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Estimate.v
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Require Import Reals.
Require Import Psatz.
Require Import SQIR.
Require Import VectorStates UnitaryOps Coq.btauto.Btauto Coq.NArith.Nnat.
Require Import Dirac.
Require Import QPE.
Require Import BasicUtility.
Require Import MathSpec.
Require Import Classical_Prop.
Require Import OQASM.
(**********************)
(** Unitary Programs **)
(**********************)
Declare Scope exp_scope.
Delimit Scope exp_scope with exp.
Local Open Scope exp_scope.
Local Open Scope nat_scope.
Inductive upexp := Haf (e:exp) (* H; U *)
| Oracle (e:exp) | QPE (e:exp) | AA (e:exp).
(* add two types: restricted Hadamard mode, and QFT^{-1} mode. *)
(* Use restricted Hadmard mode to evaluate the difference between QFT and AQFT. *)
(* Use QFT^{-1} to show that AQFT is useful, but it is not useful in calculation. *)
(* irrelavent vars. *)
Definition vars_neq (l:list var) := forall n m x y,
nth_error l m = Some x -> nth_error l n = Some y -> n <> m -> x <> y.
Inductive exp := SKIP (p:posi) | X (p:posi) | CU (p:posi) (e:exp)
| RZ (q:nat) (p:posi) (* 2 * PI * i / 2^q *)
| RRZ (q:nat) (p:posi)
| SR (q:nat) (x:var) (* a series of RZ gates for QFT mode. *)
| SRR (q:nat) (x:var) (* a series of RRZ gates for QFT mode. *)
| HCNOT (p1:posi) (p2:posi)
| Lshift (x:var)
| Rshift (x:var)
| Rev (x:var)
| QFT (x:var)
| RQFT (x:var)
| H (x:var)
| Seq (s1:exp) (s2:exp).
Notation "p1 ; p2" := (Seq p1 p2) (at level 50) : exp_scope.
Fixpoint exp_elim (p:exp) :=
match p with
| CU q p => match exp_elim p with
| SKIP a => SKIP a
| p' => CU q p'
end
| Seq p1 p2 => match exp_elim p1, exp_elim p2 with
| SKIP _, p2' => p2'
| p1', SKIP _ => p1'
| p1', p2' => Seq p1' p2'
end
| _ => p
end.
Definition Z (p:posi) := RZ 1 p.
Fixpoint inv_exp p :=
match p with
| SKIP a => SKIP a
| X n => X n
| CU n p => CU n (inv_exp p)
| SR n x => SRR n x
| SRR n x => SR n x
| Lshift x => Rshift x
| Rshift x => Lshift x
| Rev x => Rev x
| HCNOT p1 p2 => HCNOT p1 p2
| RZ q p1 => RRZ q p1
| RRZ q p1 => RZ q p1
| QFT x => RQFT x
| RQFT x => QFT x
| H x => H x
| Seq p1 p2 => inv_exp p2; inv_exp p1
end.
Fixpoint GCCX' x n size :=
match n with
| O | S O => X (x,n - 1)
| S m => CU (x,size-n) (GCCX' x m size)
end.
Definition GCCX x n := GCCX' x n n.
Fixpoint nX x n :=
match n with 0 => X (x,0)
| S m => X (x,m); nX x m
end.
(* Grover diffusion operator. *)
Definition diff_half (x c:var) (n:nat) := H x ; H c ; ((nX x n; X (c,0))).
Definition diff_1 (x c :var) (n:nat) :=
diff_half x c n ; ((GCCX x n)) ; (inv_exp (diff_half x c n)).
(*The second implementation of grover's diffusion operator.
The whole circuit is a little different, and the input for the diff_2 circuit is asssumed to in Had mode. *)
Definition diff_2 (x c :var) (n:nat) :=
H x ; ((GCCX x n)) ; H x.
Fixpoint is_all_true C n :=
match n with 0 => true
| S m => C m && is_all_true C m
end.
Definition const_u (C :nat -> bool) (n:nat) c := if is_all_true C n then ((X (c,0))) else SKIP (c,0).
Fixpoint niter_prog n (c:var) (P : exp) : exp :=
match n with
| 0 => SKIP (c,0)
| 1 => P
| S n' => niter_prog n' c P ; P
end.
Definition body (C:nat -> bool) (x c:var) (n:nat) := const_u C n c; diff_2 x c n.
Definition grover_e (i:nat) (C:nat -> bool) (x c:var) (n:nat) :=
H x; H c ; ((Z (c,0))) ; niter_prog i c (body C x c n).
(** Definition of Deutsch-Jozsa program. **)
Definition deutsch_jozsa (x c:var) (n:nat) :=
((nX x n; X (c,0))) ; H x ; H c ; ((X (c,0))); H c ; H x.
Inductive type := Had | Phi (n:nat) | Nor.
Require Import Coq.FSets.FMapList.
Require Import Coq.FSets.FMapFacts.
Require Import Coq.Structures.OrderedTypeEx.
Module Env := FMapList.Make Nat_as_OT.
Module EnvFacts := FMapFacts.Facts (Env).
Definition env := Env.t type.
Definition empty_env := @Env.empty type.
Lemma mapsto_always_same : forall k v1 v2 s,
@Env.MapsTo (type) k v1 s ->
@Env.MapsTo (type) k v2 s ->
v1 = v2.
