From 8fdf459876b999dbe184196f9abd330a3c24d560 Mon Sep 17 00:00:00 2001
From: Hannes Schulze <haschu0103@outlook.de>
Date: Thu, 4 Dec 2025 17:14:46 +0100
Subject: implement derivable equality

---
 Algebra/Derivations.lean | 136 +++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 136 insertions(+)

(limited to 'Algebra/Derivations.lean')

diff --git a/Algebra/Derivations.lean b/Algebra/Derivations.lean
index e69de29..6f81e9a 100644
--- a/Algebra/Derivations.lean
+++ b/Algebra/Derivations.lean
@@ -0,0 +1,136 @@
+import Mathlib
+import Algebra.Terms
+
+namespace Algebra
+
+section Theories
+
+variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
+variable (sig : Signature)
+
+structure Equation where
+  vars : Type
+  [vars_nominal : NominalSet 𝔸 vars]
+  [vars_orbitFinite : OrbitFinite 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
+
+instance (eq : Equation 𝔸 sig) : OrbitFinite 𝔸 eq.vars :=
+  eq.vars_orbitFinite
+
+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 EquationAtDepth (n : ℕ) :=
+  { eq : Equation 𝔸 sig // eq.HasDepth n }
+
+def UniformEquation :=
+  { eq : Equation 𝔸 sig // ∃ n, eq.HasDepth n }
+
+structure Theory where
+  sig : Signature
+  axioms : Set (UniformEquation 𝔸 sig)
+
+end Theories
+
+variable {𝔸 : Type} [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
+variable {sig : Signature}
+
+section Helpers
+
+def EquationAtDepth.left {n : ℕ} (eq : EquationAtDepth 𝔸 sig n) : TermAtDepth 𝔸 sig eq.val.vars n :=
+  ⟨eq.val.left, eq.prop.left⟩
+
+def EquationAtDepth.right {n : ℕ} (eq : EquationAtDepth 𝔸 sig n) : TermAtDepth 𝔸 sig eq.val.vars n :=
+  ⟨eq.val.right, eq.prop.right⟩
+
+def EquationAtDepth.mk
+    (X : Type) [NominalSet 𝔸 X] [OrbitFinite 𝔸 X] {n : ℕ} (t u : TermAtDepth 𝔸 sig X n) :
+    EquationAtDepth 𝔸 sig n :=
+  ⟨⟨X, t.val, u.val⟩, ⟨t.prop, u.prop⟩⟩
+
+def EquationAtDepth.toUniform {n : ℕ} (eq : EquationAtDepth 𝔸 sig n) : UniformEquation 𝔸 sig :=
+  ⟨eq.val, ⟨n, eq.prop⟩⟩
+
+notation:50 X " ⊢ " lhs " ≣ " rhs => EquationAtDepth.mk X lhs rhs
+
+end Helpers
+
+section DerivableEquations
+
+inductive Derivable (T : Theory 𝔸) : {m : ℕ} → (eq : EquationAtDepth 𝔸 T.sig m) → Prop
+  | refl {X : Type} [NominalSet 𝔸 X] [OrbitFinite 𝔸 X] {x : X} :
+      Derivable T (X ⊢ .var x ≣ .var x)
+  | symm {X : Type} [NominalSet 𝔸 X] [OrbitFinite 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] [OrbitFinite 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] [OrbitFinite 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] [OrbitFinite 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] [OrbitFinite 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] [OrbitFinite 𝔸 X] [NominalSet 𝔸 Y] [OrbitFinite 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).toUniform ∈ 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] [OrbitFinite 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)
+
+variable {X Y : Type} [NominalSet 𝔸 X] [OrbitFinite 𝔸 X] [NominalSet 𝔸 Y] [OrbitFinite 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))
+
+/-- Restated ax' rule using DerivablyEquivariant in the premiss. -/
+lemma Derivable.ax
+    {T : Theory 𝔸}
+    {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).toUniform ∈ T.axioms) →
+    σ.DerivablyEquivariant T →
+    Derivable T (X ⊢ σ.apply (τ • r) ≣ σ.apply (τ • s)) :=
+  .ax'
+
+lemma Derivable.equivariant_substitution
+    (T : Theory 𝔸)
+    {l : ℕ} (σ : UniformSubstitution 𝔸 T.sig Y X l) (hequi : σ.DerivablyEquivariant T)
+    {m : ℕ} (t : TermAtDepth 𝔸 T.sig Y m) {π : FPerm 𝔸} :
+    Derivable T (X ⊢ σ.apply (π • t) ≣ π • (σ.apply t)) := by
+  induction' t using TermAtDepth.induction <;> expose_names
+  · sorry
+  · sorry
+  · sorry
+  · sorry
+
+end DerivableEquations
+
+end Algebra
-- 
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