Documentation

Mathlib.Algebra.FreeMonoid.Basic

Free monoid over a given alphabet #

Main definitions #

def FreeAddMonoid (α : Type u_1) :
Type u_1

Free nonabelian additive monoid over a given alphabet

Equations
def FreeMonoid (α : Type u_1) :
Type u_1

Free monoid over a given alphabet.

Equations
Equations
  • FreeAddMonoid.instDecidableEqFreeAddMonoid = instDecidableEqList
Equations
  • FreeMonoid.instDecidableEqFreeMonoid = instDecidableEqList

The identity equivalence between FreeAddMonoid α and List α.

Equations
def FreeMonoid.toList {α : Type u_1} :

The identity equivalence between FreeMonoid α and List α.

Equations

The identity equivalence between List α and FreeAddMonoid α.

Equations
def FreeMonoid.ofList {α : Type u_1} :

The identity equivalence between List α and FreeMonoid α.

Equations
@[simp]
theorem FreeAddMonoid.toList_symm {α : Type u_1} :
Equiv.symm FreeAddMonoid.toList = FreeAddMonoid.ofList
@[simp]
theorem FreeMonoid.toList_symm {α : Type u_1} :
Equiv.symm FreeMonoid.toList = FreeMonoid.ofList
@[simp]
theorem FreeAddMonoid.ofList_symm {α : Type u_1} :
Equiv.symm FreeAddMonoid.ofList = FreeAddMonoid.toList
@[simp]
theorem FreeMonoid.ofList_symm {α : Type u_1} :
Equiv.symm FreeMonoid.ofList = FreeMonoid.toList
@[simp]
theorem FreeAddMonoid.toList_ofList {α : Type u_1} (l : List α) :
FreeAddMonoid.toList (FreeAddMonoid.ofList l) = l
@[simp]
theorem FreeMonoid.toList_ofList {α : Type u_1} (l : List α) :
FreeMonoid.toList (FreeMonoid.ofList l) = l
@[simp]
theorem FreeAddMonoid.ofList_toList {α : Type u_1} (xs : FreeAddMonoid α) :
FreeAddMonoid.ofList (FreeAddMonoid.toList xs) = xs
@[simp]
theorem FreeMonoid.ofList_toList {α : Type u_1} (xs : FreeMonoid α) :
FreeMonoid.ofList (FreeMonoid.toList xs) = xs
@[simp]
theorem FreeAddMonoid.toList_comp_ofList {α : Type u_1} :
FreeAddMonoid.toList FreeAddMonoid.ofList = id
@[simp]
theorem FreeMonoid.toList_comp_ofList {α : Type u_1} :
FreeMonoid.toList FreeMonoid.ofList = id
@[simp]
theorem FreeAddMonoid.ofList_comp_toList {α : Type u_1} :
FreeAddMonoid.ofList FreeAddMonoid.toList = id
@[simp]
theorem FreeMonoid.ofList_comp_toList {α : Type u_1} :
FreeMonoid.ofList FreeMonoid.toList = id
def FreeAddMonoid.instAddCancelMonoidFreeAddMonoid.proof_1 {α : Type u_1} :
∀ (x x_1 x_2 : FreeAddMonoid α), FreeAddMonoid.toList x ++ FreeAddMonoid.toList x_1 = FreeAddMonoid.toList x ++ FreeAddMonoid.toList x_2FreeAddMonoid.toList x_1 = FreeAddMonoid.toList x_2
Equations
  • One or more equations did not get rendered due to their size.
def FreeAddMonoid.instAddCancelMonoidFreeAddMonoid.proof_4 {α : Type u_1} :
∀ (x x_1 x_2 : FreeAddMonoid α), FreeAddMonoid.toList x ++ FreeAddMonoid.toList x_1 = FreeAddMonoid.toList x_2 ++ FreeAddMonoid.toList x_1FreeAddMonoid.toList x = FreeAddMonoid.toList x_2
