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https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | split_ifs | case a.refine'_2.refine'_1.hb
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
y : E
hy : y ∈ s
⊢ 0 ≤ if x = y then 1 - (↑n + 2)⁻¹ else 0 | case pos
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
y : E
hy : y ∈ s
h✝ : x = y
⊢ 0 ≤ 1 - (↑n + 2)⁻¹
case neg
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
y : E
hy : y ∈ s
h✝ : ¬x = y
⊢ 0 ≤ 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | rw [sub_nonneg] | case pos
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
y : E
hy : y ∈ s
h✝ : x = y
⊢ 0 ≤ 1 - (↑n + 2)⁻¹ | case pos
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
y : E
hy : y ∈ s
h✝ : x = y
⊢ (↑n + 2)⁻¹ ≤ 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | apply inv_le_one | case pos
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
y : E
hy : y ∈ s
h✝ : x = y
⊢ (↑n + 2)⁻¹ ≤ 1 | case pos.ha
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
y : E
hy : y ∈ s
h✝ : x = y
⊢ 1 ≤ ↑n + 2 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | norm_cast | case pos.ha
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
y : E
hy : y ∈ s
h✝ : x = y
⊢ 1 ≤ ↑n + 2 | case pos.ha
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
y : E
hy : y ∈ s
h✝ : x = y
⊢ 1 ≤ n + 2 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | apply Nat.succ_pos | case pos.ha
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
y : E
hy : y ∈ s
h✝ : x = y
⊢ 1 ≤ n + 2 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | rfl | case neg
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
y : E
hy : y ∈ s
h✝ : ¬x = y
⊢ 0 ≤ 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | apply centroid_weights | case a.refine'_2.hw₁
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | simp [Finset.sum_boole, Finset.filter_eq, hx] | case a.refine'_2.hw₂
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ (∑ i ∈ s, if x = i then 1 else 0) = 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | simp only [add_sub_cancel] | case a.refine'_2.hab
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ (↑n + 2)⁻¹ + (1 - (↑n + 2)⁻¹) = 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | rw [tendsto_iff_norm_sub_tendsto_zero] | case a.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
⊢ Filter.Tendsto (fun n => (↑n + 2)⁻¹ • Finset.centroid ℝ s id + (1 - (↑n + 2)⁻¹) • x) Filter.atTop (nhds x) | case a.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
⊢ Filter.Tendsto (fun e => ‖(↑e + 2)⁻¹ • Finset.centroid ℝ s id + (1 - (↑e + 2)⁻¹) • x - x‖) Filter.atTop (nhds 0) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | convert_to Filter.Tendsto (fun e : ℕ => ((e : ℝ) + 2)⁻¹ * ‖s.centroid ℝ id - x‖) Filter.atTop _ | case a.refine'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
⊢ Filter.Tendsto (fun e => ‖(↑e + 2)⁻¹ • Finset.centroid ℝ s id + (1 - (↑e + 2)⁻¹) • x - x‖) Filter.atTop (nhds 0) | case h.e'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
⊢ (fun e => ‖(↑e + 2)⁻¹ • Finset.centroid ℝ s id + (1 - (↑e + 2)⁻¹) • x - x‖) = fun e =>
(↑e + 2)⁻¹ * ‖Finset.centroid ℝ s id - x‖
case a.refine'_3.convert_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
⊢ Filter.Tendsto (fun e => (↑e + 2)⁻¹ * ‖Finset.centroid ℝ s id - x‖) Filter.atTop (nhds 0) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | suffices Filter.Tendsto (fun e : ℕ => (↑(e + 2) : ℝ)⁻¹) Filter.atTop (nhds 0) by
simpa using this.mul_const _ | case a.refine'_3.convert_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
⊢ Filter.Tendsto (fun e => (↑e + 2)⁻¹ * ‖Finset.centroid ℝ s id - x‖) Filter.atTop (nhds 0) | case a.refine'_3.convert_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
⊢ Filter.Tendsto (fun e => (↑(e + 2))⁻¹) Filter.atTop (nhds 0) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | refine' tendsto_inv_atTop_zero.comp _ | case a.refine'_3.convert_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
⊢ Filter.Tendsto (fun e => (↑(e + 2))⁻¹) Filter.atTop (nhds 0) | case a.refine'_3.convert_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
⊢ Filter.Tendsto (fun e => ↑(e + 2)) Filter.atTop Filter.atTop |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | rw [tendsto_natCast_atTop_iff] | case a.refine'_3.convert_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
