url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_succ_eq | [2042, 1] | [2054, 10] | rintro rfl | case h.cons.mp
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : ℕ
w : V
huw : G.Adj u w
pwv : Walk G w v
⊢ Walk.length pwv = n → ∃ i h x, Walk.length x = n ∧ Walk.cons (_ : G.Adj u i) x = Walk.cons huw pwv | case h.cons.mp
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
huw : G.Adj u w
pwv : Walk G w v
⊢ ∃ i h x, Walk.length x = Walk.length pwv ∧ Walk.cons (_ : G.Adj u i) x = Walk.cons huw pwv |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_succ_eq | [2042, 1] | [2054, 10] | exact ⟨w, huw, pwv, rfl, rfl⟩ | case h.cons.mp
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
huw : G.Adj u w
pwv : Walk G w v
⊢ ∃ i h x, Walk.length x = Walk.length pwv ∧ Walk.cons (_ : G.Adj u i) x = Walk.cons huw pwv | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_succ_eq | [2042, 1] | [2054, 10] | rintro ⟨w, huw, pwv, rfl, rfl, rfl⟩ | case h.cons.mpr
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : ℕ
w : V
huw : G.Adj u w
pwv : Walk G w v
⊢ (∃ i h x, Walk.length x = n ∧ Walk.cons (_ : G.Adj u i) x = Walk.cons huw pwv) → Walk.length pwv = n | case h.cons.mpr.intro.intro.intro.intro.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
huw✝ : G.Adj u w
pwv : Walk G w v
huw : G.Adj u w
⊢ Walk.length pwv = Walk.length pwv |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphConnectivity.lean | Digraph.set_walk_length_succ_eq | [2042, 1] | [2054, 10] | rfl | case h.cons.mpr.intro.intro.intro.intro.refl
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v w : V
huw✝ : G.Adj u w
pwv : Walk G w v
huw : G.Adj u w
⊢ Walk.length pwv = Walk.length pwv | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.iSup_adj | [110, 1] | [110, 97] | simp [iSup] | ι : Sort u_1
𝕜 : Type ?u.10671
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : ι → Digraph V
⊢ (⨆ (i : ι), f i).Adj a b ↔ ∃ i, (f i).Adj a b | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.iInf_adj | [113, 1] | [114, 14] | simp [iInf] | ι : Sort u_1
𝕜 : Type ?u.13166
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : ι → Digraph V
⊢ (⨅ (i : ι), f i).Adj a b ↔ ∀ (i : ι), (f i).Adj a b | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.edgeSet_bot | [257, 1] | [259, 7] | rw [Set.eq_empty_iff_forall_not_mem] | ι : Sort ?u.51631
𝕜 : Type ?u.51634
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
G₁ G₂ : Digraph V
⊢ edgeSet ⊥ = ∅ | ι : Sort ?u.51631
𝕜 : Type ?u.51634
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
G₁ G₂ : Digraph V
⊢ ∀ (x : V × V), ¬x ∈ edgeSet ⊥ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.edgeSet_bot | [257, 1] | [259, 7] | simp | ι : Sort ?u.51631
𝕜 : Type ?u.51634
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
G₁ G₂ : Digraph V
⊢ ∀ (x : V × V), ¬x ∈ edgeSet ⊥ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.edgeSet_sup | [262, 1] | [264, 6] | ext ⟨x, y⟩ | ι : Sort ?u.54704
𝕜 : Type ?u.54707
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
G₁ G₂ : Digraph V
⊢ edgeSet (G₁ ⊔ G₂) = edgeSet G₁ ∪ edgeSet G₂ | case h.mk
ι : Sort ?u.54704
𝕜 : Type ?u.54707
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
G₁ G₂ : Digraph V
x y : V
⊢ (x, y) ∈ edgeSet (G₁ ⊔ G₂) ↔ (x, y) ∈ edgeSet G₁ ∪ edgeSet G₂ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.edgeSet_sup | [262, 1] | [264, 6] | rfl | case h.mk
ι : Sort ?u.54704
𝕜 : Type ?u.54707
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
G₁ G₂ : Digraph V
x y : V
⊢ (x, y) ∈ edgeSet (G₁ ⊔ G₂) ↔ (x, y) ∈ edgeSet G₁ ∪ edgeSet G₂ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.edgeSet_inf | [267, 1] | [269, 6] | ext ⟨x, y⟩ | ι : Sort ?u.54886
𝕜 : Type ?u.54889
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
G₁ G₂ : Digraph V
⊢ edgeSet (G₁ ⊓ G₂) = edgeSet G₁ ∩ edgeSet G₂ | case h.mk
ι : Sort ?u.54886
𝕜 : Type ?u.54889
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
G₁ G₂ : Digraph V
x y : V
⊢ (x, y) ∈ edgeSet (G₁ ⊓ G₂) ↔ (x, y) ∈ edgeSet G₁ ∩ edgeSet G₂ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.edgeSet_inf | [267, 1] | [269, 6] | rfl | case h.mk
ι : Sort ?u.54886
𝕜 : Type ?u.54889
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
G₁ G₂ : Digraph V
x y : V
⊢ (x, y) ∈ edgeSet (G₁ ⊓ G₂) ↔ (x, y) ∈ edgeSet G₁ ∩ edgeSet G₂ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.edgeSet_sdiff | [272, 1] | [274, 6] | ext ⟨x, y⟩ | ι : Sort ?u.55069
𝕜 : Type ?u.55072
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
G₁ G₂ : Digraph V
⊢ edgeSet (G₁ \ G₂) = edgeSet G₁ \ edgeSet G₂ | case h.mk
ι : Sort ?u.55069
𝕜 : Type ?u.55072
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
G₁ G₂ : Digraph V
x y : V
⊢ (x, y) ∈ edgeSet (G₁ \ G₂) ↔ (x, y) ∈ edgeSet G₁ \ edgeSet G₂ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.edgeSet_sdiff | [272, 1] | [274, 6] | rfl | case h.mk
ι : Sort ?u.55069
𝕜 : Type ?u.55072
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
G₁ G₂ : Digraph V
x y : V
⊢ (x, y) ∈ edgeSet (G₁ \ G₂) ↔ (x, y) ∈ edgeSet G₁ \ edgeSet G₂ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.sdiff_eq_deleteEdges | [319, 1] | [321, 7] | ext | ι : Sort ?u.60271
𝕜 : Type ?u.60274
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G'✝ : Digraph W
