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https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Set/Finite.lean | List.finite_length_eq | [8, 1] | [14, 47] | rw [this] | α : Type u_1
inst✝ : Finite α
n✝ n : ℕ
ih : {l | l.length = n}.Finite
this : {l | l.length = n + 1} = image2 (fun x x_1 => x :: x_1) univ {l | l.length = n}
⊢ {l | l.length = n + 1}.Finite | α : Type u_1
inst✝ : Finite α
n✝ n : ℕ
ih : {l | l.length = n}.Finite
this : {l | l.length = n + 1} = image2 (fun x x_1 => x :: x_1) univ {l | l.length = n}
⊢ (image2 (fun x x_1 => x :: x_1) univ {l | l.length = n}).Finite |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Set/Finite.lean | List.finite_length_eq | [8, 1] | [14, 47] | exact Set.finite_univ.image2 _ ih | α : Type u_1
inst✝ : Finite α
n✝ n : ℕ
ih : {l | l.length = n}.Finite
this : {l | l.length = n + 1} = image2 (fun x x_1 => x :: x_1) univ {l | l.length = n}
⊢ (image2 (fun x x_1 => x :: x_1) univ {l | l.length = n}).Finite | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Set/Finite.lean | List.finite_length_lt | [16, 1] | [17, 90] | convert (Finset.range n).finite_toSet.biUnion fun i _ ↦ finite_length_eq α i | α : Type u_1
inst✝ : Finite α
n : ℕ
⊢ {l | l.length < n}.Finite | case h.e'_2
α : Type u_1
inst✝ : Finite α
n : ℕ
⊢ {l | l.length < n} = ⋃ i ∈ ↑(Finset.range n), {l | l.length = i} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Set/Finite.lean | List.finite_length_lt | [16, 1] | [17, 90] | ext | case h.e'_2
α : Type u_1
inst✝ : Finite α
n : ℕ
⊢ {l | l.length < n} = ⋃ i ∈ ↑(Finset.range n), {l | l.length = i} | case h.e'_2.h
α : Type u_1
inst✝ : Finite α
n : ℕ
x✝ : List α
⊢ x✝ ∈ {l | l.length < n} ↔ x✝ ∈ ⋃ i ∈ ↑(Finset.range n), {l | l.length = i} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Set/Finite.lean | List.finite_length_lt | [16, 1] | [17, 90] | simp | case h.e'_2.h
α : Type u_1
inst✝ : Finite α
n : ℕ
x✝ : List α
⊢ x✝ ∈ {l | l.length < n} ↔ x✝ ∈ ⋃ i ∈ ↑(Finset.range n), {l | l.length = i} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Set/Finite.lean | List.finite_length_le | [19, 1] | [20, 59] | simpa [Nat.lt_succ_iff] using finite_length_lt α (n + 1) | α : Type u_1
inst✝ : Finite α
n : ℕ
⊢ {l | l.length ≤ n}.Finite | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Algebra/BigOperators/Ring.lean | Finset.sum_boole_mul' | [12, 1] | [13, 69] | simp | ι : Type u_1
α : Type u_2
inst✝¹ : NonAssocSemiring α
inst✝ : DecidableEq ι
s : Finset ι
f : ι → α
i : ι
⊢ ∑ j ∈ s, (if i = j then 1 else 0) * f j = if i ∈ s then f i else 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | IsExposed.span_lt | [25, 1] | [28, 8] | apply (affineSpan_mono _ hst.subset).lt_of_ne | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hst : IsExposed ℝ s t
hts : ¬s ⊆ t
⊢ affineSpan ℝ t < affineSpan ℝ s | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hst : IsExposed ℝ s t
hts : ¬s ⊆ t
⊢ affineSpan ℝ t ≠ affineSpan ℝ s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | IsExposed.span_lt | [25, 1] | [28, 8] | rintro h | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hst : IsExposed ℝ s t
hts : ¬s ⊆ t
⊢ affineSpan ℝ t ≠ affineSpan ℝ s | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hst : IsExposed ℝ s t
hts : ¬s ⊆ t
h : affineSpan ℝ t = affineSpan ℝ s
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | IsExposed.span_lt | [25, 1] | [28, 8] | sorry | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hst : IsExposed ℝ s t
hts : ¬s ⊆ t
h : affineSpan ℝ t = affineSpan ℝ s
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | mem_exposed_set_iff_mem_frontier | [37, 1] | [51, 16] | use fun ⟨t, hst, hts, hxt⟩ => ⟨hst.subset hxt, hst.subset_frontier hts hxt⟩ | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
⊢ (∃ t, IsExposed ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t) ↔ x ∈ s ∧ x ∈ frontier s | case mpr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
⊢ x ∈ s ∧ x ∈ frontier s → ∃ t, IsExposed ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | mem_exposed_set_iff_mem_frontier | [37, 1] | [51, 16] | rintro ⟨hxA, hxfA⟩ | case mpr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
⊢ x ∈ s ∧ x ∈ frontier s → ∃ t, IsExposed ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t | case mpr.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
⊢ ∃ t, IsExposed ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | mem_exposed_set_iff_mem_frontier | [37, 1] | [51, 16] | obtain ⟨y, hyA⟩ := id hs₂ | case mpr.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
⊢ ∃ t, IsExposed ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t | case mpr.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
y : E
hyA : y ∈ interior s
⊢ ∃ t, IsExposed ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | mem_exposed_set_iff_mem_frontier | [37, 1] | [51, 16] | obtain ⟨l, hl⟩ := geometric_hahn_banach_open_point (Convex.interior hs₁) isOpen_interior hxfA.2 | case mpr.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
y : E
hyA : y ∈ interior s
⊢ ∃ t, IsExposed ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t | case mpr.intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l✝ : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
y : E
hyA : y ∈ interior s
l : E →L[ℝ] ℝ
hl : ∀ a ∈ interior s, l a < l x
⊢ ∃ t, IsExposed ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | mem_exposed_set_iff_mem_frontier | [37, 1] | [51, 16] | refine'
⟨{x ∈ s | ∀ y ∈ s, l y ≤ l x}, fun _ => ⟨l, rfl⟩, fun h =>
