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import Mathlib.GroupTheory.GroupAction.Basic
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namespace Rubin
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variable {G α β : Type _} [Group G]
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variable [MulAction G α]
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theorem smul_congr (g : G) {x y : α} (h : x = y) : g • x = g • y :=
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congr_arg ((· • ·) g) h
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#align smul_congr Rubin.smul_congr
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theorem smul_eq_iff_inv_smul_eq {x : α} {g : G} : g • x = x ↔ g⁻¹ • x = x :=
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⟨fun h => (smul_congr g⁻¹ h).symm.trans (inv_smul_smul g x), fun h =>
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(smul_congr g h).symm.trans (smul_inv_smul g x)⟩
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#align smul_eq_iff_inv_smul_eq Rubin.smul_eq_iff_inv_smul_eq
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theorem smul_pow_eq_of_smul_eq {x : α} {g : G} (n : ℕ) :
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g • x = x → g ^ n • x = x := by
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induction n with
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| zero => simp only [pow_zero, one_smul, eq_self_iff_true, imp_true_iff]
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| succ n n_ih =>
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intro h
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nth_rw 2 [← (smul_congr g (n_ih h)).trans h]
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rw [← mul_smul, ← pow_succ]
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#align smul_pow_eq_of_smul_eq Rubin.smul_pow_eq_of_smul_eq
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theorem smul_zpow_eq_of_smul_eq {x : α} {g : G} (n : ℤ) :
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g • x = x → g ^ n • x = x := by
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intro h
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cases n with
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| ofNat n => let res := smul_pow_eq_of_smul_eq n h; simp; exact res
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| negSucc n =>
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let res :=
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smul_eq_iff_inv_smul_eq.mp (smul_pow_eq_of_smul_eq (1 + n) h);
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simp
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rw [add_comm]
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exact res
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#align smul_zpow_eq_of_smul_eq Rubin.smul_zpow_eq_of_smul_eq
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-- TODO: turn this into a structure?
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def is_equivariant (G : Type _) {β : Type _} [Group G] [MulAction G α]
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[MulAction G β] (f : α → β) :=
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∀ g : G, ∀ x : α, f (g • x) = g • f x
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#align is_equivariant Rubin.is_equivariant
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lemma disjoint_not_mem {α : Type _} {U V : Set α} (disj: Disjoint U V) :
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∀ {x : α}, x ∈ U → x ∉ V :=
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by
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intro x x_in_U x_in_V
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apply disj <;> try simp
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· exact Set.singleton_subset_iff.mpr x_in_U
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· rw [Set.singleton_subset_iff]
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exact x_in_V
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· rfl
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lemma disjoint_not_mem₂ {α : Type _} {U V : Set α} (disj: Disjoint U V) :
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∀ {x : α}, x ∈ V → x ∉ U := disjoint_not_mem disj.symm
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lemma fixes_inv {G α : Type _} [Group G] [MulAction G α] {g : G} {x : α}:
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g • x = x ↔ g⁻¹ • x = x :=
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by
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constructor
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{
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intro h
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nth_rw 1 [<-h]
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rw [<-mul_smul, mul_left_inv, one_smul]
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}
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{
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intro h
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nth_rw 1 [<-h]
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rw [<-mul_smul, mul_right_inv, one_smul]
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}
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lemma exists_smul_ne {G : Type _} (α : Type _) [Group G] [MulAction G α] [h_f : FaithfulSMul G α]
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{f g : G} (f_ne_g : f ≠ g) : ∃ (x : α), f • x ≠ g • x :=
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by
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by_contra h
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rw [Classical.not_exists_not] at h
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apply f_ne_g
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apply h_f.eq_of_smul_eq_smul
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exact h
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end Rubin
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