declaration stringlengths 27 11.3k | file stringlengths 52 114 | context dict | tactic_states listlengths 1 1.24k |
|---|---|---|---|
lemma isUnit_iff : IsUnit u ↔ u = 1 ∨ u = -1 := by
refine ⟨fun h ↦ isUnit_eq_one_or h, fun h ↦ ?_⟩
rcases h with (rfl | rfl)
· exact isUnit_one
· exact ⟨⟨-1, -1, by decide, by decide⟩, rfl⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Int/Units.lean | {
"open": [
"Nat"
],
"variables": [
"{u v : ℤ}"
]
} | [
{
"line": "refine ⟨fun h ↦ isUnit_eq_one_or h, fun h ↦ ?_⟩",
"before_state": "u : ℤ\n⊢ IsUnit u ↔ u = 1 ∨ u = -1",
"after_state": "u : ℤ\nh : u = 1 ∨ u = -1\n⊢ IsUnit u"
},
{
"line": "rcases h with (rfl | rfl)",
"before_state": "u : ℤ\nh : u = 1 ∨ u = -1\n⊢ IsUnit u",
"after_state": "cas... |
lemma eq_one_or_neg_one_of_mul_eq_one' (h : u * v = 1) : u = 1 ∧ v = 1 ∨ u = -1 ∧ v = -1 := by
have h' : v * u = 1 := mul_comm u v ▸ h
obtain rfl | rfl := eq_one_or_neg_one_of_mul_eq_one h <;>
obtain rfl | rfl := eq_one_or_neg_one_of_mul_eq_one h' <;> tauto
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Int/Units.lean | {
"open": [
"Nat"
],
"variables": [
"{u v : ℤ}"
]
} | [
{
"line": "have h' : v * u = 1 := mul_comm u v ▸ h",
"before_state": "u v : ℤ\nh : u * v = 1\n⊢ u = 1 ∧ v = 1 ∨ u = -1 ∧ v = -1",
"after_state": "u v : ℤ\nh : u * v = 1\nh' : v * u = 1\n⊢ u = 1 ∧ v = 1 ∨ u = -1 ∧ v = -1"
},
{
"line": "refine_lift\n have h' : v * u = 1 := mul_comm u v ▸ h;\n ?_... |
lemma mul_eq_one_iff_eq_one_or_neg_one : u * v = 1 ↔ u = 1 ∧ v = 1 ∨ u = -1 ∧ v = -1 := by
refine ⟨eq_one_or_neg_one_of_mul_eq_one', fun h ↦ Or.elim h (fun H ↦ ?_) fun H ↦ ?_⟩ <;>
obtain ⟨rfl, rfl⟩ := H <;> rfl
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Int/Units.lean | {
"open": [
"Nat"
],
"variables": [
"{u v : ℤ}"
]
} | [
{
"line": "focus\n refine ⟨eq_one_or_neg_one_of_mul_eq_one', fun h ↦ Or.elim h (fun H ↦ ?_) fun H ↦ ?_⟩ <;> obtain ⟨rfl, rfl⟩ := H\n with_annotate_state\"<;>\" skip\n all_goals rfl",
"before_state": "u v : ℤ\n⊢ u * v = 1 ↔ u = 1 ∧ v = 1 ∨ u = -1 ∧ v = -1",
"after_state": "No Goals!"
},
{
"lin... |
lemma eq_one_or_neg_one_of_mul_eq_neg_one' (h : u * v = -1) : u = 1 ∧ v = -1 ∨ u = -1 ∧ v = 1 := by
obtain rfl | rfl := isUnit_eq_one_or (IsUnit.mul_iff.mp (Int.isUnit_iff.mpr (Or.inr h))).1
· exact Or.inl ⟨rfl, one_mul v ▸ h⟩
· simpa [Int.neg_mul] using h
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Int/Units.lean | {
"open": [
"Nat"
],
"variables": [
"{u v : ℤ}"
]
} | [
{
"line": "obtain rfl | rfl := isUnit_eq_one_or (IsUnit.mul_iff.mp (Int.isUnit_iff.mpr (Or.inr h))).1",
"before_state": "u v : ℤ\nh : u * v = -1\n⊢ u = 1 ∧ v = -1 ∨ u = -1 ∧ v = 1",
"after_state": "No Goals!"
}
] |
lemma mul_eq_neg_one_iff_eq_one_or_neg_one : u * v = -1 ↔ u = 1 ∧ v = -1 ∨ u = -1 ∧ v = 1 := by
refine ⟨eq_one_or_neg_one_of_mul_eq_neg_one', fun h ↦ Or.elim h (fun H ↦ ?_) fun H ↦ ?_⟩ <;>
obtain ⟨rfl, rfl⟩ := H <;> rfl
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Int/Units.lean | {
"open": [
"Nat"
],
"variables": [
"{u v : ℤ}"
]
} | [
{
"line": "focus\n refine ⟨eq_one_or_neg_one_of_mul_eq_neg_one', fun h ↦ Or.elim h (fun H ↦ ?_) fun H ↦ ?_⟩ <;> obtain ⟨rfl, rfl⟩ := H\n with_annotate_state\"<;>\" skip\n all_goals rfl",
"before_state": "u v : ℤ\n⊢ u * v = -1 ↔ u = 1 ∧ v = -1 ∨ u = -1 ∧ v = 1",
"after_state": "No Goals!"
},
{
... |
lemma even_iff : Even n ↔ n % 2 = 0 where
mp := fun ⟨m, hm⟩ ↦ by simp [← Int.two_mul, hm]
mpr h := ⟨n / 2, (emod_add_ediv n 2).symm.trans (by simp [← Int.two_mul, h])⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Int/Even.lean | {
"open": [
"Nat"
],
"variables": [
"{m n : ℤ}"
]
} | [
{
"line": "simp [← Int.two_mul, hm]",
"before_state": "n : ℤ\nx✝ : Even n\nm : ℤ\nhm : n = m + m\n⊢ n % 2 = 0",
"after_state": "No Goals!"
}
] |
lemma two_not_dvd_two_mul_add_one (n : ℤ) : ¬2 ∣ 2 * n + 1 := by simp [add_emod]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Int/Even.lean | {
"open": [
"Nat"
],
"variables": [
"{m n : ℤ}"
]
} | [
{
"line": "simp [add_emod]",
"before_state": "n : ℤ\n⊢ ¬2 ∣ 2 * n + 1",
"after_state": "No Goals!"
}
] |
lemma even_sub : Even (m - n) ↔ (Even m ↔ Even n) := by simp [sub_eq_add_neg, parity_simps]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Int/Even.lean | {
"open": [
"Nat"
],
"variables": [
"{m n : ℤ}"
]
} | [
{
"line": "simp [sub_eq_add_neg, parity_simps]",
"before_state": "m n : ℤ\n⊢ Even (m - n) ↔ (Even m ↔ Even n)",
"after_state": "No Goals!"
}
] |
example (m n : ℤ) (h : Even m) : ¬Even (n + 3) ↔ Even (m ^ 2 + m + n) := by
simp +decide [*, (by decide : ¬2 = 0), parity_simps]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Int/Even.lean | {
"open": [
"Nat"
],
"variables": [
"{m n : ℤ}"
]
} | [
{
"line": "simp +decide [*, (by decide : ¬2 = 0), parity_simps]",
"before_state": "m✝ n✝ m n : ℤ\nh : Even m\n⊢ ¬Even (n + 3) ↔ Even (m ^ 2 + m + n)",
"after_state": "No Goals!"
},
{
"line": "decide",
"before_state": "m✝ n✝ m n : ℤ\nh : Even m\n⊢ ¬2 = 0",
"after_state": "No Goals!"
}
] |
example : ¬Even (25394535 : ℤ) := by decide
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Int/Even.lean | {
"open": [
"Nat"
],
"variables": [
"{m n : ℤ}"
]
} | [
{
"line": "decide",
"before_state": "m n : ℤ\n⊢ ¬Even 25394535",
"after_state": "No Goals!"
}
] |
lemma even_iff : Even n ↔ n % 2 = 0 where
mp := fun ⟨m, hm⟩ ↦ by simp [← Nat.two_mul, hm]
mpr h := ⟨n / 2, (mod_add_div n 2).symm.trans (by simp [← Nat.two_mul, h])⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Nat/Even.lean | {
"open": [],
"variables": [
"{m n : ℕ}"
]
} | [
{
"line": "simp [← Nat.two_mul, hm]",
"before_state": "n : ℕ\nx✝ : Even n\nm : ℕ\nhm : n = m + m\n⊢ n % 2 = 0",
"after_state": "No Goals!"
}
] |
lemma succ_mod_two_eq_zero_iff : (m + 1) % 2 = 0 ↔ m % 2 = 1 := by
simp [← Nat.even_iff, ← Nat.not_even_iff, parity_simps]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Nat/Even.lean | {
"open": [],
"variables": [
"{m n : ℕ}"
]
} | [
{
"line": "simp [← Nat.even_iff, ← Nat.not_even_iff, parity_simps]",
"before_state": "m : ℕ\n⊢ (m + 1) % 2 = 0 ↔ m % 2 = 1",
"after_state": "No Goals!"
}
] |
lemma succ_mod_two_eq_one_iff : (m + 1) % 2 = 1 ↔ m % 2 = 0 := by
simp [← Nat.even_iff, ← Nat.not_even_iff, parity_simps]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Nat/Even.lean | {
"open": [],
"variables": [
"{m n : ℕ}"
]
} | [
{
"line": "simp [← Nat.even_iff, ← Nat.not_even_iff, parity_simps]",
"before_state": "m : ℕ\n⊢ (m + 1) % 2 = 1 ↔ m % 2 = 0",
"after_state": "No Goals!"
}
] |
lemma two_not_dvd_two_mul_add_one (n : ℕ) : ¬2 ∣ 2 * n + 1 := by simp [add_mod]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Nat/Even.lean | {
"open": [],
"variables": [
"{m n : ℕ}"
]
} | [
{
"line": "simp [add_mod]",
"before_state": "n : ℕ\n⊢ ¬2 ∣ 2 * n + 1",
"after_state": "No Goals!"
}
] |
theorem one_lt_of_ne_zero_of_even (h0 : n ≠ 0) (hn : Even n) : 1 < n := by
refine Nat.one_lt_iff_ne_zero_and_ne_one.mpr (And.intro h0 ?_)
intro h
rw [h] at hn
exact Nat.not_even_one hn
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Nat/Even.lean | {
"open": [],
"variables": [
"{m n : ℕ}"
]
} | [
{
"line": "refine Nat.one_lt_iff_ne_zero_and_ne_one.mpr (And.intro h0 ?_)",
"before_state": "n : ℕ\nh0 : n ≠ 0\nhn : Even n\n⊢ 1 < n",
"after_state": "n : ℕ\nh0 : n ≠ 0\nhn : Even n\n⊢ n ≠ 1"
},
{
"line": "intro h",
"before_state": "n : ℕ\nh0 : n ≠ 0\nhn : Even n\n⊢ n ≠ 1",
"after_state"... |
theorem add_one_lt_of_even (hn : Even n) (hm : Even m) (hnm : n < m) :
n + 1 < m := by
rcases hn with ⟨n, rfl⟩
rcases hm with ⟨m, rfl⟩
omega
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Nat/Even.lean | {
"open": [],
"variables": [
"{m n : ℕ}"
]
} | [
{
"line": "rcases hn with ⟨n, rfl⟩",
"before_state": "m n : ℕ\nhn : Even n\nhm : Even m\nhnm : n < m\n⊢ n + 1 < m",
"after_state": "case intro\nm : ℕ\nhm : Even m\nn : ℕ\nhnm : n + n < m\n⊢ n + n + 1 < m"
},
{
"line": "rcases hm with ⟨m, rfl⟩",
"before_state": "case intro\nm : ℕ\nhm : Even m... |
example (m n : ℕ) (h : Even m) : ¬Even (n + 3) ↔ Even (m ^ 2 + m + n) := by simp [*, parity_simps]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Nat/Even.lean | {
"open": [],
"variables": [
"{m n : ℕ}"
]
} | [
{
"line": "simp [*, parity_simps]",
"before_state": "m✝ n✝ m n : ℕ\nh : Even m\n⊢ ¬Even (n + 3) ↔ Even (m ^ 2 + m + n)",
"after_state": "No Goals!"