Proof.
intros.
apply Env.find_1 in H0.
apply Env.find_1 in H1.
rewrite H0 in H1.
injection H1.
easy.
Qed.
Lemma find_add : forall k v m,
@Env.find (type) k (Env.add k v m) = Some v.
Proof.
intros.
apply Env.find_1.
apply Env.add_1.
easy.
Qed.
Lemma mapsto_add1 : forall k v1 v2 s,
@Env.MapsTo (type) k v1 (Env.add k v2 s) -> v1 = v2.
Proof.
intros.
apply Env.find_1 in H0.
rewrite find_add in H0.
inversion H0.
reflexivity.
Qed.
Lemma mapsto_equal : forall k v s1 s2,
@Env.MapsTo type k v s1 ->
Env.Equal s1 s2 ->
Env.MapsTo k v s2.
Proof.
intros.
unfold Env.Equal in H1.
apply Env.find_2. rewrite <- H1.
apply Env.find_1.
assumption.
Qed.
Lemma map_find_add : forall x v env, @Env.find (type) x (Env.add x v env) = Some v.
Proof.
intros.
apply Env.find_1.
apply Env.add_1. easy.
Qed.
Lemma map_find_neq : forall x y v env, x <> y -> @Env.find (type) x (Env.add y v env) = Env.find x env.
Proof.
intros.
destruct (Env.find (elt:=type) x env0) eqn:eq1.
apply Env.find_1. apply Env.add_2. lia.
apply Env.find_2 in eq1. easy.
destruct (Env.find (elt:=type) x (Env.add y v env0)) eqn:eq2.
apply Env.find_2 in eq2. apply Env.add_3 in eq2.
apply Env.find_1 in eq2. rewrite eq1 in eq2. inv eq2. lia.
easy.
Qed.
Definition or_not_r (x y:var) (n1 n2:nat) := x <> y \/ n1 < n2.
Definition or_not_eq (x y:var) (n1 n2:nat) := x <> y \/ n1 <= n2.
Inductive exp_fresh (aenv:var->nat): posi -> exp -> Prop :=
| skip_fresh : forall p p1, p <> p1 -> exp_fresh aenv p (SKIP p1)
| x_fresh : forall p p' , p <> p' -> exp_fresh aenv p (X p')
| sr_fresh : forall p x n, or_not_r (fst p) x n (snd p) -> exp_fresh aenv p (SR n x)
| srr_fresh : forall p x n, or_not_r (fst p) x n (snd p) -> exp_fresh aenv p (SRR n x)
| lshift_fresh : forall p x, or_not_eq (fst p) x (aenv x) (snd p) -> exp_fresh aenv p (Lshift x)
| rshift_fresh : forall p x, or_not_eq (fst p) x (aenv x) (snd p) -> exp_fresh aenv p (Rshift x)
| rev_fresh : forall p x, or_not_eq (fst p) x (aenv x) (snd p) -> exp_fresh aenv p (Rev x)
| cu_fresh : forall p p' e, p <> p' -> exp_fresh aenv p e -> exp_fresh aenv p (CU p' e)
| cnot_fresh : forall p p1 p2, p <> p1 -> p <> p2 -> exp_fresh aenv p (HCNOT p1 p2)
| rz_fresh : forall p p' q, p <> p' -> exp_fresh aenv p (RZ q p')
| rrz_fresh : forall p p' q, p <> p' -> exp_fresh aenv p (RRZ q p')
| qft_fresh : forall p x , or_not_eq (fst p) x (aenv x) (snd p) -> exp_fresh aenv p (QFT x)
| rqft_fresh : forall p x , or_not_eq (fst p) x (aenv x) (snd p) -> exp_fresh aenv p (RQFT x)
| h_fresh : forall p x , or_not_eq (fst p) x (aenv x) (snd p) -> exp_fresh aenv p (H x)
| seq_fresh : forall p e1 e2, exp_fresh aenv p e1 -> exp_fresh aenv p e2 -> exp_fresh aenv p (Seq e1 e2).
(* Defining matching shifting stack. *)
Inductive sexp := Ls | Rs | Re.
Definition opp_ls (s : sexp) := match s with Ls => Rs | Rs => Ls | Re => Re end.
Definition fst_not_opp (s:sexp) (l : list sexp) :=
match l with [] => True
| (a::al) => s <> opp_ls a
end.