a b c u v w : V
G G' : Digraph V
⊢ G \ G' = deleteEdges G (edgeSet G') | case Adj.h.h.a
ι : Sort ?u.60271
𝕜 : Type ?u.60274
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G'✝ : Digraph W
a b c u v w : V
G G' : Digraph V
x✝¹ x✝ : V
⊢ (G \ G').Adj x✝¹ x✝ ↔ (deleteEdges G (edgeSet G')).Adj x✝¹ x✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.sdiff_eq_deleteEdges | [319, 1] | [321, 7] | simp | case Adj.h.h.a
ι : Sort ?u.60271
𝕜 : Type ?u.60274
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G'✝ : Digraph W
a b c u v w : V
G G' : Digraph V
x✝¹ x✝ : V
⊢ (G \ G').Adj x✝¹ x✝ ↔ (deleteEdges G (edgeSet G')).Adj x✝¹ x✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.compl_eq_deleteEdges | [323, 1] | [325, 7] | ext | ι : Sort ?u.60648
𝕜 : Type ?u.60651
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
⊢ Gᶜ = deleteEdges ⊤ (edgeSet G) | case Adj.h.h.a
ι : Sort ?u.60648
𝕜 : Type ?u.60651
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w x✝¹ x✝ : V
⊢ Gᶜ.Adj x✝¹ x✝ ↔ (deleteEdges ⊤ (edgeSet G)).Adj x✝¹ x✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.compl_eq_deleteEdges | [323, 1] | [325, 7] | simp | case Adj.h.h.a
ι : Sort ?u.60648
𝕜 : Type ?u.60651
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w x✝¹ x✝ : V
⊢ Gᶜ.Adj x✝¹ x✝ ↔ (deleteEdges ⊤ (edgeSet G)).Adj x✝¹ x✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.deleteEdges_deleteEdges | [329, 1] | [332, 27] | ext | ι : Sort ?u.61070
𝕜 : Type ?u.61073
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
s s' : Set (V × V)
⊢ deleteEdges (deleteEdges G s) s' = deleteEdges G (s ∪ s') | case Adj.h.h.a
ι : Sort ?u.61070
𝕜 : Type ?u.61073
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
s s' : Set (V × V)
x✝¹ x✝ : V
⊢ (deleteEdges (deleteEdges G s) s').Adj x✝¹ x✝ ↔ (deleteEdges G (s ∪ s')).Adj x✝¹ x✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.deleteEdges_deleteEdges | [329, 1] | [332, 27] | simp [and_assoc, not_or] | case Adj.h.h.a
ι : Sort ?u.61070
𝕜 : Type ?u.61073
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
s s' : Set (V × V)
x✝¹ x✝ : V
⊢ (deleteEdges (deleteEdges G s) s').Adj x✝¹ x✝ ↔ (deleteEdges G (s ∪ s')).Adj x✝¹ x✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.deleteEdges_empty_eq | [335, 1] | [337, 7] | ext | ι : Sort ?u.61543
𝕜 : Type ?u.61546
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
⊢ deleteEdges G ∅ = G | case Adj.h.h.a
ι : Sort ?u.61543
𝕜 : Type ?u.61546
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w x✝¹ x✝ : V
⊢ (deleteEdges G ∅).Adj x✝¹ x✝ ↔ G.Adj x✝¹ x✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.deleteEdges_empty_eq | [335, 1] | [337, 7] | simp | case Adj.h.h.a
ι : Sort ?u.61543
𝕜 : Type ?u.61546
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w x✝¹ x✝ : V
⊢ (deleteEdges G ∅).Adj x✝¹ x✝ ↔ G.Adj x✝¹ x✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.deleteEdges_univ_eq | [340, 1] | [342, 7] | ext | ι : Sort ?u.61865
𝕜 : Type ?u.61868
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
⊢ deleteEdges G Set.univ = ⊥ | case Adj.h.h.a
ι : Sort ?u.61865
𝕜 : Type ?u.61868
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w x✝¹ x✝ : V
⊢ (deleteEdges G Set.univ).Adj x✝¹ x✝ ↔ ⊥.Adj x✝¹ x✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.deleteEdges_univ_eq | [340, 1] | [342, 7] | simp | case Adj.h.h.a
ι : Sort ?u.61865
𝕜 : Type ?u.61868
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w x✝¹ x✝ : V
⊢ (deleteEdges G Set.univ).Adj x✝¹ x✝ ↔ ⊥.Adj x✝¹ x✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.deleteEdges_le | [344, 1] | [346, 42] | intro | ι : Sort ?u.62316
𝕜 : Type ?u.62319
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
s : Set (V × V)
⊢ deleteEdges G s ≤ G | ι : Sort ?u.62316
𝕜 : Type ?u.62319
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
s : Set (V × V)
v✝ : V
⊢ ∀ ⦃w : V⦄, (deleteEdges G s).Adj v✝ w → G.Adj v✝ w |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.deleteEdges_le | [344, 1] | [346, 42] | simp (config := { contextual := true }) | ι : Sort ?u.62316
𝕜 : Type ?u.62319
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
s : Set (V × V)
v✝ : V
⊢ ∀ ⦃w : V⦄, (deleteEdges G s).Adj v✝ w → G.Adj v✝ w | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.deleteEdges_le_of_le | [349, 1] | [352, 33] | simp (config := { contextual := true }) only [deleteEdges_adj, and_imp, true_and_iff] | ι : Sort ?u.64664
𝕜 : Type ?u.64667
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v✝ w✝ : V
s s' : Set (V × V)
h : s ⊆ s'
v w : V
⊢ (deleteEdges G s').Adj v w → (deleteEdges G s).Adj v w | ι : Sort ?u.64664
𝕜 : Type ?u.64667
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v✝ w✝ : V
s s' : Set (V × V)
h : s ⊆ s'
v w : V
⊢ G.Adj v w → ¬(v, w) ∈ s' → ¬(v, w) ∈ s |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.deleteEdges_le_of_le | [349, 1] | [352, 33] | exact fun _ hn hs => hn (h hs) | ι : Sort ?u.64664
𝕜 : Type ?u.64667
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v✝ w✝ : V
s s' : Set (V × V)
h : s ⊆ s'
v w : V
⊢ G.Adj v w → ¬(v, w) ∈ s' → ¬(v, w) ∈ s | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.deleteEdges_eq_inter_edgeSet | [354, 1] | [357, 54] | ext | ι : Sort ?u.65081
