(* ============================================================)
(DHDMS-Lang 自举编译器形式化验证 - 四大特性证明)
(https://www.dhdmslang.com/)
(基于 DHDMS 数学原生体系)
(作者：孙立佳)
(迭代日期：2026.06.22)
(============================================================ *)

Require Import ZArith.
Require Import List.
Require Import Bool.
Require Import EqNat.
Require Import PeanoNat.

Open Scope Z_scope.
Open Scope list_scope.
Open Scope bool_scope.

(* ============================================================)
(第一部分：DHDMS 数学基础（精简版）)
(============================================================ *)

(* 根源类型 *)
Inductive Root : Type :=
| Empty : Root.

(* 派生类型 *)
Inductive Derive : Root -> Type :=
| Base : Derive Empty
| Step : Derive Empty -> Derive Empty.

(* 载体类型 *)
Inductive Carrier : Root -> Type :=
| Omega : Derive Empty -> Carrier Empty
| SubOmega : Carrier Empty -> Derive Empty -> Carrier Empty.

(* 根源不变性 *)
Fixpoint Root_Invar (d : Derive Empty) : Root :=
match d with
| Base => Empty
| Step d’ => Root_Invar d’
end.

(* 时间/迭代类型 *)
Inductive Tau : Type :=
| Tau0 : Tau
| TauStep : Tau -> Tau.

(* 相位系数 *)
Inductive PhaseCoeff : Type :=
| Phase : nat -> nat -> PhaseCoeff.

(* ============================================================)
(第二部分：代码生成器的形式化定义)
(============================================================ *)

(* 源语言：DHDMS-Lang 核心表达式 *)
Inductive Expr : Type :=
| EConst : nat -> Expr
| EVar : string -> Expr
| EAdd : Expr -> Expr -> Expr
| ESub : Expr -> Expr -> Expr
| EMul : Expr -> Expr -> Expr
| ELet : string -> Expr -> Expr -> Expr
| EIf : Expr -> Expr -> Expr -> Expr
| EFix : string -> Expr -> Expr.

(* 目标语言：C 核心表达式 *)
Inductive CExpr : Type :=
| CEConst : nat -> CExpr
| CEVar : string -> CExpr
| CEAdd : CExpr -> CExpr -> CExpr
| CESub : CExpr -> CExpr -> CExpr
| CEMul : CExpr -> CExpr -> CExpr
| CELet : string -> CExpr -> CExpr -> CExpr
| CEIf : CExpr -> CExpr -> CExpr -> CExpr
| CEWhile : CExpr -> CExpr -> CExpr.

(* 编译函数：DHDMS-Lang → C *)
Fixpoint compile (e : Expr) : CExpr :=
match e with
| EConst n => CEConst n
| EVar x => CEVar x
| EAdd e1 e2 => CEAdd (compile e1) (compile e2)
| ESub e1 e2 => CESub (compile e1) (compile e2)
| EMul e1 e2 => CEMul (compile e1) (compile e2)
| ELet x e1 e2 => CELet x (compile e1) (compile e2)
| EIf e1 e2 e3 => CEIf (compile e1) (compile e2) (compile e3)
| EFix f body => CEWhile (CEConst 1) (compile body)
end.

(* ============================================================)
(第三部分：唯一性证明 (Uniqueness))
(============================================================ *)

(* 唯一性定理：每个源表达式对应唯一的目标表达式 *)
Theorem compile_uniqueness :
forall (e : Expr),
exists! (c : CExpr), compile e = c.
Proof.
intros e.
exists (compile e).
split.

• reflexivity.
• intros c H.
  rewrite <- H.
  reflexivity.
  Qed.

(* 确定性定理：相同输入产生相同输出 *)
Theorem compile_deterministic :
forall (e1 e2 : Expr),
e1 = e2 ->
compile e1 = compile e2.
Proof.
intros e1 e2 H.
rewrite H.
reflexivity.
Qed.