⊢ Filter.Tendsto (fun e => ↑(e + 2)) Filter.atTop Filter.atTop | case a.refine'_3.convert_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
⊢ Filter.Tendsto (fun e => e + 2) Filter.atTop Filter.atTop |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | apply Filter.tendsto_add_atTop_nat | case a.refine'_3.convert_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
⊢ Filter.Tendsto (fun e => e + 2) Filter.atTop Filter.atTop | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | ext n | case h.e'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
⊢ (fun e => ‖(↑e + 2)⁻¹ • Finset.centroid ℝ s id + (1 - (↑e + 2)⁻¹) • x - x‖) = fun e =>
(↑e + 2)⁻¹ * ‖Finset.centroid ℝ s id - x‖ | case h.e'_3.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ ‖(↑n + 2)⁻¹ • Finset.centroid ℝ s id + (1 - (↑n + 2)⁻¹) • x - x‖ = (↑n + 2)⁻¹ * ‖Finset.centroid ℝ s id - x‖ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | rw [add_sub_assoc, sub_smul, sub_right_comm, one_smul, sub_self, zero_sub, ← smul_neg, ←
smul_add, norm_smul_of_nonneg, ← sub_eq_add_neg] | case h.e'_3.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ ‖(↑n + 2)⁻¹ • Finset.centroid ℝ s id + (1 - (↑n + 2)⁻¹) • x - x‖ = (↑n + 2)⁻¹ * ‖Finset.centroid ℝ s id - x‖ | case h.e'_3.h.ht
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ 0 ≤ (↑n + 2)⁻¹ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | rw [inv_nonneg] | case h.e'_3.h.ht
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ 0 ≤ (↑n + 2)⁻¹ | case h.e'_3.h.ht
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ 0 ≤ ↑n + 2 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | norm_cast | case h.e'_3.h.ht
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ 0 ≤ ↑n + 2 | case h.e'_3.h.ht
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ 0 ≤ n + 2 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | apply Nat.zero_le | case h.e'_3.h.ht
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ 0 ≤ n + 2 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | simpa using this.mul_const _ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
this : Filter.Tendsto (fun e => (↑(e + 2))⁻¹) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun e => (↑e + 2)⁻¹ * ‖Finset.centroid ℝ s id - x‖) Filter.atTop (nhds 0) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | closure_combiInterior_eq_convexHull | [293, 1] | [299, 40] | refine'
Set.Subset.antisymm _
(convexHull_min (subset_closure_combiInterior hs) (convex_combiInterior hs).closure) | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
⊢ closure (combiInterior ℝ s) = (convexHull ℝ) ↑s | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
⊢ closure (combiInterior ℝ s) ⊆ (convexHull ℝ) ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | closure_combiInterior_eq_convexHull | [293, 1] | [299, 40] | rw [s.isClosed_convexHull.closure_subset_iff] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
⊢ closure (combiInterior ℝ s) ⊆ (convexHull ℝ) ↑s | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
⊢ combiInterior ℝ s ⊆ (convexHull ℝ) ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | closure_combiInterior_eq_convexHull | [293, 1] | [299, 40] | exact combiInterior_subset_convexHull | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
⊢ combiInterior ℝ s ⊆ (convexHull ℝ) ↑s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | convexHull_subset_convexHull_of_combiInterior_subset_combiInterior | [301, 1] | [308, 23] | rw [← closure_combiInterior_eq_convexHull hs] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
ht : AffineIndependent ℝ Subtype.val
⊢ combiInterior ℝ s ⊆ combiInterior ℝ t → (convexHull ℝ) ↑s ⊆ (convexHull ℝ) ↑t | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
ht : AffineIndependent ℝ Subtype.val
⊢ combiInterior ℝ s ⊆ combiInterior ℝ t → closure (combiInterior ℝ s) ⊆ (convexHull ℝ) ↑t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | convexHull_subset_convexHull_of_combiInterior_subset_combiInterior | [301, 1] | [308, 23] | rw [← closure_combiInterior_eq_convexHull ht] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
ht : AffineIndependent ℝ Subtype.val
⊢ combiInterior ℝ s ⊆ combiInterior ℝ t → closure (combiInterior ℝ s) ⊆ (convexHull ℝ) ↑t | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
ht : AffineIndependent ℝ Subtype.val