Equations
  • One or more equations did not get rendered due to their size.
def FreeAddMonoid.instAddCancelMonoidFreeAddMonoid.proof_3 {α : Type u_1} :
∀ (n : ) (x : FreeAddMonoid α), nsmulRec (n + 1) x = nsmulRec (n + 1) x
Equations
  • (_ : nsmulRec (n + 1) x = nsmulRec (n + 1) x) = (_ : nsmulRec (n + 1) x = nsmulRec (n + 1) x)
Equations
  • One or more equations did not get rendered due to their size.
def FreeAddMonoid.instAddCancelMonoidFreeAddMonoid.proof_2 {α : Type u_1} :
∀ (x : FreeAddMonoid α), nsmulRec 0 x = nsmulRec 0 x
Equations
  • (_ : nsmulRec 0 x = nsmulRec 0 x) = (_ : nsmulRec 0 x = nsmulRec 0 x)
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  • One or more equations did not get rendered due to their size.
Equations
  • FreeAddMonoid.instInhabitedFreeAddMonoid = { default := 0 }
Equations
  • FreeMonoid.instInhabitedFreeMonoid = { default := 1 }
@[simp]
theorem FreeAddMonoid.toList_zero {α : Type u_1} :
FreeAddMonoid.toList 0 = []
@[simp]
theorem FreeMonoid.toList_one {α : Type u_1} :
FreeMonoid.toList 1 = []
@[simp]
theorem FreeAddMonoid.ofList_nil {α : Type u_1} :
FreeAddMonoid.ofList [] = 0
@[simp]
theorem FreeMonoid.ofList_nil {α : Type u_1} :
FreeMonoid.ofList [] = 1
@[simp]
theorem FreeAddMonoid.toList_add {α : Type u_1} (xs : FreeAddMonoid α) (ys : FreeAddMonoid α) :
FreeAddMonoid.toList (xs + ys) = FreeAddMonoid.toList xs ++ FreeAddMonoid.toList ys
@[simp]
theorem FreeMonoid.toList_mul {α : Type u_1} (xs : FreeMonoid α) (ys : FreeMonoid α) :
FreeMonoid.toList (xs * ys) = FreeMonoid.toList xs ++ FreeMonoid.toList ys
@[simp]
theorem FreeAddMonoid.ofList_append {α : Type u_1} (xs : List α) (ys : List α) :
FreeAddMonoid.ofList (xs ++ ys) = FreeAddMonoid.ofList xs + FreeAddMonoid.ofList ys
@[simp]
theorem FreeMonoid.ofList_append {α : Type u_1} (xs : List α) (ys : List α) :
FreeMonoid.ofList (xs ++ ys) = FreeMonoid.ofList xs * FreeMonoid.ofList ys
@[simp]
theorem FreeAddMonoid.toList_sum {α : Type u_1} (xs : List (FreeAddMonoid α)) :
FreeAddMonoid.toList (List.sum xs) = List.join (List.map (FreeAddMonoid.toList) xs)
@[simp]
theorem FreeMonoid.toList_prod {α : Type u_1} (xs : List (FreeMonoid α)) :
FreeMonoid.toList (List.prod xs) = List.join (List.map (FreeMonoid.toList) xs)
@[simp]
theorem FreeAddMonoid.ofList_join {α : Type u_1} (xs : List (List α)) :
FreeAddMonoid.ofList (List.join xs) = List.sum (List.map (FreeAddMonoid.ofList) xs)
@[simp]
theorem FreeMonoid.ofList_join {α : Type u_1} (xs : List (List α)) :
FreeMonoid.ofList (List.join xs) = List.prod (List.map (FreeMonoid.ofList) xs)
def FreeAddMonoid.of {α : Type u_1} (x : α) :

Embeds an element of α into FreeAddMonoid α as a singleton list.

Equations
def FreeMonoid.of {α : Type u_1} (x : α) :

Embeds an element of α into FreeMonoid α as a singleton list.