Inductive exp_neu_l (x:var) : list sexp -> exp -> list sexp -> Prop :=
| skip_neul : forall l p, exp_neu_l x l (SKIP p) l
| x_neul : forall l p, exp_neu_l x l (X p) l
| sr_neul : forall l y n, exp_neu_l x l (SR n y) l
| srr_neul : forall l y n, exp_neu_l x l (SRR n y) l
| cu_neul : forall l p e, exp_neu_l x [] e [] -> exp_neu_l x l (CU p e) l
| hcnot_neul : forall l p1 p2, exp_neu_l x l (HCNOT p1 p2) l
| rz_neul : forall l p q, exp_neu_l x l (RZ q p) l
| rrz_neul : forall l p q, exp_neu_l x l (RRZ q p) l
| qft_neul : forall l y, exp_neu_l x l (QFT y) l
| rqft_neul : forall l y , exp_neu_l x l (RQFT y) l
| h_neul : forall l y, exp_neu_l x l (H y) l
| lshift_neul_a : forall l, exp_neu_l x (Rs::l) (Lshift x) l
| lshift_neul_b : forall l, fst_not_opp Ls l -> exp_neu_l x l (Lshift x) (Ls::l)
| lshift_neul_ne : forall l y, x <> y -> exp_neu_l x l (Lshift y) l
| rshift_neul_a : forall l, exp_neu_l x (Ls::l) (Rshift x) l
| rshift_neul_b : forall l, fst_not_opp Rs l -> exp_neu_l x l (Rshift x) (Rs::l)
| rshift_neul_ne : forall l y, x <> y -> exp_neu_l x l (Rshift y) l
| rev_neul_a : forall l, exp_neu_l x (Re::l) (Rev x) l
| rev_neul_b : forall l, fst_not_opp Re l -> exp_neu_l x l (Rev x) (Re::l)
| rev_neul_ne : forall l y, x <> y -> exp_neu_l x l (Rev y) l
| seq_neul : forall l l' l'' e1 e2, exp_neu_l x l e1 l' -> exp_neu_l x l' e2 l'' -> exp_neu_l x l (Seq e1 e2) l''.
Definition exp_neu (xl : list var) (e:exp) : Prop :=
forall x, In x xl -> exp_neu_l x [] e [].
Definition exp_neu_prop (l:list sexp) :=
(forall i a, i + 1 < length l -> nth_error l i = Some a -> nth_error l (i+1) <> Some (opp_ls a)).
Lemma exp_neu_t_prop : forall p x l l', exp_neu_l x l p l' -> exp_neu_prop l -> exp_neu_prop l'.
Proof.
induction p; intros; try easy.
1-8:inv H0; easy.
unfold exp_neu_prop in *.
intros. inv H0.
destruct l'. simpl in *. lia.
destruct i. simpl in *.
destruct l'. easy.
specialize (H1 1 a).
assert (1 + 1 < S (S (length (s0 :: l')))) by lia.
apply H1 in H0. simpl in *. easy.
simpl in *. easy.
specialize (H1 (S (S i)) a).
assert (S (S i) + 1 < length (Rs :: s :: l')).
simpl in *. lia.
apply H1 in H0.
simpl in *. easy.
simpl in *. easy.
unfold fst_not_opp in H6. destruct l. simpl in *. lia.
destruct i. simpl in *. inv H3.
unfold opp_ls in *. intros R. inv R. easy.
specialize (H1 i a).
assert (i + 1 < length (s :: l)). simpl in *. lia.
apply H1 in H0. simpl in *. easy. simpl in *. easy.
apply H1; try easy.
unfold exp_neu_prop in *.
intros. inv H0.
destruct l'. simpl in *. lia.
destruct i. simpl in *.
destruct l'. easy.
specialize (H1 1 a).
assert (1 + 1 < S (S (length (s0 :: l')))) by lia.
apply H1 in H0. simpl in *. easy.
simpl in *. easy.
specialize (H1 (S (S i)) a).
assert (S (S i) + 1 < length (Ls :: s :: l')).
simpl in *. lia.
apply H1 in H0.
simpl in *. easy.
simpl in *. easy.
unfold fst_not_opp in *. destruct l. simpl in *. lia.
destruct i. simpl in *. inv H3.
unfold opp_ls. intros R. inv R. easy.
specialize (H1 i a).
assert (i + 1 < length (s :: l)). simpl in *. lia.
apply H1 in H0. simpl in *. easy. simpl in *. easy.
apply H1; try easy.
unfold exp_neu_prop in *.
intros. inv H0.
destruct i. simpl in *.
destruct l'. easy.
specialize (H1 1 a).
assert (1 + 1 < S (length (s :: l'))) by lia.
apply H1 in H0. simpl in *. easy.
simpl in *. easy.
specialize (H1 (S (S i)) a).
assert (S (S i) + 1 < length (Re :: l')).
simpl in *. lia.
apply H1 in H0.
simpl in *. easy.
simpl in *. easy.
unfold fst_not_opp in *. destruct l. simpl in *. lia.
destruct i. simpl in *. inv H3.
unfold opp_ls. intros R. inv R. easy.
specialize (H1 i a).
assert (i + 1 < length (s :: l)). simpl in *. lia.
apply H1 in H0. simpl in *. easy. simpl in *. easy.
apply H1; try easy.
1-3:inv H0; easy.
inv H0.
apply IHp2 with (x := x) (l := l'0); try easy.
apply IHp1 with (x:=x) (l := l); try easy.
Qed.