⊢ combiInterior ℝ s ⊆ combiInterior ℝ t → closure (combiInterior ℝ s) ⊆ closure (combiInterior ℝ t) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | convexHull_subset_convexHull_of_combiInterior_subset_combiInterior | [301, 1] | [308, 23] | intro h | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
ht : AffineIndependent ℝ Subtype.val
⊢ combiInterior ℝ s ⊆ combiInterior ℝ t → closure (combiInterior ℝ s) ⊆ closure (combiInterior ℝ t) | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
ht : AffineIndependent ℝ Subtype.val
h : combiInterior ℝ s ⊆ combiInterior ℝ t
⊢ closure (combiInterior ℝ s) ⊆ closure (combiInterior ℝ t) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | convexHull_subset_convexHull_of_combiInterior_subset_combiInterior | [301, 1] | [308, 23] | apply closure_mono h | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
ht : AffineIndependent ℝ Subtype.val
h : combiInterior ℝ s ⊆ combiInterior ℝ t
⊢ closure (combiInterior ℝ s) ⊆ closure (combiInterior ℝ t) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.isContained_iff_exists_subgraph | [90, 1] | [95, 49] | constructor | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ G ⊑ H ↔ ∃ H', Nonempty (G ≃g H'.coe) | case mp
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ G ⊑ H → ∃ H', Nonempty (G ≃g H'.coe)
case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ (∃ H', Nonempty (G ≃g H'.coe)) → G ⊑ H |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.isContained_iff_exists_subgraph | [90, 1] | [95, 49] | rintro ⟨f, hf⟩ | case mp
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ G ⊑ H → ∃ H', Nonempty (G ≃g H'.coe) | case mp.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
f : G →g H
hf : Injective ⇑f
⊢ ∃ H', Nonempty (G ≃g H'.coe) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.isContained_iff_exists_subgraph | [90, 1] | [95, 49] | exact ⟨Subgraph.map f ⊤, ⟨(Subgraph.isoMap _ hf _).comp Subgraph.topIso.symm⟩⟩ | case mp.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
f : G →g H
hf : Injective ⇑f
⊢ ∃ H', Nonempty (G ≃g H'.coe) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.isContained_iff_exists_subgraph | [90, 1] | [95, 49] | rintro ⟨H', ⟨e⟩⟩ | case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ (∃ H', Nonempty (G ≃g H'.coe)) → G ⊑ H | case mpr.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
H' : H.Subgraph
e : G ≃g H'.coe
⊢ G ⊑ H |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.isContained_iff_exists_subgraph | [90, 1] | [95, 49] | exact e.isContained.trans H'.coe_isContained | case mpr.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
H' : H.Subgraph
e : G ≃g H'.coe
⊢ G ⊑ H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.isIndContained_iff_exists_subgraph | [133, 1] | [142, 52] | constructor | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ G ⊴ H ↔ ∃ H' _e, H'.IsInduced' | case mp
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ G ⊴ H → ∃ H' _e, H'.IsInduced'
case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ (∃ H' _e, H'.IsInduced') → G ⊴ H |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.isIndContained_iff_exists_subgraph | [133, 1] | [142, 52] | rintro ⟨f⟩ | case mp
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ G ⊴ H → ∃ H' _e, H'.IsInduced' | case mp.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
f : G ↪g H
⊢ ∃ H' _e, H'.IsInduced' |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.isIndContained_iff_exists_subgraph | [133, 1] | [142, 52] | refine' ⟨Subgraph.map f.toHom ⊤,
(Subgraph.isoMap f.toHom f.injective _).comp Subgraph.topIso.symm, _⟩ | case mp.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
f : G ↪g H
⊢ ∃ H' _e, H'.IsInduced' | case mp.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
f : G ↪g H
⊢ (Subgraph.map f.toHom ⊤).IsInduced' |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.isIndContained_iff_exists_subgraph | [133, 1] | [142, 52] | rintro _ _ ⟨a, -, rfl⟩ ⟨b, -, rfl⟩ | case mp.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
f : G ↪g H
⊢ (Subgraph.map f.toHom ⊤).IsInduced' | case mp.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
f : G ↪g H
a b : α
⊢ H.Adj (f.toHom a) (f.toHom b) → (Subgraph.map f.toHom ⊤).Adj (f.toHom a) (f.toHom b) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.isIndContained_iff_exists_subgraph | [133, 1] | [142, 52] | simp [Relation.map_apply_apply, f.injective] | case mp.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
f : G ↪g H
a b : α