Equations
@[simp]
theorem FreeAddMonoid.toList_of {α : Type u_1} (x : α) :
FreeAddMonoid.toList (FreeAddMonoid.of x) = [x]
@[simp]
theorem FreeMonoid.toList_of {α : Type u_1} (x : α) :
FreeMonoid.toList (FreeMonoid.of x) = [x]
theorem FreeAddMonoid.ofList_singleton {α : Type u_1} (x : α) :
FreeAddMonoid.ofList [x] = FreeAddMonoid.of x
theorem FreeMonoid.ofList_singleton {α : Type u_1} (x : α) :
FreeMonoid.ofList [x] = FreeMonoid.of x
@[simp]
theorem FreeAddMonoid.ofList_cons {α : Type u_1} (x : α) (xs : List α) :
FreeAddMonoid.ofList (x :: xs) = FreeAddMonoid.of x + FreeAddMonoid.ofList xs
@[simp]
theorem FreeMonoid.ofList_cons {α : Type u_1} (x : α) (xs : List α) :
FreeMonoid.ofList (x :: xs) = FreeMonoid.of x * FreeMonoid.ofList xs
theorem FreeAddMonoid.toList_of_add {α : Type u_1} (x : α) (xs : FreeAddMonoid α) :
FreeAddMonoid.toList (FreeAddMonoid.of x + xs) = x :: FreeAddMonoid.toList xs
theorem FreeMonoid.toList_of_mul {α : Type u_1} (x : α) (xs : FreeMonoid α) :
FreeMonoid.toList (FreeMonoid.of x * xs) = x :: FreeMonoid.toList xs
theorem FreeAddMonoid.of_injective {α : Type u_1} :
Function.Injective FreeAddMonoid.of
theorem FreeMonoid.of_injective {α : Type u_1} :
Function.Injective FreeMonoid.of
def FreeAddMonoid.recOn {α : Type u_1} {C : FreeAddMonoid αSort u_2} (xs : FreeAddMonoid α) (h0 : C 0) (ih : (x : α) → (xs : FreeAddMonoid α) → C xsC (FreeAddMonoid.of x + xs)) :
C xs

Recursor for FreeAddMonoid using 0 and FreeAddMonoid.of x + xs instead of [] and x :: xs.

Equations
def FreeMonoid.recOn {α : Type u_1} {C : FreeMonoid αSort u_2} (xs : FreeMonoid α) (h0 : C 1) (ih : (x : α) → (xs : FreeMonoid α) → C xsC (FreeMonoid.of x * xs)) :
C xs

Recursor for FreeMonoid using 1 and FreeMonoid.of x * xs instead of [] and x :: xs.

Equations
@[simp]
theorem FreeAddMonoid.recOn_zero {α : Type u_1} {C : FreeAddMonoid αSort u_2} (h0 : C 0) (ih : (x : α) → (xs : FreeAddMonoid α) → C xsC (FreeAddMonoid.of x + xs)) :
@[simp]
theorem FreeMonoid.recOn_one {α : Type u_1} {C : FreeMonoid αSort u_2} (h0 : C 1) (ih : (x : α) → (xs : FreeMonoid α) → C xsC (FreeMonoid.of x * xs)) :
FreeMonoid.recOn 1 h0 ih = h0
@[simp]
theorem FreeAddMonoid.recOn_of_add {α : Type u_1} {C : FreeAddMonoid αSort u_2} (x : α) (xs : FreeAddMonoid α) (h0 : C 0) (ih : (x : α) → (xs : FreeAddMonoid α) → C xsC (FreeAddMonoid.of x + xs)) :
@[simp]
theorem FreeMonoid.recOn_of_mul {α : Type u_1} {C : FreeMonoid αSort u_2} (x : α) (xs : FreeMonoid α) (h0 : C 1) (ih : (x : α) → (xs : FreeMonoid α) → C xsC (FreeMonoid.of x * xs)) :
FreeMonoid.recOn (FreeMonoid.of x * xs) h0 ih = ih x xs (FreeMonoid.recOn xs h0 ih)
def FreeAddMonoid.casesOn {α : Type u_1} {C : FreeAddMonoid αSort u_2} (xs : FreeAddMonoid α) (h0 : C 0) (ih : (x : α) → (xs : FreeAddMonoid α) → C (FreeAddMonoid.of x + xs)) :
C xs

A version of List.casesOn for FreeAddMonoid using 0 and FreeAddMonoid.of x + xs instead of [] and x :: xs.

