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 | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.two_cliqueFree_imp_empty | [54, 1] | [58, 45] | obtain ⟨v, w, had⟩ := edge_of_not_empty h | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ¬G = ⊥
⊢ ¬CliqueFree G 2 | case intro.intro
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ¬G = ⊥
v w : α
had : v ≠ w ∧ Adj G v w
⊢ ¬CliqueFree G 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ¬G = ⊥
⊢ ¬CliqueFree G 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.two_cliqueFree_imp_empty | [54, 1] | [58, 45] | intro hf | case intro.intro
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ¬G = ⊥
v w : α
had : v ≠ w ∧ Adj G v w
⊢ ¬CliqueFree G 2 | case intro.intro
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ¬G = ⊥
v w : α
had : v ≠ w ∧ Adj G v w
hf : CliqueFree G 2
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ¬G = ⊥
v w : α
had : v ≠ w ∧ Adj G v w
⊢ ¬CliqueFree G 2
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.two_cliqueFree_imp_empty | [54, 1] | [58, 45] | apply hf _ (adj_is2Clique had.2) | case intro.intro
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ¬G = ⊥
v w : α
had : v ≠ w ∧ Adj G v w
hf : CliqueFree G 2
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : ¬G = ⊥
v w : α
had : v ≠ w ∧ Adj G v w
hf : CliqueFree G 2
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.disjoint_edges_iff_meet_empty | [61, 1] | [63, 94] | rw [disjoint_iff,← edgeFinset_inj,edgeFinset_inf,inf_eq_inter,edgeFinset_bot,bot_eq_empty] | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ Disjoint (edgeFinset G) (edgeFinset H) ↔ G ⊓ H = ⊥ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ Disjoint (edgeFinset G) (edgeFinset H) ↔ G ⊓ H = ⊥
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.card_edges_add_of_meet_empty | [66, 1] | [70, 39] | rw [← disjoint_edges_iff_meet_empty,edgeFinset_sup] | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ G ⊓ H = ⊥ → card (edgeFinset (G ⊔ H)) = card (edgeFinset G) + card (edgeFinset H) | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ Disjoint (edgeFinset G) (edgeFinset H) →
card (edgeFinset G ∪ edgeFinset H) = card (edgeFinset G) + card (edgeFinset H) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ G ⊓ H = ⊥ → card (edgeFinset (G ⊔ H)) = card (edgeFinset G) + card (edgeFinset H)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.card_edges_add_of_meet_empty | [66, 1] | [70, 39] | intro h | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ Disjoint (edgeFinset G) (edgeFinset H) →
card (edgeFinset G ∪ edgeFinset H) = card (edgeFinset G) + card (edgeFinset H) | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : Disjoint (edgeFinset G) (edgeFinset H)
⊢ card (edgeFinset G ∪ edgeFinset H) = card (edgeFinset G) + card (edgeFinset H) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
⊢ Disjoint (edgeFinset G) (edgeFinset H) →
card (edgeFinset G ∪ edgeFinset H) = card (edgeFinset G) + card (edgeFinset H)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.card_edges_add_of_meet_empty | [66, 1] | [70, 39] | exact card_disjoint_union h | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : Disjoint (edgeFinset G) (edgeFinset H)
⊢ card (edgeFinset G ∪ edgeFinset H) = card (edgeFinset G) + card (edgeFinset H) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G H : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
h : Disjoint (edgeFinset G) (edgeFinset H)
⊢ card (edgeFinset G ∪ edgeFinset H) = card (edgeFinset G) + card (edgeFinset H)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.delFedges_is_sdiff | [82, 1] | [86, 6] | ext u v | α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
inst✝ : DecidableRel H.Adj
⊢ delFedges G (edgeFinset H) = G \ H | case Adj.h.h.a
α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
inst✝ : DecidableRel H.Adj
u v : α
⊢ Adj (delFedges G (edgeFinset H)) u v ↔ Adj (G \ H) u v | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
inst✝ : DecidableRel H.Adj
⊢ delFedges G (edgeFinset H) = G \ H
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.delFedges_is_sdiff | [82, 1] | [86, 6] | simp only [delFedges, sdiff_adj, Set.coe_toFinset, Sym2.toRel_prop, mem_edgeSet] | case Adj.h.h.a
α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
inst✝ : DecidableRel H.Adj
u v : α
⊢ Adj (delFedges G (edgeFinset H)) u v ↔ Adj (G \ H) u v | case Adj.h.h.a
α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
inst✝ : DecidableRel H.Adj
u v : α
⊢ Adj G u v \ Adj H u v ↔ Adj G u v ∧ ¬Adj H u v | Please generate a tactic in lean4 to solve the state.
STATE:
case Adj.h.h.a
α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
inst✝ : DecidableRel H.Adj
u v : α
⊢ Adj (delFedges G (edgeFinset H)) u v ↔ Adj (G \ H) u v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.delFedges_is_sdiff | [82, 1] | [86, 6] | rfl | case Adj.h.h.a
α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
inst✝ : DecidableRel H.Adj
u v : α
⊢ Adj G u v \ Adj H u v ↔ Adj G u v ∧ ¬Adj H u v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case Adj.h.h.a
α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
inst✝ : DecidableRel H.Adj
u v : α
⊢ Adj G u v \ Adj H u v ↔ Adj G u v ∧ ¬Adj H u v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.isFar_le | [93, 1] | [96, 30] | intro h1 | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝¹ : DecidableRel G✝.Adj
inst✝ : DecidableRel H✝.Adj
s t : ℕ
G H : SimpleGraph α
h : s ≤ t
⊢ IsFar G H s → IsFar G H t | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝¹ : DecidableRel G✝.Adj
inst✝ : DecidableRel H✝.Adj
s t : ℕ
G H : SimpleGraph α
h : s ≤ t
h1 : IsFar G H s
⊢ IsFar G H t | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝¹ : DecidableRel G✝.Adj
inst✝ : DecidableRel H✝.Adj
s t : ℕ
G H : SimpleGraph α
h : s ≤ t
⊢ IsFar G H s → IsFar G H t
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.isFar_le | [93, 1] | [96, 30] | obtain ⟨S, hS1, hS2⟩ := h1 | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝¹ : DecidableRel G✝.Adj