𝕜 : Type ?u.65084
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
s : Set (V × V)
⊢ deleteEdges G s = deleteEdges G (s ∩ edgeSet G) | case Adj.h.h.a
ι : Sort ?u.65081
𝕜 : Type ?u.65084
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
s : Set (V × V)
x✝¹ x✝ : V
⊢ (deleteEdges G s).Adj x✝¹ x✝ ↔ (deleteEdges G (s ∩ edgeSet G)).Adj x✝¹ x✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.deleteEdges_eq_inter_edgeSet | [354, 1] | [357, 54] | simp (config := { contextual := true }) [imp_false] | case Adj.h.h.a
ι : Sort ?u.65081
𝕜 : Type ?u.65084
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
s : Set (V × V)
x✝¹ x✝ : V
⊢ (deleteEdges G s).Adj x✝¹ x✝ ↔ (deleteEdges G (s ∩ edgeSet G)).Adj x✝¹ x✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.deleteEdges_sdiff_eq_of_le | [359, 1] | [362, 67] | ext (v w) | ι : Sort ?u.66345
𝕜 : Type ?u.66348
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
H : Digraph V
h : H ≤ G
⊢ deleteEdges G (edgeSet G \ edgeSet H) = H | case Adj.h.h.a
ι : Sort ?u.66345
𝕜 : Type ?u.66348
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v✝ w✝ : V
H : Digraph V
h : H ≤ G
v w : V
⊢ (deleteEdges G (edgeSet G \ edgeSet H)).Adj v w ↔ H.Adj v w |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.deleteEdges_sdiff_eq_of_le | [359, 1] | [362, 67] | constructor <;> simp (config := { contextual := true }) [@h v w] | case Adj.h.h.a
ι : Sort ?u.66345
𝕜 : Type ?u.66348
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v✝ w✝ : V
H : Digraph V
h : H ≤ G
v w : V
⊢ (deleteEdges G (edgeSet G \ edgeSet H)).Adj v w ↔ H.Adj v w | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.edgeSet_deleteEdges | [364, 1] | [367, 7] | ext e | ι : Sort ?u.68599
𝕜 : Type ?u.68602
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
s : Set (V × V)
⊢ edgeSet (deleteEdges G s) = edgeSet G \ s | case h
ι : Sort ?u.68599
𝕜 : Type ?u.68602
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
s : Set (V × V)
e : V × V
⊢ e ∈ edgeSet (deleteEdges G s) ↔ e ∈ edgeSet G \ s |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.edgeSet_deleteEdges | [364, 1] | [367, 7] | cases e | case h
ι : Sort ?u.68599
𝕜 : Type ?u.68602
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
s : Set (V × V)
e : V × V
⊢ e ∈ edgeSet (deleteEdges G s) ↔ e ∈ edgeSet G \ s | case h.mk
ι : Sort ?u.68599
𝕜 : Type ?u.68602
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
s : Set (V × V)
fst✝ snd✝ : V
⊢ (fst✝, snd✝) ∈ edgeSet (deleteEdges G s) ↔ (fst✝, snd✝) ∈ edgeSet G \ s |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.edgeSet_deleteEdges | [364, 1] | [367, 7] | simp | case h.mk
ι : Sort ?u.68599
𝕜 : Type ?u.68602
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
s : Set (V × V)
fst✝ snd✝ : V
⊢ (fst✝, snd✝) ∈ edgeSet (deleteEdges G s) ↔ (fst✝, snd✝) ∈ edgeSet G \ s | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.map_monotone | [385, 1] | [387, 31] | rintro G G' h _ _ ⟨u, v, ha, rfl, rfl⟩ | ι : Sort ?u.69221
𝕜 : Type ?u.69224
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : V → W
⊢ Monotone (Digraph.map f) | case intro.intro.intro.intro
ι : Sort ?u.69221
𝕜 : Type ?u.69224
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G'✝ : Digraph W
a b c u✝ v✝ w : V
f : V → W
G G' : Digraph V
h : G ≤ G'
u v : V
ha : G.Adj u v
⊢ (Digraph.map f G').Adj (f u) (f v) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.map_monotone | [385, 1] | [387, 31] | exact ⟨_, _, h ha, rfl, rfl⟩ | case intro.intro.intro.intro
ι : Sort ?u.69221
𝕜 : Type ?u.69224
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G'✝ : Digraph W
a b c u✝ v✝ w : V
f : V → W
G G' : Digraph V
h : G ≤ G'
u v : V
ha : G.Adj u v
⊢ (Digraph.map f G').Adj (f u) (f v) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.comap_monotone | [398, 1] | [400, 13] | intro G G' h _ _ ha | ι : Sort ?u.70267
𝕜 : Type ?u.70270
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : V → W
⊢ Monotone (Digraph.comap f) | ι : Sort ?u.70267
𝕜 : Type ?u.70270
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G'✝ : Digraph W
a b c u v w : V
f : V → W
G G' : Digraph W
h : G ≤ G'
v✝ w✝ : V
ha : (Digraph.comap f G).Adj v✝ w✝
⊢ (Digraph.comap f G').Adj v✝ w✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.comap_monotone | [398, 1] | [400, 13] | exact h ha | ι : Sort ?u.70267
𝕜 : Type ?u.70270
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G'✝ : Digraph W
a b c u v w : V
f : V → W
G G' : Digraph W
h : G ≤ G'
v✝ w✝ : V
ha : (Digraph.comap f G).Adj v✝ w✝
⊢ (Digraph.comap f G').Adj v✝ w✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.comap_map_eq | [403, 1] | [405, 7] | ext | ι : Sort ?u.70997
𝕜 : Type ?u.71000
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G' : Digraph W
a b c u v w : V
f : V ↪ W
G : Digraph V
⊢ Digraph.comap (↑f) (Digraph.map (↑f) G) = G | case Adj.h.h.a
ι : Sort ?u.70997
𝕜 : Type ?u.71000
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G' : Digraph W
a b c u v w : V
f : V ↪ W
G : Digraph V
x✝¹ x✝ : V
⊢ (Digraph.comap (↑f) (Digraph.map (↑f) G)).Adj x✝¹ x✝ ↔ G.Adj x✝¹ x✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.comap_map_eq | [403, 1] | [405, 7] | simp | case Adj.h.h.a
ι : Sort ?u.70997
𝕜 : Type ?u.71000
