more cleanup

2 files changed, 93 insertions, 47 deletions

diff --git a/Algebra/Derivations.lean b/Algebra/Derivations.lean
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/Algebra/Derivations.lean
diff --git a/Algebra/Term.lean b/Algebra/Terms.lean
index fe7e742..aa853a8 100644
--- a/Algebra/Term.lean
+++ b/Algebra/Terms.lean
@@ -3,7 +3,7 @@ import Nominal
 
 namespace Algebra
 
-variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
+section Definitions
 
 /-- A graded signature. -/
 structure Signature where
@@ -33,17 +33,20 @@ def Signature.pureDepth (sig : Signature) (f : sig.pure) : ℕ := sig.depth (.in
 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 {sig : Signature}
-variable {X : Type} [NominalSet 𝔸 X]
-variable {Y : Type} [NominalSet 𝔸 Y]
+variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
+variable (sig : Signature)
 
-inductive Term (sig : Signature) (X : Type) [NominalSet 𝔸 X]
-  | var (x : X) : Term sig X
-  | pure (f : sig.pure) (args : sig.pureArgs f → Term sig X) : Term sig X
-  | free (a : 𝔸) (f : sig.free) (args : sig.freeArgs f → Term sig X) : Term sig X
-  | bound (a : 𝔸) (f : sig.bound) (args : sig.boundArgs f → Term sig X) : Term sig X
+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 : Term 𝔸 sig X → ℕ → Prop
+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)
@@ -52,8 +55,23 @@ inductive Term.HasDepth : Term 𝔸 sig X → ℕ → Prop
   | 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 (sig : Signature) (X : Type) [NominalSet 𝔸 X] (n : ℕ) :=
-  { t : Term 𝔸 sig X // t.HasDepth 𝔸 n }
+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⟩
 
@@ -69,8 +87,9 @@ def TermAtDepth.bound {n : ℕ} (a : 𝔸) (f : sig.bound) (args : sig.boundArgs
     TermAtDepth 𝔸 sig X (n + sig.boundDepth f) :=
   ⟨.bound a f (Subtype.val ∘ args), by constructor; intro arg; exact (args arg).prop⟩
 
-def Substitution (sig : Signature) (X : Type) [NominalSet 𝔸 X] (Y : Type) [NominalSet 𝔸 Y] :=
-  X → Term 𝔸 sig Y
+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
@@ -79,30 +98,27 @@ def Substitution.apply (σ : Substitution 𝔸 sig X Y) : Term 𝔸 sig X → Te
   | .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 𝔸 (.var x) = (σ x) := rfl
+    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 𝔸 (.pure f args) = .pure f (apply 𝔸 σ ∘ args) := rfl
+    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 𝔸 (.free a f args) = .free a f (apply 𝔸 σ ∘ args) := rfl
+    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 𝔸 (.bound a f args) = .bound a f (apply 𝔸 σ ∘ args) := rfl
-
-def UniformSubstitution (sig : Signature) (X : Type) [NominalSet 𝔸 X] (Y : Type) [NominalSet 𝔸 Y] (l : ℕ) :=
-  X → TermAtDepth 𝔸 sig Y l
+    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 𝔸 t.val, by
-    suffices ∀ t n, t.HasDepth 𝔸 n → (σ.toSubstitution.apply 𝔸 t).HasDepth 𝔸 (n + l) by
+  ⟨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
@@ -116,22 +132,28 @@ def UniformSubstitution.apply {n l : ℕ} (σ : UniformSubstitution 𝔸 sig X Y
 
 @[simp] lemma UniformSubstitution.apply_var
     {x : X} {l : ℕ} {σ : UniformSubstitution 𝔸 sig X Y l} :
-    (σ.apply 𝔸 (.var 𝔸 x)).val = (σ x).val := rfl
+    (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 𝔸 (.pure 𝔸 f args)).val = (TermAtDepth.pure 𝔸 f (apply 𝔸 σ ∘ args)).val := rfl
+    (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 𝔸 (.free 𝔸 a f args)).val = (TermAtDepth.free 𝔸 a f (apply 𝔸 σ ∘ args)).val := rfl
+    (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 𝔸 (.bound 𝔸 a f args)).val = (TermAtDepth.bound 𝔸 a f (apply 𝔸 σ ∘ args)).val := rfl
+    (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)
@@ -139,37 +161,37 @@ private def rename (π : FPerm 𝔸) : Term 𝔸 sig X → Term 𝔸 sig 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) : rename 𝔸 1 t = t := by
+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) :
-    rename 𝔸 (π * τ) t = rename 𝔸 π (rename 𝔸 τ t) := by
+    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 𝔸
-  one_smul := one_rename 𝔸
-  mul_smul := mul_rename 𝔸
+  smul := rename
+  one_smul := one_rename
+  mul_smul := mul_rename
 
 @[simp] lemma smul_var {π : FPerm 𝔸} {x : X} :
-    π • (Term.var x : Term 𝔸 sig X) = Term.var (π • x) := rfl
+    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 = Term.pure f ((π • ·) ∘ args) := rfl
+    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 = Term.free (π • a) f ((π • ·) ∘ args) := rfl
+    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 = Term.bound (π • a) f ((π • ·) ∘ args) := rfl
+    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 {atoms (args x) | x : sig.pureArgs f}
-  | .free a f args => {a} ∪ ⋃₀ {atoms (args x) | x : sig.freeArgs f}
-  | .bound a f args => {a} ∪ ⋃₀ {atoms (args x) | x : sig.boundArgs f}
+  | .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) : (atoms 𝔸 t).Finite := by
+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
   · have hfin := (sig.pureArgs_fintype f).finite
@@ -198,7 +220,7 @@ private lemma atoms_finite (t : Term 𝔸 sig X) : (atoms 𝔸 t).Finite := by
         rw [← hS]
         exact ih1 x
 
-private lemma atoms_support (t : Term 𝔸 sig X) : PermSet.HasSupport 𝔸 t (atoms 𝔸 t) := by
+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
@@ -231,13 +253,13 @@ private lemma atoms_support (t : Term 𝔸 sig X) : PermSet.HasSupport 𝔸 t (a
 
 instance : NominalSet 𝔸 (Term 𝔸 sig X) where
   fin_supp t := by
-    use atoms 𝔸 t
-    exact ⟨atoms_support 𝔸 t, atoms_finite 𝔸 t⟩
+    use t.atoms
+    exact ⟨t.atoms_support, t.atoms_finite⟩
 
-instance Term.instHasDepthEquivariant {n : ℕ} :
-    Set.Equivariant 𝔸 {t : Term 𝔸 sig X | HasDepth 𝔸 t n} := by
+instance instHasDepthEquivariant {n : ℕ} :
+    Set.Equivariant 𝔸 {t : Term 𝔸 sig X | t.HasDepth n} := by
   constructor; intro t π ht
-  suffices ∀ t n, t.HasDepth 𝔸 n → (rename 𝔸 π t).HasDepth 𝔸 n by
+  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
@@ -247,7 +269,31 @@ instance Term.instHasDepthEquivariant {n : ℕ} :
       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 | Term.HasDepth 𝔸 t 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 Algebra