inst✝ : DecidableRel H✝.Adj
s t : ℕ
G H : SimpleGraph α
h : s ≤ t
h1 : IsFar G H s
⊢ IsFar G H t | case intro.intro
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝¹ : DecidableRel G✝.Adj
inst✝ : DecidableRel H✝.Adj
s t : ℕ
G H : SimpleGraph α
h : s ≤ t
S : Finset (Sym2 α)
hS1 : delFedges G S ≤ H
hS2 : card S ≤ s
⊢ IsFar G H t | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝¹ : DecidableRel G✝.Adj
inst✝ : DecidableRel H✝.Adj
s t : ℕ
G H : SimpleGraph α
h : s ≤ t
h1 : IsFar G H s
⊢ IsFar G H t
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.isFar_le | [93, 1] | [96, 30] | exact ⟨S, hS1, hS2.trans h⟩ | case intro.intro
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝¹ : DecidableRel G✝.Adj
inst✝ : DecidableRel H✝.Adj
s t : ℕ
G H : SimpleGraph α
h : s ≤ t
S : Finset (Sym2 α)
hS1 : delFedges G S ≤ H
hS2 : card S ≤ s
⊢ IsFar G H t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝¹ : DecidableRel G✝.Adj
inst✝ : DecidableRel H✝.Adj
s t : ℕ
G H : SimpleGraph α
h : s ≤ t
S : Finset (Sym2 α)
hS1 : delFedges G S ≤ H
hS2 : card S ≤ s
⊢ IsFar G H t
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.isFar_trivial | [98, 1] | [103, 15] | intro h | α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
s : ℕ
inst✝ : DecidableRel G.Adj
⊢ card (edgeFinset G) ≤ s → IsFar G H s | α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
s : ℕ
inst✝ : DecidableRel G.Adj
h : card (edgeFinset G) ≤ s
⊢ IsFar G H s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
s : ℕ
inst✝ : DecidableRel G.Adj
⊢ card (edgeFinset G) ≤ s → IsFar G H s
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.isFar_trivial | [98, 1] | [103, 15] | refine ⟨G.edgeFinset, ?_, h⟩ | α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
s : ℕ
inst✝ : DecidableRel G.Adj
h : card (edgeFinset G) ≤ s
⊢ IsFar G H s | α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
s : ℕ
inst✝ : DecidableRel G.Adj
h : card (edgeFinset G) ≤ s
⊢ delFedges G (edgeFinset G) ≤ H | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
s : ℕ
inst✝ : DecidableRel G.Adj
h : card (edgeFinset G) ≤ s
⊢ IsFar G H s
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.isFar_trivial | [98, 1] | [103, 15] | rw [delFedges_is_sdiff,_root_.sdiff_self] | α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
s : ℕ
inst✝ : DecidableRel G.Adj
h : card (edgeFinset G) ≤ s
⊢ delFedges G (edgeFinset G) ≤ H | α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
s : ℕ
inst✝ : DecidableRel G.Adj
h : card (edgeFinset G) ≤ s
⊢ ⊥ ≤ H | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
s : ℕ
inst✝ : DecidableRel G.Adj
h : card (edgeFinset G) ≤ s
⊢ delFedges G (edgeFinset G) ≤ H
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Fedges.lean | SimpleGraph.isFar_trivial | [98, 1] | [103, 15] | exact bot_le | α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
s : ℕ
inst✝ : DecidableRel G.Adj
h : card (edgeFinset G) ≤ s
⊢ ⊥ ≤ H | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
G✝ H✝ : SimpleGraph α
inst✝² : DecidableRel G✝.Adj
inst✝¹ : DecidableRel H✝.Adj
G H : SimpleGraph α
s : ℕ
inst✝ : DecidableRel G.Adj
h : card (edgeFinset G) ≤ s
⊢ ⊥ ≤ H
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Groups.lean | inverse_involution | [3, 1] | [9, 33] | calc
g⁻¹⁻¹ = g⁻¹⁻¹ * 1 := by rw [mul_one]
_ = g⁻¹⁻¹ * (g⁻¹ * g) := by rw [inv_mul_self]
_ = (g⁻¹⁻¹ * g⁻¹) * g := by rw [mul_assoc]
_ = 1 * g := by rw [inv_mul_self]
_ = g := by rw [one_mul] | G : Type u_1
inst✝ : Group G
g : G
⊢ g⁻¹⁻¹ = g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
g : G
⊢ g⁻¹⁻¹ = g
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Groups.lean | inverse_involution | [3, 1] | [9, 33] | rw [mul_one] | G : Type u_1
inst✝ : Group G
g : G
⊢ g⁻¹⁻¹ = g⁻¹⁻¹ * 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
g : G
⊢ g⁻¹⁻¹ = g⁻¹⁻¹ * 1
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Groups.lean | inverse_involution | [3, 1] | [9, 33] | rw [inv_mul_self] | G : Type u_1
inst✝ : Group G
g : G
⊢ g⁻¹⁻¹ * 1 = g⁻¹⁻¹ * (g⁻¹ * g) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
g : G
⊢ g⁻¹⁻¹ * 1 = g⁻¹⁻¹ * (g⁻¹ * g)
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Groups.lean | inverse_involution | [3, 1] | [9, 33] | rw [mul_assoc] | G : Type u_1
inst✝ : Group G
g : G
⊢ g⁻¹⁻¹ * (g⁻¹ * g) = g⁻¹⁻¹ * g⁻¹ * g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
g : G
⊢ g⁻¹⁻¹ * (g⁻¹ * g) = g⁻¹⁻¹ * g⁻¹ * g
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Groups.lean | inverse_involution | [3, 1] | [9, 33] | rw [inv_mul_self] | G : Type u_1
inst✝ : Group G
g : G
⊢ g⁻¹⁻¹ * g⁻¹ * g = 1 * g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
g : G
⊢ g⁻¹⁻¹ * g⁻¹ * g = 1 * g
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Groups.lean | inverse_involution | [3, 1] | [9, 33] | rw [one_mul] | G : Type u_1
inst✝ : Group G
g : G
⊢ 1 * g = g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
g : G
⊢ 1 * g = g
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Groups.lean | inverse_involution' | [11, 1] | [13, 47] | rw [← mul_one g⁻¹⁻¹, ← inv_mul_self g,
← mul_assoc, inv_mul_self g⁻¹, one_mul] | G : Type u_1
inst✝ : Group G
g : G
⊢ g⁻¹⁻¹ = g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
g : G
⊢ g⁻¹⁻¹ = g
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Polynomials.lean | square_eq_two_times | [4, 1] | [10, 38] | calc
x^2 = Monoid.npow (Nat.succ (Nat.succ 0)) x :=
by simp [HPow.hPow, instHPow]
_ = x * (x * 1) :=
by rw [Monoid.npow_succ, Monoid.npow_succ, Monoid.npow_zero]
_ = x * x := by rw [Monoid.mul_one] | R : Type u_1
inst✝ : CommRing R
x : R
⊢ x ^ 2 = x * x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x : R
⊢ x ^ 2 = x * x
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Polynomials.lean | square_eq_two_times | [4, 1] | [10, 38] | simp [HPow.hPow, instHPow] | R : Type u_1
inst✝ : CommRing R
x : R