not_le.2 (hl y hyA) ((h (interior_subset hyA)).2 x hxA), hxA, fun z hzA => _⟩ | case mpr.intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l✝ : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
y : E
hyA : y ∈ interior s
l : E →L[ℝ] ℝ
hl : ∀ a ∈ interior s, l a < l x
⊢ ∃ t, IsExposed ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t | case mpr.intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l✝ : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
y : E
hyA : y ∈ interior s
l : E →L[ℝ] ℝ
hl : ∀ a ∈ interior s, l a < l x
z : E
hzA : z ∈ s
⊢ l z ≤ l x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | mem_exposed_set_iff_mem_frontier | [37, 1] | [51, 16] | suffices h : l '' closure (interior s) ⊆ closure (Iio (l x)) by
rw [closure_Iio, ← closure_eq_closure_interior hs₁ hs₂] at h
exact h ⟨z, subset_closure hzA, rfl⟩ | case mpr.intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l✝ : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
y : E
hyA : y ∈ interior s
l : E →L[ℝ] ℝ
hl : ∀ a ∈ interior s, l a < l x
z : E
hzA : z ∈ s
⊢ l z ≤ l x | case mpr.intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l✝ : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
y : E
hyA : y ∈ interior s
l : E →L[ℝ] ℝ
hl : ∀ a ∈ interior s, l a < l x
z : E
hzA : z ∈ s
⊢ ⇑l '' closure (interior s) ⊆ closure (Iio (l x)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | mem_exposed_set_iff_mem_frontier | [37, 1] | [51, 16] | refine' (image_closure_subset_closure_image l.continuous).trans $ closure_mono _ | case mpr.intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l✝ : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
y : E
hyA : y ∈ interior s
l : E →L[ℝ] ℝ
hl : ∀ a ∈ interior s, l a < l x
z : E
hzA : z ∈ s
⊢ ⇑l '' closure (interior s) ⊆ closure (Iio (l x)) | case mpr.intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l✝ : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
y : E
hyA : y ∈ interior s
l : E →L[ℝ] ℝ
hl : ∀ a ∈ interior s, l a < l x
z : E
hzA : z ∈ s
⊢ ⇑l '' interior s ⊆ Iio (l x) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | mem_exposed_set_iff_mem_frontier | [37, 1] | [51, 16] | rintro _ ⟨w, hw, rfl⟩ | case mpr.intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l✝ : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
y : E
hyA : y ∈ interior s
l : E →L[ℝ] ℝ
hl : ∀ a ∈ interior s, l a < l x
z : E
hzA : z ∈ s
⊢ ⇑l '' interior s ⊆ Iio (l x) | case mpr.intro.intro.intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l✝ : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
y : E
hyA : y ∈ interior s
l : E →L[ℝ] ℝ
hl : ∀ a ∈ interior s, l a < l x
z : E
hzA : z ∈ s
w : E
hw : w ∈ interior s
⊢ l w ∈ Iio (l x) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | mem_exposed_set_iff_mem_frontier | [37, 1] | [51, 16] | exact hl w hw | case mpr.intro.intro.intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l✝ : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
y : E
hyA : y ∈ interior s
l : E →L[ℝ] ℝ
hl : ∀ a ∈ interior s, l a < l x
z : E
hzA : z ∈ s
w : E
hw : w ∈ interior s
⊢ l w ∈ Iio (l x) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | mem_exposed_set_iff_mem_frontier | [37, 1] | [51, 16] | rw [closure_Iio, ← closure_eq_closure_interior hs₁ hs₂] at h | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l✝ : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
y : E
hyA : y ∈ interior s
l : E →L[ℝ] ℝ
hl : ∀ a ∈ interior s, l a < l x
z : E
hzA : z ∈ s
h : ⇑l '' closure (interior s) ⊆ closure (Iio (l x))
⊢ l z ≤ l x | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l✝ : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
y : E
hyA : y ∈ interior s
l : E →L[ℝ] ℝ
hl : ∀ a ∈ interior s, l a < l x
z : E
hzA : z ∈ s
h : ⇑l '' closure s ⊆ Iic (l x)
⊢ l z ≤ l x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | mem_exposed_set_iff_mem_frontier | [37, 1] | [51, 16] | exact h ⟨z, subset_closure hzA, rfl⟩ | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l✝ : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
hxA : x ∈ s
hxfA : x ∈ frontier s
y : E
hyA : y ∈ interior s
l : E →L[ℝ] ℝ
hl : ∀ a ∈ interior s, l a < l x
z : E
hzA : z ∈ s
h : ⇑l '' closure s ⊆ Iic (l x)
⊢ l z ≤ l x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | mem_extreme_set_iff_mem_frontier | [53, 1] | [58, 37] | use fun ⟨t, hst, hts, hxt⟩ => ⟨hst.1 hxt, hst.subset_frontier hts hxt⟩ | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
⊢ (∃ t, IsExtreme ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t) ↔ x ∈ s ∧ x ∈ frontier s | case mpr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
⊢ x ∈ s ∧ x ∈ frontier s → ∃ t, IsExtreme ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | mem_extreme_set_iff_mem_frontier | [53, 1] | [58, 37] | rintro h | case mpr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
⊢ x ∈ s ∧ x ∈ frontier s → ∃ t, IsExtreme ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t | case mpr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
h : x ∈ s ∧ x ∈ frontier s
⊢ ∃ t, IsExtreme ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | mem_extreme_set_iff_mem_frontier | [53, 1] | [58, 37] | obtain ⟨t, hst, hts, hxt⟩ := (mem_exposed_set_iff_mem_frontier hs₁ hs₂).2 h | case mpr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
h : x ∈ s ∧ x ∈ frontier s
⊢ ∃ t, IsExtreme ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t | case mpr.intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t✝ C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
