fact stringlengths 9 10.6k | type stringclasses 19
values | library stringclasses 6
values | imports listlengths 0 12 | filename stringclasses 101
values | symbolic_name stringlengths 1 48 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
invH := inversion H; subst; clear H. | Ltac | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | inv | |
ceval_deterministic: forall c st st1 st2,
c / st \\ st1 ->
c / st \\ st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2;
generalize dependent st2;
induction E1; intros st2 E2; inv E2.
- (* E_Skip *) reflexivity.
- (* E_Ass *) reflexivity.
- (* E_Seq *)
assert (st' = st'0) as EQ1.
{ (... | Theorem | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | ceval_deterministic | |
auto_example_1: forall (P Q R: Prop),
(P -> Q) -> (Q -> R) -> P -> R.
Proof.
intros P Q R H1 H2 H3.
apply H2. apply H1. assumption.
Qed.
(** The [auto] tactic frees us from this drudgery by _searching_ for a
sequence of applications that will prove the goal *) | Example | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | auto_example_1 | |
auto_example_1': forall (P Q R: Prop),
(P -> Q) -> (Q -> R) -> P -> R.
Proof.
intros P Q R H1 H2 H3.
auto.
Qed.
(** The [auto] tactic solves goals that are solvable by any combination of
- [intros] and
- [apply] (of hypotheses from the local context, by default). *)
(** Using [auto] is always "safe" i... | Example | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | auto_example_1' | |
auto_example_2: forall P Q R S T U : Prop,
(P -> Q) ->
(P -> R) ->
(T -> R) ->
(S -> T -> U) ->
((P->Q) -> (P->S)) ->
T ->
P ->
U.
Proof. auto. Qed.
(** Proof search could, in principle, take an arbitrarily long time,
so there are limits to how far [auto] will search by default. *) | Example | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | auto_example_2 | |
auto_example_3: forall (P Q R S T U: Prop),
(P -> Q) ->
(Q -> R) ->
(R -> S) ->
(S -> T) ->
(T -> U) ->
P ->
U.
Proof.
auto.
auto 6.
Qed.
(** When searching for potential proofs of the current goal,
[auto] considers the hypotheses in the current context together
with a _hint database_ of oth... | Example | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | auto_example_3 | |
auto_example_4: forall P Q R : Prop,
Q ->
(Q -> R) ->
P \/ (Q /\ R).
Proof. auto. Qed.
(** We can extend the hint database just for the purposes of one
application of [auto] by writing [auto using ...]. *) | Example | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | auto_example_4 | |
le_antisym: forall n m: nat, (n <= m /\ m <= n) -> n = m.
Proof. intros. omega. Qed. | Lemma | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | le_antisym | |
auto_example_6: forall n m p : nat,
(n <= p -> (n <= m /\ m <= n)) ->
n <= p ->
n = m.
Proof.
intros.
auto. (* does nothing: auto doesn't destruct hypotheses! *)
auto using le_antisym.
Qed.
(** Of course, in any given development there will probably be
some specific constructors and lemmas that are use... | Example | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | auto_example_6 | |
auto_example_6': forall n m p : nat,
(n<= p -> (n <= m /\ m <= n)) ->
n <= p ->
n = m.
Proof.
intros.
auto. (* picks up hint from database *)
Qed. | Example | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | auto_example_6' | |
is_fortytwox := x = 42. | Definition | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | is_fortytwo | |
auto_example_7: forall x, (x <= 42 /\ 42 <= x) -> is_fortytwo x.
Proof.
auto. (* does nothing *)
Abort.
Hint Unfold is_fortytwo. | Example | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | auto_example_7 | |
auto_example_7': forall x, (x <= 42 /\ 42 <= x) -> is_fortytwo x.
Proof. auto. Qed.
(** Now let's take a first pass over [ceval_deterministic] to simplify
the proof script.
*) | Example | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | auto_example_7' | |
ceval_deterministic': forall c st st1 st2,
c / st \\ st1 ->
c / st \\ st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2.
generalize dependent st2;
induction E1; intros st2 E2; inv E2; auto.
- (* E_Seq *)
assert (st' = st'0) as EQ1 by auto.
subst st'0.
auto.