⊢ x ^ 2 = Monoid.npow (Nat.succ (Nat.succ 0)) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x : R
⊢ x ^ 2 = Monoid.npow (Nat.succ (Nat.succ 0)) x
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Polynomials.lean | square_eq_two_times | [4, 1] | [10, 38] | rw [Monoid.npow_succ, Monoid.npow_succ, Monoid.npow_zero] | R : Type u_1
inst✝ : CommRing R
x : R
⊢ Monoid.npow (Nat.succ (Nat.succ 0)) x = x * (x * 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x : R
⊢ Monoid.npow (Nat.succ (Nat.succ 0)) x = x * (x * 1)
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Polynomials.lean | square_eq_two_times | [4, 1] | [10, 38] | rw [Monoid.mul_one] | R : Type u_1
inst✝ : CommRing R
x : R
⊢ x * (x * 1) = x * x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x : R
⊢ x * (x * 1) = x * x
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Polynomials.lean | double_eq_two_additions | [12, 1] | [13, 26] | ring | R : Type u_1
inst✝ : CommRing R
x : R
⊢ 2 * x = x + x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x : R
⊢ 2 * x = x + x
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Polynomials.lean | binomial_formula | [15, 1] | [25, 44] | calc
(x + y)^2 = (x + y)*(x + y) := by rw [square_eq_two_times]
_ = x*x + y*x + (x*y + y*y) := by rw [Distrib.left_distrib,
Distrib.right_distrib, Distrib.right_distrib]
_ = x*x + x*y + x*y + y*y := by simp only [add_left_comm,
add_assoc, mul_comm]
_ = x*x + (x*y + x*y) + y*y := by simp only [add_assoc]
_ = x*x + 2*(x*y) + y*y := by rw [← double_eq_two_additions]
_ = x^2 + 2*(x*y) + y^2 := by rw [square_eq_two_times,
square_eq_two_times] | R : Type u_1
inst✝ : CommRing R
x y : R
⊢ (x + y) ^ 2 = x ^ 2 + 2 * (x * y) + y ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
⊢ (x + y) ^ 2 = x ^ 2 + 2 * (x * y) + y ^ 2
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Polynomials.lean | binomial_formula | [15, 1] | [25, 44] | rw [square_eq_two_times] | R : Type u_1
inst✝ : CommRing R
x y : R
⊢ (x + y) ^ 2 = (x + y) * (x + y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
⊢ (x + y) ^ 2 = (x + y) * (x + y)
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Polynomials.lean | binomial_formula | [15, 1] | [25, 44] | rw [Distrib.left_distrib,
Distrib.right_distrib, Distrib.right_distrib] | R : Type u_1
inst✝ : CommRing R
x y : R
⊢ (x + y) * (x + y) = x * x + y * x + (x * y + y * y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
⊢ (x + y) * (x + y) = x * x + y * x + (x * y + y * y)
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Polynomials.lean | binomial_formula | [15, 1] | [25, 44] | simp only [add_left_comm,
add_assoc, mul_comm] | R : Type u_1
inst✝ : CommRing R
x y : R
⊢ x * x + y * x + (x * y + y * y) = x * x + x * y + x * y + y * y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
⊢ x * x + y * x + (x * y + y * y) = x * x + x * y + x * y + y * y
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Polynomials.lean | binomial_formula | [15, 1] | [25, 44] | simp only [add_assoc] | R : Type u_1
inst✝ : CommRing R
x y : R
⊢ x * x + x * y + x * y + y * y = x * x + (x * y + x * y) + y * y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
⊢ x * x + x * y + x * y + y * y = x * x + (x * y + x * y) + y * y
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Polynomials.lean | binomial_formula | [15, 1] | [25, 44] | rw [← double_eq_two_additions] | R : Type u_1
inst✝ : CommRing R
x y : R
⊢ x * x + (x * y + x * y) + y * y = x * x + 2 * (x * y) + y * y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
⊢ x * x + (x * y + x * y) + y * y = x * x + 2 * (x * y) + y * y
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Polynomials.lean | binomial_formula | [15, 1] | [25, 44] | rw [square_eq_two_times,
square_eq_two_times] | R : Type u_1
inst✝ : CommRing R
x y : R
⊢ x * x + 2 * (x * y) + y * y = x ^ 2 + 2 * (x * y) + y ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
⊢ x * x + 2 * (x * y) + y * y = x ^ 2 + 2 * (x * y) + y ^ 2
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Polynomials.lean | binomial_formula' | [28, 1] | [29, 44] | ring | R : Type u_1
inst✝ : CommRing R
x y : R
⊢ (x + y) ^ 2 = x ^ 2 + 2 * x * y + y ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
⊢ (x + y) ^ 2 = x ^ 2 + 2 * x * y + y ^ 2
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/Polynomials.lean | trinomial | [32, 1] | [35, 69] | ring | R : Type u_1
inst✝ : CommRing R
x y z : R
⊢ (x + y + z) ^ 3 =
x * y * z * 6 + x * y ^ 2 * 3 + x * z ^ 2 * 3 + x ^ 2 * y * 3 + x ^ 2 * z * 3 + x ^ 3 + y * z ^ 2 * 3 +
y ^ 2 * z * 3 +
y ^ 3 +
z ^ 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y z : R
⊢ (x + y + z) ^ 3 =
x * y * z * 6 + x * y ^ 2 * 3 + x * z ^ 2 * 3 + x ^ 2 * y * 3 + x ^ 2 * z * 3 + x ^ 3 + y * z ^ 2 * 3 +
y ^ 2 * z * 3 +
y ^ 3 +
z ^ 3
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/ConvexFunctions.lean | ConvexOn.add' | [9, 1] | [16, 88] | constructor | 𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
⊢ ConvexOn 𝕜 s (f + g) | case left
𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
⊢ Convex 𝕜 s
case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
⊢ ∀ ⦃x : E⦄,
x ∈ s →
∀ ⦃y : E⦄,
y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (f + g) (a • x + b • y) ≤ a • (f + g) x + b • (f + g) y | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
⊢ ConvexOn 𝕜 s (f + g)
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/ConvexFunctions.lean | ConvexOn.add' | [9, 1] | [16, 88] | exact hf.1 | case left
𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
⊢ Convex 𝕜 s
case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
⊢ ∀ ⦃x : E⦄,
x ∈ s →
∀ ⦃y : E⦄,
y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (f + g) (a • x + b • y) ≤ a • (f + g) x + b • (f + g) y | case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
⊢ ∀ ⦃x : E⦄,
x ∈ s →
∀ ⦃y : E⦄,
y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (f + g) (a • x + b • y) ≤ a • (f + g) x + b • (f + g) y | Please generate a tactic in lean4 to solve the state.