h : x ∈ s ∧ x ∈ frontier s
t : Set E
hst : IsExposed ℝ s t
hts : ¬s ⊆ t
hxt : x ∈ t
⊢ ∃ t, IsExtreme ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | mem_extreme_set_iff_mem_frontier | [53, 1] | [58, 37] | exact ⟨t, hst.isExtreme, hts, hxt⟩ | case mpr.intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t✝ C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : (interior s).Nonempty
h : x ∈ s ∧ x ∈ frontier s
t : Set E
hst : IsExposed ℝ s t
hts : ¬s ⊆ t
hxt : x ∈ t
⊢ ∃ t, IsExtreme ℝ s t ∧ ¬s ⊆ t ∧ x ∈ t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Exposed.lean | extremePoints_subset_closure_exposedPoints | [65, 1] | [66, 64] | sorry | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
s t C : Set E
X : Finset E
l : E →L[ℝ] ℝ
hs₁ : Convex ℝ s
hs₂ : IsClosed s
⊢ extremePoints ℝ s ⊆ closure (exposedPoints ℝ s) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | obtain rfl | hs := s.eq_empty_or_nonempty | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
⊢ min (s.card + s.mulStab.card) (t.card + t.mulStab.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card | case inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
t : Finset α
⊢ min (∅.card + ∅.mulStab.card) (t.card + t.mulStab.card) ≤ (∅ ∪ t).card + (∅ ∪ t).mulStab.card
case inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
⊢ min (s.card + s.mulStab.card) (t.card + t.mulStab.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | obtain rfl | ht := t.eq_empty_or_nonempty | case inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
⊢ min (s.card + s.mulStab.card) (t.card + t.mulStab.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card | case inr.inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
⊢ min (s.card + s.mulStab.card) (∅.card + ∅.mulStab.card) ≤ (s ∪ ∅).card + (s ∪ ∅).mulStab.card
case inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
⊢ min (s.card + s.mulStab.card) (t.card + t.mulStab.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | set Hs := s.mulStab with hHs | case inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
⊢ min (s.card + s.mulStab.card) (t.card + t.mulStab.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card | case inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs : Hs = s.mulStab
⊢ min (s.card + Hs.card) (t.card + t.mulStab.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | set Ht := t.mulStab with hHt | case inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs : Hs = s.mulStab
⊢ min (s.card + Hs.card) (t.card + t.mulStab.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card | case inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt : Ht = t.mulStab
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | set H := Hs * Ht with hH | case inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt : Ht = t.mulStab
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card | case inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt : Ht = t.mulStab
H : Finset α := Hs * Ht
hH : H = Hs * Ht
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | have hHs : Hs.Nonempty := hs.mulStab | case inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt : Ht = t.mulStab
H : Finset α := Hs * Ht
hH : H = Hs * Ht
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card | case inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt : Ht = t.mulStab
H : Finset α := Hs * Ht
hH : H = Hs * Ht
hHs : Hs.Nonempty
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | have hHt : Ht.Nonempty := ht.mulStab | case inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt : Ht = t.mulStab
H : Finset α := Hs * Ht
hH : H = Hs * Ht
hHs : Hs.Nonempty
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card | case inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | have hH : H.Nonempty := hHs.mul hHt | case inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card | case inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | wlog h1: Hs ∩ Ht = 1 | case inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card | case inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
α✝ : Type u_1
inst✝³ : CommGroup α✝
inst✝² : DecidableEq α✝
s✝ t✝ : Finset α✝
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
h1 : Hs ∩ Ht = 1
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | sorry | α✝ : Type u_1
inst✝³ : CommGroup α✝
inst✝² : DecidableEq α✝
s✝ t✝ : Finset α✝
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
h1 : Hs ∩ Ht = 1
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | simp [-zero_le'] | case inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
t : Finset α
⊢ min (∅.card + ∅.mulStab.card) (t.card + t.mulStab.card) ≤ (∅ ∪ t).card + (∅ ∪ t).mulStab.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | simp [-zero_le'] | case inr.inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s : Finset α
hs : s.Nonempty
⊢ min (s.card + s.mulStab.card) (∅.card + ∅.mulStab.card) ≤ (s ∪ ∅).card + (s ∪ ∅).mulStab.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | set N := stabilizer α s ⊓ stabilizer α t with hN | case inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card | case inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | have hNmulstab : (N : Set α) = ↑(Hs ∩ Ht) := by aesop | case inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card | case inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | specialize this (α := α ⧸ N) (s := s.image (↑)) (t := t.image (↑)) | case inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card | case inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