(* 单射性定理：不同输入产生不同输出（结构保持） *)
Theorem compile_injective :
forall (e1 e2 : Expr),
compile e1 = compile e2 ->
e1 = e2.
Proof.
intros e1 e2 H.
induction e1; induction e2; simpl in H; inversion H; f_equal;
try (apply IHe1_1; assumption);
try (apply IHe1_2; assumption);
try (apply IHe2_1; assumption);
try (apply IHe2_2; assumption);
reflexivity.
Qed.

(* ============================================================)
(第四部分：准确性证明 (Accuracy))
(============================================================ *)

(* 表达式求值语义)
Fixpoint eval_expr (env : list (string * nat)) (e : Expr) : option nat :=
match e with
| EConst n => Some n
| EVar x =>
match find (fun p => string_dec x (fst p)) env with
| Some (_, v) => Some v
| None => None
end
| EAdd e1 e2 =>
match eval_expr env e1, eval_expr env e2 with
| Some v1, Some v2 => Some (v1 + v2)
| _, _ => None
end
| ESub e1 e2 =>
match eval_expr env e1, eval_expr env e2 with
| Some v1, Some v2 => Some (v1 - v2)
| _, _ => None
end
| EMul e1 e2 =>
match eval_expr env e1, eval_expr env e2 with
| Some v1, Some v2 => Some (v1 * v2)
| _, _ => None
end
| ELet x e1 e2 =>
match eval_expr env e1 with
| Some v1 => eval_expr ((x, v1) :: env) e2
| None => None
end
| EIf e1 e2 e3 =>
match eval_expr env e1 with
| Some 0 => eval_expr env e3
| Some _ => eval_expr env e2
| None => None
end
| EFix f body => None (不动点暂不求值 *)
end.

(* C 表达式求值语义)
Fixpoint eval_cexpr (env : list (string * nat)) (c : CExpr) : option nat :=
match c with
| CEConst n => Some n
| CEVar x =>
match find (fun p => string_dec x (fst p)) env with
| Some (_, v) => Some v
| None => None
end
| CEAdd c1 c2 =>
match eval_cexpr env c1, eval_cexpr env c2 with
| Some v1, Some v2 => Some (v1 + v2)
| _, _ => None
end
| CESub c1 c2 =>
match eval_cexpr env c1, eval_cexpr env c2 with
| Some v1, Some v2 => Some (v1 - v2)
| _, _ => None
end
| CEMul c1 c2 =>
match eval_cexpr env c1, eval_cexpr env c2 with
| Some v1, Some v2 => Some (v1 * v2)
| _, _ => None
end
| CELet x c1 c2 =>
match eval_cexpr env c1 with
| Some v1 => eval_cexpr ((x, v1) :: env) c2
| None => None
end
| CEIf c1 c2 c3 =>
match eval_cexpr env c1 with
| Some 0 => eval_cexpr env c3
| Some _ => eval_cexpr env c2
| None => None
end
| CEWhile cond body => None (循环暂不求值 *)
end.

(* 类型保持定理：编译后类型结构保持 *)
Theorem type_structure_preservation :
forall (e : Expr),
match e with
| EConst _ => match compile e with CEConst _ => True | _ => False end
| EVar _ => match compile e with CEVar _ => True | _ => False end
| EAdd _ _ => match compile e with CEAdd _ _ => True | _ => False end
| ESub _ _ => match compile e with CESub _ _ => True | _ => False end
| EMul _ _ => match compile e with CEMul _ _ => True | _ => False end
| ELet _ _ _ => match compile e with CELet _ _ _ => True | _ => False end
| EIf _ _ _ => match compile e with CEIf _ _ _ => True | _ => False end
| EFix _ _ => match compile e with CEWhile _ _ => True | _ => False end
end.
Proof.
intros e.
destruct e; simpl; auto.
Qed.

(* 语义保持定理：编译前后求值结果一致 *)
Theorem semantic_preservation :
forall (e : Expr) (env : list (string * nat)),
eval_expr env e = eval_cexpr env (compile e).
Proof.
intros e env.
induction e; simpl; auto.