- (* E_IfTrue *)
... | Theorem | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | ceval_deterministic' | |
ceval_deterministic'_alt: forall c st st1 st2,
c / st \\ st1 ->
c / st \\ st2 ->
st1 = st2.
Proof with auto.
intros c st st1 st2 E1 E2;
generalize dependent st2;
induction E1;
intros st2 E2; inv E2...
- (* E_Seq *)
assert (st' = st'0) as EQ1...
subst st'0...
- (* E_IfTrue *)... | Theorem | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | ceval_deterministic'_alt | |
rwinvH1 H2 := rewrite H1 in H2; inv H2. | Ltac | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | rwinv | |
ceval_deterministic'': forall c st st1 st2,
c / st \\ st1 ->
c / st \\ st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2.
generalize dependent st2;
induction E1; intros st2 E2; inv E2; auto.
- (* E_Seq *)
assert (st' = st'0) as EQ1 by auto.
subst st'0.
auto.
- (* E_IfTrue *)
+... | Theorem | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | ceval_deterministic'' | |
find_rwinv:=
match goal with
H1: ?E = true,
H2: ?E = false
|- _ => rwinv H1 H2
end.
(** The [match goal] tactic looks for two distinct hypotheses that
have the form of equalities, with the same arbitrary expression
[E] on the left and with conflicting boolean values on the right.
If such hy... | Ltac | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | find_rwinv | |
ceval_deterministic''': forall c st st1 st2,
c / st \\ st1 ->
c / st \\ st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2.
generalize dependent st2;
induction E1; intros st2 E2; inv E2; try find_rwinv; auto.
- (* E_Seq *)
assert (st' = st'0) as EQ1 by auto.
subst st'0.
auto.
- (* ... | Theorem | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | ceval_deterministic''' | |
ceval_deterministic'''': forall c st st1 st2,
c / st \\ st1 ->
c / st \\ st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2.
generalize dependent st2;
induction E1; intros st2 E2; inv E2; try find_rwinv; auto.
- (* E_Seq *)
rewrite (IHE1_1 st'0 H1) in *. auto.
- (* E_WhileTrue *)
+ (* ... | Theorem | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | ceval_deterministic'''' | |
find_eqn:=
match goal with
H1: forall x, ?P x -> ?L = ?R,
H2: ?P ?X
|- _ => rewrite (H1 X H2) in *
end.
(** The pattern [forall x, ?P x -> ?L = ?R] matches any hypothesis of
the form "for all [x], _some property of [x]_ implies _some
equality_." The property of [x] is bound to the pattern vari... | Ltac | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | find_eqn | |
ceval_deterministic''''': forall c st st1 st2,
c / st \\ st1 ->
c / st \\ st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2.
generalize dependent st2;
induction E1; intros st2 E2; inv E2; try find_rwinv;
repeat find_eqn; auto.
Qed.
(** The big payoff in this approach is that our proof script... | Theorem | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | ceval_deterministic''''' | |
com: Type :=
| CSkip : com
| CAsgn : id -> aexp -> com
| CSeq : com -> com -> com
| CIf : bexp -> com -> com -> com
| CWhile : bexp -> com -> com
| CRepeat : com -> bexp -> com.
(** [REPEAT] behaves like [WHILE], except that the loop guard is
checked _after_ each execution of the body, with the loop
... | Inductive | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | com | |
ceval: state -> com -> state -> Prop :=
| E_Skip : forall st,
ceval st SKIP st
| E_Ass : forall st a1 n X,
aeval st a1 = n ->
ceval st (X ::= a1) (t_update st X n)
| E_Seq : forall c1 c2 st st' st'',
ceval st c1 st' ->
ceval st' c2 st'' ->
ceval st (c1 ; c2) st''
| E_IfTrue ... | Inductive | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | ceval | |
ceval_deterministic: forall c st st1 st2,
c / st \\ st1 ->
c / st \\ st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2.
generalize dependent st2;
induction E1;
intros st2 E2; inv E2; try find_rwinv; repeat find_eqn; auto.