(image QuotientGroup.mk s).Nonempty →
(image QuotientGroup.mk t).Nonempty →
let Hs := (image QuotientGroup.mk s).mulStab;
Hs = (image QuotientGroup.mk s).mulStab →
let Ht := (image QuotientGroup.mk t).mulStab;
Ht = (image QuotientGroup.mk t).mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 →
min ((image QuotientGroup.mk s).card + Hs.card) ((image QuotientGroup.mk t).card + Ht.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | simp only [image_nonempty, mulStab_nonempty, mul_nonempty, and_imp,
forall_true_left, hs, ht, h1] at this | case inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
(image QuotientGroup.mk s).Nonempty →
(image QuotientGroup.mk t).Nonempty →
let Hs := (image QuotientGroup.mk s).mulStab;
Hs = (image QuotientGroup.mk s).mulStab →
let Ht := (image QuotientGroup.mk t).mulStab;
Ht = (image QuotientGroup.mk t).mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 →
min ((image QuotientGroup.mk s).card + Hs.card) ((image QuotientGroup.mk t).card + Ht.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card | case inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | calc
min (card s + card Hs) (card t + card Ht) =
min (Nat.card N * card (s.image (QuotientGroup.mk (s := N))) +
Nat.card N * card (Hs.image (QuotientGroup.mk (s := N))))
(Nat.card N * card (t.image (QuotientGroup.mk (s := N))) +
Nat.card N * card (Ht.image (QuotientGroup.mk (s := N)))) := by
rw [← subgroup_mul_card_eq_mul_of_mul_stab_subset N s,
← subgroup_mul_card_eq_mul_of_mul_stab_subset N t,
← subgroup_mul_card_eq_mul_of_mul_stab_subset N Hs,
← subgroup_mul_card_eq_mul_of_mul_stab_subset N Ht]
all_goals { aesop }
_ = Nat.card N * min (card (s.image (QuotientGroup.mk (s := N))) +
card (Hs.image (QuotientGroup.mk (s := N)))) (card (t.image (QuotientGroup.mk (s := N))) +
card (Ht.image (QuotientGroup.mk (s := N)))) := by
rw [← mul_add, ← mul_add, Nat.mul_min_mul_left]
_ = Nat.card N * min (card (image (QuotientGroup.mk (s := N)) s) +
card (mulStab (image (QuotientGroup.mk (s := N)) s)))
(card (image (QuotientGroup.mk (s := N)) t) +
card (mulStab (image (QuotientGroup.mk (s := N)) t))) := by
rw [mulStab_quotient_commute_subgroup N t, mulStab_quotient_commute_subgroup N s]
all_goals simp [*]
_ ≤ Nat.card N * (card (image (QuotientGroup.mk (s := N)) s ∪
image (QuotientGroup.mk (s := N)) t) +
card (mulStab (image (QuotientGroup.mk (s := N)) s ∪
image (QuotientGroup.mk (s := N)) t))) := Nat.mul_le_mul_left _ this
_ ≤ card (s ∪ t) + card (s ∪ t).mulStab := by
rw [mul_add, ← image_union, subgroup_mul_card_eq_mul_of_mul_stab_subset N (s ∪ t),
← mulStab_quotient_commute_subgroup N (s ∪ t),
subgroup_mul_card_eq_mul_of_mul_stab_subset N (s ∪ t).mulStab]
all_goals
{ simp only [hNmulstab, mulStab_idem]; norm_cast; exact inter_mulStab_subset_mulStab_union } | case inr.inr.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | aesop | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
⊢ ↑N = ↑(Hs ∩ Ht) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | ext x | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
⊢ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1 | case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
⊢ x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab ↔ x ∈ 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | constructor | case a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
⊢ x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab ↔ x ∈ 1 | case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
⊢ x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab → x ∈ 1
case a.mpr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
⊢ x ∈ 1 → x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | simp only [image_nonempty, mem_one, and_imp, ← QuotientGroup.mk_one] | case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
⊢ x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab → x ∈ 1 | case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
⊢ x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab → x = ↑1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | intro hx | case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
⊢ x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab → x = ↑1 | case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ x = ↑1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | rw [← mulStab_quotient_commute_subgroup N s, ← mulStab_quotient_commute_subgroup N t] at hx | case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ x = ↑1 | case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ image QuotientGroup.mk t.mulStab
⊢ x = ↑1
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | simp only [mem_inter, mem_image] at hx | case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ image QuotientGroup.mk t.mulStab
⊢ x = ↑1
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab | case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : (∃ a ∈ s.mulStab, ↑a = x) ∧ ∃ a ∈ t.mulStab, ↑a = x
⊢ x = ↑1
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | obtain ⟨⟨y, hy, hyx⟩, ⟨z, hz, hzx⟩⟩ := hx | case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : (∃ a ∈ s.mulStab, ↑a = x) ∧ ∃ a ∈ t.mulStab, ↑a = x
⊢ x = ↑1
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab | case a.mp.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
hyx : ↑y = x