⊢ H.Adj (f.toHom a) (f.toHom b) → (Subgraph.map f.toHom ⊤).Adj (f.toHom a) (f.toHom b) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.isIndContained_iff_exists_subgraph | [133, 1] | [142, 52] | rintro ⟨H', e, hH'⟩ | case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ (∃ H' _e, H'.IsInduced') → G ⊴ H | case mpr.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
H' : H.Subgraph
e : G ≃g H'.coe
hH' : H'.IsInduced'
⊢ G ⊴ H |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.isIndContained_iff_exists_subgraph | [133, 1] | [142, 52] | exact e.isIndContained.trans hH'.isIndContained | case mpr.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
H' : H.Subgraph
e : G ≃g H'.coe
hH' : H'.IsInduced'
⊢ G ⊴ H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | rw [copyCount] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ ⊥.copyCount H = 1 | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ (filter (fun H' => Nonempty (⊥ ≃g H'.coe)) univ).card = 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | convert
card_singleton
({ verts := Set.univ
Adj := ⊥
adj_sub := False.elim
edge_vert := False.elim } : H.Subgraph) | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ (filter (fun H' => Nonempty (⊥ ≃g H'.coe)) univ).card = 1 | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ filter (fun H' => Nonempty (⊥ ≃g H'.coe)) univ =
{{ verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | simp only [eq_singleton_iff_unique_mem, mem_filter, mem_univ, Subgraph.coe_bot, true_and_iff,
Nonempty.forall] | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ filter (fun H' => Nonempty (⊥ ≃g H'.coe)) univ =
{{ verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }} | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ Nonempty (⊥ ≃g { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }.coe) ∧
∀ (x : H.Subgraph), ⊥ ≃g x.coe → x = { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | refine'
⟨⟨⟨(Equiv.Set.univ _).symm, by
simp only [Prop.bot_eq_false, Subgraph.coe_adj, Pi.bot_apply, bot_adj, iff_self_iff,
forall₂_true_iff]⟩⟩, fun H' e ↦
Subgraph.ext _ _ ((set_fintype_card_eq_univ_iff _).1 $ Fintype.card_congr e.toEquiv.symm) _⟩ | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ Nonempty (⊥ ≃g { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }.coe) ∧
∀ (x : H.Subgraph), ⊥ ≃g x.coe → x = { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ } | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
H' : H.Subgraph
e : ⊥ ≃g H'.coe
⊢ H'.Adj = { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }.Adj |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | ext a b | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
H' : H.Subgraph
e : ⊥ ≃g H'.coe
⊢ H'.Adj = { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }.Adj | case h.e'_2.h.e'_2.h.h.a
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ H'.Adj a b ↔ { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }.Adj a b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | simp only [Prop.bot_eq_false, Pi.bot_apply, iff_false_iff] | case h.e'_2.h.e'_2.h.h.a
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ H'.Adj a b ↔ { verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }.Adj a b | case h.e'_2.h.e'_2.h.h.a
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ ¬H'.Adj a b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | exact fun hab ↦ e.symm.map_rel_iff.2 hab.coe | case h.e'_2.h.e'_2.h.h.a
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ ¬H'.Adj a b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_bot | [160, 1] | [177, 47] | simp only [Prop.bot_eq_false, Subgraph.coe_adj, Pi.bot_apply, bot_adj, iff_self_iff,
forall₂_true_iff] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
H : SimpleGraph β
⊢ ∀ {a b : β},
{ verts := Set.univ, Adj := ⊥, adj_sub := ⋯, edge_vert := ⋯, symm := ⋯ }.coe.Adj ((Equiv.Set.univ β).symm a)
((Equiv.Set.univ β).symm b) ↔
⊥.Adj a b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | rw [copyCount] | α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ G.copyCount H = 1 | α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ (filter (fun H' => Nonempty (G ≃g H'.coe)) univ).card = 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | convert card_singleton (⊥ : H.Subgraph) | α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ (filter (fun H' => Nonempty (G ≃g H'.coe)) univ).card = 1 | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ filter (fun H' => Nonempty (G ≃g H'.coe)) univ = {⊥} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | simp only [eq_singleton_iff_unique_mem, mem_filter, mem_univ, Subgraph.coe_bot, true_and_iff,
