move wip

3 files changed, 0 insertions, 628 deletions

diff --git a/Algebra/Derivations.lean b/Algebra/Derivations.lean
deleted file mode 100644
index b6e5cb5..0000000
--- a/Algebra/Derivations.lean
+++ /dev/null
@@ -1,237 +0,0 @@
-import Mathlib
-import Algebra.Terms
-
-namespace NominalAlgebra
-
-section Theories
-
-variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
-variable (sig : Signature)
-
-@[ext]
-structure Equation where
-  vars : Type
-  [vars_nominal : NominalSet 𝔸 vars]
-  left : Term 𝔸 sig vars
-  right : Term 𝔸 sig vars
-
-namespace Equation
-
-instance (eq : Equation 𝔸 sig) : NominalSet 𝔸 eq.vars :=
-  eq.vars_nominal
-
-def HasDepth
-    {𝔸 : Type} [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸] {sig : Signature}
-    (eq : Equation 𝔸 sig) (n : ℕ) : Prop :=
-  eq.left.HasDepth n ∧ eq.right.HasDepth n
-
-end Equation
-
-def UniformEquation :=
-  { eq : Equation 𝔸 sig // ∃ n, eq.HasDepth n }
-
-structure Theory where
-  sig : Signature
-  axioms : Set (UniformEquation 𝔸 sig)
-  axioms_orbitFinite : ∀ eq ∈ axioms, OrbitFinite 𝔸 eq.val.vars
-
-end Theories
-
-variable {𝔸 : Type} [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
-variable {sig : Signature}
-
-section Helpers
-
-def UniformEquation.mk
-    (X : Type) [NominalSet 𝔸 X] {n : ℕ} (t u : TermAtDepth 𝔸 sig X n) :
-    UniformEquation 𝔸 sig :=
-  ⟨⟨X, t.val, u.val⟩, ⟨n, t.prop, u.prop⟩⟩
-
-notation:50 X " ⊢ " lhs " ≣ " rhs => UniformEquation.mk X lhs rhs
-
-end Helpers
-
-section DerivableEquations
-
-inductive Derivable (T : Theory 𝔸) : (eq : UniformEquation 𝔸 T.sig) → Prop
-  | refl {X : Type} [NominalSet 𝔸 X] {x : X} :
-      Derivable T (X ⊢ .var x ≣ .var x)
-  | symm {X : Type} [NominalSet 𝔸 X]
-      {m : ℕ} {t u : TermAtDepth 𝔸 T.sig X m} :
-      Derivable T (X ⊢ u ≣ t) →
-      Derivable T (X ⊢ t ≣ u)
-  | trans {X : Type} [NominalSet 𝔸 X]
-      {m : ℕ} {t u v : TermAtDepth 𝔸 T.sig X m} :
-      Derivable T (X ⊢ t ≣ v) →
-      Derivable T (X ⊢ v ≣ u) →
-      Derivable T (X ⊢ t ≣ u)
-  | cong_pure {X : Type} [NominalSet 𝔸 X]
-      {f : T.sig.pure} {m : ℕ} {ts us : T.sig.pureArgs f → TermAtDepth 𝔸 T.sig X m} :
-      (∀ x, Derivable T (X ⊢ ts x ≣ us x)) →
-      Derivable T (X ⊢ .pure f ts ≣ .pure f us)
-  | cong_free {X : Type} [NominalSet 𝔸 X]
-      {a : 𝔸} {f : T.sig.free} {m : ℕ} {ts us : T.sig.freeArgs f → TermAtDepth 𝔸 T.sig X m} :
-      (∀ x, Derivable T (X ⊢ ts x ≣ us x)) →
-      Derivable T (X ⊢ .free a f ts ≣ .free a f us)
-  | cong_bound {X : Type} [NominalSet 𝔸 X]
-      {a : 𝔸} {f : T.sig.bound} {m : ℕ} {ts us : T.sig.boundArgs f → TermAtDepth 𝔸 T.sig X m} :
-      (∀ x, Derivable T (X ⊢ ts x ≣ us x)) →
-      Derivable T (X ⊢ .bound a f ts ≣ .bound a f us)
-  | ax' {X Y : Type} [NominalSet 𝔸 X] [NominalSet 𝔸 Y]
-      {m : ℕ} {r s : TermAtDepth 𝔸 T.sig Y m}
-      {l : ℕ} {σ : UniformSubstitution 𝔸 T.sig Y X l} {τ : FPerm 𝔸} :
-      ((Y ⊢ r ≣ s) ∈ T.axioms) →
-      (∀ (y : Y) (π : FPerm 𝔸), Derivable T (X ⊢ σ (π • y) ≣ π • (σ y))) →
-      Derivable T (X ⊢ σ.apply (τ • r) ≣ σ.apply (τ • s))
-  | perm {X : Type} [NominalSet 𝔸 X]
