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chore(algebra/order/group/defs): split file (#17787)
Splits: * algebra/order/group/defs.lean * algebra/group/with_one.lean * algebra/order/monoid/with_zero.lean This will get us unstuck on the mathlib4 port, moving some of the files (about algebraic morphisms and equivalences) that are currently waiting for fixes in Lean 4 off the critical path. Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
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‎counterexamples/linear_order_with_pos_mul_pos_eq_zero.lean‎

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Authors: Johan Commelin, Damiano Testa, Kevin Buzzard
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-/
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import algebra.order.monoid.defs
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import algebra.order.monoid.with_zero
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import algebra.order.monoid.with_zero.defs
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/-!
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An example of a `linear_ordered_comm_monoid_with_zero` in which the product of two positive

‎src/algebra/category/Mon/adjunctions.lean‎

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-/
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import algebra.category.Mon.basic
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import algebra.category.Semigroup.basic
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import algebra.group.with_one
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import algebra.group.with_one.basic
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import algebra.free_monoid.basic
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/-!

‎src/algebra/free.lean‎

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Authors: Kenny Lau
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-/
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import algebra.hom.group
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import algebra.hom.equiv.basic
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import control.applicative
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import control.traversable.basic
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import logic.equiv.defs

‎src/algebra/group/default.lean‎

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-/
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import algebra.group.conj
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import algebra.group.type_tags
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import algebra.group.with_one
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import algebra.group.with_one.basic
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import algebra.hom.units
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/-!

‎src/algebra/group/with_one/basic.lean‎

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/-
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Copyright (c) 2018 Mario Carneiro. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Mario Carneiro, Johan Commelin
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-/
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import algebra.group.with_one.defs
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import algebra.hom.equiv.basic
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/-!
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# More operations on `with_one` and `with_zero`
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This file defines various bundled morphisms on `with_one` and `with_zero`
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that were not available in `algebra/group/with_one/defs`.
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## Main definitions
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* `with_one.lift`, `with_zero.lift`
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* `with_one.map`, `with_zero.map`
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-/
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universes u v w
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variables {α : Type u} {β : Type v} {γ : Type w}
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namespace with_one
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section
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-- workaround: we make `with_one`/`with_zero` irreducible for this definition, otherwise `simps`
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-- will unfold it in the statement of the lemma it generates.
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local attribute [irreducible] with_one with_zero
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/-- `coe` as a bundled morphism -/
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@[to_additive "`coe` as a bundled morphism", simps apply]
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def coe_mul_hom [has_mul α] : α →ₙ* (with_one α) :=
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{ to_fun := coe, map_mul' := λ x y, rfl }
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end
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section lift
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local attribute [semireducible] with_one with_zero
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variables [has_mul α] [mul_one_class β]
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/-- Lift a semigroup homomorphism `f` to a bundled monoid homorphism. -/
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@[to_additive "Lift an add_semigroup homomorphism `f` to a bundled add_monoid homorphism."]
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def lift : (α →ₙ* β) ≃ (with_one α →* β) :=
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{ to_fun := λ f,
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{ to_fun := λ x, option.cases_on x 1 f,
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map_one' := rfl,
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map_mul' := λ x y,
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with_one.cases_on x (by { rw one_mul, exact (one_mul _).symm }) $ λ x,
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with_one.cases_on y (by { rw mul_one, exact (mul_one _).symm }) $ λ y,
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f.map_mul x y },
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inv_fun := λ F, F.to_mul_hom.comp coe_mul_hom,
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left_inv := λ f, mul_hom.ext $ λ x, rfl,
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right_inv := λ F, monoid_hom.ext $ λ x, with_one.cases_on x F.map_one.symm $ λ x, rfl }
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variables (f : α →ₙ* β)
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@[simp, to_additive]
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lemma lift_coe (x : α) : lift f x = f x := rfl
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@[simp, to_additive]
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lemma lift_one : lift f 1 = 1 := rfl
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@[to_additive]
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theorem lift_unique (f : with_one α →* β) : f = lift (f.to_mul_hom.comp coe_mul_hom) :=
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(lift.apply_symm_apply f).symm
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end lift
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section map
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variables [has_mul α] [has_mul β] [has_mul γ]
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/-- Given a multiplicative map from `α → β` returns a monoid homomorphism
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from `with_one α` to `with_one β` -/
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@[to_additive "Given an additive map from `α → β` returns an add_monoid homomorphism
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from `with_zero α` to `with_zero β`"]
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def map (f : α →ₙ* β) : with_one α →* with_one β :=
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lift (coe_mul_hom.comp f)
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@[simp, to_additive] lemma map_coe (f : α →ₙ* β) (a : α) : map f (a : with_one α) = f a :=
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lift_coe _ _
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@[simp, to_additive]
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lemma map_id : map (mul_hom.id α) = monoid_hom.id (with_one α) :=
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by { ext, induction x using with_one.cases_on; refl }
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@[to_additive]
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lemma map_map (f : α →ₙ* β) (g : β →ₙ* γ) (x) :
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map g (map f x) = map (g.comp f) x :=
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by { induction x using with_one.cases_on; refl }
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@[simp, to_additive]
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lemma map_comp (f : α →ₙ* β) (g : β →ₙ* γ) :
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map (g.comp f) = (map g).comp (map f) :=
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monoid_hom.ext $ λ x, (map_map f g x).symm
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/-- A version of `equiv.option_congr` for `with_one`. -/
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@[to_additive "A version of `equiv.option_congr` for `with_zero`.", simps apply]
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def _root_.mul_equiv.with_one_congr (e : α ≃* β) : with_one α ≃* with_one β :=
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{ to_fun := map e.to_mul_hom,
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inv_fun := map e.symm.to_mul_hom,
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left_inv := λ x, (map_map _ _ _).trans $ by induction x using with_one.cases_on; { simp },
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right_inv := λ x, (map_map _ _ _).trans $ by induction x using with_one.cases_on; { simp },
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.. map e.to_mul_hom }
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@[simp]
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lemma _root_.mul_equiv.with_one_congr_refl : (mul_equiv.refl α).with_one_congr = mul_equiv.refl _ :=
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mul_equiv.to_monoid_hom_injective map_id
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@[simp]
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lemma _root_.mul_equiv.with_one_congr_symm (e : α ≃* β) :
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e.with_one_congr.symm = e.symm.with_one_congr := rfl
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@[simp]
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lemma _root_.mul_equiv.with_one_congr_trans (e₁ : α ≃* β) (e₂ : β ≃* γ) :
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e₁.with_one_congr.trans e₂.with_one_congr = (e₁.trans e₂).with_one_congr :=
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mul_equiv.to_monoid_hom_injective (map_comp _ _).symm
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end map
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end with_one