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G' : Digraph W
a b c u v w : V
f : V ↪ W
G : Digraph V
x✝¹ x✝ : V
⊢ (Digraph.comap (↑f) (Digraph.map (↑f) G)).Adj x✝¹ x✝ ↔ G.Adj x✝¹ x✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.map_le_iff_le_comap | [417, 1] | [421, 16] | rintro h _ _ ⟨u, v, ha, rfl, rfl⟩ | ι : Sort ?u.118218
𝕜 : Type ?u.118221
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G'✝ : Digraph W
a b c u v w : V
f : V ↪ W
G : Digraph V
G' : Digraph W
⊢ G ≤ Digraph.comap (↑f) G' → Digraph.map (↑f) G ≤ G' | case intro.intro.intro.intro
ι : Sort ?u.118218
𝕜 : Type ?u.118221
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G'✝ : Digraph W
a b c u✝ v✝ w : V
f : V ↪ W
G : Digraph V
G' : Digraph W
h : G ≤ Digraph.comap (↑f) G'
u v : V
ha : G.Adj u v
⊢ G'.Adj (↑f u) (↑f v) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.map_le_iff_le_comap | [417, 1] | [421, 16] | exact h ha | case intro.intro.intro.intro
ι : Sort ?u.118218
𝕜 : Type ?u.118221
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G'✝ : Digraph W
a b c u✝ v✝ w : V
f : V ↪ W
G : Digraph V
G' : Digraph W
h : G ≤ Digraph.comap (↑f) G'
u v : V
ha : G.Adj u v
⊢ G'.Adj (↑f u) (↑f v) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.map_comap_le | [423, 1] | [424, 27] | rw [map_le_iff_le_comap] | ι : Sort ?u.133674
𝕜 : Type ?u.133677
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G' : Digraph W
a b c u v w : V
f : V ↪ W
G : Digraph W
⊢ Digraph.map (↑f) (Digraph.comap (↑f) G) ≤ G | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Hom.map_mem_edgeSet | [499, 1] | [501, 21] | cases e | ι : Sort ?u.200863
𝕜 : Type ?u.200866
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
e : V × V
h : e ∈ edgeSet G
⊢ Prod.map (↑f) (↑f) e ∈ edgeSet G' | case mk
ι : Sort ?u.200863
𝕜 : Type ?u.200866
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
fst✝ snd✝ : V
h : (fst✝, snd✝) ∈ edgeSet G
⊢ Prod.map ↑f ↑f (fst✝, snd✝) ∈ edgeSet G' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Hom.map_mem_edgeSet | [499, 1] | [501, 21] | exact f.map_rel' h | case mk
ι : Sort ?u.200863
𝕜 : Type ?u.200866
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
fst✝ snd✝ : V
h : (fst✝, snd✝) ∈ edgeSet G
⊢ Prod.map ↑f ↑f (fst✝, snd✝) ∈ edgeSet G' | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Hom.mapEdgeSet.injective | [516, 1] | [524, 26] | rintro ⟨e₁, h₁⟩ ⟨e₂, h₂⟩ | ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
⊢ Injective (mapEdgeSet f) | case mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
e₁ : V × V
h₁ : e₁ ∈ edgeSet G
e₂ : V × V
h₂ : e₂ ∈ edgeSet G
⊢ mapEdgeSet f { val := e₁, property := h₁ } = mapEdgeSet f { val := e₂, property := h₂ } →
{ val := e₁, property := h₁ } = { val := e₂, property := h₂ } |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Hom.mapEdgeSet.injective | [516, 1] | [524, 26] | dsimp [Hom.mapEdgeSet] | case mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
e₁ : V × V
h₁ : e₁ ∈ edgeSet G
e₂ : V × V
h₂ : e₂ ∈ edgeSet G
⊢ mapEdgeSet f { val := e₁, property := h₁ } = mapEdgeSet f { val := e₂, property := h₂ } →
{ val := e₁, property := h₁ } = { val := e₂, property := h₂ } | case mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
e₁ : V × V
h₁ : e₁ ∈ edgeSet G
e₂ : V × V
h₂ : e₂ ∈ edgeSet G
⊢ { val := (↑f e₁.fst, ↑f e₁.snd), property := (_ : Prod.map ↑f ↑f ↑{ val := e₁, property := h₁ } ∈ edgeSet G') } =
{ val := (↑f e₂.fst, ↑f e₂.snd), property := (_ : Prod.map ↑f ↑f ↑{ val := e₂, property := h₂ } ∈ edgeSet G') } →
{ val := e₁, property := h₁ } = { val := e₂, property := h₂ } |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Hom.mapEdgeSet.injective | [516, 1] | [524, 26] | repeat' rw [Subtype.mk_eq_mk] | case mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
e₁ : V × V
h₁ : e₁ ∈ edgeSet G
e₂ : V × V
h₂ : e₂ ∈ edgeSet G
⊢ { val := (↑f e₁.fst, ↑f e₁.snd), property := (_ : Prod.map ↑f ↑f ↑{ val := e₁, property := h₁ } ∈ edgeSet G') } =
{ val := (↑f e₂.fst, ↑f e₂.snd), property := (_ : Prod.map ↑f ↑f ↑{ val := e₂, property := h₂ } ∈ edgeSet G') } →
{ val := e₁, property := h₁ } = { val := e₂, property := h₂ } | case mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
e₁ : V × V
h₁ : e₁ ∈ edgeSet G
e₂ : V × V
h₂ : e₂ ∈ edgeSet G
⊢ (↑f e₁.fst, ↑f e₁.snd) = (↑f e₂.fst, ↑f e₂.snd) → e₁ = e₂ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Hom.mapEdgeSet.injective | [516, 1] | [524, 26] | cases e₁ | case mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
e₁ : V × V
h₁ : e₁ ∈ edgeSet G
e₂ : V × V
h₂ : e₂ ∈ edgeSet G
⊢ (↑f e₁.fst, ↑f e₁.snd) = (↑f e₂.fst, ↑f e₂.snd) → e₁ = e₂ | case mk.mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
e₂ : V × V
h₂ : e₂ ∈ edgeSet G
fst✝ snd✝ : V
h₁ : (fst✝, snd✝) ∈ edgeSet G
⊢ (↑f (fst✝, snd✝).fst, ↑f (fst✝, snd✝).snd) = (↑f e₂.fst, ↑f e₂.snd) → (fst✝, snd✝) = e₂ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Hom.mapEdgeSet.injective | [516, 1] | [524, 26] | cases e₂ | case mk.mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
e₂ : V × V
h₂ : e₂ ∈ edgeSet G
fst✝ snd✝ : V
h₁ : (fst✝, snd✝) ∈ edgeSet G
⊢ (↑f (fst✝, snd✝).fst, ↑f (fst✝, snd✝).snd) = (↑f e₂.fst, ↑f e₂.snd) → (fst✝, snd✝) = e₂ | case mk.mk.mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