STATE:
case left
𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
⊢ Convex 𝕜 s
case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
⊢ ∀ ⦃x : E⦄,
x ∈ s →
∀ ⦃y : E⦄,
y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (f + g) (a • x + b • y) ≤ a • (f + g) x + b • (f + g) y
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/ConvexFunctions.lean | ConvexOn.add' | [9, 1] | [16, 88] | intro x hx y hy a b ha hb hab | case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
⊢ ∀ ⦃x : E⦄,
x ∈ s →
∀ ⦃y : E⦄,
y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (f + g) (a • x + b • y) ≤ a • (f + g) x + b • (f + g) y | case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ (f + g) (a • x + b • y) ≤ a • (f + g) x + b • (f + g) y | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
⊢ ∀ ⦃x : E⦄,
x ∈ s →
∀ ⦃y : E⦄,
y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (f + g) (a • x + b • y) ≤ a • (f + g) x + b • (f + g) y
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/ConvexFunctions.lean | ConvexOn.add' | [9, 1] | [16, 88] | calc
f (a • x + b • y) + g (a • x + b • y) ≤ a • f x + b • f y + (a • g x + b • g y) :=
add_le_add (hf.2 hx hy ha hb hab) (hg.2 hx hy ha hb hab)
_ = a • (f x + g x) + b • (f y + g y) := by rw [smul_add, smul_add, add_add_add_comm] | case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ (f + g) (a • x + b • y) ≤ a • (f + g) x + b • (f + g) y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ (f + g) (a • x + b • y) ≤ a • (f + g) x + b • (f + g) y
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/ConvexFunctions.lean | ConvexOn.add' | [9, 1] | [16, 88] | rw [smul_add, smul_add, add_add_add_comm] | 𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • f x + b • f y + (a • g x + b • g y) = a • (f x + g x) + b • (f y + g y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
F : Type ?u.692
α : Type ?u.695
β : Type u_3
ι : Type ?u.701
inst✝⁶ : OrderedSemiring 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : OrderedAddCommMonoid α
inst✝² : OrderedAddCommMonoid β
inst✝¹ : SMul 𝕜 E
inst✝ : DistribMulAction 𝕜 β
s : Set E
f g : E → β
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • f x + b • f y + (a • g x + b • g y) = a • (f x + g x) + b • (f y + g y)
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/ConvexFunctions.lean | ConvexOn.convex_le' | [19, 1] | [28, 4] | intro x hx y hy a b ha hb hab | 𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
⊢ Convex 𝕜 { x | x ∈ s ∧ f x ≤ r } | 𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • x + b • y ∈ { x | x ∈ s ∧ f x ≤ r } | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
⊢ Convex 𝕜 { x | x ∈ s ∧ f x ≤ r }
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/ConvexFunctions.lean | ConvexOn.convex_le' | [19, 1] | [28, 4] | constructor | 𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • x + b • y ∈ { x | x ∈ s ∧ f x ≤ r } | case left
𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • x + b • y ∈ s
case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ f (a • x + b • y) ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • x + b • y ∈ { x | x ∈ s ∧ f x ≤ r }
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/ConvexFunctions.lean | ConvexOn.convex_le' | [19, 1] | [28, 4] | {exact hf.1 hx.1 hy.1 ha hb hab} | case left
𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • x + b • y ∈ s
case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ f (a • x + b • y) ≤ r | case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ f (a • x + b • y) ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
case left
𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • x + b • y ∈ s
case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ f (a • x + b • y) ≤ r
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/ConvexFunctions.lean | ConvexOn.convex_le' | [19, 1] | [28, 4] | {calc
f (a • x + b • y) ≤ a • f x + b • f y := hf.2 hx.1 hy.1 ha hb hab
_ ≤ a • r + b • r :=
(add_le_add (smul_le_smul_of_nonneg hx.2 ha) (smul_le_smul_of_nonneg hy.2 hb))
_ = r := Convex.combo_self hab r
} | case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ f (a • x + b • y) ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ f (a • x + b • y) ≤ r
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/ConvexFunctions.lean | ConvexOn.convex_le' | [19, 1] | [28, 4] | exact hf.1 hx.1 hy.1 ha hb hab | case left
𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • x + b • y ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • x + b • y ∈ s
TACTIC:
|
https://github.com/goens/lean-power-calc.git | b68b0f976d150a157030b01d1437a8e9cbf660fd | PowerCalc/Examples/ConvexFunctions.lean | ConvexOn.convex_le' | [19, 1] | [28, 4] | calc
f (a • x + b • y) ≤ a • f x + b • f y := hf.2 hx.1 hy.1 ha hb hab
_ ≤ a • r + b • r :=
(add_le_add (smul_le_smul_of_nonneg hx.2 ha) (smul_le_smul_of_nonneg hy.2 hb))
_ = r := Convex.combo_self hab r | case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ f (a • x + b • y) ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type u_1
E : Type u_2
F : Type ?u.19360
α : Type ?u.19363
β : Type u_3
ι : Type ?u.19369
inst✝⁹ : OrderedSemiring 𝕜
inst✝⁸ : AddCommMonoid E
inst✝⁷ : AddCommMonoid F
inst✝⁶ : OrderedAddCommMonoid α
inst✝⁵ : OrderedAddCommMonoid β
inst✝⁴ : SMul 𝕜 E
inst✝³ : DistribMulAction 𝕜 β
s✝ : Set E
f✝ g : E → β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
r : β
x : E
hx : x ∈ { x | x ∈ s ∧ f x ≤ r }
y : E
hy : y ∈ { x | x ∈ s ∧ f x ≤ r }
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ f (a • x + b • y) ≤ r
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/degree.lean | degree_distribution | [21, 1] | [22, 129] | sorry | V : Type
Ω : Type u_3
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
p : NNReal
hp : p ≤ 1
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
x✝ : Sort u_1
Degree : x✝
h : sorryAx (Sort u_2) true
v : V
k : ℕ
⊢ ↑↑μ {ω | sorryAx ℕ true = k} = ↑(p ^ k) * ↑(p ^ (Fintype.card V - 1 - k)) * ↑(Nat.choose (Fintype.card V - 1) k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
Ω : Type u_3
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
p : NNReal
hp : p ≤ 1
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
x✝ : Sort u_1
Degree : x✝
h : sorryAx (Sort u_2) true
v : V
k : ℕ
⊢ ↑↑μ {ω | sorryAx ℕ true = k} = ↑(p ^ k) * ↑(p ^ (Fintype.card V - 1 - k)) * ↑(Nat.choose (Fintype.card V - 1) k)
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/degree.lean | expected_degree | [26, 1] | [26, 141] | sorry | V : Type
Ω : Type u_3
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
p : NNReal
hp : p ≤ 1
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
x✝ : Sort u_1
Degree : x✝
h : sorryAx (Sort u_2) true
v : V
⊢ ∫ (ω : Ω), sorryAx ℝ true ∂μ = (↑(Fintype.card V) - 1) * ↑p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
Ω : Type u_3
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
p : NNReal
hp : p ≤ 1
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
x✝ : Sort u_1
Degree : x✝
h : sorryAx (Sort u_2) true
v : V
⊢ ∫ (ω : Ω), sorryAx ℝ true ∂μ = (↑(Fintype.card V) - 1) * ↑p
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/bernoulli.lean | bernoulli_expectation | [25, 1] | [30, 43] | rw [←SimpleFunc.integral_eq_integral B (integrable B μ)] | Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
B : SimpleFunc Ω ℝ
p : ℝ≥0
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
h : Bernoulli B p μ
⊢ ∫ (ω : Ω), ↑B ω ∂μ = ↑p | Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
B : SimpleFunc Ω ℝ
p : ℝ≥0
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
h : Bernoulli B p μ
⊢ SimpleFunc.integral μ B = ↑p | Please generate a tactic in lean4 to solve the state.