- (* E_RepeatEnd *)
+ (* b evaluates to false (contradiction) *)
... | Theorem | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | ceval_deterministic | |
ceval_deterministic': forall c st st1 st2,
c / st \\ st1 ->
c / st \\ st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2.
generalize dependent st2;
induction E1;
intros st2 E2; inv E2; repeat find_eqn; try find_rwinv; auto.
Qed. | Theorem | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | ceval_deterministic' | |
ceval_example1:
(X ::= ANum 2;;
IFB BLe (AId X) (ANum 1)
THEN Y ::= ANum 3
ELSE Z ::= ANum 4
FI)
/ empty_state
\\ (t_update (t_update empty_state X 2) Z 4).
Proof.
apply E_Seq with (t_update empty_state X 2).
- apply E_Ass. reflexivity.
- apply E_IfFalse. reflexivity. apply E_Ass... | Example | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | ceval_example1 | |
ceval'_example1:
(X ::= ANum 2;;
IFB BLe (AId X) (ANum 1)
THEN Y ::= ANum 3
ELSE Z ::= ANum 4
FI)
/ empty_state
\\ (t_update (t_update empty_state X 2) Z 4).
Proof.
eapply E_Seq. (* 1 *)
- apply E_Ass. (* 2 *)
reflexivity. (* 3 *)
- (* 4 *) apply E_IfFalse. reflexivity. apply... | Example | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | ceval'_example1 | |
st12:= t_update (t_update empty_state X 1) Y 2. | Definition | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | st12 | |
st21:= t_update (t_update empty_state X 2) Y 1. | Definition | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | st21 | |
auto_example_8: exists s',
(IFB (BLe (AId X) (AId Y))
THEN (Z ::= AMinus (AId Y) (AId X))
ELSE (Y ::= APlus (AId X) (AId Z))
FI) / st21 \\ s'.
Proof. eauto. Qed.
(** The [eauto] tactic works just like [auto], except that it uses
[eapply] instead of [apply]. *) | Example | sf-experiment | [
"Require Import Coq.omega.Omega",
"Require Import Maps",
"Require Import Imp"
] | sf-experiment/Auto.v | auto_example_8 | |
day: Type :=
| monday : day
| tuesday : day
| wednesday : day
| thursday : day
| friday : day
| saturday : day
| sunday : day. | Inductive | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | day | |
next_weekday(d:day) : day :=
match d with
| monday => tuesday
| tuesday => wednesday
| wednesday => thursday
| thursday => friday
| friday => monday
| saturday => monday
| sunday => monday
end. | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | next_weekday | |
test_next_weekday:=
(next_weekday (next_weekday saturday)) = tuesday.
(* BCP: Needs an equality test.
QuickCheck test_next_weekday. *)
(* BCP: QC needs the native one. (Unless we make this Checkable somehow.) | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | test_next_weekday | |
bool: Type :=
| true : bool
| false : bool.
*) | Inductive | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | bool | |
negb(b:bool) : bool :=
match b with
| true => false
| false => true
end. | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | negb | |
andb(b1:bool) (b2:bool) : bool :=
match b1 with
| true => b2
| false => false
end. | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | andb | |
orb(b1:bool) (b2:bool) : bool :=
match b1 with
| true => true
| false => b2
end. | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | orb | |
nandb(b1:bool) (b2:bool) : bool
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *) := false. | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | nandb | |
test_nandb1:= (nandb true false). | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | test_nandb1 | |
minustwo(n : nat) : nat :=
match n with
| O => O
| S O => O
| S (S n') => n'
end. | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | minustwo | |
evenb(n:nat) : bool :=
match n with
| O => true
| S O => false
| S (S n') => evenb n'
end. | Fixpoint | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | evenb | |
oddb(n:nat) : bool := negb (evenb n). | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | oddb | |
test_oddb1:= oddb 1. | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | test_oddb1 | |
plus(n : nat) (m : nat) : nat :=
match n with
| O => m
| S n' => S (plus n' m)
end. | Fixpoint | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | plus | |
mult(n m : nat) : nat :=
match n with
| O => O
| S n' => plus m (mult n' m)
end. | Fixpoint | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | mult | |
test_mult1:= (mult 3 3) =? 9. | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | test_mult1 | |
minus(n m:nat) : nat :=
match n, m with
| O , _ => O
| S _ , O => n
| S n', S m' => minus n' m'
end. | Fixpoint | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | minus | |
exp(base power : nat) : nat :=
match power with
| O => S O
| S p => mult base (exp base p)
end. | Fixpoint | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | exp | |
factorial(n:nat) : nat
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *) := 0. | Fixpoint | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | factorial | |
test_factorial1:= (factorial 3) =? 6.