fst✝¹ snd✝¹ : V
h₁ : (fst✝¹, snd✝¹) ∈ edgeSet G
fst✝ snd✝ : V
h₂ : (fst✝, snd✝) ∈ edgeSet G
⊢ (↑f (fst✝¹, snd✝¹).fst, ↑f (fst✝¹, snd✝¹).snd) = (↑f (fst✝, snd✝).fst, ↑f (fst✝, snd✝).snd) →
(fst✝¹, snd✝¹) = (fst✝, snd✝) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Hom.mapEdgeSet.injective | [516, 1] | [524, 26] | simp | case mk.mk.mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
fst✝¹ snd✝¹ : V
h₁ : (fst✝¹, snd✝¹) ∈ edgeSet G
fst✝ snd✝ : V
h₂ : (fst✝, snd✝) ∈ edgeSet G
⊢ (↑f (fst✝¹, snd✝¹).fst, ↑f (fst✝¹, snd✝¹).snd) = (↑f (fst✝, snd✝).fst, ↑f (fst✝, snd✝).snd) →
(fst✝¹, snd✝¹) = (fst✝, snd✝) | case mk.mk.mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
fst✝¹ snd✝¹ : V
h₁ : (fst✝¹, snd✝¹) ∈ edgeSet G
fst✝ snd✝ : V
h₂ : (fst✝, snd✝) ∈ edgeSet G
⊢ ↑f fst✝¹ = ↑f fst✝ → ↑f snd✝¹ = ↑f snd✝ → fst✝¹ = fst✝ ∧ snd✝¹ = snd✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Hom.mapEdgeSet.injective | [516, 1] | [524, 26] | intro h1 h2 | case mk.mk.mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
fst✝¹ snd✝¹ : V
h₁ : (fst✝¹, snd✝¹) ∈ edgeSet G
fst✝ snd✝ : V
h₂ : (fst✝, snd✝) ∈ edgeSet G
⊢ ↑f fst✝¹ = ↑f fst✝ → ↑f snd✝¹ = ↑f snd✝ → fst✝¹ = fst✝ ∧ snd✝¹ = snd✝ | case mk.mk.mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
fst✝¹ snd✝¹ : V
h₁ : (fst✝¹, snd✝¹) ∈ edgeSet G
fst✝ snd✝ : V
h₂ : (fst✝, snd✝) ∈ edgeSet G
h1 : ↑f fst✝¹ = ↑f fst✝
h2 : ↑f snd✝¹ = ↑f snd✝
⊢ fst✝¹ = fst✝ ∧ snd✝¹ = snd✝ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Hom.mapEdgeSet.injective | [516, 1] | [524, 26] | simp [hinj h1, hinj h2] | case mk.mk.mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
fst✝¹ snd✝¹ : V
h₁ : (fst✝¹, snd✝¹) ∈ edgeSet G
fst✝ snd✝ : V
h₂ : (fst✝, snd✝) ∈ edgeSet G
h1 : ↑f fst✝¹ = ↑f fst✝
h2 : ↑f snd✝¹ = ↑f snd✝
⊢ fst✝¹ = fst✝ ∧ snd✝¹ = snd✝ | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Hom.mapEdgeSet.injective | [516, 1] | [524, 26] | rw [Subtype.mk_eq_mk] | case mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
e₁ : V × V
h₁ : e₁ ∈ edgeSet G
e₂ : V × V
h₂ : e₂ ∈ edgeSet G
⊢ (↑f e₁.fst, ↑f e₁.snd) = (↑f e₂.fst, ↑f e₂.snd) → { val := e₁, property := h₁ } = { val := e₂, property := h₂ } | case mk.mk
ι : Sort ?u.224451
𝕜 : Type ?u.224454
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G →g G'
hinj : Injective ↑f
e₁ : V × V
h₁ : e₁ ∈ edgeSet G
e₂ : V × V
h₂ : e₂ ∈ edgeSet G
⊢ (↑f e₁.fst, ↑f e₁.snd) = (↑f e₂.fst, ↑f e₂.snd) → e₁ = e₂ |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Embedding.map_mem_edgeSet_iff | [566, 1] | [568, 23] | cases e | ι : Sort ?u.263245
𝕜 : Type ?u.263248
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G ↪g G'
e : V × V
⊢ Prod.map (↑f) (↑f) e ∈ edgeSet G' ↔ e ∈ edgeSet G | case mk
ι : Sort ?u.263245
𝕜 : Type ?u.263248
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G ↪g G'
fst✝ snd✝ : V
⊢ Prod.map ↑f ↑f (fst✝, snd✝) ∈ edgeSet G' ↔ (fst✝, snd✝) ∈ edgeSet G |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Embedding.map_mem_edgeSet_iff | [566, 1] | [568, 23] | simp [f.map_adj_iff] | case mk
ι : Sort ?u.263245
𝕜 : Type ?u.263248
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G ↪g G'
fst✝ snd✝ : V
⊢ Prod.map ↑f ↑f (fst✝, snd✝) ∈ edgeSet G' ↔ (fst✝, snd✝) ∈ edgeSet G | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.induceHom_id | [652, 1] | [655, 6] | ext x | ι : Sort ?u.395639
𝕜 : Type ?u.395642
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G' : Digraph W
a b c u v w : V
G'' : Digraph X
s✝ : Set V
t : Set W
r : Set X
φ : G✝ →g G'
φst : Set.MapsTo (↑φ) s✝ t
ψ : G' →g G''
ψtr : Set.MapsTo (↑ψ) t r
G : Digraph V
s : Set V
⊢ InduceHom Hom.id (_ : Set.MapsTo id s s) = Hom.id | case h.a
ι : Sort ?u.395639
𝕜 : Type ?u.395642
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G' : Digraph W
a b c u v w : V
G'' : Digraph X
s✝ : Set V
t : Set W
r : Set X
φ : G✝ →g G'
φst : Set.MapsTo (↑φ) s✝ t
ψ : G' →g G''
ψtr : Set.MapsTo (↑ψ) t r
G : Digraph V
s : Set V
x : ↑s
⊢ ↑(↑(InduceHom Hom.id (_ : Set.MapsTo id s s)) x) = ↑(↑Hom.id x) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.induceHom_id | [652, 1] | [655, 6] | rfl | case h.a
ι : Sort ?u.395639
𝕜 : Type ?u.395642
V : Type u
W : Type v
X : Type w
G✝ : Digraph V
G' : Digraph W
a b c u v w : V
G'' : Digraph X
s✝ : Set V
t : Set W
r : Set X
φ : G✝ →g G'
φst : Set.MapsTo (↑φ) s✝ t
ψ : G' →g G''
ψtr : Set.MapsTo (↑ψ) t r
G : Digraph V
s : Set V
x : ↑s
⊢ ↑(↑(InduceHom Hom.id (_ : Set.MapsTo id s s)) x) = ↑(↑Hom.id x) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.induceHom_comp | [658, 1] | [661, 6] | ext x | ι : Sort ?u.407307
𝕜 : Type ?u.407310
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
G'' : Digraph X
s : Set V
t : Set W
r : Set X
φ : G →g G'
φst : Set.MapsTo (↑φ) s t
ψ : G' →g G''
ψtr : Set.MapsTo (↑ψ) t r
⊢ Hom.comp (InduceHom ψ ψtr) (InduceHom φ φst) = InduceHom (Hom.comp ψ φ) (_ : Set.MapsTo (↑ψ ∘ fun x => ↑φ x) s r) | case h.a
ι : Sort ?u.407307
𝕜 : Type ?u.407310
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
G'' : Digraph X
s : Set V
t : Set W
r : Set X
φ : G →g G'
φst : Set.MapsTo (↑φ) s t
ψ : G' →g G''
ψtr : Set.MapsTo (↑ψ) t r
x : ↑s
⊢ ↑(↑(Hom.comp (InduceHom ψ ψtr) (InduceHom φ φst)) x) =