-      {a b : 𝔸} {f : T.sig.bound} {m : ℕ} {ts us : T.sig.boundArgs f → TermAtDepth 𝔸 T.sig X m} :
-      a ≠ b →
-      (∀ x, a # us x) →
-      (∀ x, Derivable T (X ⊢ ts x ≣ ⸨a, b⸩ • us x)) →
-      Derivable T (X ⊢ .bound a f ts ≣ .bound b f us)
-
-def DerivableEq (T : Theory 𝔸) (X : Type) [NominalSet 𝔸 X] (n : ℕ) :
-    Rel (TermAtDepth 𝔸 T.sig X n) (TermAtDepth 𝔸 T.sig X n) :=
-  fun t u => Derivable T (X ⊢ t ≣ u)
-
-variable {X Y : Type} [NominalSet 𝔸 X] [NominalSet 𝔸 Y]
-
-def UniformSubstitution.DerivablyEquivariant
-    (T : Theory 𝔸) {l : ℕ} (σ : UniformSubstitution 𝔸 T.sig X Y l) : Prop :=
-  (∀ (x : X) (π : FPerm 𝔸), Derivable T (Y ⊢ σ (π • x) ≣ π • σ x))
-
-variable {T : Theory 𝔸} {n : ℕ}
-
-/-- Restated ax' rule using DerivablyEquivariant in the premiss. -/
-lemma Derivable.ax
-    {m : ℕ} {r s : TermAtDepth 𝔸 T.sig Y m}
-    {l : ℕ} {σ : UniformSubstitution 𝔸 T.sig Y X l} {τ : FPerm 𝔸} :
-    ((Y ⊢ r ≣ s) ∈ T.axioms) →
-    σ.DerivablyEquivariant T →
-    Derivable T (X ⊢ σ.apply (τ • r) ≣ σ.apply (τ • s)) := .ax'
-
-lemma Derivable.of_eq
-    {m : ℕ} {eq : UniformEquation 𝔸 T.sig} (t u : TermAtDepth 𝔸 T.sig eq.val.vars m)
-    (ht : t.val = eq.val.left := by rfl)
-    (hu : u.val = eq.val.right := by rfl) :
-    Derivable T (eq.val.vars ⊢ t ≣ u) →
-    Derivable T eq := by
-  intro hderiv
-  have heq : (eq.val.vars ⊢ t ≣ u) = eq := by
-    apply Subtype.ext; ext
-    · rfl
-    · rfl
-    · exact heq_of_eq ht
-    · exact heq_of_eq hu
-  rw [← heq]; exact hderiv
-
-lemma Derivable.equivariant_substitution
-    {l : ℕ} (σ : UniformSubstitution 𝔸 T.sig Y X l) :
-    σ.DerivablyEquivariant T →
-    ∀ {n : ℕ} (t : TermAtDepth 𝔸 T.sig Y n) (π : FPerm 𝔸),
-      Derivable T (X ⊢ σ.apply (π • t) ≣ π • (σ.apply t)) := by
-  intro hequi n t π
-  induction' t using TermAtDepth.induction <;> expose_names
-  · apply of_eq (σ (π • x)) (π • σ x)
-    exact hequi x π
-  · apply of_eq (.pure f (σ.apply ∘ (π • ·) ∘ args)) (.pure f ((π • ·) ∘ σ.apply ∘ args))
-    apply Derivable.cong_pure
-    exact h
-  · apply of_eq (.free (π • a) f (σ.apply ∘ (π • ·) ∘ args)) (.free (π • a) f ((π • ·) ∘ σ.apply ∘ args))
-    apply Derivable.cong_free
-    exact h
-  · apply of_eq (.bound (π • a) f (σ.apply ∘ (π • ·) ∘ args)) (.bound (π • a) f ((π • ·) ∘ σ.apply ∘ args))
-    apply Derivable.cong_bound
-    exact h
-
-lemma Derivable.equivariant {m : ℕ} {t u : TermAtDepth 𝔸 T.sig X m} :
-    ∀ (π : FPerm 𝔸), Derivable T (X ⊢ t ≣ u) → Derivable T (X ⊢ π • t ≣ π • u) := by
-  intro π
-  let rename (π : FPerm 𝔸) (eq : UniformEquation 𝔸 T.sig) : UniformEquation 𝔸 T.sig :=
-    ⟨⟨eq.val.vars, π • eq.val.left, π • eq.val.right⟩, by
-      obtain ⟨n, ⟨hn₁, hn₂⟩⟩ := eq.prop
-      let left : TermAtDepth 𝔸 T.sig eq.val.vars n := ⟨eq.val.left, hn₁⟩
-      let right : TermAtDepth 𝔸 T.sig eq.val.vars n := ⟨eq.val.right, hn₂⟩
-      use n; exact ⟨(π • left).prop, (π • right).prop⟩⟩
-  have rename_eq
-      {Y : Type} [NominalSet 𝔸 Y] {π : FPerm 𝔸} {k : ℕ}
-      {r s : TermAtDepth 𝔸 T.sig Y k} : rename π (Y ⊢ r ≣ s) = (Y ⊢ π • r ≣ π • s) := rfl
-  suffices ∀ eq, Derivable T eq → Derivable T (rename π eq) by
-    exact this (X ⊢ t ≣ u)