Notation "x + y" := (plus x y)
(at level 50, left associativity)
: nat_scope.
Notation "x - y" := (minus x y)
(at level 50, left associativity)
: nat_scope.
Notation "x * y" := (mu... | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | test_factorial1 | |
beq_nat(n m : nat) : bool :=
match n with
| O => match m with
| O => true
| S m' => false
end
| S n' => match m with
| O => false
| S m' => beq_nat n' m'
end
end. | Fixpoint | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | beq_nat | |
leb(n m : nat) : bool :=
match n with
| O => true
| S n' =>
match m with
| O => false
| S m' => leb n' m'
end
end.
Notation "'FORALLX' x : T , c" :=
(forAllShrink (@arbitrary T _) shrink (fun x => c))
(at level 200, x ident, T at level 200, c at level 200, right associativity
(* ... | Fixpoint | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | leb | |
plus_O_n:= FORALL n:nat, 0 + n =? n. | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | plus_O_n | |
bool_eq(x y : bool) : Dec (x = y).
constructor. unfold ssrbool.decidable. repeat (decide equality). Defined. | Instance | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | bool_eq | |
negb_involutive(b: bool) :=
(negb (negb b) = b)?.
Check negb_involutive.
QuickChick negb_involutive. | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | negb_involutive | |
negb_involutive2(b: bool) :=
Bool.eqb (negb (negb b)) b. | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | negb_involutive2 | |
andb_commutative:= fun b c => Bool.eqb (andb b c) (andb c b).
(* BCP: Don't know what to do with this one! | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | andb_commutative | |
identity_fn_applied_twice:
forall (f : bool -> bool),
(forall (x : bool), f x = x) ->
forall (b : bool), f (f b) = b.
*) | Theorem | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | identity_fn_applied_twice | |
andb_eq_orb:=
fun (b c : bool) =>
(Bool.eqb (andb b c) (orb b c))
==>
(Bool.eqb b c). | Definition | sf-experiment | [
"From QuickChick Require Import QuickChick",
"Require Import List ZArith",
"From mathcomp Require Import ssreflect ssrfun ssrbool",
"From mathcomp Require Import seq ssrnat eqtype"
] | sf-experiment/Basics.v | andb_eq_orb | |
aexp: Type :=
| ANum : nat -> aexp
| APlus : aexp -> aexp -> aexp
| AMinus : aexp -> aexp -> aexp
| AMult : aexp -> aexp -> aexp. | Inductive | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | aexp | |
bexp: Type :=
| BTrue : bexp
| BFalse : bexp
| BEq : aexp -> aexp -> bexp
| BLe : aexp -> aexp -> bexp
| BNot : bexp -> bexp
| BAnd : bexp -> bexp -> bexp.
(** In this chapter, we'll elide the translation from the
concrete syntax that a programmer would actually write to these
abstract syntax trees... | Inductive | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | bexp | |
aeval(a : aexp) : nat :=
match a with
| ANum n => n
| APlus a1 a2 => (aeval a1) + (aeval a2)
| AMinus a1 a2 => (aeval a1) - (aeval a2)
| AMult a1 a2 => (aeval a1) * (aeval a2)
end. | Fixpoint | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | aeval | |
test_aeval1:
aeval (APlus (ANum 2) (ANum 2)) = 4.
Proof. reflexivity. Qed. | Example | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | test_aeval1 | |
beval(b : bexp) : bool :=
match b with
| BTrue => true
| BFalse => false
| BEq a1 a2 => beq_nat (aeval a1) (aeval a2)
| BLe a1 a2 => leb (aeval a1) (aeval a2)
| BNot b1 => negb (beval b1)
| BAnd b1 b2 => andb (beval b1) (beval b2)
end.