(* Type System. *)
Inductive well_typed_exp : env -> exp -> Prop :=
| skip_refl : forall env, forall p, well_typed_exp env (SKIP p)
| x_nor : forall env p, Env.MapsTo (fst p) Nor env -> well_typed_exp env (X p)
| x_had : forall env p, Env.MapsTo (fst p) Had env -> well_typed_exp env (X p)
| cnot_had : forall env p1 p2, p1 <> p2 -> Env.MapsTo (fst p1) Had env -> Env.MapsTo (fst p2) Had env
-> well_typed_exp env (HCNOT p1 p2)
| rz_nor : forall env q p, Env.MapsTo (fst p) Nor env -> well_typed_exp env (RZ q p)
| rz_had : forall env p, Env.MapsTo (fst p) Had env -> well_typed_exp env (RZ 1 p)
| rrz_nor : forall env q p, Env.MapsTo (fst p) Nor env -> well_typed_exp env (RRZ q p)
| rrz_had : forall env p, Env.MapsTo (fst p) Had env -> well_typed_exp env (RRZ 1 p)
| sr_phi : forall env n m x, Env.MapsTo x (Phi n) env -> m < n -> well_typed_exp env (SR m x)
| srr_phi : forall env n m x, Env.MapsTo x (Phi n) env -> m < n -> well_typed_exp env (SRR m x)
| lshift_nor : forall env x, Env.MapsTo x Nor env -> well_typed_exp env (Lshift x)
| lshift_had : forall env x, Env.MapsTo x Had env -> well_typed_exp env (Lshift x)
| rshift_nor : forall env x, Env.MapsTo x Nor env -> well_typed_exp env (Rshift x)
| rshift_had : forall env x, Env.MapsTo x Had env -> well_typed_exp env (Rshift x)
| rev_nor : forall env x, Env.MapsTo x Nor env -> well_typed_exp env (Rev x)
| rev_had : forall env x, Env.MapsTo x Had env -> well_typed_exp env (Rev x).
(*
Definition opp_ls (s : sexp) := match s with Ls => Rs | Rs => Ls | Re => Re end.
*)
Fixpoint get_vars e : list var :=
match e with SKIP p => [(fst p)]
| X p => [(fst p)]
| CU p e => (fst p)::(get_vars e)
| HCNOT p1 p2 => ((fst p1)::(fst p2)::[])
| RZ q p => ((fst p)::[])
| RRZ q p => ((fst p)::[])
| SR n x => (x::[])
| SRR n x => (x::[])
| Lshift x => (x::[])
| Rshift x => (x::[])
| Rev x => (x::[])
| QFT x => (x::[])
| RQFT x => (x::[])
| H x => (x::[])
| Seq e1 e2 => get_vars e1 ++ (get_vars e2)
end.
Inductive well_typed_pexp (aenv: var -> nat) : env -> exp -> env -> Prop :=
| exp_refl : forall env e,
well_typed_exp env e -> well_typed_pexp aenv env e env
| qft_nor : forall env env' x, Env.MapsTo x Nor env -> Env.Equal env' (Env.add x (Phi (aenv x)) env)
-> well_typed_pexp aenv env (QFT x) env'
| rqft_phi : forall env env' x, Env.MapsTo x (Phi (aenv x)) env -> Env.Equal env' (Env.add x Nor env) ->
well_typed_pexp aenv env (RQFT x) env'
| h_nor : forall env env' x, Env.MapsTo x Nor env -> Env.Equal env' (Env.add x Had env) ->
well_typed_pexp aenv env (H x) env'
| h_had : forall env env' x, Env.MapsTo x Had env -> Env.Equal env' (Env.add x Nor env) ->
well_typed_pexp aenv env (H x) env'
| pcu_nor : forall env p e, Env.MapsTo (fst p) Nor env -> exp_fresh aenv p e -> exp_neu (get_vars e) e ->
well_typed_pexp aenv env e env -> well_typed_pexp aenv env (CU p e) env
| pe_seq : forall env env' env'' e1 e2, well_typed_pexp aenv env e1 env' ->
well_typed_pexp aenv env' e2 env'' -> well_typed_pexp aenv env (e1 ; e2) env''.
Inductive exp_WF (aenv:var -> nat): exp -> Prop :=
| skip_wf : forall p, snd p < aenv (fst p) -> exp_WF aenv (SKIP p)
| x_wf : forall p, snd p < aenv (fst p) -> exp_WF aenv (X p)
| cu_wf : forall p e, snd p < aenv (fst p) -> exp_WF aenv e -> exp_WF aenv (CU p e)
| hcnot_wf : forall p1 p2, snd p1 < aenv (fst p1)
-> snd p2 < aenv (fst p2) -> exp_WF aenv (HCNOT p1 p2)
| rz_wf : forall p q, snd p < aenv (fst p) -> exp_WF aenv (RZ q p)
| rrz_wf : forall p q, snd p < aenv (fst p) -> exp_WF aenv (RRZ q p)
| sr_wf : forall n x, S n < aenv x -> exp_WF aenv (SR n x)
| ssr_wf : forall n x, S n < aenv x -> exp_WF aenv (SRR n x)
| seq_wf : forall e1 e2, exp_WF aenv e1 -> exp_WF aenv e2 -> exp_WF aenv (Seq e1 e2)
| lshift_wf : forall x, 0 < aenv x -> exp_WF aenv (Lshift x)
| rshift_wf : forall x, 0 < aenv x -> exp_WF aenv (Rshift x)
| rev_wf : forall x, 0 < aenv x -> exp_WF aenv (Rev x)
| qft_wf : forall x, 0 < aenv x -> exp_WF aenv (QFT x)
| rqft_wf : forall x, 0 < aenv x -> exp_WF aenv (RQFT x)
| h_wf : forall x, 0 < aenv x -> exp_WF aenv (H x).