-  intro eq hderiv
-  induction' hderiv <;> expose_names
-  · exact .refl
-  · rw [rename_eq]
-    apply Derivable.symm
-    exact a_ih
-  · rw [rename_eq] at *
-    apply Derivable.trans
-    · exact a_ih
-    · exact a_ih_1
-  · apply of_eq (TermAtDepth.pure f ((π • ·) ∘ ts)) (TermAtDepth.pure f ((π • ·) ∘ us))
-    apply Derivable.cong_pure
-    exact a_ih
-  · apply of_eq (TermAtDepth.free (π • a) f ((π • ·) ∘ ts)) (TermAtDepth.free (π • a) f ((π • ·) ∘ us))
-    apply Derivable.cong_free
-    exact a_ih
-  · apply of_eq (TermAtDepth.bound (π • a) f ((π • ·) ∘ ts)) (TermAtDepth.bound (π • a) f ((π • ·) ∘ us))
-    apply Derivable.cong_bound
-    exact a_ih
-  · rw [rename_eq]
-    apply Derivable.trans (by apply Derivable.symm; apply equivariant_substitution; exact h_1) _
-    apply Derivable.trans _ (by apply equivariant_substitution; exact h_1)
-    rw [← SemigroupAction.mul_smul, ← SemigroupAction.mul_smul]
-    apply Derivable.ax
-    · exact h
-    · exact h_1
-  · apply of_eq (TermAtDepth.bound (π • a) f ((π • ·) ∘ ts)) (TermAtDepth.bound (π • b) f ((π • ·) ∘ us))
-    apply Derivable.perm
-    · intro hcontra
-      apply π.val.injective at hcontra
-      contradiction
-    · intro x
-      show π • a # π • (us x)
-      rw [← Rel.equivariant 𝔸 NominalSet.IsFresh]
-      exact h_1 x
-    · intro x
-      show Derivable T (X_1 ⊢ π • ts x ≣ ⸨π a, π b⸩ • π • us x)
-      rw [← SemigroupAction.mul_smul, ← FPerm.mul_swap_eq_swap_mul, SemigroupAction.mul_smul]
-      exact a_ih x
-
-instance DerivableEq.instEquivariant : Rel.Equivariant 𝔸 (DerivableEq T X n) := by
-  constructor; intro t u
-  exact Derivable.equivariant
-
-@[refl]
-lemma DerivableEq.refl (t : TermAtDepth 𝔸 T.sig X n) : DerivableEq T X n t t := by
-  induction' t using TermAtDepth.induction <;> expose_names
-  · apply Derivable.refl
-  · apply Derivable.cong_pure; assumption
-  · apply Derivable.cong_free; assumption
-  · apply Derivable.cong_bound; assumption
-
-@[symm]
-lemma DerivableEq.symm {t u : TermAtDepth 𝔸 T.sig X n} :
-    DerivableEq T X n u t → DerivableEq T X n t u := by
-  intro h
-  apply Derivable.symm
-  exact h
-
-@[trans]
-lemma DerivableEq.trans {t u v : TermAtDepth 𝔸 T.sig X n} :
-    DerivableEq T X n t v → DerivableEq T X n v u → DerivableEq T X n t u := by
-  intro htv hvu
-  apply Derivable.trans
-  · exact htv
-  · exact hvu
-
-instance TermAtDepth.setoid : Setoid (TermAtDepth 𝔸 T.sig X n) where
-  r := DerivableEq T X n
-  iseqv :=
-    { refl := DerivableEq.refl
-      symm := DerivableEq.symm
-      trans := DerivableEq.trans }
-
-end DerivableEquations
-
-end NominalAlgebra
diff --git a/Algebra/Semantics.lean b/Algebra/Semantics.lean
deleted file mode 100644
index 13b54a0..0000000
--- a/Algebra/Semantics.lean
+++ /dev/null
@@ -1,79 +0,0 @@
-import Mathlib
-import Algebra.Terms
-import Algebra.Derivations
-
-open CategoryTheory
-
-namespace NominalAlgebra
-
-section AlgebraDefinition
-
-variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
-variable (sig : Signature)
-
-structure Algebra where
-  carrier : ℕ → Nom 𝔸
-  pure (f : sig.pure) {m : ℕ} :