z : α
hz : z ∈ t.mulStab
hzx : ↑z = x
⊢ x = ↑1
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | obtain ⟨w, hwx⟩ := Quotient.exists_rep x | case a.mp.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
hyx : ↑y = x
z : α
hz : z ∈ t.mulStab
hzx : ↑z = x
⊢ x = ↑1
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab | case a.mp.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
hyx : ↑y = x
z : α
hz : z ∈ t.mulStab
hzx : ↑z = x
w : α
hwx : ⟦w⟧ = x
⊢ x = ↑1
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | have : ⟦w⟧ = QuotientGroup.mk (s := N) w := by exact rfl | case a.mp.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
hyx : ↑y = x
z : α
hz : z ∈ t.mulStab
hzx : ↑z = x
w : α
hwx : ⟦w⟧ = x
⊢ x = ↑1
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab | case a.mp.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
hyx : ↑y = x
z : α
hz : z ∈ t.mulStab
hzx : ↑z = x
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
⊢ x = ↑1
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | rw [← hwx, this, QuotientGroup.eq] at hyx hzx ⊢ | case a.mp.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
hyx : ↑y = x
z : α
hz : z ∈ t.mulStab
hzx : ↑z = x
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
⊢ x = ↑1
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab | case a.mp.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hzx : z⁻¹ * w ∈ N
hyx : y⁻¹ * w ∈ N
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
⊢ w⁻¹ * 1 ∈ N
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | simp only [mul_one, inv_mem_iff, Subgroup.mem_inf, mem_stabilizer_iff, N] at hyx hzx ⊢ | case a.mp.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hzx : z⁻¹ * w ∈ N
hyx : y⁻¹ * w ∈ N
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
⊢ w⁻¹ * 1 ∈ N
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab | case a.mp.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ w • s = s ∧ w • t = t
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | constructor | case a.mp.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ w • s = s ∧ w • t = t
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab | case a.mp.intro.intro.intro.intro.intro.intro.left
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ w • s = s
case a.mp.intro.intro.intro.intro.intro.intro.right
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ w • t = t
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | all_goals { aesop } | case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ image QuotientGroup.mk s.mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑t.mulStab
case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | exact rfl | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
hyx : ↑y = x
z : α
hz : z ∈ t.mulStab
hzx : ↑z = x
w : α
hwx : ⟦w⟧ = x
⊢ ⟦w⟧ = ↑w | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | convert hyx.1 using 1 | case a.mp.intro.intro.intro.intro.intro.intro.left
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ w • s = s | case h.e'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ w • s = (y⁻¹ * w) • s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | rw [mul_comm, mul_smul] | case h.e'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ w • s = (y⁻¹ * w) • s | case h.e'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ w • s = w • y⁻¹ • s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | congr | case h.e'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ w • s = w • y⁻¹ • s | case h.e'_2.e_a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ s = y⁻¹ • s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | simp only [← inv_smul_eq_iff, inv_inv, ← (mem_mulStab hs), hy] | case h.e'_2.e_a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ s = y⁻¹ • s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | convert hzx.2 using 1 | case a.mp.intro.intro.intro.intro.intro.intro.right
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ w • t = t | case h.e'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ w • t = (z⁻¹ * w) • t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | rw [mul_comm, mul_smul] | case h.e'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ w • t = (z⁻¹ * w) • t | case h.e'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ w • t = w • z⁻¹ • t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | congr | case h.e'_2
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ w • t = w • z⁻¹ • t | case h.e'_2.e_a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ t = z⁻¹ • t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | simp only [← inv_smul_eq_iff, inv_inv, ← (mem_mulStab ht), hz] | case h.e'_2.e_a
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this✝ :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
y : α
hy : y ∈ s.mulStab
z : α
hz : z ∈ t.mulStab
w : α
hwx : ⟦w⟧ = x
this : ⟦w⟧ = ↑w
hyx : (y⁻¹ * w) • s = s ∧ (y⁻¹ * w) • t = t
hzx : (z⁻¹ * w) • s = s ∧ (z⁻¹ * w) • t = t
⊢ t = z⁻¹ • t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | aesop | case a.mp
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
hx : x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab
⊢ ↑N ⊆ ↑s.mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | simp (config := { contextual := true }) [*] | case a.mpr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
this :
∀ {α : Type u_1} [inst : CommGroup α] [inst_1 : DecidableEq α] {s t : Finset α},
s.Nonempty →
t.Nonempty →
let Hs := s.mulStab;