Equations
def FreeMonoid.casesOn {α : Type u_1} {C : FreeMonoid αSort u_2} (xs : FreeMonoid α) (h0 : C 1) (ih : (x : α) → (xs : FreeMonoid α) → C (FreeMonoid.of x * xs)) :
C xs

A version of List.cases_on for FreeMonoid using 1 and FreeMonoid.of x * xs instead of [] and x :: xs.

Equations
@[simp]
theorem FreeAddMonoid.casesOn_zero {α : Type u_1} {C : FreeAddMonoid αSort u_2} (h0 : C 0) (ih : (x : α) → (xs : FreeAddMonoid α) → C (FreeAddMonoid.of x + xs)) :
@[simp]
theorem FreeMonoid.casesOn_one {α : Type u_1} {C : FreeMonoid αSort u_2} (h0 : C 1) (ih : (x : α) → (xs : FreeMonoid α) → C (FreeMonoid.of x * xs)) :
@[simp]
theorem FreeAddMonoid.casesOn_of_add {α : Type u_1} {C : FreeAddMonoid αSort u_2} (x : α) (xs : FreeAddMonoid α) (h0 : C 0) (ih : (x : α) → (xs : FreeAddMonoid α) → C (FreeAddMonoid.of x + xs)) :
@[simp]
theorem FreeMonoid.casesOn_of_mul {α : Type u_1} {C : FreeMonoid αSort u_2} (x : α) (xs : FreeMonoid α) (h0 : C 1) (ih : (x : α) → (xs : FreeMonoid α) → C (FreeMonoid.of x * xs)) :
FreeMonoid.casesOn (FreeMonoid.of x * xs) h0 ih = ih x xs
theorem FreeAddMonoid.hom_eq {α : Type u_1} {M : Type u_2} [inst : AddMonoid M] ⦃f : FreeAddMonoid α →+ M ⦃g : FreeAddMonoid α →+ M (h : ∀ (x : α), f (FreeAddMonoid.of x) = g (FreeAddMonoid.of x)) :
f = g
theorem FreeMonoid.hom_eq {α : Type u_1} {M : Type u_2} [inst : Monoid M] ⦃f : FreeMonoid α →* M ⦃g : FreeMonoid α →* M (h : ∀ (x : α), f (FreeMonoid.of x) = g (FreeMonoid.of x)) :
f = g
def FreeAddMonoid.sumAux {M : Type u_1} [inst : AddMonoid M] :
List MM

A variant of List.sum that has [x].sum = x true definitionally. The purpose is to make FreeAddMonoid.lift_eval_of true by rfl.

Equations
abbrev FreeAddMonoid.sumAux.match_1 {M : Type u_1} (motive : List MSort u_2) :
(x : List M) → (Unitmotive []) → ((x : M) → (xs : List M) → motive (x :: xs)) → motive x
Equations
def FreeMonoid.prodAux {M : Type u_1} [inst : Monoid M] :
List MM

A variant of List.prod that has [x].prod = x true definitionally. The purpose is to make FreeMonoid.lift_eval_of true by rfl.