Fixpoint init_v (n:nat) (x:var) (M: nat -> bool) :=
match n with 0 => (SKIP (x,0))
| S m => if M m then init_v m x M; X (x,m) else init_v m x M
end.
Inductive right_mode_val : type -> val -> Prop :=
| right_nor: forall b r, right_mode_val Nor (nval b r)
| right_had: forall b1 b2 r, r 0 = false -> right_mode_val Had (hval b1 b2 r)
| right_phi: forall n rc r, right_mode_val (Phi n) (qval rc r).
Definition right_mode_env (aenv: var -> nat) (env: env) (st: posi -> val)
:= forall t p, snd p < aenv (fst p) -> Env.MapsTo (fst p) t env -> right_mode_val t (st p).
(* helper functions/lemmas for NOR states. *)
Definition nor_mode (f : posi -> val) (x:posi) : Prop :=
match f x with nval a b => True | _ => False end.
Definition nor_modes (f:posi -> val) (x:var) (n:nat) :=
forall i, i < n -> nor_mode f (x,i).
Lemma type_nor_mode : forall f aenv env p,
Env.MapsTo (fst p) Nor env -> snd p < aenv (fst p) -> right_mode_env aenv env f -> nor_mode f p.
Proof.
intros. unfold right_mode_env in *.
apply (H2 Nor) in H1 ; try easy.
inv H1. unfold nor_mode.
rewrite <- H3. easy.
Qed.
Lemma type_nor_modes : forall f aenv env x,
Env.MapsTo x Nor env -> right_mode_env aenv env f -> nor_modes f x (aenv x).
Proof.
intros. unfold right_mode_env in *.
unfold nor_modes. intros.
specialize (H1 Nor (x,i)).
simpl in H1. apply H1 in H2; try easy.
inv H2. unfold nor_mode.
assert ((@pair var nat x i) = (@pair Env.key nat x i)) by easy.
rewrite H2 in *.
rewrite <- H3. easy.
Qed.
Lemma nor_mode_nval : forall f x, nor_mode f x -> (exists r, f x = nval true r \/ f x = nval false r).
Proof.
intros. unfold nor_mode in *. destruct (f x); inv H0.
exists r.
destruct b. left. easy. right. easy.
Qed.
Lemma neq_sym {A} : forall (x y: A), x <> y -> y <> x.
Proof.
intros. intros R.
subst. contradiction.
Qed.
Lemma nor_mode_up : forall f x y v, x <> y -> nor_mode f x -> nor_mode (f[y |-> v]) x.
Proof.
intros. unfold nor_mode in *.
assert ((f [y |-> v]) x = (f x)).
rewrite eupdate_index_neq. reflexivity. apply neq_sym. assumption.
rewrite H2.
destruct (f x); inv H1. easy.
Qed.
Definition put_cu (v:val) (b:bool) :=
match v with nval x r => nval b r | a => a end.
Definition get_cua (v:val) :=
match v with nval x r => x | _ => false end.
Lemma get_cua_eq : forall f x v, nor_mode f x -> get_cua ((f[x |-> put_cu (f x) v]) x) = v.
Proof.
intros.
unfold get_cua.
rewrite eupdate_index_eq.
unfold put_cu.
unfold nor_mode in H0.
destruct (f x). easy. inv H0. inv H0.
Qed.
Lemma double_put_cu : forall (f:posi -> val) x v v', put_cu (put_cu (f x) v) v' = put_cu (f x) v'.
Proof.
intros.
unfold put_cu.
destruct (f x). easy. easy. easy.
Qed.
Definition get_cus (n:nat) (f:posi -> val) (x:var) :=
fun i => if i <? n then (match f (x,i) with nval b r => b | _ => false end) else allfalse i.
Lemma get_cus_cua : forall n f x m, m < n -> get_cus n f x m = get_cua (f (x,m)).
Proof.
intros.
unfold get_cus,get_cua.
bdestruct (m <? n).
destruct (f (x,n)). easy. easy. easy.
lia.
Qed.
Definition put_cus (f:posi -> val) (x:var) (g:nat -> bool) (n:nat) : (posi -> val) :=
fun a => if fst a =? x then if snd a <? n then put_cu (f a) (g (snd a)) else f a else f a.
Lemma cus_get_neq : forall (f:posi -> val) (x y :var) g n i,
x <> y -> get_cua ((put_cus f y g n) (x,i)) = get_cua (f (x,i)).
Proof.
intros.
unfold get_cua.
unfold put_cus.
simpl.
bdestruct (x =? y).
lia. easy.
Qed.
Lemma put_cus_out : forall (f:posi -> val) (x y :var) g n i,
n <= i -> ((put_cus f y g n) (x,i)) = (f (x,i)).
Proof.
intros.
unfold put_cus.
simpl.
bdestruct (x =? y).
bdestruct (i <? n). lia.
easy. easy.
Qed.
Lemma nor_mode_cus_eq: forall f x g n y i,
nor_mode f (y,i) -> nor_mode (put_cus f x g n) (y,i).
Proof.
intros. unfold nor_mode in *.
unfold put_cus.
simpl.
bdestruct (y =? x).
bdestruct (i <? n).
destruct (f (y, i)).
unfold put_cu. easy.
inv H0. inv H0.
apply H0. apply H0.