-    (sig.pureArgs f → carrier m) →ₑ[𝔸] carrier (m + sig.pureDepth f)
-  free (f : sig.free) {m : ℕ} :
-    (𝔸 × (sig.freeArgs f → carrier m)) →ₑ[𝔸] carrier (m + sig.freeDepth f)
-  bound (f : sig.bound) {m : ℕ} :
-    NameAbstraction 𝔸 (sig.boundArgs f → carrier m) →ₑ[𝔸] carrier (m + sig.boundDepth f)
-
-instance : CoeFun (Algebra 𝔸 sig) (fun _ => ℕ → Nom 𝔸) where
-  coe A := A.carrier
-
-@[ext]
-structure Algebra.Hom (A B : Algebra 𝔸 sig) where
-  toFun : (n : ℕ) → (A n →ₑ[𝔸] B n)
-  preserve_pure : ∀ (f : sig.pure) {m : ℕ} (args : sig.pureArgs f → A m),
-    toFun (m + sig.pureDepth f) (A.pure f args) = B.pure f (toFun m ∘ args)
-  preserve_free : ∀ (a : 𝔸) (f : sig.free) {m : ℕ} (args : sig.freeArgs f → A m),
-    toFun (m + sig.freeDepth f) (A.free f ⟨a, args⟩) = B.free f ⟨a, toFun m ∘ args⟩
-  preserve_bound : ∀ (a : 𝔸) (f : sig.bound) {m : ℕ} (args : sig.boundArgs f → A m),
-    toFun (m + sig.boundDepth f) (A.bound f (⟪a⟫ args)) = B.bound f (⟪a⟫ toFun m ∘ args)
-
-instance {A B : Algebra 𝔸 sig} : CoeFun (Algebra.Hom 𝔸 sig A B) (fun _ => (n : ℕ) → (A n →ₑ[𝔸] B n)) where
-  coe h := h.toFun
-
-end AlgebraDefinition
-
-section Algebra
-
-variable {𝔸 : Type} [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
-variable {sig : Signature}
-
-namespace Algebra
-
-def Hom.id (A : Algebra 𝔸 sig) : Hom 𝔸 sig A A where
-  toFun _ := EquivariantMap.id 𝔸
-  preserve_pure _ _ _ := rfl
-  preserve_free _ _ _ _ := rfl
-  preserve_bound _ _ _ _ := rfl
-
-def Hom.comp {A B C : Algebra 𝔸 sig} (h : Hom 𝔸 sig B C) (g : Hom 𝔸 sig A B) : Hom 𝔸 sig A C where
-  toFun n := EquivariantMap.comp 𝔸 (h n) (g n)
-  preserve_pure f m args := by
-    rw [EquivariantMap.comp_app, g.preserve_pure, h.preserve_pure]
-    rfl
-  preserve_free a f m args := by
-    rw [EquivariantMap.comp_app, g.preserve_free, h.preserve_free]
-    rfl
-  preserve_bound a f m args := by
-    rw [EquivariantMap.comp_app, g.preserve_bound, h.preserve_bound]
-    rfl
-
-instance : Category (Algebra 𝔸 sig) where
-  Hom := Hom 𝔸 sig
-  id := Hom.id
-  comp h g := Hom.comp g h
-
-def forgetfulFunctor : Algebra 𝔸 sig ⥤ Nom 𝔸 where
-  obj X := X 0
-  map f := f.toFun 0
-
-end Algebra
-
-end Algebra
-
-end NominalAlgebra
diff --git a/Algebra/Terms.lean b/Algebra/Terms.lean
deleted file mode 100644
index 568f5cb..0000000
--- a/Algebra/Terms.lean
+++ /dev/null
@@ -1,312 +0,0 @@
-import Mathlib
-import Nominal
-
-namespace NominalAlgebra
-
-section Definitions
-
-/-- A graded signature. -/
-structure Signature where
-  pure : Type
-  free : Type
-  bound : Type
-  /-- Type representing the arguments required for an operation. -/
-  args : pure ⊕ free ⊕ bound → Type
-  args_fintype : ∀ f, Fintype (args f) := by intro; infer_instance
-  /-- Depth of an operation. -/
-  depth : pure ⊕ free ⊕ bound → ℕ
-
-def Signature.pureArgs (sig : Signature) (f : sig.pure) : Type := sig.args (.inl f)
-def Signature.freeArgs (sig : Signature) (f : sig.free) : Type := sig.args (.inr (.inl f))
-def Signature.boundArgs (sig : Signature) (f : sig.bound) : Type := sig.args (.inr (.inr f))
-
-instance Signature.instPureArgsFintype (sig : Signature) (f : sig.pure) : Fintype (sig.pureArgs f) := sig.args_fintype (.inl f)