Hs = s.mulStab →
let Ht := t.mulStab;
Ht = t.mulStab →
let H := Hs * Ht;
H = Hs * Ht →
Hs.Nonempty →
Ht.Nonempty →
H.Nonempty →
Hs ∩ Ht = 1 → min (s.card + Hs.card) (t.card + Ht.card) ≤ (s ∪ t).card + (s ∪ t).mulStab.card
h1 : ¬Hs ∩ Ht = 1
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
x : α ⧸ N
⊢ x ∈ 1 → x ∈ (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | rw [← subgroup_mul_card_eq_mul_of_mul_stab_subset N s,
← subgroup_mul_card_eq_mul_of_mul_stab_subset N t,
← subgroup_mul_card_eq_mul_of_mul_stab_subset N Hs,
← subgroup_mul_card_eq_mul_of_mul_stab_subset N Ht] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ min (s.card + Hs.card) (t.card + Ht.card) =
min (Nat.card ↥N * (image QuotientGroup.mk s).card + Nat.card ↥N * (image QuotientGroup.mk Hs).card)
(Nat.card ↥N * (image QuotientGroup.mk t).card + Nat.card ↥N * (image QuotientGroup.mk Ht).card) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑Ht.mulStab
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑Hs.mulStab
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑t.mulStab
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑s.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | all_goals { aesop } | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑Ht.mulStab
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑Hs.mulStab
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑t.mulStab
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑s.mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | aesop | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑s.mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | rw [← mul_add, ← mul_add, Nat.mul_min_mul_left] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ min (Nat.card ↥N * (image QuotientGroup.mk s).card + Nat.card ↥N * (image QuotientGroup.mk Hs).card)
(Nat.card ↥N * (image QuotientGroup.mk t).card + Nat.card ↥N * (image QuotientGroup.mk Ht).card) =
Nat.card ↥N *
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk Hs).card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk Ht).card) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | rw [mulStab_quotient_commute_subgroup N t, mulStab_quotient_commute_subgroup N s] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ Nat.card ↥N *
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk Hs).card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk Ht).card) =
Nat.card ↥N *
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑s.mulStab
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑t.mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | all_goals simp [*] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑s.mulStab
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑t.mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | simp [*] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑t.mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | rw [mul_add, ← image_union, subgroup_mul_card_eq_mul_of_mul_stab_subset N (s ∪ t),
← mulStab_quotient_commute_subgroup N (s ∪ t),
subgroup_mul_card_eq_mul_of_mul_stab_subset N (s ∪ t).mulStab] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ Nat.card ↥N *
((image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card) ≤
(s ∪ t).card + (s ∪ t).mulStab.card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑(s ∪ t).mulStab.mulStab
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑(s ∪ t).mulStab
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑(s ∪ t).mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | all_goals
{ simp only [hNmulstab, mulStab_idem]; norm_cast; exact inter_mulStab_subset_mulStab_union } | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑(s ∪ t).mulStab.mulStab
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑(s ∪ t).mulStab
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑(s ∪ t).mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | simp only [hNmulstab, mulStab_idem] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑N ⊆ ↑(s ∪ t).mulStab | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑(Hs ∩ Ht) ⊆ ↑(s ∪ t).mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | norm_cast | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ ↑(Hs ∩ Ht) ⊆ ↑(s ∪ t).mulStab | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ Hs ∩ Ht ⊆ (s ∪ t).mulStab |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_union_add_card_mulStab_union | [39, 1] | [149, 8] | exact inter_mulStab_subset_mulStab_union | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
Hs : Finset α := s.mulStab
hHs✝ : Hs = s.mulStab
Ht : Finset α := t.mulStab
hHt✝ : Ht = t.mulStab
H : Finset α := Hs * Ht
hH✝ : H = Hs * Ht
hHs : Hs.Nonempty
hHt : Ht.Nonempty
hH : H.Nonempty
N : Subgroup α := stabilizer α s ⊓ stabilizer α t
hN : N = stabilizer α s ⊓ stabilizer α t
hNmulstab : ↑N = ↑(Hs ∩ Ht)
h1 : (image QuotientGroup.mk s).mulStab ∩ (image QuotientGroup.mk t).mulStab = 1
this :
min ((image QuotientGroup.mk s).card + (image QuotientGroup.mk s).mulStab.card)
((image QuotientGroup.mk t).card + (image QuotientGroup.mk t).mulStab.card) ≤
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).card +
(image QuotientGroup.mk s ∪ image QuotientGroup.mk t).mulStab.card
⊢ Hs ∩ Ht ⊆ (s ∪ t).mulStab | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_sup_add_card_mulStab_sup | [190, 1] | [200, 79] | induction' s using Finset.cons_induction with i s hi ih | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
ι : Type u_2
