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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 => Equation.mk X lhs rhs
end Helpers
section DerivableEquations
inductive Derivable (T : Theory 𝔸) : (eq : Equation 𝔸 T.sig) → 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]
{t u : Term 𝔸 T.sig X} :
Derivable T (X ⊢ u ≣ t) →
Derivable T (X ⊢ t ≣ u)
| trans {X : Type} [NominalSet 𝔸 X] [OrbitFinite 𝔸 X]
{t u v : Term 𝔸 T.sig X} :
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} {ts us : T.sig.pureArgs f → Term 𝔸 T.sig X} :
(∀ 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} {ts us : T.sig.freeArgs f → Term 𝔸 T.sig X} :
(∀ 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} {ts us : T.sig.boundArgs f → Term 𝔸 T.sig X} :
(∀ x, Derivable T (X ⊢ ts x ≣ us x)) →
Derivable T (X ⊢ .bound a f ts ≣ .bound a f us)
| ax' {X : Type} [NominalSet 𝔸 X] [OrbitFinite 𝔸 X]
{eq : UniformEquation 𝔸 T.sig}
{l : ℕ} {σ : UniformSubstitution 𝔸 T.sig eq.val.vars X l} {τ : FPerm 𝔸} :
(eq ∈ T.axioms) →
(∀ (y : eq.val.vars) (π : FPerm 𝔸), Derivable T (X ⊢ σ.toSubstitution (π • y) ≣ π • (σ.toSubstitution y))) →
Derivable T (X ⊢ σ.toSubstitution.apply (τ • eq.val.left) ≣ σ.toSubstitution.apply (τ • eq.val.right))
| perm {X : Type} [NominalSet 𝔸 X] [OrbitFinite 𝔸 X]
{a b : 𝔸} {f : T.sig.bound} {ts us : T.sig.boundArgs f → Term 𝔸 T.sig X} :
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] [OrbitFinite 𝔸 X] (n : ℕ) :
Rel (TermAtDepth 𝔸 T.sig X n) (TermAtDepth 𝔸 T.sig X n) :=
fun t u => Derivable T (X ⊢ t.val ≣ u.val)
variable {X Y : Type} [NominalSet 𝔸 X] [OrbitFinite 𝔸 X] [NominalSet 𝔸 Y] [OrbitFinite 𝔸 Y]
def Substitution.DerivablyEquivariant
(T : Theory 𝔸) (σ : Substitution 𝔸 T.sig X Y) : 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}
{eq : UniformEquation 𝔸 T.sig}
{l : ℕ} {σ : UniformSubstitution 𝔸 T.sig eq.val.vars X l} {τ : FPerm 𝔸} :
(eq ∈ T.axioms) →
σ.toSubstitution.DerivablyEquivariant T →
Derivable T (X ⊢ σ.toSubstitution.apply (τ • eq.val.left) ≣ σ.toSubstitution.apply (τ • eq.val.right)) :=
.ax'
lemma Derivable.equivariant_substitution
(T : Theory 𝔸)
{l : ℕ} (σ : UniformSubstitution 𝔸 T.sig Y X l) :
σ.toSubstitution.DerivablyEquivariant T →
∀ (t : Term 𝔸 T.sig Y) (π : FPerm 𝔸),
Derivable T (X ⊢ σ.toSubstitution.apply (π • t) ≣ π • (σ.toSubstitution.apply t)) := by
intro hequi t π
induction' t <;> expose_names
· exact hequi x π
· simp only [Term.smul_pure, Substitution.apply_pure]
apply Derivable.cong_pure
exact args_ih
· simp only [Term.smul_free, Substitution.apply_free]
apply Derivable.cong_free
exact args_ih
· simp only [Term.smul_bound, Substitution.apply_bound]
apply Derivable.cong_bound
exact args_ih
lemma Derivable.equivariant
(T : Theory 𝔸) {t u : Term 𝔸 T.sig X} :
∀ (π : FPerm 𝔸), Derivable T (X ⊢ t ≣ u) → Derivable T (X ⊢ π • t ≣ π • u) := by
intro π
suffices ∀ eq, Derivable T eq → Derivable T (eq.vars ⊢ π • eq.left ≣ π • eq.right) by
exact this (X ⊢ t ≣ u)
intro eq hderiv
induction' hderiv <;> expose_names
· exact .refl
· apply Derivable.symm
exact a_ih
· apply Derivable.trans
· exact a_ih
· exact a_ih_1
· apply Derivable.cong_pure
exact a_ih
· apply Derivable.cong_free
exact a_ih
· apply Derivable.cong_bound
exact a_ih
· apply Derivable.trans (by apply Derivable.symm; apply equivariant_substitution; exact h_1) _
apply Derivable.trans _ (by apply equivariant_substitution; exact h_1)
rw [← MulAction.mul_smul, ← MulAction.mul_smul]
apply Derivable.ax
· exact h
· exact h_1
· 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 [← MulAction.mul_smul, ← FPerm.mul_swap_eq_swap_mul, MulAction.mul_smul]
exact a_ih x
variable {T : Theory 𝔸} {n : ℕ}
instance DerivableEq.instEquivariant : Rel.Equivariant 𝔸 (DerivableEq T X n) := by
constructor; intro t u
exact Derivable.equivariant T
@[refl]
lemma DerivableEq.refl (t : TermAtDepth 𝔸 T.sig X n) : DerivableEq T X n t t := by
suffices ∀ t, Derivable T (X ⊢ t ≣ t) by exact this t.val
intro t
induction' t <;> 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 Algebra
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