ≠ w
h : dist u v + dist v w ≤ dist u w
⊢ u ≠ w | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : dist u v + dist v u ≤ dist u u
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_mk | [36, 1] | [42, 14] | rw [dist_self] at h | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : dist u v + dist v u ≤ dist u u
⊢ False | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : dist u v + dist v u ≤ 0
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_mk | [36, 1] | [42, 14] | replace h : dist v u ≤ 0 := by linarith [dist_comm v u] | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : dist u v + dist v u ≤ 0
⊢ False | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : dist v u ≤ 0
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_mk | [36, 1] | [42, 14] | simp only [dist_le_zero] at h | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : dist v u ≤ 0
⊢ False | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : v = u
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_mk | [36, 1] | [42, 14] | exact h23 h | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : v = u
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | sbtw_mk | [36, 1] | [42, 14] | linarith [dist_comm v u] | V : Type u_1
inst✝ : MetricSpace V
u v : V
h12 : u ≠ v
h23 : v ≠ u
h : dist u v + dist v u ≤ 0
⊢ dist v u ≤ 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.right_cancel | [44, 1] | [45, 99] | linarith [h.dist, h'.dist, dist_triangle u w x, dist_triangle u v w] | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w✝ u v w x : V
h : sbtw u v x
h' : sbtw v w x
⊢ Dist.dist u v + Dist.dist v w ≤ Dist.dist u w | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.asymm_right | [47, 1] | [51, 24] | have := h'.dist | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
⊢ False | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
this : Dist.dist v u + Dist.dist u x = Dist.dist v x
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.asymm_right | [47, 1] | [51, 24] | rw [dist_comm] at this | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
this : Dist.dist v u + Dist.dist u x = Dist.dist v x
⊢ False | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
this : Dist.dist u v + Dist.dist u x = Dist.dist v x
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.asymm_right | [47, 1] | [51, 24] | have : Dist.dist u v = 0 := by linarith [h.dist] | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
this : Dist.dist u v + Dist.dist u x = Dist.dist v x
⊢ False | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
this✝ : Dist.dist u v + Dist.dist u x = Dist.dist v x
this : Dist.dist u v = 0
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.asymm_right | [47, 1] | [51, 24] | simp [h.ne12] at this | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
this✝ : Dist.dist u v + Dist.dist u x = Dist.dist v x
this : Dist.dist u v = 0
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.asymm_right | [47, 1] | [51, 24] | linarith [h.dist] | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
this : Dist.dist u v + Dist.dist u x = Dist.dist v x
⊢ Dist.dist u v = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.trans_right' | [53, 1] | [55, 77] | rintro rfl | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w✝ u v w x : V
h : sbtw u v x
h' : sbtw v w x
⊢ u ≠ w | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.trans_right' | [53, 1] | [55, 77] | exact h.asymm_right h' | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w u v x : V
h : sbtw u v x
h' : sbtw v u x
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/MetricBetween.lean | SBtw.sbtw.trans_right' | [53, 1] | [55, 77] | linarith [h.dist, h'.dist, dist_triangle u v w] | V : Type u_1
inst✝ : MetricSpace V
u✝ v✝ w✝ u v w x : V
h : sbtw u v x
h' : sbtw v w x
this : u ≠ w
⊢ Dist.dist u w + Dist.dist w x ≤ Dist.dist u x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.coe_eq_one | [20, 1] | [21, 85] | rfl | α : Type u_1
inst✝ : Group α
s : Subgroup α
a : α
⊢ ↑s = ↑⊥ ↔ ↑s = 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.smul_coe | [23, 1] | [25, 97] | ext | α : Type u_1
inst✝ : Group α
s : Subgroup α
a : α
ha : a ∈ s
⊢ a • ↑s = ↑s | case h
α : Type u_1
inst✝ : Group α
s : Subgroup α
a : α
ha : a ∈ s
x✝ : α
⊢ x✝ ∈ a • ↑s ↔ x✝ ∈ ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.smul_coe | [23, 1] | [25, 97] | rw [Set.mem_smul_set_iff_inv_smul_mem] | case h
α : Type u_1