Equations
abbrev FreeAddMonoid.sumAux_eq.match_1 {M : Type u_1} (motive : List MProp) :
(x : List M) → (Unitmotive []) → ((head : M) → (xs : List M) → motive (head :: xs)) → motive x
Equations
theorem FreeMonoid.prodAux_eq {M : Type u_1} [inst : Monoid M] (l : List M) :
def FreeAddMonoid.lift.proof_4 {α : Type u_1} {M : Type u_2} [inst : AddMonoid M] (f : FreeAddMonoid α →+ M) :
(fun f => { toZeroHom := { toFun := fun l => FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList l)), map_zero' := (_ : (fun l => FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList l))) 0 = (fun l => FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList l))) 0) }, map_add' := (_ : ∀ (x x_1 : FreeAddMonoid α), FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList x ++ FreeAddMonoid.toList x_1)) = FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList x)) + FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList x_1))) }) ((fun f x => f (FreeAddMonoid.of x)) f) = f
Equations
  • One or more equations did not get rendered due to their size.
def FreeAddMonoid.lift.proof_1 {α : Type u_2} {M : Type u_1} [inst : AddMonoid M] (f : αM) :
(fun l => FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList l))) 0 = (fun l => FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList l))) 0
Equations
  • One or more equations did not get rendered due to their size.
def FreeAddMonoid.lift.proof_2 {α : Type u_1} {M : Type u_2} [inst : AddMonoid M] (f : αM) :
∀ (x x_1 : FreeAddMonoid α), FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList x ++ FreeAddMonoid.toList x_1)) = FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList x)) + FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList x_1))
Equations
  • One or more equations did not get rendered due to their size.
def FreeAddMonoid.lift.proof_3 {α : Type u_1} {M : Type u_2} [inst : AddMonoid M] (f : αM) :
(fun f x => f (FreeAddMonoid.of x)) ((fun f => { toZeroHom := { toFun := fun l => FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList l)), map_zero' := (_ : (fun l => FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList l))) 0 = (fun l => FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList l))) 0) }, map_add' := (_ : ∀ (x x_1 : FreeAddMonoid α), FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList x ++ FreeAddMonoid.toList x_1)) = FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList x)) + FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList x_1))) }) f) = (fun f x => f (FreeAddMonoid.of x)) ((fun f => { toZeroHom := { toFun := fun l => FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList l)), map_zero' := (_ : (fun l => FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList l))) 0 = (fun l => FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList l))) 0) }, map_add' := (_ : ∀ (x x_1 : FreeAddMonoid α), FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList x ++ FreeAddMonoid.toList x_1)) = FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList x)) + FreeAddMonoid.sumAux (List.map f (FreeAddMonoid.toList x_1))) }) f)
Equations
  • One or more equations did not get rendered due to their size.
def FreeAddMonoid.lift {α : Type u_1} {M : Type u_2} [inst : AddMonoid M] :
(αM) (FreeAddMonoid α →+ M)

Equivalence between maps α → A→ A and additive monoid homomorphisms FreeAddMonoid α →+ A→+ A.

Equations
  • One or more equations did not get rendered due to their size.
def FreeMonoid.lift {α : Type u_1} {M : Type u_2} [inst : Monoid M] :
(αM) (FreeMonoid α →* M)

Equivalence between maps α → M→ M and monoid homomorphisms FreeMonoid α →* M→* M.