↑(↑(InduceHom (Hom.comp ψ φ) (_ : Set.MapsTo (↑ψ ∘ fun x => ↑φ x) s r)) x) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.induceHom_comp | [658, 1] | [661, 6] | rfl | case h.a
ι : Sort ?u.407307
𝕜 : Type ?u.407310
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
G'' : Digraph X
s : Set V
t : Set W
r : Set X
φ : G →g G'
φst : Set.MapsTo (↑φ) s t
ψ : G' →g G''
ψtr : Set.MapsTo (↑ψ) t r
x : ↑s
⊢ ↑(↑(Hom.comp (InduceHom ψ ψtr) (InduceHom φ φst)) x) =
↑(↑(InduceHom (Hom.comp ψ φ) (_ : Set.MapsTo (↑ψ ∘ fun x => ↑φ x) s r)) x) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Iso.map_mem_edgeSet_iff | [694, 1] | [696, 23] | cases e | ι : Sort ?u.420032
𝕜 : Type ?u.420035
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G ≃g G'
e : V × V
⊢ Prod.map (↑f) (↑f) e ∈ edgeSet G' ↔ e ∈ edgeSet G | case mk
ι : Sort ?u.420032
𝕜 : Type ?u.420035
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G ≃g G'
fst✝ snd✝ : V
⊢ Prod.map ↑f ↑f (fst✝, snd✝) ∈ edgeSet G' ↔ (fst✝, snd✝) ∈ edgeSet G |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Iso.map_mem_edgeSet_iff | [694, 1] | [696, 23] | simp [f.map_adj_iff] | case mk
ι : Sort ?u.420032
𝕜 : Type ?u.420035
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f : G ≃g G'
fst✝ snd✝ : V
⊢ Prod.map ↑f ↑f (fst✝, snd✝) ∈ edgeSet G' ↔ (fst✝, snd✝) ∈ edgeSet G | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Iso.card_eq_of_iso | [717, 1] | [720, 45] | rw [← Fintype.ofEquiv_card f.toEquiv] | ι : Sort ?u.420619
𝕜 : Type ?u.420622
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f✝ : G ≃g G'
inst✝¹ : Fintype V
inst✝ : Fintype W
f : G ≃g G'
⊢ Fintype.card V = Fintype.card W | ι : Sort ?u.420619
𝕜 : Type ?u.420622
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f✝ : G ≃g G'
inst✝¹ : Fintype V
inst✝ : Fintype W
f : G ≃g G'
⊢ Fintype.card W = Fintype.card W |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/Digraph.lean | Digraph.Iso.card_eq_of_iso | [717, 1] | [720, 45] | apply @Fintype.card_congr' _ _ (_) (_) rfl | ι : Sort ?u.420619
𝕜 : Type ?u.420622
V : Type u
W : Type v
X : Type w
G : Digraph V
G' : Digraph W
a b c u v w : V
f✝ : G ≃g G'
inst✝¹ : Fintype V
inst✝ : Fintype W
f : G ≃g G'
⊢ Fintype.card W = Fintype.card W | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.le_iff_eq_deleteEdges | [27, 1] | [37, 25] | constructor | V : Type u_1
G G' : Digraph V
⊢ G ≤ G' ↔ ∃ s, s ⊆ edgeSet G' ∧ G = deleteEdges G' s | case mp
V : Type u_1
G G' : Digraph V
⊢ G ≤ G' → ∃ s, s ⊆ edgeSet G' ∧ G = deleteEdges G' s
case mpr
V : Type u_1
G G' : Digraph V
⊢ (∃ s, s ⊆ edgeSet G' ∧ G = deleteEdges G' s) → G ≤ G' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.le_iff_eq_deleteEdges | [27, 1] | [37, 25] | intro h | case mp
V : Type u_1
G G' : Digraph V
⊢ G ≤ G' → ∃ s, s ⊆ edgeSet G' ∧ G = deleteEdges G' s | case mp
V : Type u_1
G G' : Digraph V
h : G ≤ G'
⊢ ∃ s, s ⊆ edgeSet G' ∧ G = deleteEdges G' s |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.le_iff_eq_deleteEdges | [27, 1] | [37, 25] | use G'.edgeSet \ G.edgeSet | case mp
V : Type u_1
G G' : Digraph V
h : G ≤ G'
⊢ ∃ s, s ⊆ edgeSet G' ∧ G = deleteEdges G' s | case mp
V : Type u_1
G G' : Digraph V
h : G ≤ G'
⊢ edgeSet G' \ edgeSet G ⊆ edgeSet G' ∧ G = deleteEdges G' (edgeSet G' \ edgeSet G) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.le_iff_eq_deleteEdges | [27, 1] | [37, 25] | refine ⟨Set.diff_subset _ _, ?_⟩ | case mp
V : Type u_1
G G' : Digraph V
h : G ≤ G'
⊢ edgeSet G' \ edgeSet G ⊆ edgeSet G' ∧ G = deleteEdges G' (edgeSet G' \ edgeSet G) | case mp
V : Type u_1
G G' : Digraph V
h : G ≤ G'
⊢ G = deleteEdges G' (edgeSet G' \ edgeSet G) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.le_iff_eq_deleteEdges | [27, 1] | [37, 25] | ext v w | case mp
V : Type u_1
G G' : Digraph V
h : G ≤ G'
⊢ G = deleteEdges G' (edgeSet G' \ edgeSet G) | case mp.Adj.h.h.a
V : Type u_1
G G' : Digraph V
h : G ≤ G'
v w : V
⊢ G.Adj v w ↔ (deleteEdges G' (edgeSet G' \ edgeSet G)).Adj v w |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.le_iff_eq_deleteEdges | [27, 1] | [37, 25] | simp only [deleteEdges_adj, Set.mem_diff, mem_edgeSet, not_and, not_not] | case mp.Adj.h.h.a
V : Type u_1
G G' : Digraph V
h : G ≤ G'
v w : V
⊢ G.Adj v w ↔ (deleteEdges G' (edgeSet G' \ edgeSet G)).Adj v w | case mp.Adj.h.h.a
V : Type u_1
G G' : Digraph V
h : G ≤ G'
v w : V
⊢ G.Adj v w ↔ G'.Adj v w ∧ (G'.Adj v w → G.Adj v w) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.le_iff_eq_deleteEdges | [27, 1] | [37, 25] | constructor <;> simp (config := {contextual := true}) [@h v w] | case mp.Adj.h.h.a
V : Type u_1
G G' : Digraph V
h : G ≤ G'
v w : V
⊢ G.Adj v w ↔ G'.Adj v w ∧ (G'.Adj v w → G.Adj v w) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.le_iff_eq_deleteEdges | [27, 1] | [37, 25] | rintro ⟨s, _, rfl⟩ | case mpr
V : Type u_1
G G' : Digraph V
⊢ (∃ s, s ⊆ edgeSet G' ∧ G = deleteEdges G' s) → G ≤ G' | case mpr.intro.intro
V : Type u_1
G' : Digraph V
s : Set (V × V)
left✝ : s ⊆ edgeSet G'
⊢ deleteEdges G' s ≤ G' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.le_iff_eq_deleteEdges | [27, 1] | [37, 25] | apply deleteEdges_le | case mpr.intro.intro
V : Type u_1
G' : Digraph V
s : Set (V × V)
left✝ : s ⊆ edgeSet G'
⊢ deleteEdges G' s ≤ G' | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | constructor | V : Type u_1