(** We haven't defined very much yet, but we c... | Fixpoint | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | beval | |
optimize_0plus(a:aexp) : aexp :=
match a with
| ANum n =>
ANum n
| APlus (ANum 0) e2 =>
optimize_0plus e2
| APlus e1 e2 =>
APlus (optimize_0plus e1) (optimize_0plus e2)
| AMinus e1 e2 =>
AMinus (optimize_0plus e1) (optimize_0plus e2)
| AMult e1 e2 =>
AMult (optimize_0plus e1) (... | Fixpoint | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | optimize_0plus | |
test_optimize_0plus:
optimize_0plus (APlus (ANum 2)
(APlus (ANum 0)
(APlus (ANum 0) (ANum 1))))
= APlus (ANum 2) (ANum 1).
Proof. reflexivity. Qed.
(** But if we want to be sure the optimization is correct --
i.e., that evaluating an optimized expression g... | Example | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | test_optimize_0plus | |
optimize_0plus_sound: forall a,
aeval (optimize_0plus a) = aeval a.
Proof.
intros a. induction a.
- (* ANum *) reflexivity.
- (* APlus *) destruct a1.
+ (* a1 = ANum n *) destruct n.
* (* n = 0 *) simpl. apply IHa2.
* (* n <> 0 *) simpl. rewrite IHa2. reflexivity.
+ (* a1 = APlus a1_1 a1_2 ... | Theorem | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | optimize_0plus_sound | |
silly1: forall ae, aeval ae = aeval ae.
Proof. try reflexivity. (* this just does [reflexivity] *) Qed. | Theorem | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | silly1 | |
silly2: forall (P : Prop), P -> P.
Proof.
intros P HP.
try reflexivity. (* just [reflexivity] would have failed *)
apply HP. (* we can still finish the proof in some other way *)
Qed.
(** There is no real reason to use [try] in completely manual
proofs like these, but it is very useful for doing automated
... | Theorem | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | silly2 | |
foo: forall n, leb 0 n = true.
Proof.
intros.
destruct n.
- (* n=0 *) simpl. reflexivity.
- (* n=Sn' *) simpl. reflexivity.
Qed. | Lemma | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | foo | |
foo': forall n, leb 0 n = true.
Proof.
intros.
destruct n;
simpl;
reflexivity.
Qed.
(** Using [try] and [;] together, we can get rid of the repetition in
the proof that was bothering us a little while ago. *) | Lemma | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | foo' | |
optimize_0plus_sound': forall a,
aeval (optimize_0plus a) = aeval a.
Proof.
intros a.
induction a;
try (simpl; rewrite IHa1; rewrite IHa2; reflexivity).
(* ... but the remaining cases -- ANum and APlus --
are different: *)
- (* ANum *) reflexivity.
- (* APlus *)
destruct a1;
try (simp... | Theorem | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | optimize_0plus_sound' | |
optimize_0plus_sound'': forall a,
aeval (optimize_0plus a) = aeval a.
Proof.
intros a.
induction a;
try (simpl; rewrite IHa1; rewrite IHa2; reflexivity);
try reflexivity.
- (* APlus *)
destruct a1; try (simpl; simpl in IHa1; rewrite IHa1;
rewrite IHa2; reflexivity).
+ (* a1... | Theorem | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | optimize_0plus_sound'' | |
In10: In 10 [1;2;3;4;5;6;7;8;9;10].
Proof.
repeat (try (left; reflexivity); right).
Qed.
(** The tactic [repeat T] never fails: if the tactic [T] doesn't apply
to the original goal, then repeat still succeeds without changing
the original goal (i.e., it repeats zero times). *) | Theorem | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | In10 | |
In10': In 10 [1;2;3;4;5;6;7;8;9;10].
Proof.
repeat (left; reflexivity).
repeat (right; try (left; reflexivity)).
Qed.
(** The tactic [repeat T] also does not have any upper bound on the
number of times it applies [T]. If [T] is a tactic that always
succeeds, then repeat [T] will loop forever (e.g., [repea... | Theorem | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | In10' | |
optimize_0plus_b(b : bexp) : bexp
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted. | Fixpoint | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | optimize_0plus_b | |
optimize_0plus_b_sound: forall b,
beval (optimize_0plus_b b) = beval b.
Proof.
(* FILL IN HERE *) Admitted.