Equations
  • One or more equations did not get rendered due to their size.
@[simp]
theorem FreeAddMonoid.lift_ofList {α : Type u_1} {M : Type u_2} [inst : AddMonoid M] (f : αM) (l : List α) :
↑(FreeAddMonoid.lift f) (FreeAddMonoid.ofList l) = List.sum (List.map f l)
@[simp]
theorem FreeMonoid.lift_ofList {α : Type u_1} {M : Type u_2} [inst : Monoid M] (f : αM) (l : List α) :
↑(FreeMonoid.lift f) (FreeMonoid.ofList l) = List.prod (List.map f l)
@[simp]
theorem FreeAddMonoid.lift_symm_apply {α : Type u_1} {M : Type u_2} [inst : AddMonoid M] (f : FreeAddMonoid α →+ M) :
↑(Equiv.symm FreeAddMonoid.lift) f = f FreeAddMonoid.of
@[simp]
theorem FreeMonoid.lift_symm_apply {α : Type u_1} {M : Type u_2} [inst : Monoid M] (f : FreeMonoid α →* M) :
↑(Equiv.symm FreeMonoid.lift) f = f FreeMonoid.of
theorem FreeAddMonoid.lift_apply {α : Type u_1} {M : Type u_2} [inst : AddMonoid M] (f : αM) (l : FreeAddMonoid α) :
↑(FreeAddMonoid.lift f) l = List.sum (List.map f (FreeAddMonoid.toList l))
theorem FreeMonoid.lift_apply {α : Type u_1} {M : Type u_2} [inst : Monoid M] (f : αM) (l : FreeMonoid α) :
↑(FreeMonoid.lift f) l = List.prod (List.map f (FreeMonoid.toList l))
theorem FreeAddMonoid.lift_comp_of {α : Type u_1} {M : Type u_2} [inst : AddMonoid M] (f : αM) :
↑(FreeAddMonoid.lift f) FreeAddMonoid.of = f
theorem FreeMonoid.lift_comp_of {α : Type u_1} {M : Type u_2} [inst : Monoid M] (f : αM) :
↑(FreeMonoid.lift f) FreeMonoid.of = f
@[simp]
theorem FreeAddMonoid.lift_eval_of {α : Type u_2} {M : Type u_1} [inst : AddMonoid M] (f : αM) (x : α) :
↑(FreeAddMonoid.lift f) (FreeAddMonoid.of x) = f x
@[simp]
theorem FreeMonoid.lift_eval_of {α : Type u_2} {M : Type u_1} [inst : Monoid M] (f : αM) (x : α) :
↑(FreeMonoid.lift f) (FreeMonoid.of x) = f x
@[simp]
theorem FreeAddMonoid.lift_restrict {α : Type u_1} {M : Type u_2} [inst : AddMonoid M] (f : FreeAddMonoid α →+ M) :
FreeAddMonoid.lift (f FreeAddMonoid.of) = f
@[simp]
theorem FreeMonoid.lift_restrict {α : Type u_1} {M : Type u_2} [inst : Monoid M] (f : FreeMonoid α →* M) :
FreeMonoid.lift (f FreeMonoid.of) = f
theorem FreeAddMonoid.comp_lift {α : Type u_3} {M : Type u_1} [inst : AddMonoid M] {N : Type u_2} [inst : AddMonoid N] (g : M →+ N) (f : αM) :
AddMonoidHom.comp g (FreeAddMonoid.lift f) = FreeAddMonoid.lift (g f)
theorem FreeMonoid.comp_lift {α : Type u_3} {M : Type u_1} [inst : Monoid M] {N : Type u_2} [inst : Monoid N] (g : M →* N) (f : αM) :
MonoidHom.comp g (FreeMonoid.lift f) = FreeMonoid.lift (g f)
theorem FreeAddMonoid.hom_map_lift {α : Type u_3} {M : Type u_1} [inst : AddMonoid M] {N : Type u_2} [inst : AddMonoid N] (g : M →+ N) (f : αM) (x : FreeAddMonoid α) :
g (↑(FreeAddMonoid.lift f) x) = ↑(FreeAddMonoid.lift (g f)) x
theorem FreeMonoid.hom_map_lift {α : Type u_3} {M : Type u_1} [inst : Monoid M] {N : Type u_2} [inst : Monoid N] (g : M →* N) (f : αM) (x : FreeMonoid α) :
g (↑(FreeMonoid.lift f) x) = ↑(FreeMonoid.lift (g f)) x
def FreeAddMonoid.mkAddAction.proof_2 {α : Type u_1} {β : Type u_2} (f : αββ) :
∀ (x x_1 : FreeAddMonoid α) (x_2 : β), List.foldr f x_2 (FreeAddMonoid.toList x ++ FreeAddMonoid.toList x_1) = List.foldr f (List.foldr f x_2 (FreeAddMonoid.toList x_1)) (FreeAddMonoid.toList x)
Equations
  • One or more equations did not get rendered due to their size.
def FreeAddMonoid.mkAddAction {α : Type u_1} {β : Type u_2} (f : αββ) :

Define an additive action of FreeAddMonoid α on β.

Equations
  • One or more equations did not get rendered due to their size.
def FreeAddMonoid.mkAddAction.proof_1 {α : Type u_2} {β : Type u_1} (f : αββ) :
∀ (x : β), 0 +ᵥ x = 0 +ᵥ x
Equations
def FreeMonoid.mkMulAction {α : Type u_1} {β : Type u_2} (f : αββ) :

Define a multiplicative action of FreeMonoid α on β.