STATE:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
B : SimpleFunc Ω ℝ
p : ℝ≥0
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
h : Bernoulli B p μ
⊢ ∫ (ω : Ω), ↑B ω ∂μ = ↑p
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/bernoulli.lean | bernoulli_expectation | [25, 1] | [30, 43] | simp only [SimpleFunc.integral_eq, h.range, Finset.mem_singleton, zero_ne_one, smul_eq_mul,
not_false_eq_true, Finset.sum_insert, mul_zero, Finset.sum_singleton, h.prob_1,
ENNReal.coe_toReal, mul_one, zero_add] | Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
B : SimpleFunc Ω ℝ
p : ℝ≥0
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
h : Bernoulli B p μ
⊢ SimpleFunc.integral μ B = ↑p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
B : SimpleFunc Ω ℝ
p : ℝ≥0
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
h : Bernoulli B p μ
⊢ SimpleFunc.integral μ B = ↑p
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/erdos_renyi.lean | edge_ind_finite_range | [22, 1] | [30, 37] | apply @Set.Finite.subset ℝ {0,1} _ (Set.range fun ω => (EdgeInd' (G ω) e : ℝ)) _ | V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
⊢ Set.Finite (Set.range fun ω => ↑(EdgeInd' (G ω) e)) | V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
⊢ Set.Finite {0, 1}
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
⊢ (Set.range fun ω => ↑(EdgeInd' (G ω) e)) ⊆ {0, 1} | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
⊢ Set.Finite (Set.range fun ω => ↑(EdgeInd' (G ω) e))
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/erdos_renyi.lean | edge_ind_finite_range | [22, 1] | [30, 37] | simp only [Set.mem_singleton_iff, zero_ne_one, Set.finite_singleton, Set.Finite.insert] | V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
⊢ Set.Finite {0, 1}
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
⊢ (Set.range fun ω => ↑(EdgeInd' (G ω) e)) ⊆ {0, 1} | V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
⊢ (Set.range fun ω => ↑(EdgeInd' (G ω) e)) ⊆ {0, 1} | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
⊢ Set.Finite {0, 1}
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
⊢ (Set.range fun ω => ↑(EdgeInd' (G ω) e)) ⊆ {0, 1}
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/erdos_renyi.lean | edge_ind_finite_range | [22, 1] | [30, 37] | simp [EdgeInd'] | V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
⊢ (Set.range fun ω => ↑(EdgeInd' (G ω) e)) ⊆ {0, 1} | V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
⊢ (Set.range fun ω => if Edge (G ω) e then 1 else 0) ⊆ {0, 1} | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
⊢ (Set.range fun ω => ↑(EdgeInd' (G ω) e)) ⊆ {0, 1}
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/erdos_renyi.lean | edge_ind_finite_range | [22, 1] | [30, 37] | rintro x ⟨ω, hx⟩ | V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
⊢ (Set.range fun ω => if Edge (G ω) e then 1 else 0) ⊆ {0, 1} | case intro
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
x : ℝ
ω : Ω
hx : (fun ω => if Edge (G ω) e then 1 else 0) ω = x
⊢ x ∈ {0, 1} | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
⊢ (Set.range fun ω => if Edge (G ω) e then 1 else 0) ⊆ {0, 1}
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/erdos_renyi.lean | edge_ind_finite_range | [22, 1] | [30, 37] | rw [← hx] | case intro
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
x : ℝ
ω : Ω
hx : (fun ω => if Edge (G ω) e then 1 else 0) ω = x
⊢ x ∈ {0, 1} | case intro
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
x : ℝ
ω : Ω
hx : (fun ω => if Edge (G ω) e then 1 else 0) ω = x
⊢ (fun ω => if Edge (G ω) e then 1 else 0) ω ∈ {0, 1} | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
x : ℝ
ω : Ω
hx : (fun ω => if Edge (G ω) e then 1 else 0) ω = x
⊢ x ∈ {0, 1}
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/erdos_renyi.lean | edge_ind_finite_range | [22, 1] | [30, 37] | simp only [Set.mem_singleton_iff, zero_ne_one, Set.mem_insert_iff, ite_eq_right_iff, one_ne_zero,
ite_eq_left_iff] | case intro
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
x : ℝ
ω : Ω
hx : (fun ω => if Edge (G ω) e then 1 else 0) ω = x
⊢ (fun ω => if Edge (G ω) e then 1 else 0) ω ∈ {0, 1} | case intro
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
x : ℝ
ω : Ω
hx : (fun ω => if Edge (G ω) e then 1 else 0) ω = x
⊢ (Edge (G ω) e → False) ∨ (¬Edge (G ω) e → False) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
x : ℝ
ω : Ω
hx : (fun ω => if Edge (G ω) e then 1 else 0) ω = x
⊢ (fun ω => if Edge (G ω) e then 1 else 0) ω ∈ {0, 1}
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/erdos_renyi.lean | edge_ind_finite_range | [22, 1] | [30, 37] | exact Classical.em (¬Edge (G ω) e) | case intro
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
x : ℝ
ω : Ω
hx : (fun ω => if Edge (G ω) e then 1 else 0) ω = x
⊢ (Edge (G ω) e → False) ∨ (¬Edge (G ω) e → False) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
V : Type
inst✝² : Fintype V
inst✝¹ : DecidableEq V
Ω : Sort u_1
G : Ω → SimpleGraph V
e : EdgeType V
inst✝ : (ω : Ω) → Decidable (Edge (G ω) e)
x : ℝ
ω : Ω
hx : (fun ω => if Edge (G ω) e then 1 else 0) ω = x
⊢ (Edge (G ω) e → False) ∨ (¬Edge (G ω) e → False)
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/erdos_renyi.lean | expected_edge_count | [57, 1] | [62, 90] | simp only [NumEdges, SimpleFunc.coe_sum, Finset.sum_apply] | V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∫ (ω : Ω), ↑(NumEdges G (_ : ∀ (e : EdgeType V) (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) e) = x})) ω ∂μ =
↑(Nat.choose n 2) * ↑p | V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∫ (ω : Ω), ∑ c : EdgeType V, ↑(EdgeInd G (_ : ∀ (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) c) = x})) ω ∂μ =
↑(Nat.choose n 2) * ↑p | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∫ (ω : Ω), ↑(NumEdges G (_ : ∀ (e : EdgeType V) (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) e) = x})) ω ∂μ =
↑(Nat.choose n 2) * ↑p
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/erdos_renyi.lean | expected_edge_count | [57, 1] | [62, 90] | rw [integral_finset_sum] | V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∫ (ω : Ω), ∑ c : EdgeType V, ↑(EdgeInd G (_ : ∀ (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) c) = x})) ω ∂μ =
