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(*
WG05: Execution Domains Theory
An execution domain defines a boundary for execution semantics,
resource visibility, and control flow.
Author: Amlal El Mahrouss
Formalization: January 2026
*)
From Coq Require Import Logic.Classical_Prop.
From Coq Require Import Logic.FunctionalExtensionality.
(** * Execution Contexts *)
(** Execution contexts are abstract semantic parameters *)
Record ExecutionContext : Type := mkExecutionContext {
Context : Type;
context_eq_dec : forall (c1 c2 : Context), {c1 = c2} + {c1 <> c2}
}.
(** * Execution Domains *)
(** An execution domain is indexed by an execution context *)
Record ExecutionDomain (EC : ExecutionContext) : Type := mkExecutionDomain {
Domain : Type;
domain_context : Context EC;
SubProgram : Type;
compose : SubProgram -> Domain
}.
Arguments Domain {EC}.
Arguments domain_context {EC}.
Arguments SubProgram {EC}.
Arguments compose {EC}.
(** * Core Theorem: Domains with Different Contexts are Distinct *)
Section DomainSeparation.
Variable EC : ExecutionContext.
(** Two domains with provably different contexts cannot be equal
in any meaningful semantic sense *)
Record DomainEquivalence (D1 D2 : ExecutionDomain EC) : Prop := mkDomainEquivalence {
context_eq : domain_context D1 = domain_context D2;
domain_iso : exists (f : Domain D1 -> Domain D2), True
}.
(** The separation theorem: different contexts imply distinct domains *)
Theorem separation_theorem :
forall (D1 D2 : ExecutionDomain EC),
domain_context D1 <> domain_context D2 ->
~ DomainEquivalence D1 D2.
Proof.
intros D1 D2 Hneq Heq.
destruct Heq as [Hctx _].
contradiction.
Qed.
End DomainSeparation.
(** * Composition Property *)
(** Domains may be composed of sub-programs within the execution context *)
Record ComposableDomain (EC : ExecutionContext) : Type := mkComposableDomain {
base : ExecutionDomain EC;
composition_preserves_context :
forall (sp : SubProgram base),
exists d, d = compose base sp
}.
(** * Context Abstraction *)
(** Execution contexts are treated as abstract semantic parameters.
This section provides the abstraction barrier. *)
Section ContextAbstraction.
(** A context family abstracts over specific context implementations *)
Record ContextFamily : Type := mkContextFamily {
Contexts : Type;
mkContext : Contexts -> ExecutionContext
}.
(** Domains can be parameterized by context families *)
Definition DomainFamily (cf : ContextFamily) : Type :=
forall (c : Contexts cf), ExecutionDomain (mkContext cf c).
End ContextAbstraction.
(** * Properties about Domain Inequality *)
Section DomainInequality.
(** If two domains belong to different execution contexts,
they cannot be structurally equal *)
Lemma domains_different_contexts_not_equal :
forall (EC1 EC2 : ExecutionContext)
(D1 : ExecutionDomain EC1)
(D2 : ExecutionDomain EC2),
EC1 <> EC2 ->
~ (exists (H : EC1 = EC2),
eq_rect EC1 ExecutionDomain D1 EC2 H = D2).
Proof.
intros EC1 EC2 D1 D2 Hneq [H _].
contradiction.
Qed.
End DomainInequality.
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