Nonempty.forall, Subsingleton.elim G ⊥] | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ filter (fun H' => Nonempty (G ≃g H'.coe)) univ = {⊥} | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ Nonempty (⊥ ≃g ⊥) ∧ ∀ (x : H.Subgraph), ⊥ ≃g x.coe → x = ⊥ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | haveI : IsEmpty (⊥ : H.Subgraph).verts := by simp | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ Nonempty (⊥ ≃g ⊥) ∧ ∀ (x : H.Subgraph), ⊥ ≃g x.coe → x = ⊥ | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
⊢ Nonempty (⊥ ≃g ⊥) ∧ ∀ (x : H.Subgraph), ⊥ ≃g x.coe → x = ⊥ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | refine' ⟨⟨⟨⟨isEmptyElim, isEmptyElim, isEmptyElim, isEmptyElim⟩, fun {a} ↦ isEmptyElim a⟩⟩,
fun H' e ↦ Subgraph.ext _ _ _ $ funext₂ fun a b ↦ _⟩ | case h.e'_2.h.e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
⊢ Nonempty (⊥ ≃g ⊥) ∧ ∀ (x : H.Subgraph), ⊥ ≃g x.coe → x = ⊥ | case h.e'_2.h.e'_2.refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
H' : H.Subgraph
e : ⊥ ≃g H'.coe
⊢ H'.verts = ⊥.verts
case h.e'_2.h.e'_2.refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ H'.Adj a b = ⊥.Adj a b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | simp | α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ IsEmpty ↑⊥.verts | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | simpa [Set.eq_empty_iff_forall_not_mem, filter_eq_empty_iff, ‹IsEmpty α›] using
e.toEquiv.symm.isEmpty_congr | case h.e'_2.h.e'_2.refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
H' : H.Subgraph
e : ⊥ ≃g H'.coe
⊢ H'.verts = ⊥.verts | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | simp only [Subgraph.not_bot_adj, eq_iff_iff, iff_false_iff] | case h.e'_2.h.e'_2.refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ H'.Adj a b = ⊥.Adj a b | case h.e'_2.h.e'_2.refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ ¬H'.Adj a b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_of_isEmpty | [179, 1] | [191, 49] | exact fun hab ↦ e.symm.map_rel_iff.2 hab.coe | case h.e'_2.h.e'_2.refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
this : IsEmpty ↑⊥.verts
H' : H.Subgraph
e : ⊥ ≃g H'.coe
a b : β
⊢ ¬H'.Adj a b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_eq_zero | [193, 1] | [194, 83] | simp [copyCount, isContained_iff_exists_subgraph, card_pos, filter_eq_empty_iff] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
⊢ G.copyCount H = 0 ↔ ¬G ⊑ H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_pos | [196, 1] | [197, 83] | simp [copyCount, isContained_iff_exists_subgraph, card_pos, filter_nonempty_iff] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝ : Fintype β
⊢ 0 < G.copyCount H ↔ G ⊑ H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.labelledCopyCount_of_isEmpty | [209, 1] | [212, 85] | convert Fintype.card_unique | α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝² : Fintype α
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ G.labelledCopyCount H = 1 | case convert_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝² : Fintype α
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ Unique { f // Injective ⇑f } |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.labelledCopyCount_of_isEmpty | [209, 1] | [212, 85] | exact { default := ⟨default, isEmptyElim⟩, uniq := fun _ ↦ Subsingleton.elim _ _ } | case convert_2
α : Type u_1
β : Type u_2
γ : Type u_3
G✝ G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
inst✝² : Fintype α
inst✝¹ : Fintype β
inst✝ : IsEmpty α
G : SimpleGraph α
H : SimpleGraph β
⊢ Unique { f // Injective ⇑f } | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.labelledCopyCount_eq_zero | [214, 1] | [215, 66] | simp [labelledCopyCount, IsContained, Fintype.card_eq_zero_iff] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype α
inst✝ : Fintype β
⊢ G.labelledCopyCount H = 0 ↔ ¬G ⊑ H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.labelledCopyCount_pos | [217, 1] | [218, 62] | simp [labelledCopyCount, IsContained, Fintype.card_pos_iff] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype α
inst✝ : Fintype β
⊢ 0 < G.labelledCopyCount H ↔ G ⊑ H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_le_labelledCopyCount | [220, 1] | [227, 8] | rw [copyCount, ← Fintype.card_coe] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype α
