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正如标题所暗示的，我的问题的关注证明在勒柯克算术表达式的惰性评估的正确性和整体性。我想证明的定理总共有三种：懒惰计算的正确性与总体（勒柯克）
计算中只给出规范 表达式作为结果
定理Only_canonical_results： （FORALL X Y：AEXP，比较x和y - >规范Y）。
正确性：计算关系 保留表达式的外延
定理correct_wrt_semantics： （forall的x和y：AEXP，小样X Y - > I N（外延X）（外延Y））。
每个输入导致了一些结果。定理Comp_is_total：（forall x：Aexp， （Sigma Aexp（fun y => prod（Comp x y）（Canonical y））））。
必要的定义见下面的代码。当谈到Coq时，我应该说清楚我是一个新手;更有经验的用户可能会马上注意到。在大多数情况下，或者甚至所有我编写的背景材料都可以在标准库中找到，这是绝对的情况。但是，如果我确切地知道从标准库导入什么以证明理想结果，我很可能不会在这里困扰你。这就是为什么我向你提供我迄今为止的材料，希望有一些善良的人可以帮助我。谢谢！
(* Sigma types *)
Inductive Sigma (A:Set)(B:A -> Set) :Set :=
Spair: forall a:A, forall b : B a,Sigma A B.
Definition E (A:Set)(B:A -> Set)
(C: Sigma A B -> Set)
(c: Sigma A B)
(d: (forall x:A, forall y:B x,
C (Spair A B x y))): C c :=
match c as c0 return (C c0) with
| Spair a b => d a b
end.
Definition project1 (A:Set)(B:A -> Set)(c: Sigma A B):=
E A B (fun z => A) c (fun x y => x).
(* Binary sum type *)
Inductive sum' (A B:Set):Set :=
inl': A -> sum' A B | inr': B -> sum' A B.
Print sum'_rect.
Definition D (A B : Set)(C: sum' A B -> Set)
(c: sum' A B)
(d: (forall x:A, C (inl' A B x)))
(e: (forall y:B, C (inr' A B y))): C c :=
match c as c0 return C c0 with
| inl' x => d x
| inr' y => e y
end.
(* Three useful finite sets *)
Inductive N_0: Set :=.
Definition R_0
(C:N_0 -> Set)
(c: N_0): C c :=
match c as c0 return (C c0) with
end.
Inductive N_1: Set := zero_1:N_1.
Definition R_1
(C:N_1 -> Set)
(c: N_1)
(d_zero: C zero_1): C c :=
match c as c0 return (C c0) with
| zero_1 => d_zero
end.
Inductive N_2: Set := zero_2:N_2 | one_2:N_2.
Definition R_2
(C:N_2 -> Set)
(c: N_2)
(d_zero: C zero_2)
(d_one: C one_2): C c :=
match c as c0 return (C c0) with
| zero_2 => d_zero
| one_2 => d_one
end.
(* Natural numbers *)
Inductive N:Set :=
zero: N | succ : N -> N.
Print N.
Print N_rect.
Definition R
(C:N -> Set)
(d: C zero)
(e: (forall x:N, C x -> C (succ x))):
(forall n:N, C n) :=
fix F (n: N): C n :=
match n as n0 return (C n0) with
| zero => d
| succ n0 => e n0 (F n0)
end.
(* Boolean to truth-value converter *)
Definition Tr (c:N_2) : Set :=
match c as c0 with
| zero_2 => N_0
| one_2 => N_1
end.
(* Identity type *)
Inductive I (A: Set)(x: A) : A -> Set :=
r : I A x x.
Print I_rect.
Theorem J
(A:Set)
(C: (forall x y:A,
forall z: I A x y, Set))
(d: (forall x:A, C x x (r A x)))
(a:A)(b:A)(c:I A a b): C a b c.
induction c.
apply d.
Defined.
(* functions are extensional wrt
identity types *)
Theorem I_I_extensionality (A B: Set)(f: A -> B):
(forall x y:A, I A x y -> I B (f x) (f y)).
Proof.
intros x y P.
induction P.
apply r.
Defined.
(* addition *)
Definition add (m n:N) : N
:= R (fun z=> N) m (fun x y => succ y) n.
(* multiplication *)
Definition mul (m n:N) : N
:= R (fun z=> N) zero (fun x y => add y m) n.