↑(Nat.choose n 2) * ↑p | V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∑ i : EdgeType V, ∫ (a : Ω), ↑(EdgeInd G (_ : ∀ (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) i) = x})) a ∂μ =
↑(Nat.choose n 2) * ↑p
case hf
V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∀ i ∈ Finset.univ, Integrable fun ω => ↑(EdgeInd G (_ : ∀ (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) i) = x})) ω | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∫ (ω : Ω), ∑ c : EdgeType V, ↑(EdgeInd G (_ : ∀ (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) c) = x})) ω ∂μ =
↑(Nat.choose n 2) * ↑p
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/erdos_renyi.lean | expected_edge_count | [57, 1] | [62, 90] | simp only [bernoulli_expectation (h.bernoulli_edges), Finset.sum_const, Finset.card_univ, Fintype.card, nsmul_eq_mul] | V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∑ i : EdgeType V, ∫ (a : Ω), ↑(EdgeInd G (_ : ∀ (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) i) = x})) a ∂μ =
↑(Nat.choose n 2) * ↑p
case hf
V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∀ i ∈ Finset.univ, Integrable fun ω => ↑(EdgeInd G (_ : ∀ (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) i) = x})) ω | V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ↑(Fintype.card (EdgeType V)) * ↑p = ↑(Nat.choose n 2) * ↑p
case hf
V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∀ i ∈ Finset.univ, Integrable fun ω => ↑(EdgeInd G (_ : ∀ (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) i) = x})) ω | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∑ i : EdgeType V, ∫ (a : Ω), ↑(EdgeInd G (_ : ∀ (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) i) = x})) a ∂μ =
↑(Nat.choose n 2) * ↑p
case hf
V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∀ i ∈ Finset.univ, Integrable fun ω => ↑(EdgeInd G (_ : ∀ (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) i) = x})) ω
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/erdos_renyi.lean | expected_edge_count | [57, 1] | [62, 90] | rw [Sym2.card_subtype_not_diag, hn] | V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ↑(Fintype.card (EdgeType V)) * ↑p = ↑(Nat.choose n 2) * ↑p
case hf
V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∀ i ∈ Finset.univ, Integrable fun ω => ↑(EdgeInd G (_ : ∀ (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) i) = x})) ω | case hf
V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∀ i ∈ Finset.univ, Integrable fun ω => ↑(EdgeInd G (_ : ∀ (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) i) = x})) ω | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ↑(Fintype.card (EdgeType V)) * ↑p = ↑(Nat.choose n 2) * ↑p
case hf
V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∀ i ∈ Finset.univ, Integrable fun ω => ↑(EdgeInd G (_ : ∀ (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) i) = x})) ω
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/erdos_renyi.lean | expected_edge_count | [57, 1] | [62, 90] | simp only [Finset.mem_univ, integrable, forall_true_left, Subtype.forall, implies_true] | case hf
V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∀ i ∈ Finset.univ, Integrable fun ω => ↑(EdgeInd G (_ : ∀ (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) i) = x})) ω | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
V : Type
Ω : Type u_1
inst✝⁴ : Fintype V
inst✝³ : DecidableEq V
inst✝² : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
G : Ω → SimpleGraph V
inst✝ : (ω : Ω) → (e : EdgeType V) → Decidable (Edge (G ω) e)
p : NNReal
n : ℕ
h : ErdosRenyi G p μ
hn : Fintype.card V = n
⊢ ∀ i ∈ Finset.univ, Integrable fun ω => ↑(EdgeInd G (_ : ∀ (x : ℝ), MeasurableSet {ω | ↑(EdgeInd' (G ω) i) = x})) ω
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/simple_func.lean | SimpleFunc.coe_prod | [9, 1] | [11, 55] | induction s using Finset.cons_induction <;> simp [*] | ι : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : CommMonoid β
f : ι → SimpleFunc α β
s : Finset ι
⊢ ↑(∏ i in s, f i) = ∏ i in s, ↑(f i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : CommMonoid β
f : ι → SimpleFunc α β
s : Finset ι
⊢ ↑(∏ i in s, f i) = ∏ i in s, ↑(f i)
TACTIC:
|
https://github.com/Vilin97/random_graphs.git | 1a403704fb98abab33c848a08ca8c570585f24b0 | RandomGraphs/simple_func.lean | SimpleFunc.coe_sum' | [13, 1] | [14, 22] | simp only [coe_sum] | ι : Type u_1
α : Type u_2
β : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : CommMonoid β
I : Type u_4
f : I → SimpleFunc α ℝ
inst✝ : Fintype I
⊢ ↑(∑ i : I, f i) = ∑ i : I, ↑(f i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : CommMonoid β
I : Type u_4
f : I → SimpleFunc α ℝ
inst✝ : Fintype I
⊢ ↑(∑ i : I, f i) = ∑ i : I, ↑(f i)
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | l1_1_3_i | [18, 1] | [19, 39] | rw [← @self_eq_mul_left _ _ _ f, h1] | M : Type u_1
A : Type u_2
G : Type u_3
inst✝ : Group G
a b c f : G
h1 : ∀ (g : G), f * g = g
_h2 : ∀ (g : G), g * f = g
⊢ f = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
A : Type u_2
G : Type u_3
inst✝ : Group G
a b c f : G
h1 : ∀ (g : G), f * g = g
_h2 : ∀ (g : G), g * f = g
⊢ f = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | l1_1_5_i | [27, 1] | [29, 30] | intro h | M : Type u_1
A : Type u_2
G : Type u_3
inst✝ : Group G
a b c x y z : G
⊢ x * y = x * z → y = z | M : Type u_1
A : Type u_2
G : Type u_3
inst✝ : Group G
a b c x y z : G
h : x * y = x * z
⊢ y = z | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
A : Type u_2
G : Type u_3
inst✝ : Group G
a b c x y z : G
⊢ x * y = x * z → y = z
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | l1_1_5_i | [27, 1] | [29, 30] | rwa [← mul_left_cancel_iff] | M : Type u_1
A : Type u_2
G : Type u_3
inst✝ : Group G
a b c x y z : G
h : x * y = x * z
⊢ y = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
A : Type u_2
G : Type u_3
inst✝ : Group G
a b c x y z : G
h : x * y = x * z
⊢ y = z
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | l1_1_5_ii | [31, 1] | [33, 31] | intro h | M : Type u_1
A : Type u_2
G : Type u_3
inst✝ : Group G
a b c x y z : G
⊢ x * z = y * z → x = y | M : Type u_1
A : Type u_2
G : Type u_3
inst✝ : Group G
a b c x y z : G
h : x * z = y * z
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
A : Type u_2
G : Type u_3
inst✝ : Group G
a b c x y z : G
⊢ x * z = y * z → x = y
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | l1_1_5_ii | [31, 1] | [33, 31] | rwa [← mul_right_cancel_iff] | M : Type u_1