‎src/algebra/group/with_one.lean‎ renamed to ‎src/algebra/group/with_one/defs.lean‎

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Authors: Mario Carneiro, Johan Commelin
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-/
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import order.with_bot
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import algebra.hom.equiv.basic
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import algebra.group_with_zero.units.basic
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import algebra.ring.defs
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/-!
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{ mul_comm := (option.lift_or_get_comm _).1,
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..with_one.monoid }
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section
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-- workaround: we make `with_one`/`with_zero` irreducible for this definition, otherwise `simps`
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-- will unfold it in the statement of the lemma it generates.
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local attribute [irreducible] with_one with_zero
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/-- `coe` as a bundled morphism -/
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@[to_additive "`coe` as a bundled morphism", simps apply]
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def coe_mul_hom [has_mul α] : α →ₙ* (with_one α) :=
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{ to_fun := coe, map_mul' := λ x y, rfl }
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end
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section lift
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variables [has_mul α] [mul_one_class β]
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/-- Lift a semigroup homomorphism `f` to a bundled monoid homorphism. -/
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@[to_additive "Lift an add_semigroup homomorphism `f` to a bundled add_monoid homorphism."]
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def lift : (α →ₙ* β) ≃ (with_one α →* β) :=
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{ to_fun := λ f,
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{ to_fun := λ x, option.cases_on x 1 f,
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map_one' := rfl,
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map_mul' := λ x y,
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with_one.cases_on x (by { rw one_mul, exact (one_mul _).symm }) $ λ x,
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with_one.cases_on y (by { rw mul_one, exact (mul_one _).symm }) $ λ y,
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f.map_mul x y },
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inv_fun := λ F, F.to_mul_hom.comp coe_mul_hom,
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left_inv := λ f, mul_hom.ext $ λ x, rfl,
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right_inv := λ F, monoid_hom.ext $ λ x, with_one.cases_on x F.map_one.symm $ λ x, rfl }
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variables (f : α →ₙ* β)
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@[simp, to_additive]
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lemma lift_coe (x : α) : lift f x = f x := rfl
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@[simp, to_additive]
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lemma lift_one : lift f 1 = 1 := rfl
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@[to_additive]
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theorem lift_unique (f : with_one α →* β) : f = lift (f.to_mul_hom.comp coe_mul_hom) :=
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(lift.apply_symm_apply f).symm
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end lift
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attribute [irreducible] with_one
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section map
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variables [has_mul α] [has_mul β] [has_mul γ]
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/-- Given a multiplicative map from `α → β` returns a monoid homomorphism
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from `with_one α` to `with_one β` -/
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@[to_additive "Given an additive map from `α → β` returns an add_monoid homomorphism
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from `with_zero α` to `with_zero β`"]
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def map (f : α →ₙ* β) : with_one α →* with_one β :=
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lift (coe_mul_hom.comp f)
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@[simp, to_additive] lemma map_coe (f : α →ₙ* β) (a : α) : map f (a : with_one α) = f a :=
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lift_coe _ _
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@[simp, to_additive]
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lemma map_id : map (mul_hom.id α) = monoid_hom.id (with_one α) :=
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by { ext, induction x using with_one.cases_on; refl }
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@[to_additive]
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lemma map_map (f : α →ₙ* β) (g : β →ₙ* γ) (x) :
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map g (map f x) = map (g.comp f) x :=
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by { induction x using with_one.cases_on; refl }
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@[simp, to_additive]
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lemma map_comp (f : α →ₙ* β) (g : β →ₙ* γ) :
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map (g.comp f) = (map g).comp (map f) :=
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monoid_hom.ext $ λ x, (map_map f g x).symm
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/-- A version of `equiv.option_congr` for `with_one`. -/