(* Axioms of Peano verified *)
Theorem P1a: (forall x: N, I N (add x zero) x).
intro x.
(* force use of definitional equality
by applying reflexivity *)
apply r.
Defined.
Theorem P1b: (forall x y: N,
I N (add x (succ y)) (succ (add x y))).
intros.
apply r.
Defined.
Theorem P2a: (forall x: N, I N (mul x zero) zero).
intros.
apply r.
Defined.
Theorem P2b: (forall x y: N,
I N (mul x (succ y)) (add (mul x y) x)).
intros.
apply r.
Defined.
Definition pd (n: N): N :=
R (fun _=> N) zero (fun x y=> x) n.
(* alternatively
Definition pd (x: N): N :=
match x as x0 with
| zero => zero
| succ n0 => n0
end.
*)
Theorem P3: (forall x y:N,
I N (succ x) (succ y) -> I N x y).
intros x y p.
apply (I_I_extensionality N N pd (succ x) (succ y)).
apply p.
Defined.
Definition not (A:Set): Set:= (A -> N_0).
Definition isnonzero (n: N): N_2:=
R (fun _ => N_2) zero_2 (fun x y => one_2) n.
Theorem P4 : (forall x:N,
not (I N (succ x) zero)).
intro x.
intro p.
apply (J N (fun x y z =>
Tr (isnonzero x) -> Tr (isnonzero y))
(fun x => (fun t => t)) (succ x) zero)
.
apply p.
simpl.
apply zero_1.
Defined.
Theorem P5 (P:N -> Set):
P zero -> (forall x:N, P x -> P (succ x))
-> (forall x:N, P x).
intros base step n.
apply R.
apply base.
apply step.
Defined.
(* I(A,-,-) is an equivalence relation *)
Lemma Ireflexive (A:Set): (forall x:A, I A x x).
intro x.
apply r.
Defined.
Lemma Isymmetric (A:Set): (forall x y:A, I A x y -> I A y x).
intros x y P.
induction P.
apply r.
Defined.
Lemma Itransitive (A:Set):
(forall x y z:A, I A x y -> I A y z -> I A x z).
intros x y z P Q.
induction P.
assumption.
Defined.
Definition or (A B : Set):= sum' A B.
(* arithmetical expressions *)
Inductive Aexp :Set :=
zer: Aexp
| suc: Aexp -> Aexp
| pls: Aexp -> Aexp -> Aexp.
(* denotation of an expression *)
Definition denotation: Aexp->N:=
fix F (a: Aexp): N :=
match a as a0 with
| zer => zero
| suc a1 => succ (F a1)
| pls a1 a2 => add (F a1) (F a2)
end.
(* predicate for distinguishing
canonical expressions *)
Definition Canonical (x:Aexp):Set :=
or (I Aexp x zer)
(Sigma Aexp (fun y =>
I Aexp x (suc y))).
(* the computation relation is
an inductively defined relation *)
Inductive Comp : Aexp -> Aexp -> Set
:=
refrule: forall a: Aexp,
forall p: Canonical a, Comp a a
| zerrule: forall a b c:Aexp,
forall p: Comp b zer,
forall q: Comp a c,
Comp (pls a b) c
| sucrule: forall a b c:Aexp,
forall p: Comp b (suc c),
Comp (pls a b) (suc (pls a c)).
(* Computations only give canonical
expressions as results *)
Theorem Only_canonical_results:
(forall x y: Aexp, Comp x y -> Canonical y).
admit.
Defined.
(* Here is where help is needed *)
(* Correctness: the computation relation
preserves denotation of expressions *)
Theorem correct_wrt_semantics:
(forall x y: Aexp, Comp x y ->
I N (denotation x) (denotation y)).
admit.
(* Here is where help is need*)
Defined.
(* every input leads to some result *)
Theorem Comp_is_total: (forall x:Aexp,
(Sigma Aexp (fun y =>
prod (Comp x y) (Canonical y)))).
admit.
(* Proof required *)
Defined.
看来你知道'_rect'消除器Coq为你的归纳数据类型自动定义，但是你重新实现了它们。这是为了教育目的吗？ – Ptival
@Ptival正确！我正在学习Coq，所以我认为这将是一个很好的练习，可以自己写出定义，而不是简单地导入它们。 – user111731