Equations
  • One or more equations did not get rendered due to their size.
theorem FreeAddMonoid.vadd_def {α : Type u_1} {β : Type u_2} (f : αββ) (l : FreeAddMonoid α) (b : β) :
l +ᵥ b = List.foldr f b (FreeAddMonoid.toList l)
theorem FreeMonoid.smul_def {α : Type u_1} {β : Type u_2} (f : αββ) (l : FreeMonoid α) (b : β) :
l b = List.foldr f b (FreeMonoid.toList l)
theorem FreeAddMonoid.ofList_vadd {α : Type u_1} {β : Type u_2} (f : αββ) (l : List α) (b : β) :
FreeAddMonoid.ofList l +ᵥ b = List.foldr f b l
theorem FreeMonoid.ofList_smul {α : Type u_1} {β : Type u_2} (f : αββ) (l : List α) (b : β) :
FreeMonoid.ofList l b = List.foldr f b l
@[simp]
theorem FreeAddMonoid.of_vadd {α : Type u_2} {β : Type u_1} (f : αββ) (x : α) (y : β) :
@[simp]
theorem FreeMonoid.of_smul {α : Type u_2} {β : Type u_1} (f : αββ) (x : α) (y : β) :
FreeMonoid.of x y = f x y
def FreeAddMonoid.map.proof_2 {α : Type u_1} {β : Type u_2} (f : αβ) :
∀ (x x_1 : FreeAddMonoid α), List.map f (FreeAddMonoid.toList x ++ FreeAddMonoid.toList x_1) = List.map f (FreeAddMonoid.toList x) ++ List.map f (FreeAddMonoid.toList x_1)
Equations
  • One or more equations did not get rendered due to their size.
def FreeAddMonoid.map {α : Type u_1} {β : Type u_2} (f : αβ) :

The unique additive monoid homomorphism FreeAddMonoid α →+ FreeAddMonoid β→+ FreeAddMonoid β that sends each of x to of (f x).

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def FreeAddMonoid.map.proof_1 {α : Type u_2} {β : Type u_1} (f : αβ) :
(fun l => FreeAddMonoid.ofList (List.map f (FreeAddMonoid.toList l))) 0 = (fun l => FreeAddMonoid.ofList (List.map f (FreeAddMonoid.toList l))) 0
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def FreeMonoid.map {α : Type u_1} {β : Type u_2} (f : αβ) :

The unique monoid homomorphism FreeMonoid α →* FreeMonoid β→* FreeMonoid β that sends each of x to of (f x).

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@[simp]
theorem FreeAddMonoid.map_of {α : Type u_2} {β : Type u_1} (f : αβ) (x : α) :
@[simp]
theorem FreeMonoid.map_of {α : Type u_2} {β : Type u_1} (f : αβ) (x : α) :
theorem FreeAddMonoid.toList_map {α : Type u_1} {β : Type u_2} (f : αβ) (xs : FreeAddMonoid α) :
FreeAddMonoid.toList (↑(FreeAddMonoid.map f) xs) = List.map f (FreeAddMonoid.toList xs)
theorem FreeMonoid.toList_map {α : Type u_1} {β : Type u_2} (f : αβ) (xs : FreeMonoid α) :
FreeMonoid.toList (↑(FreeMonoid.map f) xs) = List.map f (FreeMonoid.toList xs)
theorem FreeAddMonoid.ofList_map {α : Type u_1} {β : Type u_2} (f : αβ) (xs : List α) :
FreeAddMonoid.ofList (List.map f xs) = ↑(FreeAddMonoid.map f) (FreeAddMonoid.ofList xs)
theorem FreeMonoid.ofList_map {α : Type u_1} {β : Type u_2} (f : αβ) (xs : List α) :
FreeMonoid.ofList (List.map f xs) = ↑(FreeMonoid.map f) (FreeMonoid.ofList xs)
theorem FreeAddMonoid.lift_of_comp_eq_map {α : Type u_1} {β : Type u_2} (f : αβ) :
(FreeAddMonoid.lift fun x => FreeAddMonoid.of (f x)) = FreeAddMonoid.map f
theorem FreeMonoid.lift_of_comp_eq_map {α : Type u_1} {β : Type u_2} (f : αβ) :
(FreeMonoid.lift fun x => FreeMonoid.of (f x)) = FreeMonoid.map f
theorem FreeAddMonoid.map_comp {α : Type u_1} {β : Type u_3} {γ : Type u_2} (g : βγ) (f : αβ) :
theorem FreeMonoid.map_comp {α : Type u_1} {β : Type u_3} {γ : Type u_2} (g : βγ) (f : αβ) :