-instance Signature.instFreeArgsFintype (sig : Signature) (f : sig.free) : Fintype (sig.freeArgs f) := sig.args_fintype (.inr (.inl f))
-instance Signature.instBoundArgsFintype (sig : Signature) (f : sig.bound) : Fintype (sig.boundArgs f) := sig.args_fintype (.inr (.inr f))
-
-def Signature.arity (sig : Signature) (f : sig.pure ⊕ sig.free ⊕ sig.bound) : ℕ := (sig.args_fintype f).card
-def Signature.pureArity (sig : Signature) (f : sig.pure) : ℕ := sig.arity (.inl f)
-def Signature.freeArity (sig : Signature) (f : sig.free) : ℕ := sig.arity (.inr (.inl f))
-def Signature.boundArity (sig : Signature) (f : sig.bound) : ℕ := sig.arity (.inr (.inr f))
-
-def Signature.pureDepth (sig : Signature) (f : sig.pure) : ℕ := sig.depth (.inl f)
-def Signature.freeDepth (sig : Signature) (f : sig.free) : ℕ := sig.depth (.inr (.inl f))
-def Signature.boundDepth (sig : Signature) (f : sig.bound) : ℕ := sig.depth (.inr (.inr f))
-
-variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
-variable (sig : Signature)
-
-inductive Term (X : Type) [NominalSet 𝔸 X]
-  | var (x : X) : Term X
-  | pure (f : sig.pure) (args : sig.pureArgs f → Term X) : Term X
-  | free (a : 𝔸) (f : sig.free) (args : sig.freeArgs f → Term X) : Term X
-  | bound (a : 𝔸) (f : sig.bound) (args : sig.boundArgs f → Term X) : Term X
-
-inductive Term.HasDepth
-    {𝔸 : Type} [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
-    {sig : Signature}
-    {X : Type} [NominalSet 𝔸 X]
-    : Term 𝔸 sig X → ℕ → Prop
-  | var (x : X) : HasDepth (.var x) 0
-  | pure (f : sig.pure) (args : sig.pureArgs f → Term 𝔸 sig X) (k : ℕ) :
-      (∀ arg : sig.pureArgs f, HasDepth (args arg) k) → HasDepth (.pure f args) (k + sig.pureDepth f)
-  | free (a : 𝔸) (f : sig.free) (args : sig.freeArgs f → Term 𝔸 sig X) (k : ℕ) :
-      (∀ arg : sig.freeArgs f, HasDepth (args arg) k) → HasDepth (.free a f args) (k + sig.freeDepth f)
-  | bound (a : 𝔸) (f : sig.bound) (args : sig.boundArgs f → Term 𝔸 sig X) (k : ℕ) :
-      (∀ arg : sig.boundArgs f, HasDepth (args arg) k) → HasDepth (.bound a f args) (k + sig.boundDepth f)
-
-def TermAtDepth (X : Type) [NominalSet 𝔸 X] (n : ℕ) :=
-  { t : Term 𝔸 sig X // t.HasDepth n }
-
-def Substitution (X : Type) [NominalSet 𝔸 X] (Y : Type) [NominalSet 𝔸 Y] :=
-  X → Term 𝔸 sig Y
-
-def UniformSubstitution (X : Type) [NominalSet 𝔸 X] (Y : Type) [NominalSet 𝔸 Y] (l : ℕ) :=
-  X → TermAtDepth 𝔸 sig Y l
-
-end Definitions
-
-variable {𝔸 : Type} [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
-variable {sig : Signature}
-variable {X : Type} [NominalSet 𝔸 X]
-variable {Y : Type} [NominalSet 𝔸 Y]
-
-section ConvenienceConstructors
-
-def TermAtDepth.var (x : X) : TermAtDepth 𝔸 sig X 0 := ⟨.var x, by constructor⟩
-
-def TermAtDepth.pure {n : ℕ} (f : sig.pure) (args : sig.pureArgs f → TermAtDepth 𝔸 sig X n) :
-    TermAtDepth 𝔸 sig X (n + sig.pureDepth f) :=
-  ⟨.pure f (Subtype.val ∘ args), by constructor; intro arg; exact (args arg).prop⟩
-
-def TermAtDepth.free {n : ℕ} (a : 𝔸) (f : sig.free) (args : sig.freeArgs f → TermAtDepth 𝔸 sig X n) :