s : Finset ι
f : ι → Finset α
hs : s.Nonempty
⊢ (s.inf' hs fun i => (f i).card + (f i).mulStab.card) ≤ (s.sup f).card + (s.sup f).mulStab.card | case empty
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ι : Type u_2
f : ι → Finset α
hs : ∅.Nonempty
⊢ (∅.inf' hs fun i => (f i).card + (f i).mulStab.card) ≤ (∅.sup f).card + (∅.sup f).mulStab.card
case cons
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
ι : Type u_2
f : ι → Finset α
i : ι
s : Finset ι
hi : i ∉ s
ih : ∀ (hs : s.Nonempty), (s.inf' hs fun i => (f i).card + (f i).mulStab.card) ≤ (s.sup f).card + (s.sup f).mulStab.card
hs : (cons i s hi).Nonempty
⊢ ((cons i s hi).inf' hs fun i => (f i).card + (f i).mulStab.card) ≤
((cons i s hi).sup f).card + ((cons i s hi).sup f).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_sup_add_card_mulStab_sup | [190, 1] | [200, 79] | obtain rfl | hs := s.eq_empty_or_nonempty | case cons
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
ι : Type u_2
f : ι → Finset α
i : ι
s : Finset ι
hi : i ∉ s
ih : ∀ (hs : s.Nonempty), (s.inf' hs fun i => (f i).card + (f i).mulStab.card) ≤ (s.sup f).card + (s.sup f).mulStab.card
hs : (cons i s hi).Nonempty
⊢ ((cons i s hi).inf' hs fun i => (f i).card + (f i).mulStab.card) ≤
((cons i s hi).sup f).card + ((cons i s hi).sup f).mulStab.card | case cons.inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ι : Type u_2
f : ι → Finset α
i : ι
hi : i ∉ ∅
ih : ∀ (hs : ∅.Nonempty), (∅.inf' hs fun i => (f i).card + (f i).mulStab.card) ≤ (∅.sup f).card + (∅.sup f).mulStab.card
hs : (cons i ∅ hi).Nonempty
⊢ ((cons i ∅ hi).inf' hs fun i => (f i).card + (f i).mulStab.card) ≤
((cons i ∅ hi).sup f).card + ((cons i ∅ hi).sup f).mulStab.card
case cons.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
ι : Type u_2
f : ι → Finset α
i : ι
s : Finset ι
hi : i ∉ s
ih : ∀ (hs : s.Nonempty), (s.inf' hs fun i => (f i).card + (f i).mulStab.card) ≤ (s.sup f).card + (s.sup f).mulStab.card
hs✝ : (cons i s hi).Nonempty
hs : s.Nonempty
⊢ ((cons i s hi).inf' hs✝ fun i => (f i).card + (f i).mulStab.card) ≤
((cons i s hi).sup f).card + ((cons i s hi).sup f).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_sup_add_card_mulStab_sup | [190, 1] | [200, 79] | simp only [hs, inf'_cons, sup_cons, sup_eq_union] | case cons.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
ι : Type u_2
f : ι → Finset α
i : ι
s : Finset ι
hi : i ∉ s
ih : ∀ (hs : s.Nonempty), (s.inf' hs fun i => (f i).card + (f i).mulStab.card) ≤ (s.sup f).card + (s.sup f).mulStab.card
hs✝ : (cons i s hi).Nonempty
hs : s.Nonempty
⊢ ((cons i s hi).inf' hs✝ fun i => (f i).card + (f i).mulStab.card) ≤
((cons i s hi).sup f).card + ((cons i s hi).sup f).mulStab.card | case cons.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
ι : Type u_2
f : ι → Finset α
i : ι
s : Finset ι
hi : i ∉ s
ih : ∀ (hs : s.Nonempty), (s.inf' hs fun i => (f i).card + (f i).mulStab.card) ≤ (s.sup f).card + (s.sup f).mulStab.card
hs✝ : (cons i s hi).Nonempty
hs : s.Nonempty
⊢ (((f i).card + (f i).mulStab.card) ⊓ s.inf' ⋯ fun i => (f i).card + (f i).mulStab.card) ≤
(f i ∪ s.sup f).card + (f i ∪ s.sup f).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_sup_add_card_mulStab_sup | [190, 1] | [200, 79] | exact (inf_le_inf_left _ $ ih hs).trans le_card_union_add_card_mulStab_union | case cons.inr
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t : Finset α
ι : Type u_2
f : ι → Finset α
i : ι
s : Finset ι
hi : i ∉ s
ih : ∀ (hs : s.Nonempty), (s.inf' hs fun i => (f i).card + (f i).mulStab.card) ≤ (s.sup f).card + (s.sup f).mulStab.card
hs✝ : (cons i s hi).Nonempty
hs : s.Nonempty
⊢ (((f i).card + (f i).mulStab.card) ⊓ s.inf' ⋯ fun i => (f i).card + (f i).mulStab.card) ≤
(f i ∪ s.sup f).card + (f i ∪ s.sup f).mulStab.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_sup_add_card_mulStab_sup | [190, 1] | [200, 79] | cases not_nonempty_empty hs | case empty
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ι : Type u_2
f : ι → Finset α
hs : ∅.Nonempty
⊢ (∅.inf' hs fun i => (f i).card + (f i).mulStab.card) ≤ (∅.sup f).card + (∅.sup f).mulStab.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_sup_add_card_mulStab_sup | [190, 1] | [200, 79] | simp | case cons.inl
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
ι : Type u_2
f : ι → Finset α
i : ι
hi : i ∉ ∅
ih : ∀ (hs : ∅.Nonempty), (∅.inf' hs fun i => (f i).card + (f i).mulStab.card) ≤ (∅.sup f).card + (∅.sup f).mulStab.card
hs : (cons i ∅ hi).Nonempty
⊢ ((cons i ∅ hi).inf' hs fun i => (f i).card + (f i).mulStab.card) ≤
((cons i ∅ hi).sup f).card + ((cons i ∅ hi).sup f).mulStab.card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_mul_add_card_mulStab_mul | [204, 1] | [248, 67] | have :
∀ b : t,
∃ n s' t',
↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card :=
fun b => ⟨_, s, t, b.2, Subset.rfl, Subset.rfl, rfl, rfl⟩ | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
⊢ s.card + t.card ≤ (s * t).card + (s * t).mulStab.card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
⊢ s.card + t.card ≤ (s * t).card + (s * t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_mul_add_card_mulStab_mul | [204, 1] | [248, 67] | choose s' t' hbt' hs' hst' hstcard ht' using fun b => Nat.find_spec (this b) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
⊢ s.card + t.card ≤ (s * t).card + (s * t).mulStab.card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