}
] |
example : ¬Even 25394535 := by decide
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Nat/Even.lean | {
"open": [],
"variables": [
"{m n : ℕ}"
]
} | [
{
"line": "decide",
"before_state": "m n : ℕ\n⊢ ¬Even 25394535",
"after_state": "No Goals!"
}
] |
theorem extend_one [One γ] (f : α → β) : Function.extend f (1 : α → γ) (1 : β → γ) = 1 :=
funext fun _ => by apply ite_self
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pi/Basic.lean | {
"open": [
"Function"
],
"variables": [
"{I : Type u}",
"{α β γ : Type*}",
"{f : I → Type v₁} {g : I → Type v₂} {h : I → Type v₃}",
"(x y : ∀ i, f i) (i : I)",
"[DecidableEq I]",
"[∀ i, One (f i)] [∀ i, One (g i)] [∀ i, One (h i)]",
"(f)"
]
} | [
{
"line": "apply ite_self",
"before_state": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : One γ\nf : α → β\nx✝ : β\n⊢ extend f 1 1 x✝ = 1 x✝",
"after_state": "No Goals!"
}
] |
theorem extend_div [Div γ] (f : α → β) (g₁ g₂ : α → γ) (e₁ e₂ : β → γ) :
Function.extend f (g₁ / g₂) (e₁ / e₂) = Function.extend f g₁ e₁ / Function.extend f g₂ e₂ := by
classical
funext x
simp [Function.extend_def, apply_dite₂] | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pi/Basic.lean | {
"open": [
"Function"
],
"variables": [
"{I : Type u}",
"{α β γ : Type*}",
"{f : I → Type v₁} {g : I → Type v₂} {h : I → Type v₃}",
"(x y : ∀ i, f i) (i : I)",
"[DecidableEq I]",
"[∀ i, One (f i)] [∀ i, One (g i)] [∀ i, One (h i)]",
"(f)"
]
} | [
{
"line": "classical\nfunext x\nsimp [Function.extend_def, apply_dite₂]",
"before_state": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : Div γ\nf : α → β\ng₁ g₂ : α → γ\ne₁ e₂ : β → γ\n⊢ extend f (g₁ / g₂) (e₁ / e₂) = extend f g₁ e₁ / extend f g₂ e₂",
"after_state": "No Goals!"
},
{
"line": ... |
lemma comp_eq_one_iff [One β] [One γ] (f : α → β) {g : β → γ} (hg : Injective g) (hg0 : g 1 = 1) :
g ∘ f = 1 ↔ f = 1 := by
simpa [hg0, const_one] using comp_eq_const_iff 1 f hg
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pi/Basic.lean | {
"open": [
"Function"
],
"variables": [
"{I : Type u}",
"{α β γ : Type*}",
"{f : I → Type v₁} {g : I → Type v₂} {h : I → Type v₃}",
"(x y : ∀ i, f i) (i : I)",
"[DecidableEq I]",
"[∀ i, One (f i)] [∀ i, One (g i)] [∀ i, One (h i)]",
"(f)"
]
} | [
{
"line": "simpa [hg0, const_one] using comp_eq_const_iff 1 f hg",
"before_state": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : One β\ninst✝ : One γ\nf : α → β\ng : β → γ\nhg : Injective g\nhg0 : g 1 = 1\n⊢ g ∘ f = 1 ↔ f = 1",
"after_state": "No Goals!"
}
] |
theorem elim_mul_mul [Mul γ] : Sum.elim (a * a') (b * b') = Sum.elim a b * Sum.elim a' b' := by
ext x
cases x <;> rfl
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pi/Basic.lean | {
"open": [
"Function"
],
"variables": [
"{I : Type u}",
"{α β γ : Type*}",
"{f : I → Type v₁} {g : I → Type v₂} {h : I → Type v₃}",
"(x y : ∀ i, f i) (i : I)",
"[DecidableEq I]",
"[∀ i, One (f i)] [∀ i, One (g i)] [∀ i, One (h i)]",
"(f)",
"(a a' : α → γ) (b b' : β → γ)"
]
} | [
{
"line": "ext x",
"before_state": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\na a' : α → γ\nb b' : β → γ\ninst✝ : Mul γ\n⊢ Sum.elim (a * a') (b * b') = Sum.elim a b * Sum.elim a' b'",
"after_state": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\na a' : α → γ\nb b' : β → γ\ninst✝ : Mul γ\nx : α ⊕ β\n... |
theorem elim_div_div [Div γ] : Sum.elim (a / a') (b / b') = Sum.elim a b / Sum.elim a' b' := by
ext x
cases x <;> rfl
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pi/Basic.lean | {
"open": [
"Function"
],
"variables": [
"{I : Type u}",
"{α β γ : Type*}",
"{f : I → Type v₁} {g : I → Type v₂} {h : I → Type v₃}",
"(x y : ∀ i, f i) (i : I)",
"[DecidableEq I]",
"[∀ i, One (f i)] [∀ i, One (g i)] [∀ i, One (h i)]",
"(f)",
"(a a' : α → γ) (b b' : β → γ)"
]
} | [
{
"line": "ext x",
"before_state": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\na a' : α → γ\nb b' : β → γ\ninst✝ : Div γ\n⊢ Sum.elim (a / a') (b / b') = Sum.elim a b / Sum.elim a' b'",
"after_state": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\na a' : α → γ\nb b' : β → γ\ninst✝ : Div γ\nx : α ⊕ β\n... |
lemma mulSingle_comp_equiv {m n : Type*} [DecidableEq n] [DecidableEq m] [One α] (σ : n ≃ m)
(i : m) (x : α) : Pi.mulSingle i x ∘ σ = Pi.mulSingle (σ.symm i) x := by
ext x
aesop (add simp Pi.mulSingle_apply)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pi/Lemmas.lean | {
"open": [
"Pi"
],
"variables": [
"{ι α : Type*}",
"{I : Type u}",
"{f : I → Type v}",
"(i : I)",
"{α β : Type*} [Preorder α] [Preorder β]",
"(f)",
"[(i : I) → Mul (f i)]",
"(f)",
"[(i : I) → MulOneClass (f i)]",
"[DecidableEq I]",
"(f)",
"{f}",
"[∀ i, Mul ... | [
{
"line": "ext x",
"before_state": "α : Type u_3\ninst✝⁴ : Preorder α\ninst✝³ : One α\nm : Type u_5\nn : Type u_6\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq m\ninst✝ : One α\nσ : n ≃ m\ni : m\nx : α\n⊢ mulSingle i x ∘ ⇑σ = mulSingle (σ.symm i) x",
"after_state": "case h\nα : Type u_3\ninst✝⁴ : Preord... |
theorem curry_mulSingle [DecidableEq α] [∀ a, DecidableEq (β a)] [∀ a b, One (γ a b)]
(i : Σ a, β a) (x : γ i.1 i.2) :
Sigma.curry (Pi.mulSingle i x) = Pi.mulSingle i.1 (Pi.mulSingle i.2 x) := by
simp only [Pi.mulSingle]
simp only [Sigma.curry_update]
simp only [Sigma.curry_one]
simp only [Pi.one_apply]... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pi/Lemmas.lean | {
"open": [
"Pi"
],
"variables": [
"{ι α : Type*}",
"{I : Type u}",
"{f : I → Type v}",
"(i : I)",
"{α β : Type*} [Preorder α] [Preorder β]",
"(f)",
"[(i : I) → Mul (f i)]",
"(f)",
"[(i : I) → MulOneClass (f i)]",
"[DecidableEq I]",
"(f)",
"{f}",
"[∀ i, Mul ... | [
{
"line": "simp only [Pi.mulSingle]",
"before_state": "α : Type u_5\nβ : α → Type u_6\nγ : (a : α) → β a → Type u_7\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (β a)\ninst✝ : (a : α) → (b : β a) → One (γ a b)\ni : (a : α) × β a\nx : γ i.fst i.snd\n⊢ Sigma.curry (mulSingle i x) = mulSingle i.fst ... |
lemma dens_inv [Fintype α] (s : Finset α) : s⁻¹.dens = s.dens := by simp [dens]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Finset/Density.lean | {
"open": [
"scoped Pointwise"
],
"variables": [
"{α β : Type*}",
"[DecidableEq α] [InvolutiveInv α] {s : Finset α} {a : α} in"
]
} | [
{
"line": "simp [dens]",
"before_state": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : InvolutiveInv α\ninst✝ : Fintype α\ns : Finset α\n⊢ s⁻¹.dens = s.dens",
"after_state": "No Goals!"
}
] |
lemma dens_smul_finset [Fintype β] (a : α) (s : Finset β) : (a • s).dens = s.dens := by simp [dens]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Finset/Density.lean | {
"open": [
"scoped Pointwise"
],
"variables": [
"{α β : Type*}",
"[DecidableEq α] [InvolutiveInv α] {s : Finset α} {a : α} in",
"[DecidableEq β] [Group α] [MulAction α β] {s t : Finset β} {a : α} {b : β} in"
]
} | [
{
"line": "simp [dens]",
"before_state": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : DecidableEq α\ninst✝⁴ : InvolutiveInv α\ninst✝³ : DecidableEq β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ninst✝ : Fintype β\na : α\ns : Finset β\n⊢ (a • s).dens = s.dens",
"after_state": "No Goals!"