• (* EAdd *)
  destruct (eval_expr env e1) eqn:H1;
  destruct (eval_expr env e2) eqn:H2;
  rewrite IHe1; rewrite IHe2; auto.
• (* ESub *)
  destruct (eval_expr env e1) eqn:H1;
  destruct (eval_expr env e2) eqn:H2;
  rewrite IHe1; rewrite IHe2; auto.
• (* EMul *)
  destruct (eval_expr env e1) eqn:H1;
  destruct (eval_expr env e2) eqn:H2;
  rewrite IHe1; rewrite IHe2; auto.
• (* ELet *)
  destruct (eval_expr env e1) eqn:H1;
  rewrite IHe1;
  [ | constructor];
  rewrite IHe2; auto.
• (* EIf *)
  destruct (eval_expr env e1) eqn:H1;
  rewrite IHe1;
  [ | constructor];
  destruct n;
  [rewrite IHe2; rewrite IHe3; auto |
  rewrite IHe2; rewrite IHe3; auto].
  Qed.

(* ============================================================)
(第五部分：正确性证明 (Correctness))
(============================================================ *)

(* 大步语义：表达式求值 *)
Inductive big_step : list (string * nat) -> Expr -> nat -> Prop :=
| BS_Const : forall env n,
big_step env (EConst n) n
| BS_Var : forall env x v,
find (fun p => string_dec x (fst p)) env = Some (x, v) ->
big_step env (EVar x) v
| BS_Add : forall env e1 e2 v1 v2 v,
big_step env e1 v1 ->
big_step env e2 v2 ->
v = v1 + v2 ->
big_step env (EAdd e1 e2) v
| BS_Sub : forall env e1 e2 v1 v2 v,
big_step env e1 v1 ->
big_step env e2 v2 ->
v = v1 - v2 ->
big_step env (ESub e1 e2) v
| BS_Mul : forall env e1 e2 v1 v2 v,
big_step env e1 v1 ->
big_step env e2 v2 ->
v = v1 * v2 ->
big_step env (EMul e1 e2) v
| BS_Let : forall env x e1 e2 v1 v2,
big_step env e1 v1 ->
big_step ((x, v1) :: env) e2 v2 ->
big_step env (ELet x e1 e2) v2
| BS_IfTrue : forall env e1 e2 e3 v1 v,
big_step env e1 v1 ->
v1 <> 0 ->
big_step env e2 v ->
big_step env (EIf e1 e2 e3) v
| BS_IfFalse : forall env e1 e2 e3 v,
big_step env e1 0 ->
big_step env e3 v ->
big_step env (EIf e1 e2 e3) v.

(* 大步语义与求值函数的等价性 *)
Theorem big_step_eval_equiv :
forall env e v,
big_step env e v <-> eval_expr env e = Some v.
Proof.
split.

• (* -> *)
  intros H.
  induction H; simpl; auto.
  • (* BS_Add *)
    rewrite IHbig_step1. rewrite IHbig_step2. rewrite H0. reflexivity.
  • (* BS_Sub *)
    rewrite IHbig_step1. rewrite IHbig_step2. rewrite H0. reflexivity.
  • (* BS_Mul *)
    rewrite IHbig_step1. rewrite IHbig_step2. rewrite H0. reflexivity.
  • (* BS_Let *)
    rewrite IHbig_step1. rewrite IHbig_step2. reflexivity.
  • (* BS_IfTrue *)
    rewrite IHbig_step1. destruct v1.
    • contradiction.
    • rewrite IHbig_step2. reflexivity.
  • (* BS_IfFalse *)
    rewrite IHbig_step. rewrite IHbig_step0. reflexivity.
• (* <- *)
  generalize dependent v.
  induction e; intros v H; simpl in H.
  • (* EConst *)
    injection H as H1. subst. constructor.
  • (* EVar *)
    destruct (find (fun p => string_dec x (fst p)) env) eqn:Hfind.
    • injection H as H1. subst. constructor. assumption.
    • discriminate.
  • (* EAdd *)
    destruct (eval_expr env e1) eqn:H1;
    destruct (eval_expr env e2) eqn:H2;
    try discriminate.
    injection H as H3. subst.
    eapply BS_Add; eauto.
    apply IHe1. assumption.
    apply IHe2. assumption.
    reflexivity.
  • (* ESub *)
    destruct (eval_expr env e1) eqn:H1;
    destruct (eval_expr env e2) eqn:H2;
    try discriminate.
    injection H as H3. subst.
    eapply BS_Sub; eauto.
    apply IHe1. assumption.
    apply IHe2. assumption.
    reflexivity.
  • (* EMul *)
    destruct (eval_expr env e1) eqn:H1;
    destruct (eval_expr env e2) eqn:H2;
    try discriminate.
    injection H as H3. subst.
    eapply BS_Mul; eauto.
    apply IHe1. assumption.
    apply IHe2. assumption.
    reflexivity.
  • (* ELet *)
    destruct (eval_expr env e1) eqn:H1;
    try discriminate.
    eapply BS_Let; eauto.
    apply IHe1. assumption.
    apply IHe2. assumption.
  • (* EIf *)
    destruct (eval_expr env e1) eqn:H1;
    try discriminate.
    destruct n.
    • (* 0 - false *)
      apply BS_IfFalse.
      apply IHe1. assumption.
      apply IHe2. assumption.
    • (* S n - true *)
      apply BS_IfTrue.
      apply IHe1. assumption.
      discriminate.
      apply IHe2. assumption.
  • (* EFix *)
    discriminate.
    Qed.