inst✝ : Fintype β
⊢ G.copyCount H ≤ G.labelledCopyCount H | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype α
inst✝ : Fintype β
⊢ Fintype.card { x // x ∈ filter (fun H' => Nonempty (G ≃g H'.coe)) univ } ≤ G.labelledCopyCount H |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_le_labelledCopyCount | [220, 1] | [227, 8] | refine Fintype.card_le_of_injective (fun H' ↦
⟨H'.val.hom.comp (mem_filter.1 H'.2).2.some.toHom,
Subtype.coe_injective.comp (mem_filter.1 H'.2).2.some.injective⟩) ?_ | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype α
inst✝ : Fintype β
⊢ Fintype.card { x // x ∈ filter (fun H' => Nonempty (G ≃g H'.coe)) univ } ≤ G.labelledCopyCount H | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype α
inst✝ : Fintype β
⊢ Injective fun H' => ⟨(↑H').hom.comp ⋯.some.toHom, ⋯⟩ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.copyCount_le_labelledCopyCount | [220, 1] | [227, 8] | sorry | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
inst✝¹ : Fintype α
inst✝ : Fintype β
⊢ Injective fun H' => ⟨(↑H').hom.comp ⋯.some.toHom, ⋯⟩ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.aux | [242, 1] | [246, 71] | obtain ⟨e, he⟩ := edgeSet_nonempty.2 hG | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : H.Subgraph
⊢ Nonempty (G ≃g H'.coe) → H'.edgeSet.Nonempty | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : H.Subgraph
e : Sym2 α
he : e ∈ G.edgeSet
⊢ Nonempty (G ≃g H'.coe) → H'.edgeSet.Nonempty |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.aux | [242, 1] | [246, 71] | rw [← Subgraph.image_coe_edgeSet_coe] | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : H.Subgraph
e : Sym2 α
he : e ∈ G.edgeSet
⊢ Nonempty (G ≃g H'.coe) → H'.edgeSet.Nonempty | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : H.Subgraph
e : Sym2 α
he : e ∈ G.edgeSet
⊢ Nonempty (G ≃g H'.coe) → (Sym2.map Subtype.val '' H'.coe.edgeSet).Nonempty |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.aux | [242, 1] | [246, 71] | exact fun ⟨f⟩ ↦ Set.Nonempty.image _ ⟨_, f.map_mem_edgeSet_iff.2 he⟩ | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : H.Subgraph
e : Sym2 α
he : e ∈ G.edgeSet
⊢ Nonempty (G ≃g H'.coe) → (Sym2.map Subtype.val '' H'.coe.edgeSet).Nonempty | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_le | [255, 1] | [256, 91] | rw [kill] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ G.kill H ≤ H | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ (if hG : G = ⊥ then H else H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})) ≤ H |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_le | [255, 1] | [256, 91] | split_ifs | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
⊢ (if hG : G = ⊥ then H else H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})) ≤ H | case pos
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
h✝ : G = ⊥
⊢ H ≤ H
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
h✝ : ¬G = ⊥
⊢ H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some}) ≤ H |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_le | [255, 1] | [256, 91] | exacts [le_rfl, deleteEdges_le _] | case pos
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
h✝ : G = ⊥
⊢ H ≤ H
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
h✝ : ¬G = ⊥
⊢ H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some}) ≤ H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.bot_kill | [258, 1] | [259, 31] | rw [kill] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
H : SimpleGraph β
⊢ ⊥.kill H = H | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
H : SimpleGraph β
⊢ (if hG : ⊥ = ⊥ then H else H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (⊥ ≃g H'.coe)), {⋯.some})) = H |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.bot_kill | [258, 1] | [259, 31] | exact dif_pos rfl | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
H : SimpleGraph β
⊢ (if hG : ⊥ = ⊥ then H else H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (⊥ ≃g H'.coe)), {⋯.some})) = H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_of_ne_bot | [261, 1] | [264, 30] | rw [kill] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H : SimpleGraph β
⊢ G.kill H = H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some}) | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H : SimpleGraph β