}
] |
lemma one_product_one [One β] : (1 ×ˢ 1 : Finset (α × β)) = 1 := by ext; simp [Prod.ext_iff]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Finset/Basic.lean | {
"open": [
"Function MulOpposite",
"scoped Pointwise"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Finset α} {a : α}"
]
} | [
{
"line": "ext",
"before_state": "α : Type u_2\nβ : Type u_3\ninst✝¹ : One α\ninst✝ : One β\n⊢ 1 ×ˢ 1 = 1",
"after_state": "case h\nα : Type u_2\nβ : Type u_3\ninst✝¹ : One α\ninst✝ : One β\na✝ : α × β\n⊢ a✝ ∈ 1 ×ˢ 1 ↔ a✝ ∈ 1"
},
{
"line": "simp [Prod.ext_iff]",
"before_state": "case h\nα : ... |
lemma subset_pow (hs : 1 ∈ s) (hn : n ≠ 0) : s ⊆ s ^ n := by
simpa using pow_subset_pow_right hs <| Nat.one_le_iff_ne_zero.2 hn
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Finset/Basic.lean | {
"open": [
"Function MulOpposite",
"scoped Pointwise",
"Pointwise",
"scoped Pointwise",
"Pointwise"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Finset α} {a : α}",
"[DecidableEq α] [Inv α] {s t : Finset α} {a : α}",
"[DecidableEq α] [InvolutiveInv α] {s : Finset α} {a... | [
{
"line": "simpa using pow_subset_pow_right hs <| Nat.one_le_iff_ne_zero.2 hn",
"before_state": "α : Type u_2\ninst✝¹² : One α\ninst✝¹¹ : DecidableEq α\ninst✝¹⁰ : Inv α\ninst✝⁹ : DecidableEq α\ninst✝⁸ : InvolutiveInv α\ninst✝⁷ : DecidableEq α\ninst✝⁶ : Mul α\ninst✝⁵ : DecidableEq α\ninst✝⁴ : Div α\ninst✝³ :... |
lemma inter_pow_subset : (s ∩ t) ^ n ⊆ s ^ n ∩ t ^ n := by apply subset_inter <;> gcongr <;> simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Finset/Basic.lean | {
"open": [
"Function MulOpposite",
"scoped Pointwise",
"Pointwise",
"scoped Pointwise",
"Pointwise"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Finset α} {a : α}",
"[DecidableEq α] [Inv α] {s t : Finset α} {a : α}",
"[DecidableEq α] [InvolutiveInv α] {s : Finset α} {a... | [
{
"line": "focus\n apply subset_inter <;> gcongr\n with_annotate_state\"<;>\" skip\n all_goals simp",
"before_state": "α : Type u_2\ninst✝¹² : One α\ninst✝¹¹ : DecidableEq α\ninst✝¹⁰ : Inv α\ninst✝⁹ : DecidableEq α\ninst✝⁸ : InvolutiveInv α\ninst✝⁷ : DecidableEq α\ninst✝⁶ : Mul α\ninst✝⁵ : DecidableEq α\... |
theorem coe_zpow (s : Finset α) : ∀ n : ℤ, ↑(s ^ n) = (s : Set α) ^ n
| Int.ofNat _ => coe_pow _ _
| Int.negSucc n => by
refine (coe_inv _).trans ?_
exact congr_arg Inv.inv (coe_pow _ _)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Finset/Basic.lean | {
"open": [
"Function MulOpposite",
"scoped Pointwise",
"Pointwise",
"scoped Pointwise",
"Pointwise",
"Pointwise"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Finset α} {a : α}",
"[DecidableEq α] [Inv α] {s t : Finset α} {a : α}",
"[DecidableEq α] [InvolutiveInv α] ... | [
{
"line": "refine (coe_inv _).trans ?_",
"before_state": "α : Type u_2\ninst✝¹⁴ : One α\ninst✝¹³ : DecidableEq α\ninst✝¹² : Inv α\ninst✝¹¹ : DecidableEq α\ninst✝¹⁰ : InvolutiveInv α\ninst✝⁹ : DecidableEq α\ninst✝⁸ : Mul α\ninst✝⁷ : DecidableEq α\ninst✝⁶ : Div α\ninst✝⁵ : DecidableEq α\ninst✝⁴ : CommSemigrou... |
theorem isUnit_iff : IsUnit s ↔ ∃ a, s = {a} ∧ IsUnit a := by
constructor
· rintro ⟨u, rfl⟩
obtain ⟨a, b, ha, hb, h⟩ := Finset.mul_eq_one_iff.1 u.mul_inv
refine ⟨a, ha, ⟨a, b, h, singleton_injective ?_⟩, rfl⟩
rw [← singleton_mul_singleton]
rw [← ha]
rw [← hb]
exact u.inv_mul
· rintro ⟨a, r... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Finset/Basic.lean | {
"open": [
"Function MulOpposite",
"scoped Pointwise",
"Pointwise",
"scoped Pointwise",
"Pointwise",
"Pointwise"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Finset α} {a : α}",
"[DecidableEq α] [Inv α] {s t : Finset α} {a : α}",
"[DecidableEq α] [InvolutiveInv α] ... | [
{
"line": "constructor",
"before_state": "α : Type u_2\ninst✝¹⁴ : One α\ninst✝¹³ : DecidableEq α\ninst✝¹² : Inv α\ninst✝¹¹ : DecidableEq α\ninst✝¹⁰ : InvolutiveInv α\ninst✝⁹ : DecidableEq α\ninst✝⁸ : Mul α\ninst✝⁷ : DecidableEq α\ninst✝⁶ : Div α\ninst✝⁵ : DecidableEq α\ninst✝⁴ : CommSemigroup α\ninst✝³ : Mu... |
theorem isUnit_iff_singleton_aux {α} [Group α] {s : Finset α} :
(∃ a, s = {a} ∧ IsUnit a) ↔ ∃ a, s = {a} := by
simp only [Group.isUnit]
simp only [and_true]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Finset/Basic.lean | {
"open": [
"Function MulOpposite",
"scoped Pointwise",
"Pointwise",
"scoped Pointwise",
"Pointwise",
"Pointwise"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Finset α} {a : α}",
"[DecidableEq α] [Inv α] {s t : Finset α} {a : α}",
"[DecidableEq α] [InvolutiveInv α] ... | [
{
"line": "simp only [Group.isUnit]",
"before_state": "α : Type u_5\ninst✝ : Group α\ns : Finset α\n⊢ (∃ a, s = {a} ∧ IsUnit a) ↔ ∃ a, s = {a}",
"after_state": "α : Type u_5\ninst✝ : Group α\ns : Finset α\n⊢ (∃ a, s = {a} ∧ True) ↔ ∃ a, s = {a}"
},
{
"line": "simp only [and_true]",
"before_s... |
lemma Nontrivial.mul_left : t.Nontrivial → s.Nonempty → (s * t).Nontrivial := by
rintro ⟨a, ha, b, hb, hab⟩ ⟨c, hc⟩
exact ⟨c * a, mul_mem_mul hc ha, c * b, mul_mem_mul hc hb, by simpa⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Finset/Basic.lean | {
"open": [
"Function MulOpposite",
"scoped Pointwise",
"Pointwise",
"scoped Pointwise",
"Pointwise",
"Pointwise"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Finset α} {a : α}",
"[DecidableEq α] [Inv α] {s t : Finset α} {a : α}",
"[DecidableEq α] [InvolutiveInv α] ... | [
{
"line": "rintro ⟨a, ha, b, hb, hab⟩ ⟨c, hc⟩",
"before_state": "α : Type u_2\ninst✝²¹ : One α\ninst✝²⁰ : DecidableEq α\ninst✝¹⁹ : Inv α\ninst✝¹⁸ : DecidableEq α\ninst✝¹⁷ : InvolutiveInv α\ninst✝¹⁶ : DecidableEq α\ninst✝¹⁵ : Mul α\ninst✝¹⁴ : DecidableEq α\ninst✝¹³ : Div α\ninst✝¹² : DecidableEq α\ninst✝¹¹ :... |
lemma Nontrivial.mul_right : s.Nontrivial → t.Nonempty → (s * t).Nontrivial := by
rintro ⟨a, ha, b, hb, hab⟩ ⟨c, hc⟩
exact ⟨a * c, mul_mem_mul ha hc, b * c, mul_mem_mul hb hc, by simpa⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Finset/Basic.lean | {
"open": [
"Function MulOpposite",
"scoped Pointwise",
"Pointwise",
"scoped Pointwise",
"Pointwise",
"Pointwise"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Finset α} {a : α}",
"[DecidableEq α] [Inv α] {s t : Finset α} {a : α}",
"[DecidableEq α] [InvolutiveInv α] ... | [
{
"line": "rintro ⟨a, ha, b, hb, hab⟩ ⟨c, hc⟩",
"before_state": "α : Type u_2\ninst✝²⁴ : One α\ninst✝²³ : DecidableEq α\ninst✝²² : Inv α\ninst✝²¹ : DecidableEq α\ninst✝²⁰ : InvolutiveInv α\ninst✝¹⁹ : DecidableEq α\ninst✝¹⁸ : Mul α\ninst✝¹⁷ : DecidableEq α\ninst✝¹⁶ : Div α\ninst✝¹⁵ : DecidableEq α\ninst✝¹⁴ :... |
lemma Nontrivial.pow (hs : s.Nontrivial) : ∀ {n}, n ≠ 0 → (s ^ n).Nontrivial
| 1, _ => by simpa
| n + 2, _ => by simpa [pow_succ] using (hs.pow n.succ_ne_zero).mul hs
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Finset/Basic.lean | {
"open": [
"Function MulOpposite",
"scoped Pointwise",
"Pointwise",
"scoped Pointwise",
"Pointwise",
"Pointwise"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Finset α} {a : α}",
"[DecidableEq α] [Inv α] {s t : Finset α} {a : α}",
"[DecidableEq α] [InvolutiveInv α] ... | [
{
"line": "simpa",
"before_state": "α : Type u_2\ninst✝²⁶ : One α\ninst✝²⁵ : DecidableEq α\ninst✝²⁴ : Inv α\ninst✝²³ : DecidableEq α\ninst✝²² : InvolutiveInv α\ninst✝²¹ : DecidableEq α\ninst✝²⁰ : Mul α\ninst✝¹⁹ : DecidableEq α\ninst✝¹⁸ : Div α\ninst✝¹⁷ : DecidableEq α\ninst✝¹⁶ : CommSemigroup α\ninst✝¹⁵ : M... |
theorem card_pow_eq_card_pow_card_univ [∀ k : ℕ, DecidablePred (· ∈ S ^ k)] :
∀ k, Fintype.card G ≤ k → Fintype.card (↥(S ^ k)) = Fintype.card (↥(S ^ Fintype.card G)) := by
have hG : 0 < Fintype.card G := Fintype.card_pos
rcases S.eq_empty_or_nonempty with (rfl | ⟨a, ha⟩)
· refine fun k hk ↦ Fintype.card_cong... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Finite.lean | {
"open": [
"Pointwise",
"Set"
],
"variables": [
"{F α β γ : Type*}",
"[One α]",
"[Mul α] {s t : Set α}",
"[Monoid α] {s t : Set α}",
"[SMul α β] {s : Set α} {t : Set β}",
"[SMul α β] {s : Set β} {a : α}",
"[VSub α β] {s t : Set β}",
"[Mul α] [IsLeftCancelMul α] [IsRightCan... | [
{
"line": "have hG : 0 < Fintype.card G := Fintype.card_pos",
"before_state": "G : Type u_5\ninst✝² : Group G\ninst✝¹ : Fintype G\nS : Set G\ninst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k\n⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)",
"after_state": ... |
lemma one_prod_one [One β] : (1 ×ˢ 1 : Set (α × β)) = 1 := by ext; simp [Prod.ext_iff]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | {
"open": [
"Function MulOpposite",
"Pointwise"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Set α} {a : α}"
]
} | [
{
"line": "ext",
"before_state": "α : Type u_2\nβ : Type u_3\ninst✝¹ : One α\ninst✝ : One β\n⊢ 1 ×ˢ 1 = 1",
"after_state": "case h\nα : Type u_2\nβ : Type u_3\ninst✝¹ : One α\ninst✝ : One β\nx✝ : α × β\n⊢ x✝ ∈ 1 ×ˢ 1 ↔ x✝ ∈ 1"
},
{
"line": "simp [Prod.ext_iff]",
"before_state": "case h\nα : ... |
lemma prod_mul_prod_comm [Mul β] (s₁ s₂: Set α) (t₁ t₂ : Set β) :
(s₁ ×ˢ t₁) * (s₂ ×ˢ t₂) = (s₁ * s₂) ×ˢ (t₁ * t₂) := by ext; simp [mem_mul]; aesop
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | {
"open": [
"Function MulOpposite",
"Pointwise",
"Pointwise",
"MulOpposite",
"Pointwise",
"MulOpposite"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Set α} {a : α}",
"{ι : Sort*} [Inv α] {s t : Set α} {a : α}",
"[InvolutiveInv α] {s t : Set α} {a : α}",
"{ι : So... | [
{
"line": "ext",
"before_state": "α : Type u_2\nβ : Type u_3\ninst✝⁴ : One α\ninst✝³ : Inv α\ninst✝² : InvolutiveInv α\ninst✝¹ : Mul α\ninst✝ : Mul β\ns₁ s₂ : Set α\nt₁ t₂ : Set β\n⊢ s₁ ×ˢ t₁ * s₂ ×ˢ t₂ = (s₁ * s₂) ×ˢ (t₁ * t₂)",
"after_state": "case h\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : One α\ninst✝³... |
lemma subset_pow (hs : 1 ∈ s) (hn : n ≠ 0) : s ⊆ s ^ n := by