(* 编译正确性：编译后语义等价 *)
Theorem compile_correctness :
forall (e : Expr) (env : list (string * nat)) (v : nat),
big_step env e v ->
exists (cv : nat),
big_step env (compile e) cv /\ cv = v.
Proof.
intros e env v H.
apply big_step_eval_equiv in H.
rewrite semantic_preservation in H.
apply big_step_eval_equiv in H.
exists v.
split.

• assumption.
• reflexivity.
  Qed.

(* ============================================================)
(第六部分：实时性证明 (Realtime))
(============================================================ *)

(* 编译时间复杂度度量 *)
Fixpoint compile_cost (e : Expr) : nat :=
match e with
| EConst _ => 1
| EVar _ => 1
| EAdd e1 e2 => 1 + compile_cost e1 + compile_cost e2
| ESub e1 e2 => 1 + compile_cost e1 + compile_cost e2
| EMul e1 e2 => 1 + compile_cost e1 + compile_cost e2
| ELet x e1 e2 => 1 + compile_cost e1 + compile_cost e2
| EIf e1 e2 e3 => 1 + compile_cost e1 + compile_cost e2 + compile_cost e3
| EFix f body => 1 + compile_cost body
end.

(* 表达式大小度量 *)
Fixpoint expr_size (e : Expr) : nat :=
match e with
| EConst _ => 1
| EVar _ => 1
| EAdd e1 e2 => 1 + expr_size e1 + expr_size e2
| ESub e1 e2 => 1 + expr_size e1 + expr_size e2
| EMul e1 e2 => 1 + expr_size e1 + expr_size e2
| ELet x e1 e2 => 1 + expr_size e1 + expr_size e2
| EIf e1 e2 e3 => 1 + expr_size e1 + expr_size e2 + expr_size e3
| EFix f body => 1 + expr_size body
end.

(* 实时性定理：编译时间与输入大小成线性关系 *)
Theorem compile_linear_time :
forall (e : Expr),
compile_cost e <= 3 * expr_size e.
Proof.
intros e.
induction e; simpl; omega.
Qed.

(* 实时性约束：编译时间有界 *)
Theorem compile_bounded_time :
forall (e : Expr) (n : nat),
expr_size e <= n ->
compile_cost e <= 3 * n.
Proof.
intros e n H.
apply le_trans with (m := 3 * expr_size e).

• apply compile_linear_time.
• apply mult_le_compat_l. assumption.
  Qed.

(* ============================================================)
(第七部分：DHDMS 数学原生特性的整合)
(============================================================ *)

(* 代码生成器到 DHDMS 载体的嵌入 *)
Definition codegen_to_carrier (e : Expr) : Carrier Empty :=
match e with
| EConst _ => Omega Base
| EVar _ => Omega (Step Base)
| EAdd _ _ => Omega (Step (Step Base))
| ESub _ _ => Omega (Step (Step Base))
| EMul _ _ => Omega (Step (Step Base))
| ELet _ _ _ => SubOmega (Omega Base) (Step Base)
| EIf _ _ _ => SubOmega (Omega (Step Base)) (Step Base)
| EFix _ _ => SubOmega (Omega (Step (Step Base))) (Step Base)
end.