G G' : Digraph V
⊢ G < G' ↔ ∃ s, Set.Nonempty s ∧ s ⊆ edgeSet G' ∧ G = deleteEdges G' s | case mp
V : Type u_1
G G' : Digraph V
⊢ G < G' → ∃ s, Set.Nonempty s ∧ s ⊆ edgeSet G' ∧ G = deleteEdges G' s
case mpr
V : Type u_1
G G' : Digraph V
⊢ (∃ s, Set.Nonempty s ∧ s ⊆ edgeSet G' ∧ G = deleteEdges G' s) → G < G' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | intro h | case mp
V : Type u_1
G G' : Digraph V
⊢ G < G' → ∃ s, Set.Nonempty s ∧ s ⊆ edgeSet G' ∧ G = deleteEdges G' s | case mp
V : Type u_1
G G' : Digraph V
h : G < G'
⊢ ∃ s, Set.Nonempty s ∧ s ⊆ edgeSet G' ∧ G = deleteEdges G' s |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | use G'.edgeSet \ G.edgeSet | case mp
V : Type u_1
G G' : Digraph V
h : G < G'
⊢ ∃ s, Set.Nonempty s ∧ s ⊆ edgeSet G' ∧ G = deleteEdges G' s | case mp
V : Type u_1
G G' : Digraph V
h : G < G'
⊢ Set.Nonempty (edgeSet G' \ edgeSet G) ∧
edgeSet G' \ edgeSet G ⊆ edgeSet G' ∧ G = deleteEdges G' (edgeSet G' \ edgeSet G) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | constructor | case mp
V : Type u_1
G G' : Digraph V
h : G < G'
⊢ Set.Nonempty (edgeSet G' \ edgeSet G) ∧
edgeSet G' \ edgeSet G ⊆ edgeSet G' ∧ G = deleteEdges G' (edgeSet G' \ edgeSet G) | case mp.left
V : Type u_1
G G' : Digraph V
h : G < G'
⊢ Set.Nonempty (edgeSet G' \ edgeSet G)
case mp.right
V : Type u_1
G G' : Digraph V
h : G < G'
⊢ edgeSet G' \ edgeSet G ⊆ edgeSet G' ∧ G = deleteEdges G' (edgeSet G' \ edgeSet G) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | constructor | case mp.right
V : Type u_1
G G' : Digraph V
h : G < G'
⊢ edgeSet G' \ edgeSet G ⊆ edgeSet G' ∧ G = deleteEdges G' (edgeSet G' \ edgeSet G) | case mp.right.left
V : Type u_1
G G' : Digraph V
h : G < G'
⊢ edgeSet G' \ edgeSet G ⊆ edgeSet G'
case mp.right.right
V : Type u_1
G G' : Digraph V
h : G < G'
⊢ G = deleteEdges G' (edgeSet G' \ edgeSet G) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | by_contra hn | case mp.left
V : Type u_1
G G' : Digraph V
h : G < G'
⊢ Set.Nonempty (edgeSet G' \ edgeSet G) | case mp.left
V : Type u_1
G G' : Digraph V
h : G < G'
hn : ¬Set.Nonempty (edgeSet G' \ edgeSet G)
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | rw [Set.not_nonempty_iff_eq_empty, Set.diff_eq_empty, edgeSet_subset_edgeSet] at hn | case mp.left
V : Type u_1
G G' : Digraph V
h : G < G'
hn : ¬Set.Nonempty (edgeSet G' \ edgeSet G)
⊢ False | case mp.left
V : Type u_1
G G' : Digraph V
h : G < G'
hn : G' ≤ G
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | exact lt_irrefl _ (trans h hn) | case mp.left
V : Type u_1
G G' : Digraph V
h : G < G'
hn : G' ≤ G
⊢ False | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | apply Set.diff_subset | case mp.right.left
V : Type u_1
G G' : Digraph V
h : G < G'
⊢ edgeSet G' \ edgeSet G ⊆ edgeSet G' | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | ext v w | case mp.right.right
V : Type u_1
G G' : Digraph V
h : G < G'
⊢ G = deleteEdges G' (edgeSet G' \ edgeSet G) | case mp.right.right.Adj.h.h.a
V : Type u_1
G G' : Digraph V
h : G < G'
v w : V
⊢ G.Adj v w ↔ (deleteEdges G' (edgeSet G' \ edgeSet G)).Adj v w |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | simp only [deleteEdges_adj, Set.mem_diff, mem_edgeSet, not_and, not_not] | case mp.right.right.Adj.h.h.a
V : Type u_1
G G' : Digraph V
h : G < G'
v w : V
⊢ G.Adj v w ↔ (deleteEdges G' (edgeSet G' \ edgeSet G)).Adj v w | case mp.right.right.Adj.h.h.a
V : Type u_1
G G' : Digraph V
h : G < G'
v w : V
⊢ G.Adj v w ↔ G'.Adj v w ∧ (G'.Adj v w → G.Adj v w) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | have : G.Adj v w → G'.Adj v w := fun e => h.le e | case mp.right.right.Adj.h.h.a
V : Type u_1
G G' : Digraph V
h : G < G'
v w : V
⊢ G.Adj v w ↔ G'.Adj v w ∧ (G'.Adj v w → G.Adj v w) | case mp.right.right.Adj.h.h.a
V : Type u_1
G G' : Digraph V
h : G < G'
v w : V
this : G.Adj v w → G'.Adj v w
⊢ G.Adj v w ↔ G'.Adj v w ∧ (G'.Adj v w → G.Adj v w) |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | constructor <;> simp (config := {contextual := true}) [this] | case mp.right.right.Adj.h.h.a
V : Type u_1
G G' : Digraph V
h : G < G'
v w : V
this : G.Adj v w → G'.Adj v w
⊢ G.Adj v w ↔ G'.Adj v w ∧ (G'.Adj v w → G.Adj v w) | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | rintro ⟨s, ⟨⟨v, w⟩, h⟩, hs, rfl⟩ | case mpr
V : Type u_1
G G' : Digraph V
⊢ (∃ s, Set.Nonempty s ∧ s ⊆ edgeSet G' ∧ G = deleteEdges G' s) → G < G' | case mpr.intro.intro.intro.mk.intro
V : Type u_1
G' : Digraph V
s : Set (V × V)
v w : V
h : (v, w) ∈ s
hs : s ⊆ edgeSet G'
⊢ deleteEdges G' s < G' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | rw [lt_iff_le_and_ne] | case mpr.intro.intro.intro.mk.intro
V : Type u_1
G' : Digraph V
s : Set (V × V)
v w : V
h : (v, w) ∈ s
hs : s ⊆ edgeSet G'
⊢ deleteEdges G' s < G' | case mpr.intro.intro.intro.mk.intro
V : Type u_1
G' : Digraph V
s : Set (V × V)
v w : V
h : (v, w) ∈ s
hs : s ⊆ edgeSet G'
⊢ deleteEdges G' s ≤ G' ∧ deleteEdges G' s ≠ G' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | constructor | case mpr.intro.intro.intro.mk.intro
V : Type u_1
G' : Digraph V
s : Set (V × V)
v w : V
h : (v, w) ∈ s
hs : s ⊆ edgeSet G'
⊢ deleteEdges G' s ≤ G' ∧ deleteEdges G' s ≠ G' | case mpr.intro.intro.intro.mk.intro.left