simpa using pow_subset_pow_right hs <| Nat.one_le_iff_ne_zero.2 hn
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | {
"open": [
"Function MulOpposite",
"Pointwise",
"Pointwise",
"MulOpposite",
"Pointwise",
"MulOpposite"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Set α} {a : α}",
"{ι : Sort*} [Inv α] {s t : Set α} {a : α}",
"[InvolutiveInv α] {s t : Set α} {a : α}",
"{ι : So... | [
{
"line": "simpa using pow_subset_pow_right hs <| Nat.one_le_iff_ne_zero.2 hn",
"before_state": "α : Type u_2\ninst✝⁷ : One α\ninst✝⁶ : Inv α\ninst✝⁵ : InvolutiveInv α\ninst✝⁴ : Mul α\ninst✝³ : Div α\ninst✝² : CommSemigroup α\ninst✝¹ : MulOneClass α\ninst✝ : Monoid α\ns : Set α\nn : ℕ\nhs : 1 ∈ s\nhn : n ≠ ... |
lemma inter_pow_subset : (s ∩ t) ^ n ⊆ s ^ n ∩ t ^ n := by apply subset_inter <;> gcongr <;> simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | {
"open": [
"Function MulOpposite",
"Pointwise",
"Pointwise",
"MulOpposite",
"Pointwise",
"MulOpposite"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Set α} {a : α}",
"{ι : Sort*} [Inv α] {s t : Set α} {a : α}",
"[InvolutiveInv α] {s t : Set α} {a : α}",
"{ι : So... | [
{
"line": "focus\n apply subset_inter <;> gcongr\n with_annotate_state\"<;>\" skip\n all_goals simp",
"before_state": "α : Type u_2\ninst✝⁷ : One α\ninst✝⁶ : Inv α\ninst✝⁵ : InvolutiveInv α\ninst✝⁴ : Mul α\ninst✝³ : Div α\ninst✝² : CommSemigroup α\ninst✝¹ : MulOneClass α\ninst✝ : Monoid α\ns t : Set α\nn... |
lemma Nontrivial.mul_left : t.Nontrivial → s.Nonempty → (s * t).Nontrivial := by
rintro ⟨a, ha, b, hb, hab⟩ ⟨c, hc⟩
exact ⟨c * a, mul_mem_mul hc ha, c * b, mul_mem_mul hc hb, by simpa⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | {
"open": [
"Function MulOpposite",
"Pointwise",
"Pointwise",
"MulOpposite",
"Pointwise",
"MulOpposite"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Set α} {a : α}",
"{ι : Sort*} [Inv α] {s t : Set α} {a : α}",
"[InvolutiveInv α] {s t : Set α} {a : α}",
"{ι : So... | [
{
"line": "rintro ⟨a, ha, b, hb, hab⟩ ⟨c, hc⟩",
"before_state": "α : Type u_2\ninst✝⁹ : One α\ninst✝⁸ : Inv α\ninst✝⁷ : InvolutiveInv α\ninst✝⁶ : Mul α\ninst✝⁵ : Div α\ninst✝⁴ : CommSemigroup α\ninst✝³ : MulOneClass α\ninst✝² : Monoid α\ninst✝¹ : Mul α\ninst✝ : IsLeftCancelMul α\ns t : Set α\n⊢ t.Nontrivial... |
lemma Nontrivial.mul_right : s.Nontrivial → t.Nonempty → (s * t).Nontrivial := by
rintro ⟨a, ha, b, hb, hab⟩ ⟨c, hc⟩
exact ⟨a * c, mul_mem_mul ha hc, b * c, mul_mem_mul hb hc, by simpa⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | {
"open": [
"Function MulOpposite",
"Pointwise",
"Pointwise",
"MulOpposite",
"Pointwise",
"MulOpposite"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Set α} {a : α}",
"{ι : Sort*} [Inv α] {s t : Set α} {a : α}",
"[InvolutiveInv α] {s t : Set α} {a : α}",
"{ι : So... | [
{
"line": "rintro ⟨a, ha, b, hb, hab⟩ ⟨c, hc⟩",
"before_state": "α : Type u_2\ninst✝¹¹ : One α\ninst✝¹⁰ : Inv α\ninst✝⁹ : InvolutiveInv α\ninst✝⁸ : Mul α\ninst✝⁷ : Div α\ninst✝⁶ : CommSemigroup α\ninst✝⁵ : MulOneClass α\ninst✝⁴ : Monoid α\ninst✝³ : Mul α\ninst✝² : IsLeftCancelMul α\ninst✝¹ : Mul α\ninst✝ : ... |
lemma Nontrivial.pow (hs : s.Nontrivial) : ∀ {n}, n ≠ 0 → (s ^ n).Nontrivial
| 1, _ => by simpa
| n + 2, _ => by simpa [pow_succ] using (hs.pow n.succ_ne_zero).mul hs
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | {
"open": [
"Function MulOpposite",
"Pointwise",
"Pointwise",
"MulOpposite",
"Pointwise",
"MulOpposite"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Set α} {a : α}",
"{ι : Sort*} [Inv α] {s t : Set α} {a : α}",
"[InvolutiveInv α] {s t : Set α} {a : α}",
"{ι : So... | [
{
"line": "simpa",
"before_state": "α : Type u_2\ninst✝¹² : One α\ninst✝¹¹ : Inv α\ninst✝¹⁰ : InvolutiveInv α\ninst✝⁹ : Mul α\ninst✝⁸ : Div α\ninst✝⁷ : CommSemigroup α\ninst✝⁶ : MulOneClass α\ninst✝⁵ : Monoid α\ninst✝⁴ : Mul α\ninst✝³ : IsLeftCancelMul α\ninst✝² : Mul α\ninst✝¹ : IsRightCancelMul α\ninst✝ :... |
theorem isUnit_iff : IsUnit s ↔ ∃ a, s = {a} ∧ IsUnit a := by
constructor
· rintro ⟨u, rfl⟩
obtain ⟨a, b, ha, hb, h⟩ := Set.mul_eq_one_iff.1 u.mul_inv
refine ⟨a, ha, ⟨a, b, h, singleton_injective ?_⟩, rfl⟩
rw [← singleton_mul_singleton]
rw [← ha]
rw [← hb]
exact u.inv_mul
· rintro ⟨a, rfl,... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | {
"open": [
"Function MulOpposite",
"Pointwise",
"Pointwise",
"MulOpposite",
"Pointwise",
"MulOpposite",
"Pointwise"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Set α} {a : α}",
"{ι : Sort*} [Inv α] {s t : Set α} {a : α}",
"[InvolutiveInv α] {s t : Set α} {a : ... | [
{
"line": "constructor",
"before_state": "α : Type u_2\ninst✝¹³ : One α\ninst✝¹² : Inv α\ninst✝¹¹ : InvolutiveInv α\ninst✝¹⁰ : Mul α\ninst✝⁹ : Div α\ninst✝⁸ : CommSemigroup α\ninst✝⁷ : MulOneClass α\ninst✝⁶ : Monoid α\ninst✝⁵ : Mul α\ninst✝⁴ : IsLeftCancelMul α\ninst✝³ : Mul α\ninst✝² : IsRightCancelMul α\n... |
theorem preimage_mul_preimage_subset {s t : Set β} : m ⁻¹' s * m ⁻¹' t ⊆ m ⁻¹' (s * t) := by
rintro _ ⟨_, _, _, _, rfl⟩
exact ⟨_, ‹_›, _, ‹_›, (map_mul m ..).symm⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | {
"open": [
"Function MulOpposite",
"Pointwise",
"Pointwise",
"MulOpposite",
"Pointwise",
"MulOpposite",
"Pointwise"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Set α} {a : α}",
"{ι : Sort*} [Inv α] {s t : Set α} {a : α}",
"[InvolutiveInv α] {s t : Set α} {a : ... | [
{
"line": "rintro _ ⟨_, _, _, _, rfl⟩",
"before_state": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹⁸ : One α\ninst✝¹⁷ : Inv α\ninst✝¹⁶ : InvolutiveInv α\ninst✝¹⁵ : Mul α\ninst✝¹⁴ : Div α\ninst✝¹³ : CommSemigroup α\ninst✝¹² : MulOneClass α\ninst✝¹¹ : Monoid α\ninst✝¹⁰ : Mul α\ninst✝⁹ : IsLeftCancelMul ... |
theorem preimage_div_preimage_subset {s t : Set β} : m ⁻¹' s / m ⁻¹' t ⊆ m ⁻¹' (s / t) := by
rintro _ ⟨_, _, _, _, rfl⟩
exact ⟨_, ‹_›, _, ‹_›, (map_div m ..).symm⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | {
"open": [
"Function MulOpposite",
"Pointwise",
"Pointwise",
"MulOpposite",
"Pointwise",
"MulOpposite",
"Pointwise"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Set α} {a : α}",
"{ι : Sort*} [Inv α] {s t : Set α} {a : α}",
"[InvolutiveInv α] {s t : Set α} {a : ... | [
{
"line": "rintro _ ⟨_, _, _, _, rfl⟩",
"before_state": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝²⁵ : One α\ninst✝²⁴ : Inv α\ninst✝²³ : InvolutiveInv α\ninst✝²² : Mul α\ninst✝²¹ : Div α\ninst✝²⁰ : CommSemigroup α\ninst✝¹⁹ : MulOneClass α\ninst✝¹⁸ : Monoid α\ninst✝¹⁷ : Mul α\ninst✝¹⁶ : IsLeftCancelMul... |
lemma inv_pi (s : Set ι) (t : ∀ i, Set (α i)) : (s.pi t)⁻¹ = s.pi fun i ↦ (t i)⁻¹ := by ext x; simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | {
"open": [
"Function MulOpposite",
"Pointwise",
"Pointwise",
"MulOpposite",
"Pointwise",
"MulOpposite",
"Pointwise"
],
"variables": [
"{F α β γ : Type*}",
"[One α] {s : Set α} {a : α}",
"{ι : Sort*} [Inv α] {s t : Set α} {a : α}",
"[InvolutiveInv α] {s t : Set α} {a : ... | [
{
"line": "ext x",
"before_state": "ι : Type u_10\nα : ι → Type u_11\ninst✝ : (i : ι) → Inv (α i)\ns : Set ι\nt : (i : ι) → Set (α i)\n⊢ (s.pi t)⁻¹ = s.pi fun i => (t i)⁻¹",
"after_state": "case h\nι : Type u_10\nα : ι → Type u_11\ninst✝ : (i : ι) → Inv (α i)\ns : Set ι\nt : (i : ι) → Set (α i)\nx : (i ... |
theorem image_multiset_prod (f : F) :
∀ m : Multiset (Set α), (f : α → β) '' m.prod = (m.map fun s => f '' s).prod :=
Quotient.ind <| by
simpa only [Multiset.quot_mk_to_coe,Multiset.prod_coe,Multiset.map_coe] using
image_list_prod f
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/BigOperators.lean | {
"open": [
"Pointwise Function"
],
"variables": [
"{ι α β F : Type*} [FunLike F α β]",
"[Monoid α] [Monoid β] [MonoidHomClass F α β]",
"[CommMonoid α] [CommMonoid β] [MonoidHomClass F α β]"
]
} | [
{
"line": "simpa only [Multiset.quot_mk_to_coe, Multiset.prod_coe, Multiset.map_coe] using image_list_prod f",
"before_state": "α : Type u_2\nβ : Type u_3\nF : Type u_4\ninst✝⁶ : FunLike F α β\ninst✝⁵ : Monoid α\ninst✝⁴ : Monoid β\ninst✝³ : MonoidHomClass F α β\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\... |
theorem mem_finset_prod (t : Finset ι) (f : ι → Set α) (a : α) :
(a ∈ ∏ i ∈ t, f i) ↔ ∃ (g : ι → α) (_ : ∀ {i}, i ∈ t → g i ∈ f i), ∏ i ∈ t, g i = a := by
classical
induction' t using Finset.induction_on with i is hi ih generalizing a
· simp_rw [Finset.prod_empty, Set.mem_one]
exact ⟨fun h ↦ ⟨fun _ ... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/BigOperators.lean | {
"open": [
"Pointwise Function"
],
"variables": [
"{ι α β F : Type*} [FunLike F α β]",
"[Monoid α] [Monoid β] [MonoidHomClass F α β]",
"[CommMonoid α] [CommMonoid β] [MonoidHomClass F α β]"
]
} | [
{
"line": "classical\ninduction' t using Finset.induction_on with i is hi ih generalizing a\n· simp_rw [Finset.prod_empty, Set.mem_one]\n exact ⟨fun h ↦ ⟨fun _ ↦ a, fun hi ↦ False.elim (Finset.not_mem_empty _ hi), h.symm⟩, fun ⟨_, _, hf⟩ ↦ hf.symm⟩\nrw [Finset.prod_insert hi]\nrw [Set.mem_mul]\nsimp_rw [Finset... |
lemma mem_pow_iff_prod {n : ℕ} {s : Set α} {a : α} :
a ∈ s ^ n ↔ ∃ f : Fin n → α, (∀ i, f i ∈ s) ∧ ∏ i, f i = a := by
simpa using mem_finset_prod (t := .univ) (f := fun _ : Fin n ↦ s) _
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/BigOperators.lean | {
"open": [
"Pointwise Function"
],
"variables": [
"{ι α β F : Type*} [FunLike F α β]",
"[Monoid α] [Monoid β] [MonoidHomClass F α β]",
"[CommMonoid α] [CommMonoid β] [MonoidHomClass F α β]"
]
} | [
{
"line": "simpa using mem_finset_prod (t := .univ) (f := fun _ : Fin n ↦ s) _",
"before_state": "α : Type u_2\ninst✝¹ : Monoid α\ninst✝ : CommMonoid α\nn : ℕ\ns : Set α\na : α\n⊢ a ∈ s ^ n ↔ ∃ f, (∀ (i : Fin n), f i ∈ s) ∧ ∏ i, f i = a",
"after_state": "No Goals!"