Qed.
Lemma cus_get_eq : forall (f:posi -> val) (x :var) g n i,
i < n -> nor_modes f x n -> get_cua ((put_cus f x g n) (x,i)) = g i.
Proof.
intros.
unfold get_cua.
unfold put_cus.
simpl.
bdestruct (x =? x).
bdestruct (i <? n).
unfold put_cu.
specialize (H1 i H3). unfold nor_mode in *.
destruct (f (x, i)) eqn:eq1. easy.
inv H1. inv H1.
lia. lia.
Qed.
Lemma put_cus_eq : forall (f:posi -> val) (x:var) g n i,
i < n -> (put_cus f x g n) (x,i) = put_cu (f (x,i)) (g i).
Proof.
intros.
unfold put_cus.
simpl.
bdestruct (x =? x).
bdestruct (i <? n). easy. lia. lia.
Qed.
Lemma put_cus_neq : forall (f:posi -> val) (x y:var) g n i,
x <> y -> (put_cus f x g n) (y,i) = f (y,i).
Proof.
intros.
unfold put_cus.
simpl.
bdestruct (y =? x). lia. easy.
Qed.
Lemma put_cus_neq_1 : forall (f:posi -> val) (x:var) g n c,
x <> fst c -> (put_cus f x g n) c = f c.
Proof.
intros.
destruct c. apply put_cus_neq.
simpl in H0. assumption.
Qed.
Lemma put_cus_neq_2 : forall (f:posi -> val) (x y:var) g n i,
n <= i -> (put_cus f x g n) (y,i) = f (y,i).
Proof.
intros.
unfold put_cus.
simpl.
bdestruct (y =? x).
bdestruct (i <? n). lia. easy.
easy.
Qed.
Lemma put_cu_cus : forall (f:posi -> val) x y g i n v, put_cu (put_cus f y g n (x,i)) v = put_cu (f (x,i)) v.
Proof.
intros.
unfold put_cus,put_cu.
simpl.
bdestruct (x =? y).
bdestruct (i <? n).
destruct (f (x,i)). easy. easy. easy. easy. easy.
Qed.
Lemma nor_mode_up_1 : forall f x v, nor_mode f x -> nor_mode (f[x |-> put_cu (f x) v]) x.
Proof.
intros.
unfold nor_mode in *.
rewrite eupdate_index_eq.
unfold put_cu.
destruct (f x).
easy. inv H0. inv H0.
Qed.
Lemma nor_mode_ups : forall f f' x v, f x = f' x -> nor_mode f x ->
nor_mode (f[x |-> put_cu (f' x) v]) x.
Proof.
intros. unfold nor_mode in *.
rewrite eupdate_index_eq.
unfold put_cu. rewrite <- H0.
destruct (f x); inv H1. easy.
Qed.
Lemma nor_mode_get : forall f x, nor_mode f x -> (exists b, get_cua (f x) = b).
Proof.
intros. unfold nor_mode in *. destruct (f x); inv H0.
exists b. unfold get_cua. reflexivity.
Qed.
Lemma put_get_cus_eq :
forall f x n, nor_modes f x n -> (put_cus f x (get_cus n f x) n) = f.
Proof.
intros.
unfold put_cus,get_cus,put_cu.
apply functional_extensionality. intros.
destruct x0. simpl.
bdestruct (v =? x). bdestruct (n0 <? n).
subst.
unfold nor_modes in H0.
specialize (H0 n0 H2) as eq1. unfold nor_mode in eq1.
destruct (f (x,n0)). easy. inv eq1. inv eq1.
easy. easy.
Qed.
Lemma get_cus_put_eq :
forall f x v n, v < 2^n -> nor_modes f x n -> get_cus n (put_cus f x (nat2fb v) n) x = (nat2fb v).
Proof.
intros.
unfold get_cus.
apply functional_extensionality. intros.
bdestruct (x0 <? n).
simpl.
unfold nor_modes in H0.
assert (nor_mode (put_cus f x (nat2fb v) n) (x, x0)).
apply nor_mode_cus_eq. apply H1. easy.
unfold nor_mode in H3.
destruct (put_cus f x ((nat2fb v)) n (x, x0)) eqn:eq2.
unfold put_cus in eq2. simpl in eq2.
bdestruct (x =? x).
bdestruct (x0 <? n).
unfold put_cu in eq2.
assert (nor_mode f (x,x0)).
apply H1. easy.
unfold nor_mode in H6.
destruct (f (x, x0)). inv eq2. easy. inv H6. inv H6. lia. lia.
inv H3. inv H3.
unfold allfalse.
rewrite nat2fb_bound with (n := n); try easy.
Qed.
Lemma put_cus_same : forall f x g n, nor_modes f x n
-> get_cus n f x = g -> put_cus f x g n = f.
Proof.
intros.
rewrite <- H1.
rewrite put_get_cus_eq. easy. easy.
Qed.
Lemma get_cus_put_neq :
forall f x y g n, x <> y -> get_cus n (put_cus f x g n) y = get_cus n f y.
Proof.
intros. unfold get_cus,put_cus.
apply functional_extensionality. intros.
simpl.
bdestruct ( y =? x). lia.
destruct (f (y, x0)). easy. easy. easy.
Qed.