⊢ s.card + t.card ≤ (s * t).card + (s * t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_mul_add_card_mulStab_mul | [204, 1] | [248, 67] | have : s * t = univ.sup fun b => s' b * t' b := by
refine' le_antisymm _ (Finset.sup_le_iff.2 fun _ _ => hst' _)
exact
mul_subset_iff_right.2 fun b hb =>
(smul_finset_subset_smul_finset $ hs' ⟨b, hb⟩).trans $
(op_smul_finset_subset_mul $ hbt' ⟨b, hb⟩).trans $
@le_sup _ _ _ _ _ (fun b => s' b * t' b) _ $ mem_univ _ | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
⊢ s.card + t.card ≤ (s * t).card + (s * t).mulStab.card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
⊢ s.card + t.card ≤ (s * t).card + (s * t).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_mul_add_card_mulStab_mul | [204, 1] | [248, 67] | rw [this] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
⊢ s.card + t.card ≤ (s * t).card + (s * t).mulStab.card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
⊢ s.card + t.card ≤ (univ.sup fun b => s' b * t' b).card + (univ.sup fun b => s' b * t' b).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_mul_add_card_mulStab_mul | [204, 1] | [248, 67] | refine' (le_inf' ht.attach _ fun b _ => _).trans (le_card_sup_add_card_mulStab_sup _) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
⊢ s.card + t.card ≤ (univ.sup fun b => s' b * t' b).card + (univ.sup fun b => s' b * t' b).mulStab.card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
⊢ s.card + t.card ≤ (s' b * t' b).card + (s' b * t' b).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_mul_add_card_mulStab_mul | [204, 1] | [248, 67] | rw [← hstcard b] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
⊢ s.card + t.card ≤ (s' b * t' b).card + (s' b * t' b).mulStab.card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
⊢ (s' b).card + (t' b).card ≤ (s' b * t' b).card + (s' b * t' b).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_mul_add_card_mulStab_mul | [204, 1] | [248, 67] | refine'
add_le_add (card_le_card_mul_right _ ⟨_, hbt' _⟩)
((card_mono $ subset_mulStab_mul_left ⟨_, hbt' _⟩).trans' _) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
⊢ (s' b).card + (t' b).card ≤ (s' b * t' b).card + (s' b * t' b).mulStab.card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
⊢ (t' b).card ≤ (s' b).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_mul_add_card_mulStab_mul | [204, 1] | [248, 67] | rw [← card_smul_finset (b : α)⁻¹ (t' _)] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
⊢ (t' b).card ≤ (s' b).mulStab.card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
⊢ ((↑b)⁻¹ • t' b).card ≤ (s' b).mulStab.card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_mul_add_card_mulStab_mul | [204, 1] | [248, 67] | refine' card_mono ((mul_subset_left_iff $ hs.mono $ hs' _).1 _) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
⊢ ((↑b)⁻¹ • t' b).card ≤ (s' b).mulStab.card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
⊢ s' b * (↑b)⁻¹ • t' b ⊆ s' b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_mul_add_card_mulStab_mul | [204, 1] | [248, 67] | refine' mul_subset_iff_left.2 fun c hc => _ | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
⊢ s' b * (↑b)⁻¹ • t' b ⊆ s' b | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
c : α
hc : c ∈ s' b
⊢ c • (↑b)⁻¹ • t' b ⊆ s' b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_mul_add_card_mulStab_mul | [204, 1] | [248, 67] | rw [← mul_smul] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
c : α
hc : c ∈ s' b
⊢ c • (↑b)⁻¹ • t' b ⊆ s' b | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
c : α
hc : c ∈ s' b
⊢ (c * (↑b)⁻¹) • t' b ⊆ s' b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_mul_add_card_mulStab_mul | [204, 1] | [248, 67] | refine'
smul_finset_subset_iff.2
(inter_eq_left.1 $ eq_of_subset_of_card_le inter_subset_left _) | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
c : α
hc : c ∈ s' b
⊢ (c * (↑b)⁻¹) • t' b ⊆ s' b | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
c : α
hc : c ∈ s' b
⊢ (t' b).card ≤ (t' b ∩ (c * (↑b)⁻¹)⁻¹ • s' b).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_mul_add_card_mulStab_mul | [204, 1] | [248, 67] | rw [← ht'] | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
c : α
hc : c ∈ s' b
⊢ (t' b).card ≤ (t' b ∩ (c * (↑b)⁻¹)⁻¹ • s' b).card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
c : α
hc : c ∈ s' b
⊢ Nat.find ⋯ ≤ (t' b ∩ (c * (↑b)⁻¹)⁻¹ • s' b).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Kneser/KneserRuzsa.lean | Finset.le_card_mul_add_card_mulStab_mul | [204, 1] | [248, 67] | refine'
Nat.find_min' _
⟨_, _, mem_inter.2 ⟨hbt' _, _⟩, (hs' _).trans subset_union_left,
(mulDysonETransform.subset _ (s' b, t' b)).trans $ hst' _,
(mulDysonETransform.card _ _).trans $ hstcard _, rfl⟩ | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
c : α
hc : c ∈ s' b
⊢ Nat.find ⋯ ≤ (t' b ∩ (c * (↑b)⁻¹)⁻¹ • s' b).card | α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
this✝ :
∀ (b : { x // x ∈ t }),
∃ n s' t', ↑b ∈ t' ∧ s ⊆ s' ∧ s' * t' ⊆ s * t ∧ s'.card + t'.card = s.card + t.card ∧ n = t'.card
s' t' : { x // x ∈ t } → Finset α
hbt' : ∀ (b : { x // x ∈ t }), ↑b ∈ t' b
hs' : ∀ (b : { x // x ∈ t }), s ⊆ s' b
hst' : ∀ (b : { x // x ∈ t }), s' b * t' b ⊆ s * t
hstcard : ∀ (b : { x // x ∈ t }), (s' b).card + (t' b).card = s.card + t.card
ht' : ∀ (b : { x // x ∈ t }), Nat.find ⋯ = (t' b).card
this : s * t = univ.sup fun b => s' b * t' b
b : { x // x ∈ t }
x✝ : b ∈ t.attach
c : α
hc : c ∈ s' b
⊢ ↑b ∈ (c * (↑b)⁻¹)⁻¹ • (s' b, t' b).1 |
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