⊢ (if hG : G = ⊥ then H else H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})) =
H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some}) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_of_ne_bot | [261, 1] | [264, 30] | exact dif_neg hG | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H✝ : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H : SimpleGraph β
⊢ (if hG : G = ⊥ then H else H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})) =
H.deleteEdges (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some}) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_eq_right | [266, 1] | [271, 87] | simp only [kill_of_ne_bot hG, Set.disjoint_left, isContained_iff_exists_subgraph,
@forall_swap _ H.Subgraph, Set.iUnion_singleton_eq_range, deleteEdges_eq_self, Set.mem_iUnion,
Set.mem_range, not_exists, not_nonempty_iff, Nonempty.forall] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
⊢ G.kill H = H ↔ ¬G ⊑ H | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
⊢ (∀ (y : H.Subgraph), ∀ x ∈ H.edgeSet, ∀ (a : G ≃g y.coe), x ∉ {⋯.some}) ↔ ∀ (x : H.Subgraph), IsEmpty (G ≃g x.coe) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_eq_right | [266, 1] | [271, 87] | exact forall_congr' fun H' ↦ ⟨fun h ↦ ⟨fun f ↦ h _
(Subgraph.edgeSet_subset _ $ (aux hG ⟨f⟩).choose_spec) f rfl⟩, fun h _ _ ↦ h.elim⟩ | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
⊢ (∀ (y : H.Subgraph), ∀ x ∈ H.edgeSet, ∀ (a : G ≃g y.coe), x ∉ {⋯.some}) ↔ ∀ (x : H.Subgraph), IsEmpty (G ≃g x.coe) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_of_not_isContained | [273, 1] | [276, 35] | obtain rfl | hG := eq_or_ne G ⊥ | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hGH : ¬G ⊑ H
⊢ G.kill H = H | case inl
α : Type u_1
β : Type u_2
γ : Type u_3
G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hGH : ¬⊥ ⊑ H
⊢ ⊥.kill H = H
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hGH : ¬G ⊑ H
hG : G ≠ ⊥
⊢ G.kill H = H |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_of_not_isContained | [273, 1] | [276, 35] | exact bot_kill _ | case inl
α : Type u_1
β : Type u_2
γ : Type u_3
G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hGH : ¬⊥ ⊑ H
⊢ ⊥.kill H = H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.kill_of_not_isContained | [273, 1] | [276, 35] | exact (kill_eq_right hG).2 hGH | case inr
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hGH : ¬G ⊑ H
hG : G ≠ ⊥
⊢ G.kill H = H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | rw [kill_of_ne_bot hG, deleteEdges, isContained_iff_exists_subgraph] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
⊢ ¬G ⊑ G.kill H | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
⊢ ¬∃ H', Nonempty (G ≃g H'.coe) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | rintro ⟨H', hGH'⟩ | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
⊢ ¬∃ H', Nonempty (G ≃g H'.coe) | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | have hH' : (H'.map $ Hom.ofLE (sdiff_le : H \ _ ≤ H)).edgeSet.Nonempty := by
rw [Subgraph.edgeSet_map]
exact (aux hG hGH').image _ | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
⊢ False | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | set e := hH'.some with he | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
⊢ False | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β := hH'.some
he : e = hH'.some
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | have : e ∈ _ := hH'.some_mem | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β := hH'.some
he : e = hH'.some
⊢ False | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β := hH'.some
he : e = hH'.some
this : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').edgeSet
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | clear_value e | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β := hH'.some
he : e = hH'.some
this : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').edgeSet
⊢ False | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β
he : e = hH'.some
this : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').edgeSet
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | rw [← Subgraph.image_coe_edgeSet_coe] at this | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β
he : e = hH'.some
this : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').edgeSet
⊢ False | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β
he : e = hH'.some