Lemma get_put_cus_cut_n : forall n f x g, nor_modes f x n
-> (get_cus n (put_cus f x g n) x) = cut_n g n.
Proof.
intros. unfold get_cus,put_cus,cut_n.
apply functional_extensionality. intros.
bdestruct (x0 <? n). simpl.
bdestruct (x =? x).
bdestruct (x0 <? n).
unfold put_cu.
unfold nor_modes in H0.
specialize (H0 x0 H3). unfold nor_mode in H0.
destruct (f (x,x0)). easy. lia. lia.
lia. lia. easy.
Qed.
Definition get_cu (v : val) :=
match v with nval b r => Some b
| hval b1 b2 r => Some b1
| _ => None
end.
Lemma put_get_cu : forall f x, nor_mode f x -> put_cu (f x) (get_cua (f x)) = f x.
Proof.
intros. unfold put_cu, get_cua.
unfold nor_mode in H0. destruct (f x); inv H0.
reflexivity.
Qed.
Lemma get_put_cu : forall f x v, nor_mode f x -> get_cua (put_cu (f x) v) = v.
Proof.
intros. unfold put_cu, get_cua.
unfold nor_mode in H0. destruct (f x); inv H0.
reflexivity.
Qed.
Lemma get_cua_t : forall b r, get_cua (nval b r) = b.
Proof.
intros. unfold get_cua. reflexivity.
Qed.
(* Proofs of types and syntax. *)
Ltac nor_sym := try (apply neq_sym; assumption) ; try assumption.
Lemma iner_neq : forall (x y:var) (a b:nat), x <> y -> (x,a) <> (y,b).
Proof.
intros. intros R.
inv R. contradiction.
Qed.
Lemma iner_neq_1 : forall (x:var) (c:posi) a, x <> fst c -> (x,a) <> c.
Proof.
intros. intros R.
destruct c.
inv R. contradiction.
Qed.
Lemma iner_neq_2 : forall (x:var) (c:posi) a, x <> fst c -> c <> (x,a).
Proof.
intros. intros R.
destruct c.
inv R. contradiction.
Qed.
Ltac tuple_eq := intros R; inv R; lia.
Ltac iner_p := try nor_sym; try tuple_eq ; try (apply iner_neq ; assumption)
; try (apply iner_neq_1 ; assumption) ; try (apply iner_neq_2 ; assumption).
Lemma eupdates_twice_neq : forall f x g n c v, x <> fst c
-> (put_cus f x g n)[c |-> v] = put_cus (f[c |-> v]) x g n.
Proof.
intros. unfold put_cus.
apply functional_extensionality.
intros.
bdestruct (x0 ==? c).
subst.
rewrite eupdate_index_eq.
bdestruct (fst c =? x).
subst. contradiction.
rewrite eupdate_index_eq. easy.
rewrite eupdate_index_neq.
bdestruct (fst x0 =? x).
rewrite eupdate_index_neq.
easy. nor_sym.
rewrite eupdate_index_neq by nor_sym.
easy. nor_sym.
Qed.
Lemma get_cus_up : forall n f x c v, fst c <> x -> get_cus n (f[c |-> v]) x = get_cus n f x.
Proof.
intros.
apply functional_extensionality; intro.
unfold get_cus.
bdestruct (x0 <? n). destruct c. simpl in *. rewrite eupdate_index_neq by iner_p.
easy. easy.
Qed.
Lemma get_cus_up_ge : forall n f x c v, n <= snd c -> get_cus n (f[c |-> v]) x = get_cus n f x.
Proof.
intros.
apply functional_extensionality; intro.
unfold get_cus.
bdestruct (x0 <? n). destruct c. simpl in *. rewrite eupdate_index_neq by iner_p.
easy. easy.
Qed.
(*A function to get the rotation angle of a state. *)
Definition get_r (v:val) :=
match v with nval x r => r
| qval rc r => rc
| hval x y r => r
end.
Lemma get_r_put_same : forall (f:posi -> val) x y g n i,
get_r (put_cus f x g n (y,i)) = get_r (f (y,i)).
Proof.
intros.
unfold put_cus.
simpl.
bdestruct (y =? x).
bdestruct (i <? n).
unfold put_cu.
destruct (f (y, i)).
unfold get_r. easy. easy. easy. easy. easy.
Qed.
Lemma get_r_put_cu_same : forall (f:posi -> val) p v,
nor_mode f p ->
get_r (put_cu (f p) v) = get_r (f p).
Proof.
intros.
unfold put_cu,nor_mode in *.
destruct (f p). easy. easy. easy.
Qed.
Lemma put_cu_mid_eq : forall (f g:posi -> val) a v,
nor_mode f a -> nor_mode g a -> get_r (f a) = get_r (g a) -> (put_cu (f a) v) = (put_cu (g a) v).
Proof.
intros.
unfold put_cu. unfold nor_mode in *.
destruct (f a). destruct (g a).
unfold get_r in H2. subst.
easy. inv H1. inv H1.
inv H0. inv H0.
Qed.
Lemma put_cus_twice_neq : forall (f:posi -> val) (x y:var) g1 g2 n,
x <> y -> (put_cus (put_cus f x g1 n) y g2 n)
= (put_cus (put_cus f y g2 n) x g1 n).