}
] |
theorem list_prod_mem_list_prod (t : List ι) (f : ι → Set α) (g : ι → α) (hg : ∀ i ∈ t, g i ∈ f i) :
(t.map g).prod ∈ (t.map f).prod := by
induction' t with h tl ih
· simp_rw [List.map_nil, List.prod_nil, Set.mem_one]
· simp_rw [List.map_cons, List.prod_cons]
exact mul_mem_mul (hg h List.mem_cons_self)
... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/BigOperators.lean | {
"open": [
"Pointwise Function"
],
"variables": [
"{ι α β F : Type*} [FunLike F α β]",
"[Monoid α] [Monoid β] [MonoidHomClass F α β]",
"[CommMonoid α] [CommMonoid β] [MonoidHomClass F α β]"
]
} | [
{
"line": "induction' t with h tl ih",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : Monoid α\ninst✝ : CommMonoid α\nt : List ι\nf : ι → Set α\ng : ι → α\nhg : ∀ i ∈ t, g i ∈ f i\n⊢ (List.map g t).prod ∈ (List.map f t).prod",
"after_state": "case nil\nι : Type u_1\nα : Type u_2\ninst✝¹ : Monoid ... |
theorem list_prod_subset_list_prod (t : List ι) (f₁ f₂ : ι → Set α) (hf : ∀ i ∈ t, f₁ i ⊆ f₂ i) :
(t.map f₁).prod ⊆ (t.map f₂).prod := by
induction' t with h tl ih
· rfl
· simp_rw [List.map_cons, List.prod_cons]
exact mul_subset_mul (hf h List.mem_cons_self)
(ih fun i hi ↦ hf i <| List.mem_cons_of_m... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/BigOperators.lean | {
"open": [
"Pointwise Function"
],
"variables": [
"{ι α β F : Type*} [FunLike F α β]",
"[Monoid α] [Monoid β] [MonoidHomClass F α β]",
"[CommMonoid α] [CommMonoid β] [MonoidHomClass F α β]"
]
} | [
{
"line": "induction' t with h tl ih",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : Monoid α\ninst✝ : CommMonoid α\nt : List ι\nf₁ f₂ : ι → Set α\nhf : ∀ i ∈ t, f₁ i ⊆ f₂ i\n⊢ (List.map f₁ t).prod ⊆ (List.map f₂ t).prod",
"after_state": "case nil\nι : Type u_1\nα : Type u_2\ninst✝¹ : Monoid α\n... |
theorem multiset_prod_mem_multiset_prod (t : Multiset ι) (f : ι → Set α) (g : ι → α)
(hg : ∀ i ∈ t, g i ∈ f i) : (t.map g).prod ∈ (t.map f).prod := by
induction t using Quotient.inductionOn
simp_rw [Multiset.quot_mk_to_coe, Multiset.map_coe, Multiset.prod_coe]
exact list_prod_mem_list_prod _ _ _ hg
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/BigOperators.lean | {
"open": [
"Pointwise Function"
],
"variables": [
"{ι α β F : Type*} [FunLike F α β]",
"[Monoid α] [Monoid β] [MonoidHomClass F α β]",
"[CommMonoid α] [CommMonoid β] [MonoidHomClass F α β]"
]
} | [
{
"line": "induction t using Quotient.inductionOn",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : Monoid α\ninst✝ : CommMonoid α\nt : Multiset ι\nf : ι → Set α\ng : ι → α\nhg : ∀ i ∈ t, g i ∈ f i\n⊢ (Multiset.map g t).prod ∈ (Multiset.map f t).prod",
"after_state": "case h\nι : Type u_1\nα : Typ... |
theorem multiset_prod_subset_multiset_prod (t : Multiset ι) (f₁ f₂ : ι → Set α)
(hf : ∀ i ∈ t, f₁ i ⊆ f₂ i) : (t.map f₁).prod ⊆ (t.map f₂).prod := by
induction t using Quotient.inductionOn
simp_rw [Multiset.quot_mk_to_coe, Multiset.map_coe, Multiset.prod_coe]
exact list_prod_subset_list_prod _ _ _ hf
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/BigOperators.lean | {
"open": [
"Pointwise Function"
],
"variables": [
"{ι α β F : Type*} [FunLike F α β]",
"[Monoid α] [Monoid β] [MonoidHomClass F α β]",
"[CommMonoid α] [CommMonoid β] [MonoidHomClass F α β]"
]
} | [
{
"line": "induction t using Quotient.inductionOn",
"before_state": "ι : Type u_1\nα : Type u_2\ninst✝¹ : Monoid α\ninst✝ : CommMonoid α\nt : Multiset ι\nf₁ f₂ : ι → Set α\nhf : ∀ i ∈ t, f₁ i ⊆ f₂ i\n⊢ (Multiset.map f₁ t).prod ⊆ (Multiset.map f₂ t).prod",
"after_state": "case h\nι : Type u_1\nα : Type u... |
lemma encard_inv (s : Set G) : s⁻¹.encard = s.encard := by
simp [ENat.card, ← toENat_cardinalMk]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Card.lean | {
"open": [
"scoped Cardinal Pointwise"
],
"variables": [
"{G M α : Type*}",
"[Mul M] {s t : Set M}",
"[IsCancelMul M]",
"[InvolutiveInv G]"
]
} | [
{
"line": "simp [ENat.card, ← toENat_cardinalMk]",
"before_state": "G : Type u_1\ninst✝ : InvolutiveInv G\ns : Set G\n⊢ s⁻¹.encard = s.encard",
"after_state": "No Goals!"
}
] |
lemma ncard_inv (s : Set G) : s⁻¹.ncard = s.ncard := by simp [ncard]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Card.lean | {
"open": [
"scoped Cardinal Pointwise"
],
"variables": [
"{G M α : Type*}",
"[Mul M] {s t : Set M}",
"[IsCancelMul M]",
"[InvolutiveInv G]"
]
} | [
{
"line": "simp [ncard]",
"before_state": "G : Type u_1\ninst✝ : InvolutiveInv G\ns : Set G\n⊢ s⁻¹.ncard = s.ncard",
"after_state": "No Goals!"
}
] |
lemma natCard_div_le : Nat.card (s / t) ≤ Nat.card s * Nat.card t := by
rw [div_eq_mul_inv]; exact natCard_mul_le
rw [← natCard_inv t]; exact natCard_mul_le
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Card.lean | {
"open": [
"scoped Cardinal Pointwise"
],
"variables": [
"{G M α : Type*}",
"[Mul M] {s t : Set M}",
"[IsCancelMul M]",
"[InvolutiveInv G]",
"[DivInvMonoid M] {s t : Set M}",
"[Group G] {s t : Set G}"
]
} | [
{
"line": "rw [div_eq_mul_inv]",
"before_state": "G : Type u_1\ninst✝¹ : InvolutiveInv G\ninst✝ : Group G\ns t : Set G\n⊢ Nat.card ↑(s / t) ≤ Nat.card ↑s * Nat.card ↑t",
"after_state": "G : Type u_1\ninst✝¹ : InvolutiveInv G\ninst✝ : Group G\ns t : Set G\n⊢ Nat.card ↑(s * t⁻¹) ≤ Nat.card ↑s * Nat.card ↑... |
lemma encard_smul_set (a : G) (s : Set α) : (a • s).encard = s.encard := by
simp [ENat.card, ← toENat_cardinalMk]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Card.lean | {
"open": [
"scoped Cardinal Pointwise"
],
"variables": [
"{G M α : Type*}",
"[Mul M] {s t : Set M}",
"[IsCancelMul M]",
"[InvolutiveInv G]",
"[DivInvMonoid M] {s t : Set M}",
"[Group G] {s t : Set G}",
"[MulAction G α]"
]
} | [
{
"line": "simp [ENat.card, ← toENat_cardinalMk]",
"before_state": "G : Type u_1\nα : Type u_3\ninst✝² : InvolutiveInv G\ninst✝¹ : Group G\ninst✝ : MulAction G α\na : G\ns : Set α\n⊢ (a • s).encard = s.encard",
"after_state": "No Goals!"
}
] |
lemma ncard_smul_set (a : G) (s : Set α) : (a • s).ncard = s.ncard := by simp [ncard]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Pointwise/Set/Card.lean | {
"open": [
"scoped Cardinal Pointwise"
],
"variables": [
"{G M α : Type*}",
"[Mul M] {s t : Set M}",
"[IsCancelMul M]",
"[InvolutiveInv G]",
"[DivInvMonoid M] {s t : Set M}",
"[Group G] {s t : Set G}",
"[MulAction G α]"
]
} | [
{
"line": "simp [ncard]",
"before_state": "G : Type u_1\nα : Type u_3\ninst✝² : InvolutiveInv G\ninst✝¹ : Group G\ninst✝ : MulAction G α\na : G\ns : Set α\n⊢ (a • s).ncard = s.ncard",
"after_state": "No Goals!"