-    TermAtDepth 𝔸 sig X (n + sig.freeDepth f) :=
-  ⟨.free a f (Subtype.val ∘ args), by constructor; intro arg; exact (args arg).prop⟩
-
-def TermAtDepth.bound {n : ℕ} (a : 𝔸) (f : sig.bound) (args : sig.boundArgs f → TermAtDepth 𝔸 sig X n) :
-    TermAtDepth 𝔸 sig X (n + sig.boundDepth f) :=
-  ⟨.bound a f (Subtype.val ∘ args), by constructor; intro arg; exact (args arg).prop⟩
-
-theorem TermAtDepth.induction
-  {motive : {n : ℕ} → TermAtDepth 𝔸 sig X n → Prop}
-  (var : ∀ (x : X), motive (.var x))
-  (pure : ∀ {n : ℕ} (f : sig.pure) (args : sig.pureArgs f → TermAtDepth 𝔸 sig X n),
-    (∀ x, motive (args x)) → motive (.pure f args))
-  (free : ∀ {n : ℕ} (a : 𝔸) (f : sig.free) (args : sig.freeArgs f → TermAtDepth 𝔸 sig X n),
-    (∀ x, motive (args x)) → motive (.free a f args))
-  (bound : ∀ {n : ℕ} (a : 𝔸) (f : sig.bound) (args : sig.boundArgs f → TermAtDepth 𝔸 sig X n),
-    (∀ x, motive (args x)) → motive (.bound a f args))
-  : ∀ {n : ℕ} (t : TermAtDepth 𝔸 sig X n), motive t := by
-  intro n ⟨t, ht⟩
-  induction' t generalizing n <;> cases ht <;> expose_names
-  · exact var x
-  · let args' : sig.pureArgs f → TermAtDepth 𝔸 sig X k := fun x => ⟨args x, h x⟩
-    have args_ih' : ∀ x, motive (args' x) := fun x => args_ih x (h x)
-    exact pure f args' args_ih'
-  · let args' : sig.freeArgs f → TermAtDepth 𝔸 sig X k := fun x => ⟨args x, h x⟩
-    have args_ih' : ∀ x, motive (args' x) := fun x => args_ih x (h x)
-    exact free a f args' args_ih'
-  · let args' : sig.boundArgs f → TermAtDepth 𝔸 sig X k := fun x => ⟨args x, h x⟩
-    have args_ih' : ∀ x, motive (args' x) := fun x => args_ih x (h x)
-    exact bound a f args' args_ih'
-
-end ConvenienceConstructors
-
-section Substitution
-
-def Substitution.apply (σ : Substitution 𝔸 sig X Y) : Term 𝔸 sig X → Term 𝔸 sig Y
-  | .var x => σ x
-  | .pure f args => .pure f (fun x => σ.apply (args x))
-  | .free a f args => .free a f (fun x => σ.apply (args x))
-  | .bound a f args => .bound a f (fun x => σ.apply (args x))
-
-@[simp] lemma Substitution.apply_var {x : X} {σ : Substitution 𝔸 sig X Y} :
-    σ.apply (.var x) = (σ x) := rfl
-
-@[simp] lemma Substitution.apply_pure
-    {f : sig.pure} {args : sig.pureArgs f → Term 𝔸 sig X} {σ : Substitution 𝔸 sig X Y} :
-    σ.apply (.pure f args) = .pure f (σ.apply ∘ args) := rfl
-
-@[simp] lemma Substitution.apply_free
-    {a : 𝔸} {f : sig.free} {args : sig.freeArgs f → Term 𝔸 sig X} {σ : Substitution 𝔸 sig X Y} :
-    σ.apply (.free a f args) = .free a f (σ.apply ∘ args) := rfl
-
-@[simp] lemma Substitution.apply_bound
-    {a : 𝔸} {f : sig.bound} {args : sig.boundArgs f → Term 𝔸 sig X} {σ : Substitution 𝔸 sig X Y} :
-    σ.apply (.bound a f args) = .bound a f (σ.apply ∘ args) := rfl
-
-def UniformSubstitution.toSubstitution {l : ℕ} (σ : UniformSubstitution 𝔸 sig X Y l) : Substitution 𝔸 sig X Y :=
-  fun x => (σ x).val
-
-def UniformSubstitution.apply {n l : ℕ} (σ : UniformSubstitution 𝔸 sig X Y l) (t : TermAtDepth 𝔸 sig X n) :
-    TermAtDepth 𝔸 sig Y (n + l) :=
-  ⟨σ.toSubstitution.apply t.val, by
-    suffices ∀ t n, t.HasDepth n → (σ.toSubstitution.apply t).HasDepth (n + l) by
-      exact this t.val n t.prop