V : Type u_1
G' : Digraph V
s : Set (V × V)
v w : V
h : (v, w) ∈ s
hs : s ⊆ edgeSet G'
⊢ deleteEdges G' s ≤ G'
case mpr.intro.intro.intro.mk.intro.right
V : Type u_1
G' : Digraph V
s : Set (V × V)
v w : V
h : (v, w) ∈ s
hs : s ⊆ edgeSet G'
⊢ deleteEdges G' s ≠ G' |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | apply deleteEdges_le | case mpr.intro.intro.intro.mk.intro.left
V : Type u_1
G' : Digraph V
s : Set (V × V)
v w : V
h : (v, w) ∈ s
hs : s ⊆ edgeSet G'
⊢ deleteEdges G' s ≤ G' | no goals |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | intro hd | case mpr.intro.intro.intro.mk.intro.right
V : Type u_1
G' : Digraph V
s : Set (V × V)
v w : V
h : (v, w) ∈ s
hs : s ⊆ edgeSet G'
⊢ deleteEdges G' s ≠ G' | case mpr.intro.intro.intro.mk.intro.right
V : Type u_1
G' : Digraph V
s : Set (V × V)
v w : V
h : (v, w) ∈ s
hs : s ⊆ edgeSet G'
hd : deleteEdges G' s = G'
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | have := hs h | case mpr.intro.intro.intro.mk.intro.right
V : Type u_1
G' : Digraph V
s : Set (V × V)
v w : V
h : (v, w) ∈ s
hs : s ⊆ edgeSet G'
hd : (deleteEdges G' s).Adj v w = G'.Adj v w
⊢ False | case mpr.intro.intro.intro.mk.intro.right
V : Type u_1
G' : Digraph V
s : Set (V × V)
v w : V
h : (v, w) ∈ s
hs : s ⊆ edgeSet G'
hd : (deleteEdges G' s).Adj v w = G'.Adj v w
this : (v, w) ∈ edgeSet G'
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | rw [mem_edgeSet] at this | case mpr.intro.intro.intro.mk.intro.right
V : Type u_1
G' : Digraph V
s : Set (V × V)
v w : V
h : (v, w) ∈ s
hs : s ⊆ edgeSet G'
hd : (deleteEdges G' s).Adj v w = G'.Adj v w
this : (v, w) ∈ edgeSet G'
⊢ False | case mpr.intro.intro.intro.mk.intro.right
V : Type u_1
G' : Digraph V
s : Set (V × V)
v w : V
h : (v, w) ∈ s
hs : s ⊆ edgeSet G'
hd : (deleteEdges G' s).Adj v w = G'.Adj v w
this : G'.Adj v w
⊢ False |
https://github.com/kmill/msri2023_graphs.git | c87c12d835cabc843f7f9c83d4c947c11d8ff043 | GraphProjects/DigraphExtra.lean | Digraph.lt_iff_eq_deleteEdges | [40, 1] | [63, 27] | simp [this, h] at hd | case mpr.intro.intro.intro.mk.intro.right
V : Type u_1
G' : Digraph V
s : Set (V × V)
v w : V
h : (v, w) ∈ s
hs : s ⊆ edgeSet G'
hd : (deleteEdges G' s).Adj v w = G'.Adj v w
this : G'.Adj v w
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Algebra/Order/Sub/Canonical.lean | tsub_tsub_eq_min | [11, 1] | [12, 70] | rw [tsub_eq_tsub_min _ b, tsub_tsub_cancel_of_le (min_le_left a _)] | α : Type u_1
inst✝³ : CanonicallyLinearOrderedAddCommMonoid α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
a b : α
⊢ a - (a - b) = min a b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Finite.lean | Geometry.SimplicialComplex.LocallyFiniteAt.mono | [47, 1] | [50, 43] | apply hX.subset | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x t✝ : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hX : K.LocallyFiniteAt s
hXY : s ⊆ t
⊢ K.LocallyFiniteAt t | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x t✝ : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hX : K.LocallyFiniteAt s
hXY : s ⊆ t
⊢ {t_1 | t_1 ∈ K.faces ∧ t ⊆ t_1} ⊆ {t | t ∈ K.faces ∧ s ⊆ t} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Finite.lean | Geometry.SimplicialComplex.LocallyFiniteAt.mono | [47, 1] | [50, 43] | rintro u ⟨_, _⟩ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x t✝ : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hX : K.LocallyFiniteAt s
hXY : s ⊆ t
⊢ {t_1 | t_1 ∈ K.faces ∧ t ⊆ t_1} ⊆ {t | t ∈ K.faces ∧ s ⊆ t} | case intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x t✝ : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hX : K.LocallyFiniteAt s
hXY : s ⊆ t
u : Finset E
left✝ : u ∈ K.faces
right✝ : t ⊆ u
⊢ u ∈ {t | t ∈ K.faces ∧ s ⊆ t} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Finite.lean | Geometry.SimplicialComplex.LocallyFiniteAt.mono | [47, 1] | [50, 43] | exact ⟨‹u ∈ K.faces›, hXY.trans ‹t ⊆ u›⟩ | case intro
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x t✝ : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hX : K.LocallyFiniteAt s
hXY : s ⊆ t
u : Finset E
left✝ : u ∈ K.faces
right✝ : t ⊆ u
⊢ u ∈ {t | t ∈ K.faces ∧ s ⊆ t} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Finite.lean | Geometry.SimplicialComplex.locallyFinite_iff_mem_finitely_many_faces | [58, 1] | [82, 88] | constructor | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : OrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x t✝ : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : DecidableEq E
⊢ K.LocallyFinite ↔ ∀ (x : E), {s | s ∈ K.faces ∧ x ∈ (convexHull 𝕜) ↑s}.Finite | case mp
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : OrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x t✝ : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : DecidableEq E
⊢ K.LocallyFinite → ∀ (x : E), {s | s ∈ K.faces ∧ x ∈ (convexHull 𝕜) ↑s}.Finite
case mpr
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : OrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x t✝ : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
inst✝ : DecidableEq E
⊢ (∀ (x : E), {s | s ∈ K.faces ∧ x ∈ (convexHull 𝕜) ↑s}.Finite) → K.LocallyFinite |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.