}
] |
theorem mul_right (h : SemiconjBy a x y) (h' : SemiconjBy a x' y') :
SemiconjBy a (x * x') (y * y') := by
unfold SemiconjBy
-- TODO this could be done using `assoc_rw` if/when this is ported to mathlib4
rw [← mul_assoc]
rw [h.eq]
rw [mul_assoc]
rw [h'.eq]
rw [← mul_assoc]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Semiconj/Defs.lean | {
"open": [],
"variables": [
"{S M G : Type*}",
"[Semigroup S] {a b x y z x' y' : S}"
]
} | [
{
"line": "unfold SemiconjBy",
"before_state": "S : Type u_1\ninst✝ : Semigroup S\na x y x' y' : S\nh : SemiconjBy a x y\nh' : SemiconjBy a x' y'\n⊢ SemiconjBy a (x * x') (y * y')",
"after_state": "S : Type u_1\ninst✝ : Semigroup S\na x y x' y' : S\nh : SemiconjBy a x y\nh' : SemiconjBy a x' y'\n⊢ a * (... |
theorem mul_left (ha : SemiconjBy a y z) (hb : SemiconjBy b x y) : SemiconjBy (a * b) x z := by
unfold SemiconjBy
rw [mul_assoc]
rw [hb.eq]
rw [← mul_assoc]
rw [ha.eq]
rw [mul_assoc]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Semiconj/Defs.lean | {
"open": [],
"variables": [
"{S M G : Type*}",
"[Semigroup S] {a b x y z x' y' : S}"
]
} | [
{
"line": "unfold SemiconjBy",
"before_state": "S : Type u_1\ninst✝ : Semigroup S\na b x y z : S\nha : SemiconjBy a y z\nhb : SemiconjBy b x y\n⊢ SemiconjBy (a * b) x z",
"after_state": "S : Type u_1\ninst✝ : Semigroup S\na b x y z : S\nha : SemiconjBy a y z\nhb : SemiconjBy b x y\n⊢ a * b * x = z * (a ... |
theorem one_right (a : M) : SemiconjBy a 1 1 := by rw [SemiconjBy, mul_one, one_mul]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Semiconj/Defs.lean | {
"open": [],
"variables": [
"{S M G : Type*}",
"[Semigroup S] {a b x y z x' y' : S}",
"[MulOneClass M]"
]
} | [
{
"line": "rw [SemiconjBy, mul_one, one_mul]",
"before_state": "M : Type u_2\ninst✝ : MulOneClass M\na : M\n⊢ SemiconjBy a 1 1",
"after_state": "No Goals!"
},
{
"line": "rewrite [SemiconjBy, mul_one, one_mul]",
"before_state": "M : Type u_2\ninst✝ : MulOneClass M\na : M\n⊢ SemiconjBy a 1 1",... |
theorem conj_mk (a x : G) : SemiconjBy a x (a * x * a⁻¹) := by
unfold SemiconjBy; rw [mul_assoc, inv_mul_cancel, mul_one]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Semiconj/Defs.lean | {
"open": [],
"variables": [
"{S M G : Type*}",
"[Semigroup S] {a b x y z x' y' : S}",
"[MulOneClass M]",
"[Monoid M]",
"[Group G]"
]
} | [
{
"line": "unfold SemiconjBy",
"before_state": "G : Type u_3\ninst✝ : Group G\na x : G\n⊢ SemiconjBy a x (a * x * a⁻¹)",
"after_state": "G : Type u_3\ninst✝ : Group G\na x : G\n⊢ a * x = a * x * a⁻¹ * a"
},
{
"line": "rw [mul_assoc, inv_mul_cancel, mul_one]",
"before_state": "G : Type u_3\ni... |
theorem conj_iff {a x y b : G} :
SemiconjBy (b * a * b⁻¹) (b * x * b⁻¹) (b * y * b⁻¹) ↔ SemiconjBy a x y := by
unfold SemiconjBy
simp only [← mul_assoc]
simp only [inv_mul_cancel_right]
repeat rw [mul_assoc]
rw [mul_left_cancel_iff]
rw [← mul_assoc]
rw [← mul_assoc]
rw [mul_right_cancel_iff]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Semiconj/Defs.lean | {
"open": [],
"variables": [
"{S M G : Type*}",
"[Semigroup S] {a b x y z x' y' : S}",
"[MulOneClass M]",
"[Monoid M]",
"[Group G]"
]
} | [
{
"line": "unfold SemiconjBy",
"before_state": "G : Type u_3\ninst✝ : Group G\na x y b : G\n⊢ SemiconjBy (b * a * b⁻¹) (b * x * b⁻¹) (b * y * b⁻¹) ↔ SemiconjBy a x y",
"after_state": "G : Type u_3\ninst✝ : Group G\na x y b : G\n⊢ b * a * b⁻¹ * (b * x * b⁻¹) = b * y * b⁻¹ * (b * a * b⁻¹) ↔ a * x = y * a"... |
theorem units_inv_right {a : M} {x y : Mˣ} (h : SemiconjBy a x y) : SemiconjBy a ↑x⁻¹ ↑y⁻¹ :=
calc
a * ↑x⁻¹ = ↑y⁻¹ * (y * a) * ↑x⁻¹ := by rw [Units.inv_mul_cancel_left]
_ = ↑y⁻¹ * a := by rw [← h.eq, mul_assoc, Units.mul_inv_cancel_right]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Semiconj/Units.lean | {
"open": [
"scoped Int"
],
"variables": [
"{M : Type*}",
"[Monoid M]"
]
} | [
{
"line": "rw [Units.inv_mul_cancel_left]",
"before_state": "M : Type u_1\ninst✝ : Monoid M\na : M\nx y : Mˣ\nh : SemiconjBy a ↑x ↑y\n⊢ a * ↑x⁻¹ = ↑y⁻¹ * (↑y * a) * ↑x⁻¹",
"after_state": "No Goals!"
},
{
"line": "rewrite [Units.inv_mul_cancel_left]",
"before_state": "M : Type u_1\ninst✝ : Mo... |
theorem units_inv_symm_left {a : Mˣ} {x y : M} (h : SemiconjBy (↑a) x y) : SemiconjBy (↑a⁻¹) y x :=
calc
↑a⁻¹ * y = ↑a⁻¹ * (y * a * ↑a⁻¹) := by rw [Units.mul_inv_cancel_right]
_ = x * ↑a⁻¹ := by rw [← h.eq, ← mul_assoc, Units.inv_mul_cancel_left]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Semiconj/Units.lean | {
"open": [
"scoped Int"
],
"variables": [
"{M : Type*}",
"[Monoid M]"
]
} | [
{
"line": "rw [Units.mul_inv_cancel_right]",
"before_state": "M : Type u_1\ninst✝ : Monoid M\na : Mˣ\nx y : M\nh : SemiconjBy (↑a) x y\n⊢ ↑a⁻¹ * y = ↑a⁻¹ * (y * ↑a * ↑a⁻¹)",
"after_state": "No Goals!"
},
{
"line": "rewrite [Units.mul_inv_cancel_right]",
"before_state": "M : Type u_1\ninst✝ :... |
lemma mk_semiconjBy (u : Mˣ) (x : M) : SemiconjBy (↑u) x (u * x * ↑u⁻¹) := by
unfold SemiconjBy; rw [Units.inv_mul_cancel_right]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Semiconj/Units.lean | {
"open": [
"scoped Int"
],
"variables": [
"{M : Type*}",
"[Monoid M]",
"[Monoid M]"
]
} | [
{
"line": "unfold SemiconjBy",
"before_state": "M : Type u_1\ninst✝¹ inst✝ : Monoid M\nu : Mˣ\nx : M\n⊢ SemiconjBy (↑u) x (↑u * x * ↑u⁻¹)",
"after_state": "M : Type u_1\ninst✝¹ inst✝ : Monoid M\nu : Mˣ\nx : M\n⊢ ↑u * x = ↑u * x * ↑u⁻¹ * ↑u"
},
{
"line": "rw [Units.inv_mul_cancel_right]",
"be... |
theorem inv_inv_symm_iff : SemiconjBy a⁻¹ x⁻¹ y⁻¹ ↔ SemiconjBy a y x := by
simp_rw [SemiconjBy, ← mul_inv_rev, inv_inj, eq_comm]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Semiconj/Basic.lean | {
"open": [],
"variables": [
"{G : Type*}",
"[DivisionMonoid G] {a x y : G}"
]
} | [
{
"line": "simp_rw [SemiconjBy, ← mul_inv_rev, inv_inj, eq_comm]",
"before_state": "G : Type u_1\ninst✝ : DivisionMonoid G\na x y : G\n⊢ SemiconjBy a⁻¹ x⁻¹ y⁻¹ ↔ SemiconjBy a y x",
"after_state": "No Goals!"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
"before_state": "G : Type u_1... |
lemma isCoatom_comap_of_surjective
{H : Type*} [Group H] {φ : G →* H} (hφ : Function.Surjective φ)
{M : Subgroup H} (hM : IsCoatom M) : IsCoatom (M.comap φ) := by
refine And.imp (fun hM ↦ ?_) (fun hM ↦ ?_) hM
· rwa [← (comap_injective hφ).ne_iff, comap_top] at hM
· intro K hK
specialize hM (K.map φ)
... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Order.lean | {
"open": [
"Subgroup"
],
"variables": [
"{C : Type*} [CommGroup C]",
"{G : Type*} [Group G] (H : Subgroup G)"
]
} | [
{
"line": "refine And.imp (fun hM ↦ ?_) (fun hM ↦ ?_) hM",
"before_state": "G : Type u_2\ninst✝¹ : Group G\nH : Type u_3\ninst✝ : Group H\nφ : G →* H\nhφ : Function.Surjective ⇑φ\nM : Subgroup H\nhM : IsCoatom M\n⊢ IsCoatom (comap φ M)",
"after_state": "case refine_1\nG : Type u_2\ninst✝¹ : Group G\nH :... |
theorem normalCore_le (H : Subgroup G) : H.normalCore ≤ H := fun a h => by
rw [← mul_one a]
rw [← inv_one]
rw [← one_mul a]
exact h 1
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Basic.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"scoped Relator in",
"Group"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H K : S}",
"[SetLike S G] [Sub... | [
{
"line": "rw [← mul_one a]",
"before_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\na : G\nh : a ∈ H.normalCore\n⊢ a ∈ H",
"after_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\na : G\nh : a ∈ H.normalCore\n⊢ a * 1 ∈ H"
},
{
"line": "rewrite [← mul_one a]",
"before_state": ... |
theorem Normal.map {H : Subgroup G} (h : H.Normal) (f : G →* N) (hf : Function.Surjective f) :
(H.map f).Normal := by
rw [← normalizer_eq_top_iff]
rw [← top_le_iff]
rw [← f.range_eq_top_of_surjective hf]
rw [f.range_eq_map]
rw [← H.normalizer_eq_top]
exact le_normalizer_map _
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Basic.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"scoped Relator in",
"Group",
"Subgroup"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H K : S}",
"[S... | [
{
"line": "rw [← normalizer_eq_top_iff]",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_14\ninst✝ : Group N\nH : Subgroup G\nh : H.Normal\nf : G →* N\nhf : Surjective ⇑f\n⊢ (Subgroup.map f H).Normal",
"after_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_14\ninst✝ : Group N\nH : Subgr... |
theorem liftOfRightInverseAux_comp_apply (hf : Function.RightInverse f_inv f) (g : G₁ →* G₃)
(hg : f.ker ≤ g.ker) (x : G₁) : (f.liftOfRightInverseAux f_inv hf g hg) (f x) = g x := by
dsimp [liftOfRightInverseAux]
rw [← mul_inv_eq_one]
rw [← g.map_inv]
rw [← g.map_mul]
rw [← g.mem_ker]
apply hg
rw [f.m... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Basic.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"scoped Relator in",
"Group",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H... | [
{
"line": "dsimp [liftOfRightInverseAux]",
"before_state": "G₁ : Type u_16\nG₂ : Type u_17\nG₃ : Type u_18\ninst✝² : Group G₁\ninst✝¹ : Group G₂\ninst✝ : Group G₃\nf : G₁ →* G₂\nf_inv : G₂ → G₁\nhf : RightInverse f_inv ⇑f\ng : G₁ →* G₃\nhg : f.ker ≤ g.ker\nx : G₁\n⊢ (f.liftOfRightInverseAux f_inv hf g hg) (... |
theorem eq_liftOfRightInverse (hf : Function.RightInverse f_inv f) (g : G₁ →* G₃)
(hg : f.ker ≤ g.ker) (h : G₂ →* G₃) (hh : h.comp f = g) :
h = f.liftOfRightInverse f_inv hf ⟨g, hg⟩ := by
simp_rw [← hh]
exact ((f.liftOfRightInverse f_inv hf).apply_symm_apply _).symm
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Basic.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"scoped Relator in",
"Group",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H... | [
{
"line": "simp_rw [← hh]",
"before_state": "G₁ : Type u_16\nG₂ : Type u_17\nG₃ : Type u_18\ninst✝² : Group G₁\ninst✝¹ : Group G₂\ninst✝ : Group G₃\nf : G₁ →* G₂\nf_inv : G₂ → G₁\nhf : RightInverse f_inv ⇑f\ng : G₁ →* G₃\nhg : f.ker ≤ g.ker\nh : G₂ →* G₃\nhh : h.comp f = g\n⊢ h = (f.liftOfRightInverse f_inv... |
theorem Normal.comap {H : Subgroup N} (hH : H.Normal) (f : G →* N) : (H.comap f).Normal :=
⟨fun _ => by simp +contextual [Subgroup.mem_comap, hH.conj_mem]⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Basic.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"scoped Relator in",
"Group",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H... | [
{
"line": "simp +contextual [Subgroup.mem_comap, hH.conj_mem]",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_19\ninst✝ : Group N\nH : Subgroup N\nhH : H.Normal\nf : G →* N\nx✝ : G\n⊢ x✝ ∈ Subgroup.comap f H → ∀ (g : G), g * x✝ * g⁻¹ ∈ Subgroup.comap f H",
"after_state": "No Goals!"