A : Type u_2
G : Type u_3
inst✝ : Group G
a b c x y z : G
h : x * z = y * z
⊢ x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
A : Type u_2
G : Type u_3
inst✝ : Group G
a b c x y z : G
h : x * z = y * z
⊢ x = y
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex112.one_mul | [132, 1] | [150, 49] | rw [h2, h1] at this | G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
this✝ : a * (a⁻¹ * a) * a⁻¹ = a * (a * a⁻¹) * a⁻¹
this : a * (a⁻¹ * a) = a * (a * a⁻¹)
⊢ 1 * a = a | G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
this✝ : a * (a⁻¹ * a) * a⁻¹ = a * (a * a⁻¹) * a⁻¹
this : a * (a⁻¹ * a) = a
⊢ 1 * a = a | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
this✝ : a * (a⁻¹ * a) * a⁻¹ = a * (a * a⁻¹) * a⁻¹
this : a * (a⁻¹ * a) = a * (a * a⁻¹)
⊢ 1 * a = a
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex112.one_mul | [132, 1] | [150, 49] | calc
a * (a⁻¹ * a) * a⁻¹ = a * a⁻¹ * (a * a⁻¹) := ?_
_ = a * a⁻¹ * 1 := ?_
_ = a * 1 * a⁻¹ := ?_
_ = a * (a * a⁻¹) * a⁻¹ := ?_ | G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * (a⁻¹ * a) * a⁻¹ = a * (a * a⁻¹) * a⁻¹ | case calc_1
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * (a⁻¹ * a) * a⁻¹ = a * a⁻¹ * (a * a⁻¹)
case calc_2
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * a⁻¹ * (a * a⁻¹) = a * a⁻¹ * 1
case calc_3
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * a⁻¹ * 1 = a * 1 * a⁻¹
case calc_4
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * 1 * a⁻¹ = a * (a * a⁻¹) * a⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * (a⁻¹ * a) * a⁻¹ = a * (a * a⁻¹) * a⁻¹
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex112.one_mul | [132, 1] | [150, 49] | rw [h2] | case calc_2
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * a⁻¹ * (a * a⁻¹) = a * a⁻¹ * 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case calc_2
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * a⁻¹ * (a * a⁻¹) = a * a⁻¹ * 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex112.one_mul | [132, 1] | [150, 49] | rw [h1, h1] | case calc_3
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * a⁻¹ * 1 = a * 1 * a⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case calc_3
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * a⁻¹ * 1 = a * 1 * a⁻¹
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex112.one_mul | [132, 1] | [150, 49] | rw [h2] | case calc_4
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * 1 * a⁻¹ = a * (a * a⁻¹) * a⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case calc_4
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * 1 * a⁻¹ = a * (a * a⁻¹) * a⁻¹
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex112.one_mul | [132, 1] | [150, 49] | simp only at this | G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
this✝ : a * (a⁻¹ * a) * a⁻¹ = a * (a * a⁻¹) * a⁻¹
this : (fun x => x * a⁻¹⁻¹) (a * (a⁻¹ * a) * a⁻¹) = (fun x => x * a⁻¹⁻¹) (a * (a * a⁻¹) * a⁻¹)
⊢ a * (a⁻¹ * a) = a * (a * a⁻¹) | G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
this✝ : a * (a⁻¹ * a) * a⁻¹ = a * (a * a⁻¹) * a⁻¹
this : a * (a⁻¹ * a) * a⁻¹ * a⁻¹⁻¹ = a * (a * a⁻¹) * a⁻¹ * a⁻¹⁻¹
⊢ a * (a⁻¹ * a) = a * (a * a⁻¹) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
this✝ : a * (a⁻¹ * a) * a⁻¹ = a * (a * a⁻¹) * a⁻¹
this : (fun x => x * a⁻¹⁻¹) (a * (a⁻¹ * a) * a⁻¹) = (fun x => x * a⁻¹⁻¹) (a * (a * a⁻¹) * a⁻¹)
⊢ a * (a⁻¹ * a) = a * (a * a⁻¹)
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex112.one_mul | [132, 1] | [150, 49] | exact this | G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
this✝ : a * (a⁻¹ * a) * a⁻¹ = a * (a * a⁻¹) * a⁻¹
this : a * (a⁻¹ * a) = a * (a * a⁻¹)
⊢ a * (a⁻¹ * a) = a * (a * a⁻¹) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
this✝ : a * (a⁻¹ * a) * a⁻¹ = a * (a * a⁻¹) * a⁻¹
this : a * (a⁻¹ * a) = a * (a * a⁻¹)
⊢ a * (a⁻¹ * a) = a * (a * a⁻¹)
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex112.mul_left_inv | [152, 1] | [162, 37] | calc
a⁻¹ * a = a⁻¹ * a⁻¹⁻¹ := ?_
_ = 1 := h2 a⁻¹ | G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a⁻¹ * a = 1 | G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a⁻¹ * a = a⁻¹ * a⁻¹⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a⁻¹ * a = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex112.mul_left_inv | [152, 1] | [162, 37] | apply congr_arg | G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a⁻¹ * a = a⁻¹ * a⁻¹⁻¹ | case h
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a = a⁻¹⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a⁻¹ * a = a⁻¹ * a⁻¹⁻¹
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex112.mul_left_inv | [152, 1] | [162, 37] | calc
a = a * 1 := (h1 a).symm
_ = a * (a⁻¹ * a⁻¹⁻¹) := by rw [h2]
_ = a * a⁻¹ * a⁻¹⁻¹ := (Std.Associative.assoc a a⁻¹ a⁻¹⁻¹).symm
_ = 1 * a⁻¹⁻¹ := by rw [h2]
_ = a⁻¹⁻¹ := one_mul h1 h2 a⁻¹⁻¹ | case h
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a = a⁻¹⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a = a⁻¹⁻¹
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex112.mul_left_inv | [152, 1] | [162, 37] | rw [h2] | G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * 1 = a * (a⁻¹ * a⁻¹⁻¹) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * 1 = a * (a⁻¹ * a⁻¹⁻¹)
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex112.mul_left_inv | [152, 1] | [162, 37] | rw [h2] | G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * a⁻¹ * a⁻¹⁻¹ = 1 * a⁻¹⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : One G
inst✝² : Mul G
inst✝¹ : Inv G
inst✝ : Std.Associative fun x x_1 => x * x_1
h1 : ∀ (a : G), a * 1 = a
h2 : ∀ (a : G), a * a⁻¹ = 1
a : G
⊢ a * a⁻¹ * a⁻¹⁻¹ = 1 * a⁻¹⁻¹
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex113.mul_def | [179, 1] | [181, 6] | rw [norm_mul, a.prop, b.prop, one_mul] | a b : { z // ‖z‖ = 1 }
⊢ ‖↑a * ↑b‖ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : { z // ‖z‖ = 1 }
⊢ ‖↑a * ↑b‖ = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex113.inv_def | [189, 1] | [190, 6] | rw [norm_inv, a.prop, inv_one] | a : { z // ‖z‖ = 1 }
⊢ ‖(↑a)⁻¹‖ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : { z // ‖z‖ = 1 }
⊢ ‖(↑a)⁻¹‖ = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex113.ex1_1_3' | [209, 1] | [211, 76] | ext z | ⊢ {z | ‖z‖ = 1} = {x | ∃ θ, Complex.exp (↑θ * Complex.I) = x} | case h