-    intro t; fun_induction Substitution.apply
-    · intro n h; cases h
-      rw [Nat.zero_add]
-      exact (σ _).prop
-    all_goals
-      intro n h; cases h; expose_names
-      rw [Nat.add_right_comm]
-      constructor; intro arg
-      apply ih1; exact h arg⟩
-
-@[simp] lemma UniformSubstitution.apply_var
-    {x : X} {l : ℕ} {σ : UniformSubstitution 𝔸 sig X Y l} :
-    (σ.apply (.var x)).val = (σ x).val := rfl
-
-@[simp] lemma UniformSubstitution.apply_pure
-    {n : ℕ} {f : sig.pure} {args : sig.pureArgs f → TermAtDepth 𝔸 sig X n}
-    {l : ℕ} {σ : UniformSubstitution 𝔸 sig X Y l} :
-    (σ.apply (.pure f args)).val = (TermAtDepth.pure f (σ.apply ∘ args)).val := rfl
-
-@[simp] lemma UniformSubstitution.apply_free
-    {n : ℕ} {a : 𝔸} {f : sig.free} {args : sig.freeArgs f → TermAtDepth 𝔸 sig X n}
-    {l : ℕ} {σ : UniformSubstitution 𝔸 sig X Y l} :
-    (σ.apply (.free a f args)).val = (TermAtDepth.free a f (σ.apply ∘ args)).val := rfl
-
-@[simp] lemma UniformSubstitution.apply_bound
-    {n : ℕ} {a : 𝔸} {f : sig.bound} {args : sig.boundArgs f → TermAtDepth 𝔸 sig X n}
-    {l : ℕ} {σ : UniformSubstitution 𝔸 sig X Y l} :
-    (σ.apply (.bound a f args)).val = (TermAtDepth.bound a f (σ.apply ∘ args)).val := rfl
-
-end Substitution
-
-section NominalSet
-
-namespace Term
-
-private def rename (π : FPerm 𝔸) : Term 𝔸 sig X → Term 𝔸 sig X
-  | .var x => .var (π • x)
-  | .pure f args => .pure f (fun x => rename π (args x))
-  | .free a f args => .free (π • a) f (fun x => rename π (args x))
-  | .bound a f args => .bound (π • a) f (fun x => rename π (args x))
-
-private lemma one_rename (t : Term 𝔸 sig X) : t.rename 1 = t := by
-  induction' t <;> all_goals simp [rename, *]
-
-private lemma mul_rename (π τ : FPerm 𝔸) (t : Term 𝔸 sig X) :
-    t.rename (π * τ) = (t.rename τ).rename π := by
-  induction' t <;> all_goals simp [rename, mul_smul, *]
-
-instance : PermSet 𝔸 (Term 𝔸 sig X) where
-  smul := rename
-  one_smul := one_rename
-  mul_smul := mul_rename
-
-@[simp] lemma smul_var {π : FPerm 𝔸} {x : X} :
-    π • (.var x : Term 𝔸 sig X) = .var (π • x) := rfl
-
-@[simp] lemma smul_pure {π : FPerm 𝔸} {f : sig.pure} {args : sig.pureArgs f → Term 𝔸 sig X} :
-    π • Term.pure f args = .pure f ((π • ·) ∘ args) := rfl
-
-@[simp] lemma smul_free {π : FPerm 𝔸} {a : 𝔸} {f : sig.free} {args : sig.freeArgs f → Term 𝔸 sig X} :
-    π • Term.free a f args = .free (π • a) f ((π • ·) ∘ args) := rfl
-
-@[simp] lemma smul_bound {π : FPerm 𝔸} {a : 𝔸} {f : sig.bound} {args : sig.boundArgs f → Term 𝔸 sig X} :
-    π • Term.bound a f args = .bound (π • a) f ((π • ·) ∘ args) := rfl
-
-private def atoms : Term 𝔸 sig X → Set 𝔸
-  | .var x => NominalSet.supp 𝔸 x
-  | .pure f args => ⋃₀ {(args x).atoms | x : sig.pureArgs f}
-  | .free a f args => {a} ∪ ⋃₀ {(args x).atoms | x : sig.freeArgs f}
-  | .bound a f args => {a} ∪ ⋃₀ {(args x).atoms | x : sig.boundArgs f}
-
-private lemma atoms_finite (t : Term 𝔸 sig X) : t.atoms.Finite := by
-  fun_induction atoms <;> all_goals expose_names
-  · exact NominalSet.supp_finite 𝔸
-  · apply Set.Finite.sUnion