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@[to_additive "A version of `equiv.option_congr` for `with_zero`.", simps apply]
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def _root_.mul_equiv.with_one_congr (e : α ≃* β) : with_one α ≃* with_one β :=
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{ to_fun := map e.to_mul_hom,
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inv_fun := map e.symm.to_mul_hom,
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left_inv := λ x, (map_map _ _ _).trans $ by induction x using with_one.cases_on; { simp },
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right_inv := λ x, (map_map _ _ _).trans $ by induction x using with_one.cases_on; { simp },
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.. map e.to_mul_hom }
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@[simp]
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lemma _root_.mul_equiv.with_one_congr_refl : (mul_equiv.refl α).with_one_congr = mul_equiv.refl _ :=
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mul_equiv.to_monoid_hom_injective map_id
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@[simp]
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lemma _root_.mul_equiv.with_one_congr_symm (e : α ≃* β) :
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e.with_one_congr.symm = e.symm.with_one_congr := rfl
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@[simp]
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lemma _root_.mul_equiv.with_one_congr_trans (e₁ : α ≃* β) (e₂ : β ≃* γ) :
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e₁.with_one_congr.trans e₂.with_one_congr = (e₁.trans e₂).with_one_congr :=
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mul_equiv.to_monoid_hom_injective (map_comp _ _).symm
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end map
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@[simp, norm_cast, to_additive]
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lemma coe_mul [has_mul α] (a b : α) : ((a * b : α) : with_one α) = a * b := rfl
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..with_zero.mul_zero_class,
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..with_zero.monoid_with_zero }
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/-- Any group is isomorphic to the units of itself adjoined with `0`. -/
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def units_with_zero_equiv [group α] : (with_zero α)ˣ ≃* α :=
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{ to_fun := λ a, unzero a.ne_zero,
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inv_fun := λ a, units.mk0 a coe_ne_zero,
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left_inv := λ _, units.ext $ by simpa only [coe_unzero],
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right_inv := λ _, rfl,
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map_mul' := λ _ _, coe_inj.mp $ by simpa only [coe_unzero, coe_mul] }
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attribute [irreducible] with_zero
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end with_zero

‎src/algebra/group/with_one/units.lean‎

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/-
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Copyright (c) 2018 Mario Carneiro. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Mario Carneiro, Johan Commelin
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-/
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import algebra.group.with_one.basic
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import algebra.group_with_zero.units.basic
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/-!
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# Isomorphism between a group and the units of itself adjoined with `0`
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## Notes
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This is here to keep `algebra.group_with_zero.units.basic` out of the import requirements of
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`algebra.order.field.defs`.
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-/
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namespace with_zero
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/-- Any group is isomorphic to the units of itself adjoined with `0`. -/
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def units_with_zero_equiv {α : Type*} [group α] : (with_zero α)ˣ ≃* α :=
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{ to_fun := λ a, unzero a.ne_zero,
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inv_fun := λ a, units.mk0 a coe_ne_zero,
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left_inv := λ _, units.ext $ by simpa only [coe_unzero],
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right_inv := λ _, rfl,
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map_mul' := λ _ _, coe_inj.mp $ by simpa only [coe_unzero, coe_mul] }
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end with_zero

‎src/algebra/order/field/defs.lean‎

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instance linear_ordered_field.to_linear_ordered_semifield [linear_ordered_field α] :
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linear_ordered_semifield α :=
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{ ..linear_ordered_ring.to_linear_ordered_semiring, ..‹linear_ordered_field α› }
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-- Guard against import creep.
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assert_not_exists monoid_hom

‎src/algebra/order/group/abs.lean‎

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Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
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import algebra.abs
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import algebra.order.group.defs
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import algebra.order.group.order_iso
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import order.min_max
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/-!

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