From 27ff9d5b7a21ae54aed1c7a23a23723903ab2fa9 Mon Sep 17 00:00:00 2001
From: Hannes Schulze <contact@haschu.me>
Date: Wed, 15 Apr 2026 15:17:51 +0200
Subject: Revert "move wip"

This reverts commit 37f7c86a7591da1233cdf1d293bf851f43c65ba1.
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 Algebra/Derivations.lean | 237 +++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 237 insertions(+)
 create mode 100644 Algebra/Derivations.lean

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diff --git a/Algebra/Derivations.lean b/Algebra/Derivations.lean
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+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
-- 
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