inst✝ : Group α
s : Subgroup α
a : α
ha : a ∈ s
x✝ : α
⊢ x✝ ∈ a • ↑s ↔ x✝ ∈ ↑s | case h
α : Type u_1
inst✝ : Group α
s : Subgroup α
a : α
ha : a ∈ s
x✝ : α
⊢ a⁻¹ • x✝ ∈ ↑s ↔ x✝ ∈ ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.smul_coe | [23, 1] | [25, 97] | exact Subgroup.mul_mem_cancel_left _ (inv_mem ha) | case h
α : Type u_1
inst✝ : Group α
s : Subgroup α
a : α
ha : a ∈ s
x✝ : α
⊢ a⁻¹ • x✝ ∈ ↑s ↔ x✝ ∈ ↑s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | rw [← Nat.card_prod, Nat.card_congr] | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ Nat.card ↥s * Nat.card ↑(QuotientGroup.mk '' t) = Nat.card ↑(t * ↑s) | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ ↥s × ↑(QuotientGroup.mk '' t) ≃ ↑(t * ↑s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | apply Equiv.trans (QuotientGroup.preimageMkEquivSubgroupProdSet _ _).symm | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ ↥s × ↑(QuotientGroup.mk '' t) ≃ ↑(t * ↑s) | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' t)) ≃ ↑(t * ↑s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | rw [QuotientGroup.preimage_image_mk] | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' t)) ≃ ↑(t * ↑s) | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ ↑(⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t) ≃ ↑(t * ↑s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | apply Set.BijOn.equiv id | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ ↑(⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t) ≃ ↑(t * ↑s) | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ Set.BijOn id (⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t) (t * ↑s) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | convert Set.bijOn_id _ | α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ Set.BijOn id (⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t) (t * ↑s) | case h.e'_5
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ t * ↑s = ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | ext x | case h.e'_5
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
⊢ t * ↑s = ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t | case h.e'_5.h
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ x ∈ t * ↑s ↔ x ∈ ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | constructor | case h.e'_5.h
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ x ∈ t * ↑s ↔ x ∈ ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t | case h.e'_5.h.mp
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ x ∈ t * ↑s → x ∈ ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t
case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ x ∈ ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t → x ∈ t * ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | simp only [Set.mem_iUnion, Set.mem_preimage, Subtype.exists, exists_prop, Set.mem_mul] | case h.e'_5.h.mp
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ x ∈ t * ↑s → x ∈ ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t | case h.e'_5.h.mp
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ (∃ x_1 ∈ t, ∃ y ∈ ↑s, x_1 * y = x) → ∃ a ∈ s, x * a ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | aesop | case h.e'_5.h.mp
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ (∃ x_1 ∈ t, ∃ y ∈ ↑s, x_1 * y = x) → ∃ a ∈ s, x * a ∈ t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | simp only [Set.mem_iUnion, Set.mem_preimage, Subtype.exists, exists_prop, forall_exists_index,
and_imp] | case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ x ∈ ⋃ x, (fun x_1 => x_1 * ↑x) ⁻¹' t → x ∈ t * ↑s | case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ ∀ x_1 ∈ s, x * x_1 ∈ t → x ∈ t * ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | intro y hys hxy | case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x : α
⊢ ∀ x_1 ∈ s, x * x_1 ∈ t → x ∈ t * ↑s | case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x y : α
hys : y ∈ s
hxy : x * y ∈ t
⊢ x ∈ t * ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/Mathlib.lean | Subgroup.subgroup_mul_card_eq_mul | [28, 1] | [44, 41] | rw [← mul_inv_cancel_right x y] | case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x y : α
hys : y ∈ s
hxy : x * y ∈ t
⊢ x ∈ t * ↑s | case h.e'_5.h.mpr
α : Type u_1
inst✝ : Group α
s✝ : Subgroup α
a : α
s : Subgroup α
t : Set α
x y : α
hys : y ∈ s
hxy : x * y ∈ t
⊢ x * y * y⁻¹ ∈ t * ↑s |
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