z : ℂ
⊢ z ∈ {z | ‖z‖ = 1} ↔ z ∈ {x | ∃ θ, Complex.exp (↑θ * Complex.I) = x} | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ {z | ‖z‖ = 1} = {x | ∃ θ, Complex.exp (↑θ * Complex.I) = x}
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Groups.lean | Ex113.ex1_1_3' | [209, 1] | [211, 76] | simp only [Complex.norm_eq_abs, Set.mem_setOf_eq, Complex.abs_eq_one_iff] | case h
z : ℂ
⊢ z ∈ {z | ‖z‖ = 1} ↔ z ∈ {x | ∃ θ, Complex.exp (↑θ * Complex.I) = x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : ℂ
⊢ z ∈ {z | ‖z‖ = 1} ↔ z ∈ {x | ∃ θ, Complex.exp (↑θ * Complex.I) = x}
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_2_ba' | [23, 1] | [27, 94] | refine' ⟨_, Subgroup.coe_one _⟩ | G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ (∀ (a b : ↥H), ↑(a * b) = ↑a * ↑b) ∧ ↑1 = 1 | G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ ∀ (a b : ↥H), ↑(a * b) = ↑a * ↑b | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ (∀ (a b : ↥H), ↑(a * b) = ↑a * ↑b) ∧ ↑1 = 1
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_2_ba' | [23, 1] | [27, 94] | rintro ⟨a, ha⟩ ⟨b, hb⟩ | G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ ∀ (a b : ↥H), ↑(a * b) = ↑a * ↑b | case mk.mk
G : Type u_1
inst✝ : Group G
H : Subgroup G
a : G
ha : a ∈ H
b : G
hb : b ∈ H
⊢ ↑({ val := a, property := ha } * { val := b, property := hb }) =
↑{ val := a, property := ha } * ↑{ val := b, property := hb } | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ ∀ (a b : ↥H), ↑(a * b) = ↑a * ↑b
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_2_ba' | [23, 1] | [27, 94] | rw [MulMemClass.mk_mul_mk, Subgroup.coe_mk, Subgroup.coe_mk, Subgroup.coe_mk, mul_left_inj] | case mk.mk
G : Type u_1
inst✝ : Group G
H : Subgroup G
a : G
ha : a ∈ H
b : G
hb : b ∈ H
⊢ ↑({ val := a, property := ha } * { val := b, property := hb }) =
↑{ val := a, property := ha } * ↑{ val := b, property := hb } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
G : Type u_1
inst✝ : Group G
H : Subgroup G
a : G
ha : a ∈ H
b : G
hb : b ∈ H
⊢ ↑({ val := a, property := ha } * { val := b, property := hb }) =
↑{ val := a, property := ha } * ↑{ val := b, property := hb }
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_2_bc | [44, 1] | [49, 60] | refine' ⟨⟨1, SetLike.mem_coe.mpr (one_mem _)⟩, fun h₁ h₂ h₁mem h₂mem => _⟩ | G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ Set.Nonempty ↑H ∧ ∀ (h₁ h₂ : G), h₁ ∈ ↑H → h₂ ∈ ↑H → h₁ * h₂⁻¹ ∈ ↑H | G : Type u_1
inst✝ : Group G
H : Subgroup G
h₁ h₂ : G
h₁mem : h₁ ∈ ↑H
h₂mem : h₂ ∈ ↑H
⊢ h₁ * h₂⁻¹ ∈ ↑H | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ Set.Nonempty ↑H ∧ ∀ (h₁ h₂ : G), h₁ ∈ ↑H → h₂ ∈ ↑H → h₁ * h₂⁻¹ ∈ ↑H
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_2_bc | [44, 1] | [49, 60] | rw [SetLike.mem_coe] at h₁mem h₂mem ⊢ | G : Type u_1
inst✝ : Group G
H : Subgroup G
h₁ h₂ : G
h₁mem : h₁ ∈ ↑H
h₂mem : h₂ ∈ ↑H
⊢ h₁ * h₂⁻¹ ∈ ↑H | G : Type u_1
inst✝ : Group G
H : Subgroup G
h₁ h₂ : G
h₁mem : h₁ ∈ H
h₂mem : h₂ ∈ H
⊢ h₁ * h₂⁻¹ ∈ H | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
H : Subgroup G
h₁ h₂ : G
h₁mem : h₁ ∈ ↑H
h₂mem : h₂ ∈ ↑H
⊢ h₁ * h₂⁻¹ ∈ ↑H
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_2_bc | [44, 1] | [49, 60] | exact Subgroup.mul_mem _ h₁mem (Subgroup.inv_mem _ h₂mem) | G : Type u_1
inst✝ : Group G
H : Subgroup G
h₁ h₂ : G
h₁mem : h₁ ∈ H
h₂mem : h₂ ∈ H
⊢ h₁ * h₂⁻¹ ∈ H | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
H : Subgroup G
h₁ h₂ : G
h₁mem : h₁ ∈ H
h₂mem : h₂ ∈ H
⊢ h₁ * h₂⁻¹ ∈ H
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_3_2 | [81, 1] | [83, 40] | ext | n : ℤ
⊢ ↑(ex1232 n) = {k | n ∣ k} | case h
n x✝ : ℤ
⊢ x✝ ∈ ↑(ex1232 n) ↔ x✝ ∈ {k | n ∣ k} | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℤ
⊢ ↑(ex1232 n) = {k | n ∣ k}
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | l1_2_3_2 | [81, 1] | [83, 40] | simp [ex1232, Int.mem_zmultiples_iff] | case h
n x✝ : ℤ
⊢ x✝ ∈ ↑(ex1232 n) ↔ x✝ ∈ {k | n ∣ k} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n x✝ : ℤ
⊢ x✝ ∈ ↑(ex1232 n) ↔ x✝ ∈ {k | n ∣ k}
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_5' | [142, 1] | [171, 64] | simp only [Subgroup.closure, Subgroup.coe_sInf] | G : Type u_1
inst✝ : Group G
X : Set G
⊢ ↑(Subgroup.closure X) = {x | ∃ x_1 ∈ n125Set X, List.prod x_1 = x} | G : Type u_1
inst✝ : Group G
X : Set G
⊢ ⋂ s ∈ {K | X ⊆ ↑K}, ↑s = {x | ∃ x_1 ∈ n125Set X, List.prod x_1 = x} | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
X : Set G
⊢ ↑(Subgroup.closure X) = {x | ∃ x_1 ∈ n125Set X, List.prod x_1 = x}
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_5' | [142, 1] | [171, 64] | apply subset_antisymm | G : Type u_1
inst✝ : Group G
X : Set G
⊢ ⋂ s ∈ {K | X ⊆ ↑K}, ↑s = {x | ∃ x_1 ∈ n125Set X, List.prod x_1 = x} | case a
G : Type u_1
inst✝ : Group G
X : Set G
⊢ ⋂ s ∈ {K | X ⊆ ↑K}, ↑s ⊆ {x | ∃ x_1 ∈ n125Set X, List.prod x_1 = x}
case a
G : Type u_1
inst✝ : Group G
X : Set G
⊢ {x | ∃ x_1 ∈ n125Set X, List.prod x_1 = x} ⊆ ⋂ s ∈ {K | X ⊆ ↑K}, ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
X : Set G
⊢ ⋂ s ∈ {K | X ⊆ ↑K}, ↑s = {x | ∃ x_1 ∈ n125Set X, List.prod x_1 = x}
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_5' | [142, 1] | [171, 64] | intro x hx | case a
G : Type u_1
inst✝ : Group G
X : Set G
⊢ ⋂ s ∈ {K | X ⊆ ↑K}, ↑s ⊆ {x | ∃ x_1 ∈ n125Set X, List.prod x_1 = x} | case a
G : Type u_1
inst✝ : Group G
X : Set G
x : G
hx : x ∈ ⋂ s ∈ {K | X ⊆ ↑K}, ↑s
⊢ x ∈ {x | ∃ x_1 ∈ n125Set X, List.prod x_1 = x} | Please generate a tactic in lean4 to solve the state.
STATE:
case a
G : Type u_1
inst✝ : Group G
X : Set G
⊢ ⋂ s ∈ {K | X ⊆ ↑K}, ↑s ⊆ {x | ∃ x_1 ∈ n125Set X, List.prod x_1 = x}
TACTIC:
|
https://github.com/Ruben-VandeVelde/algebra-i.git | b4852b5538060d7d33a834a4c349d255e7e643dc | AlgebraI/Subgroups.lean | n1_2_5' | [142, 1] | [171, 64] | simp only [Set.mem_iInter] at hx | case a
G : Type u_1
inst✝ : Group G
X : Set G
x : G
hx : x ∈ ⋂ s ∈ {K | X ⊆ ↑K}, ↑s
⊢ x ∈ {x | ∃ x_1 ∈ n125Set X, List.prod x_1 = x} | case a
G : Type u_1
inst✝ : Group G
X : Set G
x : G
hx : ∀ i ∈ {K | X ⊆ ↑K}, x ∈ ↑i
⊢ x ∈ {x | ∃ x_1 ∈ n125Set X, List.prod x_1 = x} | Please generate a tactic in lean4 to solve the state.
STATE:
case a
G : Type u_1
inst✝ : Group G
X : Set G
x : G
hx : x ∈ ⋂ s ∈ {K | X ⊆ ↑K}, ↑s
⊢ x ∈ {x | ∃ x_1 ∈ n125Set X, List.prod x_1 = x}
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.