(* )(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 Noneend| EAdd e1 e2 match eval_expr env e1, eval_expr env e2 with| Some v1, Some v2 Some (v1 v2)| _, _ Noneend| ESub e1 e2 match eval_expr env e1, eval_expr env e2 with| Some v1, Some v2 Some (v1 - v2)| _, _ Noneend| EMul e1 e2 match eval_expr env e1, eval_expr env e2 with| Some v1, Some v2 Some (v1 * v2)| _, _ Noneend| ELet x e1 e2 match eval_expr env e1 with| Some v1 eval_expr ((x, v1) :: env) e2| None Noneend| EIf e1 e2 e3 match eval_expr env e1 with| Some 0 eval_expr env e3| Some _ eval_expr env e2| None Noneend| 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 Noneend| CEAdd c1 c2 match eval_cexpr env c1, eval_cexpr env c2 with| Some v1, Some v2 Some (v1 v2)| _, _ Noneend| CESub c1 c2 match eval_cexpr env c1, eval_cexpr env c2 with| Some v1, Some v2 Some (v1 - v2)| _, _ Noneend| CEMul c1 c2 match eval_cexpr env c1, eval_cexpr env c2 with| Some v1, Some v2 Some (v1 * v2)| _, _ Noneend| CELet x c1 c2 match eval_cexpr env c1 with| Some v1 eval_cexpr ((x, v1) :: env) c2| None Noneend| CEIf c1 c2 c3 match eval_cexpr env c1 with| Some 0 eval_cexpr env c3| Some _ eval_cexpr env c2| None Noneend| 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 endend.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 bodyend.(* 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 bodyend.(* 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| _ Trueend.(* 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 dend) 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 restend.(* )(文件结束)( *)