Proof.
intros.
apply functional_extensionality.
unfold put_cus. intros.
destruct x0. simpl.
bdestruct (v =? y).
bdestruct (v =? x). lia. easy.
easy.
Qed.
Lemma put_cu_twice_eq : forall (f:posi -> val) (x:var) v1 v2 i,
put_cu (put_cu (f (x,i)) v1) v2 = put_cu (f (x,i)) v2.
Proof.
intros. unfold put_cu.
destruct (f (x, i)). easy. easy. easy.
Qed.
Lemma put_cu_twice_eq_1 : forall (f:posi -> val) c v1 v2,
put_cu (put_cu (f c) v1) v2 = put_cu (f c) v2.
Proof.
intros. unfold put_cu.
destruct (f c). easy. easy. easy.
Qed.
Lemma put_cus_twice_eq : forall (f:posi -> val) (x:var) g1 g2 n,
(put_cus (put_cus f x g1 n) x g2 n)
= (put_cus f x g2 n).
Proof.
intros.
apply functional_extensionality.
unfold put_cus. intros.
destruct x0. simpl.
bdestruct (v =? x).
bdestruct (n0 <? n). rewrite put_cu_twice_eq. easy.
easy.
easy.
Qed.
Lemma put_cus_sem_eq : forall (f:posi -> val) (x:var) g1 g2 n,
(forall m, m < n -> g1 m = g2 m) ->
(put_cus f x g1 n) = (put_cus f x g2 n).
Proof.
intros.
apply functional_extensionality.
unfold put_cus. intros.
destruct x0. simpl.
bdestruct (v =? x).
bdestruct (n0 <? n). rewrite H0. easy.
lia. easy. easy.
Qed.
(* Defining program semantic functions. *)
Definition rotate (r :rz_val) (q:nat) := addto r q.
Definition times_rotate (v : val) (q:nat) :=
match v with nval b r => if b then nval b (rotate r q) else nval b r
| hval b1 b2 r => hval b1 (¬ b2) r
| qval rc r => qval rc (rotate r q)
end.
Fixpoint sr_rotate' (st: posi -> val) (x:var) (n:nat) (size:nat) :=
match n with 0 => st
| S m => (sr_rotate' st x m size)[(x,m) |-> times_rotate (st (x,m)) (size - m)]
end.
Definition sr_rotate st x n := sr_rotate' st x (S n) (S n).
Definition r_rotate (r :rz_val) (q:nat) := addto_n r q.
Definition times_r_rotate (v : val) (q:nat) :=
match v with nval b r => if b then nval b (r_rotate r q) else nval b r
| hval b1 b2 r => hval b1 (¬ b2) r
| qval rc r => qval rc (r_rotate r q)
end.
Fixpoint srr_rotate' (st: posi -> val) (x:var) (n:nat) (size:nat) :=
match n with 0 => st
| S m => (srr_rotate' st x m size)[(x,m) |-> times_r_rotate (st (x,m)) (size - m)]
end.
Definition srr_rotate st x n := srr_rotate' st x (S n) (S n).
Definition exchange (v: val) :=
match v with nval b r => nval (¬ b) r
| hval b1 b2 r => hval b2 b1 r
| a => a
end.
Fixpoint lshift' (n:nat) (size:nat) (f:posi -> val) (x:var) :=
match n with 0 => f[(x,0) |-> f(x,size)]
| S m => ((lshift' m size f x)[ (x,n) |-> f(x,m)])
end.
Definition lshift (f:posi -> val) (x:var) (n:nat) := lshift' (n-1) (n-1) f x.
Fixpoint rshift' (n:nat) (size:nat) (f:posi -> val) (x:var) :=
match n with 0 => f[(x,size) |-> f(x,0)]
| S m => ((rshift' m size f x)[(x,m) |-> f (x,n)])
end.
Definition rshift (f:posi -> val) (x:var) (n:nat) := rshift' (n-1) (n-1) f x.
(*
Inductive varType := SType (n1:nat) (n2:nat).
Definition inter_env (enva: var -> nat) (x:var) :=
match (enva x) with SType n1 n2 => n1 + n2 end.
*)
Definition hexchange (v1:val) (v2:val) :=
match v1 with hval b1 b2 r1 =>
match v2 with hval b3 b4 r2 => if eqb b3 b4 then v1 else hval b1 (¬ b2) r1
| _ => v1
end
| _ => v1
end.
Definition reverse (f:posi -> val) (x:var) (n:nat) := fun (a: var * nat) =>
if ((fst a) =? x) && ((snd a) <? n) then f (x, (n-1) - (snd a)) else f a.
(* Semantic definition for H gate. *)
Definition h_case (v : val) :=
match v with nval b r => if r 0 then hval false b (rotate r 1) else hval true (¬ b) r
| hval true true r => nval false r
| hval true false r => nval true r
| hval false true r => nval true (rotate r 1)
| hval false false r => nval false (rotate r 1)
| a => a
end.
Fixpoint h_sem (f:posi -> val) (x:var) (n : nat) :=
match n with 0 => f
| S m => (h_sem f x m)[(x,m) |-> h_case (f (x,m))]
end.
(* Semantics function for QFT gate. *)
Definition seq_val (v:val) :=
match v with nval b r => b
| _ => false
end.