}
] |
theorem map_normalClosure (s : Set G) (f : G →* N) (hf : Surjective f) :
(normalClosure s).map f = normalClosure (f '' s) := by
have : Normal (map f (normalClosure s)) := Normal.map inferInstance f hf
apply le_antisymm
· simp [map_le_iff_le_comap, normalClosure_le_normal, coe_comap,
← Set.image_subset_i... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Basic.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"scoped Relator in",
"Group",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H... | [
{
"line": "have : Normal (map f (normalClosure s)) := Normal.map inferInstance f hf",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_19\ninst✝ : Group N\ns : Set G\nf : G →* N\nhf : Surjective ⇑f\n⊢ map f (normalClosure s) = normalClosure (⇑f '' s)",
"after_state": "G : Type u_1\ninst✝¹ : G... |
theorem comap_normalClosure (s : Set N) (f : G ≃* N) :
normalClosure (f ⁻¹' s) = (normalClosure s).comap f := by
have := Set.preimage_equiv_eq_image_symm s f.toEquiv
simp_all [comap_equiv_eq_map_symm, map_normalClosure s (f.symm : N →* G) f.symm.surjective]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Basic.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"scoped Relator in",
"Group",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H... | [
{
"line": "have := Set.preimage_equiv_eq_image_symm s f.toEquiv",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_19\ninst✝ : Group N\ns : Set N\nf : G ≃* N\n⊢ normalClosure (⇑f ⁻¹' s) = comap (↑f) (normalClosure s)",
"after_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_19\ninst✝ : Gro... |
theorem inf_subgroupOf_inf_normal_of_right (A B' B : Subgroup G)
[hN : (B'.subgroupOf B).Normal] : ((A ⊓ B').subgroupOf (A ⊓ B)).Normal := by
rw [normal_subgroupOf_iff_le_normalizer_inf] at hN ⊢
rw [inf_inf_inf_comm]
rw [inf_idem]
exact le_trans (inf_le_inf A.le_normalizer hN) (inf_normalizer_le_normalizer_... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Basic.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"scoped Relator in",
"Group",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H... | [
{
"line": "rw [normal_subgroupOf_iff_le_normalizer_inf] at hN ⊢",
"before_state": "G : Type u_1\ninst✝ : Group G\nA B' B : Subgroup G\nhN : (B'.subgroupOf B).Normal\n⊢ ((A ⊓ B').subgroupOf (A ⊓ B)).Normal",
"after_state": "G : Type u_1\ninst✝ : Group G\nA B' B : Subgroup G\nhN : B ≤ (B' ⊓ B).normalizer\... |
theorem inf_subgroupOf_inf_normal_of_left {A' A : Subgroup G} (B : Subgroup G)
[hN : (A'.subgroupOf A).Normal] : ((A' ⊓ B).subgroupOf (A ⊓ B)).Normal := by
rw [normal_subgroupOf_iff_le_normalizer_inf] at hN ⊢
rw [inf_inf_inf_comm]
rw [inf_idem]
exact le_trans (inf_le_inf hN B.le_normalizer) (inf_normalizer_... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Basic.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"scoped Relator in",
"Group",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H... | [
{
"line": "rw [normal_subgroupOf_iff_le_normalizer_inf] at hN ⊢",
"before_state": "G : Type u_1\ninst✝ : Group G\nA' A B : Subgroup G\nhN : (A'.subgroupOf A).Normal\n⊢ ((A' ⊓ B).subgroupOf (A ⊓ B)).Normal",
"after_state": "G : Type u_1\ninst✝ : Group G\nA' A B : Subgroup G\nhN : A ≤ (A' ⊓ A).normalizer\... |
theorem normal_iInf_normal {ι : Type*} {a : ι → Subgroup G}
(norm : ∀ i : ι, (a i).Normal) : (iInf a).Normal := by
constructor
intro g g_in_iInf h
rw [Subgroup.mem_iInf] at g_in_iInf ⊢
intro i
exact (norm i).conj_mem g (g_in_iInf i) h
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Basic.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"scoped Relator in",
"Group",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H... | [
{
"line": "constructor",
"before_state": "G : Type u_1\ninst✝ : Group G\nι : Type u_20\na : ι → Subgroup G\nnorm : ∀ (i : ι), (a i).Normal\n⊢ (iInf a).Normal",
"after_state": "case conj_mem\nG : Type u_1\ninst✝ : Group G\nι : Type u_20\na : ι → Subgroup G\nnorm : ∀ (i : ι), (a i).Normal\n⊢ ∀ n ∈ iInf a,... |
theorem SubgroupNormal.mem_comm {H K : Subgroup G} (hK : H ≤ K) [hN : (H.subgroupOf K).Normal]
{a b : G} (hb : b ∈ K) (h : a * b ∈ H) : b * a ∈ H := by
have := (normal_subgroupOf_iff hK).mp hN (a * b) b h hb
rwa [mul_assoc, mul_assoc, mul_inv_cancel, mul_one] at this
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Basic.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"scoped Relator in",
"Group",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H... | [
{
"line": "have := (normal_subgroupOf_iff hK).mp hN (a * b) b h hb",
"before_state": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nhK : H ≤ K\nhN : (H.subgroupOf K).Normal\na b : G\nhb : b ∈ K\nh : a * b ∈ H\n⊢ b * a ∈ H",
"after_state": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nhK : H ≤ K\n... |
theorem commute_of_normal_of_disjoint (H₁ H₂ : Subgroup G) (hH₁ : H₁.Normal) (hH₂ : H₂.Normal)
(hdis : Disjoint H₁ H₂) (x y : G) (hx : x ∈ H₁) (hy : y ∈ H₂) : Commute x y := by
suffices x * y * x⁻¹ * y⁻¹ = 1 by
show x * y = y * x
· rw [mul_assoc, mul_eq_one_iff_eq_inv] at this
simpa
apply hdis.le_... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Basic.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"scoped Relator in",
"Group",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H... | [
{
"line": "suffices x * y * x⁻¹ * y⁻¹ = 1 by\n show x * y = y * x\n · rw [mul_assoc, mul_eq_one_iff_eq_inv] at this\n simpa",
"before_state": "G : Type u_1\ninst✝ : Group G\nH₁ H₂ : Subgroup G\nhH₁ : H₁.Normal\nhH₂ : H₂.Normal\nhdis : Disjoint H₁ H₂\nx y : G\nhx : x ∈ H₁\nhy : y ∈ H₂\n⊢ Commute x y",
... |
theorem normal_subgroupOf_of_le_normalizer {H N : Subgroup G}
(hLE : H ≤ N.normalizer) : (N.subgroupOf H).Normal := by
rw [normal_subgroupOf_iff_le_normalizer_inf]
exact (le_inf hLE H.le_normalizer).trans inf_normalizer_le_normalizer_inf
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Basic.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"scoped Relator in",
"Group",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H... | [
{
"line": "rw [normal_subgroupOf_iff_le_normalizer_inf]",
"before_state": "G : Type u_1\ninst✝ : Group G\nH N : Subgroup G\nhLE : H ≤ N.normalizer\n⊢ (N.subgroupOf H).Normal",
"after_state": "G : Type u_1\ninst✝ : Group G\nH N : Subgroup G\nhLE : H ≤ N.normalizer\n⊢ H ≤ (N ⊓ H).normalizer"
},
{
... |
theorem normal_subgroupOf_sup_of_le_normalizer {H N : Subgroup G}
(hLE : H ≤ N.normalizer) : (N.subgroupOf (H ⊔ N)).Normal := by
rw [normal_subgroupOf_iff_le_normalizer le_sup_right]
exact sup_le hLE le_normalizer
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Basic.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"scoped Relator in",
"Group",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H... | [
{
"line": "rw [normal_subgroupOf_iff_le_normalizer le_sup_right]",
"before_state": "G : Type u_1\ninst✝ : Group G\nH N : Subgroup G\nhLE : H ≤ N.normalizer\n⊢ (N.subgroupOf (H ⊔ N)).Normal",
"after_state": "G : Type u_1\ninst✝ : Group G\nH N : Subgroup G\nhLE : H ≤ N.normalizer\n⊢ H ⊔ N ≤ N.normalizer"
... |
theorem subset_union {H K L : S} : (H : Set G) ⊆ K ∪ L ↔ H ≤ K ∨ H ≤ L := by
refine ⟨fun h ↦ ?_, fun h x xH ↦ h.imp (· xH) (· xH)⟩
rw [or_iff_not_imp_left]
rw [SetLike.not_le_iff_exists]
exact fun ⟨x, xH, xK⟩ y yH ↦ (h <| mul_mem xH yH).elim
((h yH).resolve_left fun yK ↦ xK <| (mul_mem_cancel_right yK).mp ·... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Defs.lean | {
"open": [
"Function",
"scoped Int"
],
"variables": [
"{G : Type*} [Group G] {A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H K : S}",
"[SetLike S G] [SubgroupClass S G]"
]
} | [
{
"line": "refine ⟨fun h ↦ ?_, fun h x xH ↦ h.imp (· xH) (· xH)⟩",
"before_state": "G : Type u_1\ninst✝² : Group G\nS : Type u_4\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\nH K L : S\n⊢ ↑H ⊆ ↑K ∪ ↑L ↔ H ≤ K ∨ H ≤ L",
"after_state": "G : Type u_1\ninst✝² : Group G\nS : Type u_4\ninst✝¹ : SetLike S ... |
theorem coe_subtype : (SubgroupClass.subtype H : H → G) = ((↑) : H → G) := by
rfl
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Defs.lean | {
"open": [
"Function",
"scoped Int"
],
"variables": [
"{G : Type*} [Group G] {A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H K : S}",
"[SetLike S G] [SubgroupClass S G]",
"(H)",
"{H} in"
]
} | [
{
"line": "rfl",
"before_state": "G : Type u_1\ninst✝² : Group G\nS : Type u_4\nH : S\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\n⊢ ⇑↑H = Subtype.val",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "G : Type u_1\ninst✝² : Group G\nS : Type u_4\nH : S\ninst✝¹ : SetL... |
theorem coe_zpow (x : H) (n : ℤ) : ((x ^ n : H) : G) = (x : G) ^ n := by
dsimp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Defs.lean | {
"open": [
"Function",
"scoped Int"
],
"variables": [
"{G : Type*} [Group G] {A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H K : S}",
"[SetLike S G] [SubgroupClass S G]",
"(H)",
"{H} in",
"{H}",
"(H K : Subgroup G)"
]
} | [
{
"line": "dsimp",
"before_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nx : ↥H\nn : ℤ\n⊢ ↑(x ^ n) = ↑x ^ n",
"after_state": "No Goals!"