this : e ∈ Sym2.map Subtype.val '' (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | subst he | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 β
he : e = hH'.some
this : e ∈ Sym2.map Subtype.val '' (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
⊢ False | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
this : hH'.some ∈ Sym2.map Subtype.val '' (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | obtain ⟨e, he₀, he₁⟩ := this | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
this : hH'.some ∈ Sym2.map Subtype.val '' (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
⊢ False | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | let e' : Sym2 H'.verts := Sym2.map (Subgraph.isoMap (Hom.ofLE _) injective_id _).symm e | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
⊢ False | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | have he' : e' ∈ H'.coe.edgeSet := (Iso.map_mem_edgeSet_iff _).2 he₀ | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
⊢ False | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ H'.coe.edgeSet
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | rw [Subgraph.edgeSet_coe] at he' | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ H'.coe.edgeSet
⊢ False | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | have := Subgraph.edgeSet_subset _ he' | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
⊢ False | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this : Sym2.map Subtype.val e' ∈ (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).edgeSet
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | simp only [edgeSet_sdiff, edgeSet_fromEdgeSet, edgeSet_sdiff_sdiff_isDiag, Set.mem_diff,
Set.mem_iUnion, not_exists] at this | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this : Sym2.map Subtype.val e' ∈ (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).edgeSet
⊢ False | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | refine' this.2 (H'.map $ Hom.ofLE sdiff_le)
⟨(Subgraph.isoMap (Hom.ofLE _) injective_id _).comp hGH'.some⟩ _ | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
⊢ False | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
⊢ Sym2.map Subtype.val e' ∈ {⋯.some} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | rw [Sym2.map_map, Set.mem_singleton_iff, ← he₁] | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
⊢ Sym2.map Subtype.val e' ∈ {⋯.some} | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
⊢ Sym2.map (Subtype.val ∘ ⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e = Sym2.map Subtype.val e |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | congr 1 with x | case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
⊢ Sym2.map (Subtype.val ∘ ⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e = Sym2.map Subtype.val e | case intro.intro.intro.e_f.h
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
x : ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
⊢ (Subtype.val ∘ ⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) x = ↑x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | refine' congr_arg (↑) (Equiv.Set.image_symm_apply _ _ injective_id _ _) | case intro.intro.intro.e_f.h
α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
hH' : (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty
e : Sym2 ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
he₀ : e ∈ (Subgraph.map (Hom.ofLE ⋯) H').coe.edgeSet
he₁ : Sym2.map Subtype.val e = hH'.some
e' : Sym2 ↑H'.verts := Sym2.map (⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) e
he' : e' ∈ Sym2.map Subtype.val ⁻¹' H'.edgeSet
this :
Sym2.map Subtype.val e' ∈ H.edgeSet ∧
∀ (x : H.Subgraph) (x_1 : Nonempty (G ≃g x.coe)), Sym2.map Subtype.val e' ∉ {⋯.some}
x : ↑(Subgraph.map (Hom.ofLE ⋯) H').verts
⊢ (Subtype.val ∘ ⇑(Subgraph.isoMap (Hom.ofLE ⋯) ⋯ H').symm) x = ↑x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Containment.lean | SimpleGraph.not_isContained_kill | [278, 1] | [302, 74] | rw [Subgraph.edgeSet_map] | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
⊢ (Subgraph.map (Hom.ofLE ⋯) H').edgeSet.Nonempty | α : Type u_1
β : Type u_2
γ : Type u_3
G G₁ G₂ G₃ : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
hG : G ≠ ⊥
H' : (H \ fromEdgeSet (⋃ H', ⋃ (hH' : Nonempty (G ≃g H'.coe)), {⋯.some})).Subgraph
hGH' : Nonempty (G ≃g H'.coe)
⊢ (Sym2.map ⇑(Hom.ofLE ⋯) '' H'.edgeSet).Nonempty |
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