-    · simp only [Set.setOf_exists, Set.setOf_eq_eq_singleton', Set.iUnion_singleton_eq_range]
-      apply Set.finite_range
-    · intro S ⟨x, hS⟩
-      rw [← hS]
-      exact ih1 x
-  all_goals
-    apply Set.Finite.union
-    · exact Set.finite_singleton a
-    · apply Set.Finite.sUnion
-      · simp only [Set.setOf_exists, Set.setOf_eq_eq_singleton', Set.iUnion_singleton_eq_range]
-        apply Set.finite_range
-      · intro S ⟨x, hS⟩
-        rw [← hS]
-        exact ih1 x
-
-private lemma atoms_support (t : Term 𝔸 sig X) : PermSet.HasSupport 𝔸 t t.atoms := by
-  intro π
-  fun_induction atoms <;> all_goals expose_names
-  · have hsupp : PermSet.HasSupport 𝔸 x (NominalSet.supp 𝔸 x) := NominalSet.supp_is_support 𝔸
-    intro hπ
-    simp only [PermSet.fix', smul_var, Term.var.injEq, Set.mem_setOf_eq]
-    exact hsupp hπ
-  · intro hπ
-    simp only [PermSet.fix', smul_pure, Term.pure.injEq, heq_eq_eq, true_and, Set.mem_setOf_eq]
-    ext x
-    rw [Function.comp_apply]
-    apply ih1
-    apply PermSet.fix_antitone_mem 𝔸 hπ
-    apply Set.subset_sUnion_of_mem
-    use x
-  all_goals
-    intro hπ
-    simp only [PermSet.fix', smul_free, smul_bound, Term.free.injEq, Term.bound.injEq, heq_eq_eq,
-      true_and, Set.mem_setOf_eq]
-    constructor
-    · apply PermSet.fix_antitone_mem 𝔸 hπ
-      apply Set.subset_union_left
-      rfl
-    · ext x
-      rw [Function.comp_apply]
-      apply ih1
-      apply PermSet.fix_antitone_mem 𝔸 hπ
-      apply Set.subset_union_of_subset_right
-      apply Set.subset_sUnion_of_mem
-      use x
-
-instance : NominalSet 𝔸 (Term 𝔸 sig X) where
-  fin_supp t := by
-    use t.atoms
-    exact ⟨t.atoms_support, t.atoms_finite⟩
-
-instance instHasDepthEquivariant {n : ℕ} :
-    Set.Equivariant 𝔸 {t : Term 𝔸 sig X | t.HasDepth n} := by
-  constructor; intro t π ht
-  suffices ∀ (t : Term 𝔸 sig X) n, t.HasDepth n → (t.rename π).HasDepth n by
-    exact this t n ht
-  intro t; fun_induction rename
-  · intro n h; cases h; constructor
-  all_goals
-      intro n h; cases h
-      expose_names
-      constructor; intro arg
-      apply ih1; exact h arg
-
-end Term
-
-namespace TermAtDepth
-
-instance {n : ℕ} : NominalSet 𝔸 (TermAtDepth 𝔸 sig X n) :=
-  (Subtype.instNominalSetElem 𝔸 : NominalSet 𝔸 {t : Term 𝔸 sig X | t.HasDepth n})
-
-@[simp] lemma smul_var
-    {π : FPerm 𝔸} {x : X} :
-    (π • (.var x : TermAtDepth 𝔸 sig X 0)).val = (TermAtDepth.var (π • x)).val := rfl
-
-@[simp] lemma smul_pure
-    {π : FPerm 𝔸} {n : ℕ} {f : sig.pure} {args : sig.pureArgs f → TermAtDepth 𝔸 sig X n} :
-    (π • TermAtDepth.pure f args).val = (TermAtDepth.pure f ((π • ·) ∘ args)).val := rfl
-
-@[simp] lemma smul_free
-    {π : FPerm 𝔸} {n : ℕ} {a : 𝔸} {f : sig.free} {args : sig.freeArgs f → TermAtDepth 𝔸 sig X n} :
-    (π • TermAtDepth.free a f args).val = (TermAtDepth.free (π • a) f ((π • ·) ∘ args)).val := rfl
-
-@[simp] lemma smul_bound
-    {π : FPerm 𝔸} {n : ℕ} {a : 𝔸} {f : sig.bound} {args : sig.boundArgs f → TermAtDepth 𝔸 sig X n} :
-    (π • TermAtDepth.bound a f args).val = (TermAtDepth.bound (π • a) f ((π • ·) ∘ args)).val := rfl
-
-end TermAtDepth
-
-end NominalSet
-
-end NominalAlgebra