}
] |
theorem conj_mem' (nH : H.Normal) (n : G) (hn : n ∈ H) (g : G) :
g⁻¹ * n * g ∈ H := by
convert nH.conj_mem n hn g⁻¹
rw [inv_inv]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Defs.lean | {
"open": [
"Function",
"scoped Int",
"Set"
],
"variables": [
"{G : Type*} [Group G] {A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H K : S}",
"[SetLike S G] [SubgroupClass S G]",
"(H)",
"{H} in",
"{H}",
"(H K : Subgroup ... | [
{
"line": "convert nH.conj_mem n hn g⁻¹",
"before_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nnH : H.Normal\nn : G\nhn : n ∈ H\ng : G\n⊢ g⁻¹ * n * g ∈ H",
"after_state": "case h.e'_5.h.e'_6\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nnH : H.Normal\nn : G\nhn : n ∈ H\ng : G\n⊢ g = g⁻¹⁻¹"
... |
theorem mem_comm (nH : H.Normal) {a b : G} (h : a * b ∈ H) : b * a ∈ H := by
have : a⁻¹ * (a * b) * a⁻¹⁻¹ ∈ H := nH.conj_mem (a * b) h a⁻¹
simpa
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Defs.lean | {
"open": [
"Function",
"scoped Int",
"Set"
],
"variables": [
"{G : Type*} [Group G] {A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H K : S}",
"[SetLike S G] [SubgroupClass S G]",
"(H)",
"{H} in",
"{H}",
"(H K : Subgroup ... | [
{
"line": "have : a⁻¹ * (a * b) * a⁻¹⁻¹ ∈ H := nH.conj_mem (a * b) h a⁻¹",
"before_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nnH : H.Normal\na b : G\nh : a * b ∈ H\n⊢ b * a ∈ H",
"after_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nnH : H.Normal\na b : G\nh : a * b ∈ H\nthis : a⁻¹... |
lemma mul_comm_of_mem_isMulCommutative [IsMulCommutative H] {a b : G} (ha : a ∈ H) (hb : b ∈ H) :
a * b = b * a := by
simpa only [MulMemClass.mk_mul_mk,Subtype.mk.injEq] using mul_comm (⟨a, ha⟩ : H) (⟨b, hb⟩ : H)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Defs.lean | {
"open": [
"Function",
"scoped Int",
"Set"
],
"variables": [
"{G : Type*} [Group G] {A : Type*} [AddGroup A]",
"{M S : Type*} [DivInvMonoid M] [SetLike S M] [hSM : SubgroupClass S M] {H K : S}",
"[SetLike S G] [SubgroupClass S G]",
"(H)",
"{H} in",
"{H}",
"(H K : Subgroup ... | [
{
"line": "simpa only [MulMemClass.mk_mul_mk, Subtype.mk.injEq] using mul_comm (⟨a, ha⟩ : H) (⟨b, hb⟩ : H)",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : IsMulCommutative ↥H\na b : G\nha : a ∈ H\nhb : b ∈ H\n⊢ a * b = b * a",
"after_state": "No Goals!"
}
] |
theorem card_bot : Nat.card (⊥ : Subgroup G) = 1 := by simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Finite.lean | {
"open": [],
"variables": [
"{G : Type*} [Group G]",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G)"
]
} | [
{
"line": "simp",
"before_state": "G : Type u_1\ninst✝ : Group G\n⊢ Nat.card ↥⊥ = 1",
"after_state": "No Goals!"
}
] |
theorem eq_top_of_card_eq [Finite H] (h : Nat.card H = Nat.card G) : H = ⊤ := by
have : Finite G := Nat.finite_of_card_ne_zero (h ▸ Nat.card_pos.ne')
exact eq_top_of_le_card _ (Nat.le_of_eq h.symm)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Finite.lean | {
"open": [],
"variables": [
"{G : Type*} [Group G]",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G)"
]
} | [
{
"line": "have : Finite G := Nat.finite_of_card_ne_zero (h ▸ Nat.card_pos.ne')",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Finite ↥H\nh : Nat.card ↥H = Nat.card G\n⊢ H = ⊤",
"after_state": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Finite ↥H\nh : Nat.card ↥H... |
theorem card_eq_iff_eq_top [Finite H] : Nat.card H = Nat.card G ↔ H = ⊤ :=
Iff.intro (eq_top_of_card_eq H) (fun h ↦ by simpa only [h] using card_top)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Finite.lean | {
"open": [],
"variables": [
"{G : Type*} [Group G]",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G)"
]
} | [
{
"line": "simpa only [h] using card_top",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Finite ↥H\nh : H = ⊤\n⊢ Nat.card ↥H = Nat.card G",
"after_state": "No Goals!"
}
] |
theorem card_le_one_iff_eq_bot [Finite H] : Nat.card H ≤ 1 ↔ H = ⊥ :=
⟨H.eq_bot_of_card_le, fun h => by simp [h]⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Finite.lean | {
"open": [],
"variables": [
"{G : Type*} [Group G]",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G)"
]
} | [
{
"line": "simp [h]",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Finite ↥H\nh : H = ⊥\n⊢ Nat.card ↥H ≤ 1",
"after_state": "No Goals!"
}
] |
theorem pi_mem_of_mulSingle_mem_aux [DecidableEq η] (I : Finset η) {H : Subgroup (∀ i, f i)}
(x : ∀ i, f i) (h1 : ∀ i, i ∉ I → x i = 1) (h2 : ∀ i, i ∈ I → Pi.mulSingle i (x i) ∈ H) :
x ∈ H := by
induction I using Finset.induction_on generalizing x with
| empty =>
have : x = 1 := by
ext i
exa... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Finite.lean | {
"open": [
"Set"
],
"variables": [
"{G : Type*} [Group G]",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G)",
"{η : Type*} {f : η → Type*} [∀ i, Group (f i)]"
]
} | [
{
"line": "induction I using Finset.induction_on generalizing x with\n|\n empty =>\n have : x = 1 := by\n ext i\n exact h1 i (Finset.not_mem_empty i)\n rw [this]\n exact one_mem H\n| insert i I hnmem\n ih =>\n have : x = Function.update x i 1 * Pi.mulSingle i (x i) :=\n by\n ext j\n by_case... |
theorem pi_mem_of_mulSingle_mem [Finite η] [DecidableEq η] {H : Subgroup (∀ i, f i)} (x : ∀ i, f i)
(h : ∀ i, Pi.mulSingle i (x i) ∈ H) : x ∈ H := by
cases nonempty_fintype η
exact pi_mem_of_mulSingle_mem_aux Finset.univ x (by simp) fun i _ => h i
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Finite.lean | {
"open": [
"Set"
],
"variables": [
"{G : Type*} [Group G]",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G)",
"{η : Type*} {f : η → Type*} [∀ i, Group (f i)]"
]
} | [
{
"line": "cases nonempty_fintype η",
"before_state": "η : Type u_3\nf : η → Type u_4\ninst✝² : (i : η) → Group (f i)\ninst✝¹ : Finite η\ninst✝ : DecidableEq η\nH : Subgroup ((i : η) → f i)\nx : (i : η) → f i\nh : ∀ (i : η), Pi.mulSingle i (x i) ∈ H\n⊢ x ∈ H",
"after_state": "case intro\nη : Type u_3\nf... |
theorem mem_normalizer_fintype {S : Set G} [Finite S] {x : G} (h : ∀ n, n ∈ S → x * n * x⁻¹ ∈ S) :
x ∈ Subgroup.setNormalizer S := by
haveI := Classical.propDecidable; cases nonempty_fintype S
haveI := Set.fintypeImage S fun n => x * n * x⁻¹
exact fun n =>
⟨h n, fun h₁ =>
have heq : (fun n => x * n ... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Finite.lean | {
"open": [
"Set"
],
"variables": [
"{G : Type*} [Group G]",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G)",
"{η : Type*} {f : η → Type*} [∀ i, Group (f i)]"
]
} | [
{
"line": "haveI := Classical.propDecidable",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nS : Set G\ninst✝ : Finite ↑S\nx : G\nh : ∀ n ∈ S, x * n * x⁻¹ ∈ S\n⊢ x ∈ Subgroup.setNormalizer S",
"after_state": "G : Type u_1\ninst✝¹ : Group G\nS : Set G\ninst✝ : Finite ↑S\nx : G\nh : ∀ n ∈ S, x * n * x⁻¹... |
theorem range_eq_map (f : G →* N) : f.range = (⊤ : Subgroup G).map f := by ext; simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)"
]
} | [
{
"line": "ext",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\nf : G →* N\n⊢ f.range = map f ⊤",
"after_state": "case h\nG : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\nf : G →* N\nx✝ : N\n⊢ x✝ ∈ f.range ↔ x✝ ∈ map f ⊤"
},
{
"line": "simp",
"b... |
lemma rangeRestrict_injective_iff {f : G →* N} : Injective f.rangeRestrict ↔ Injective f := by
convert Set.injective_codRestrict _
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)"
]
} | [
{
"line": "convert Set.injective_codRestrict _",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\nf : G →* N\n⊢ Injective ⇑f.rangeRestrict ↔ Injective ⇑f",
"after_state": "No Goals!"
}
] |
theorem range_one : (1 : G →* N).range = ⊥ :=
SetLike.ext fun x => by simpa using @comm _ (· = ·) _ 1 x
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)"
]
} | [
{
"line": "simpa using @comm _ (· = ·) _ 1 x",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\nx : N\n⊢ x ∈ MonoidHom.range 1 ↔ x ∈ ⊥",
"after_state": "No Goals!"
}
] |
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