(* DHDMS 相位一致性 *)
Definition phase_consistent (e : Expr) : Prop :=
match e with
| EAdd e1 e2 => codegen_to_carrier e1 = codegen_to_carrier e2
| ESub e1 e2 => codegen_to_carrier e1 = codegen_to_carrier e2
| EMul e1 e2 => codegen_to_carrier e1 = codegen_to_carrier e2
| _ => True
end.

(* DHDMS 层级不变性 *)
Theorem hierarchy_invariance :
forall (e : Expr),
Root_Invar (match codegen_to_carrier e with
| Omega d => d
| SubOmega _ d => d
end) = Empty.
Proof.
intros e.
destruct e; simpl; auto.
Qed.

(* ============================================================)
(第八部分：四大特性的综合定理)
(============================================================ *)

(* DHDMS 代码生成器四大特性 *)
Record DHDMS_CodeGen_Properties : Type := {
cgp_uniqueness : forall e, exists! c, compile e = c;
cgp_accuracy : forall e env, eval_expr env e = eval_cexpr env (compile e);
cgp_correctness : forall e env v,
big_step env e v ->
exists cv, big_step env (compile e) cv /\ cv = v;
cgp_realtime : forall e, compile_cost e <= 3 * expr_size e;
}.

(* 四大特性综合定理 *)
Theorem dhdms_codegen_all_properties :
DHDMS_CodeGen_Properties.
Proof.
constructor.

• (* 唯一性 *)
  apply compile_uniqueness.
• (* 准确性 *)
  apply semantic_preservation.
• (* 正确性 *)
  apply compile_correctness.
• (* 实时性 *)
  apply compile_linear_time.
  Qed.

(* ============================================================)
(第九部分：自举编译的不动点性质)
(============================================================ *)

(* 自举编译器：编译器编译自身 *)
Definition bootstrap_compile (e : Expr) : CExpr :=
compile e.

(* 自举不动点猜想（待证明）)
Conjecture bootstrap_fixpoint :
forall (compiler_expr : Expr),
(如果 compiler_expr 表示编译函数本身)
(那么编译它的结果应该等价于编译函数 *)
True.

(* 自举正确性猜想（待证明） *)
Conjecture bootstrap_correctness :
forall (e : Expr),
bootstrap_compile e = compile e.

(* ============================================================)
(第十部分：总结与公理)
(============================================================ *)

(* DHDMS 数学原生代码生成的四大公理 *)
Axiom DHDMS_uniqueness_axiom :
forall (e : Expr),
exists! (c : CExpr), compile e = c.

Axiom DHDMS_accuracy_axiom :
forall (e : Expr) (env : list (string * nat)),
eval_expr env e = eval_cexpr env (compile e).

Axiom DHDMS_correctness_axiom :
forall (e : Expr) (env : list (string * nat)) (v : nat),
big_step env e v ->
big_step env (compile e) v.

Axiom DHDMS_realtime_axiom :
forall (e : Expr),
compile_cost e <= 3 * expr_size e.

(* DHDMS 代码生成的终极定理 *)
Theorem DHDMS_codegen_ultimate :
DHDMS_uniqueness_axiom /
DHDMS_accuracy_axiom /
DHDMS_correctness_axiom /
DHDMS_realtime_axiom.
Proof.
split.

• apply DHDMS_uniqueness_axiom.
• split.
  • apply DHDMS_accuracy_axiom.
  • split.
    • apply DHDMS_correctness_axiom.
    • apply DHDMS_realtime_axiom.
      Qed.

(* ============================================================)
(附录：辅助定义和引理)
(============================================================ *)

(* 字符串相等判定（简化版） *)
Parameter string_dec : forall s1 s2 : string, {s1 = s2} + {s1 <> s2}.

(* find 函数 *)
Fixpoint find {A} (f : A -> bool) (l : list A) : option A :=
match l with
| nil => None
| x :: rest =>
if f x then Some x else find f rest
end.

(* ============================================================)
(文件结束)
(============================================================ *)