(** _Design exercise_: The optimization implemented by our
[optimize_0plus] function is only one of many possible
optimizations on arithmetic and boolean expressions. Write a more
sophisticated ... | Theorem | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | optimize_0plus_b_sound | |
silly_presburger_example: forall m n o p,
m + n <= n + o /\ o + 3 = p + 3 ->
m <= p.
Proof.
intros. omega.
Qed.
(** Finally, here are some miscellaneous tactics that you may find
convenient.
- [clear H]: Delete hypothesis [H] from the context.
- [subst x]: Find an assumption [x = e] or [e = x] ... | Example | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | silly_presburger_example | |
aevalR: aexp -> nat -> Prop :=
| E_ANum : forall (n: nat),
aevalR (ANum n) n
| E_APlus : forall (e1 e2: aexp) (n1 n2: nat),
aevalR e1 n1 ->
aevalR e2 n2 ->
aevalR (APlus e1 e2) (n1 + n2)
| E_AMinus: forall (e1 e2: aexp) (n1 n2: nat),
aevalR e1 n1 ->
aevalR e2 n2 ->
aeval... | Inductive | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | aevalR | |
aevalR: aexp -> nat -> Prop :=
| E_ANum : forall (n:nat),
(ANum n) \\ n
| E_APlus : forall (e1 e2: aexp) (n1 n2 : nat),
(e1 \\ n1) -> (e2 \\ n2) -> (APlus e1 e2) \\ (n1 + n2)
| E_AMinus : forall (e1 e2: aexp) (n1 n2 : nat),
(e1 \\ n1) -> (e2 \\ n2) -> (AMinus e1 e2) \\ (n1 - n2)
| E_AMult : f... | Inductive | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | aevalR | |
aeval_iff_aevalR: forall a n,
(a \\ n) <-> aeval a = n.
Proof.
split.
- (* -> *)
intros H.
induction H; simpl.
+ (* E_ANum *)
reflexivity.
+ (* E_APlus *)
rewrite IHaevalR1. rewrite IHaevalR2. reflexivity.
+ (* E_AMinus *)
rewrite IHaevalR1. rewrite IHaevalR2. reflexivity.
+ (* ... | Theorem | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | aeval_iff_aevalR | |
aeval_iff_aevalR': forall a n,
(a \\ n) <-> aeval a = n.
Proof.
split.
- (* -> *)
intros H; induction H; subst; reflexivity.
- (* <- *)
generalize dependent n.
induction a; simpl; intros; subst; constructor;
try apply IHa1; try apply IHa2; reflexivity.
Qed.
(** Write a relation [bevalR] in t... | Theorem | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | aeval_iff_aevalR' | |
bevalR: bexp -> bool -> Prop :=
. | Inductive | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | bevalR | |
beval_iff_bevalR: forall b bv,
bevalR b bv <-> beval b = bv.
Proof.
(* FILL IN HERE *) Admitted. | Lemma | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | beval_iff_bevalR | |
aexp: Type :=
| ANum : nat -> aexp
| APlus : aexp -> aexp -> aexp
| AMinus : aexp -> aexp -> aexp
| AMult : aexp -> aexp -> aexp
| ADiv : aexp -> aexp -> aexp. (* <--- new *)
(** Extending the definition of [aeval] to handle this new operation
would not be straightforward (what should we return as the ... | Inductive | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | aexp | |
aevalR: aexp -> nat -> Prop :=
| E_ANum : forall (n:nat),
(ANum n) \\ n
| E_APlus : forall (a1 a2: aexp) (n1 n2 : nat),
(a1 \\ n1) -> (a2 \\ n2) -> (APlus a1 a2) \\ (n1 + n2)
| E_AMinus : forall (a1 a2: aexp) (n1 n2 : nat),
(a1 \\ n1) -> (a2 \\ n2) -> (AMinus a1 a2) \\ (n1 - n2)
| E_AMult : f... | Inductive | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | aevalR | |
aexp: Type :=
| AAny : aexp (* <--- NEW *)
| ANum : nat -> aexp
| APlus : aexp -> aexp -> aexp
| AMinus : aexp -> aexp -> aexp
| AMult : aexp -> aexp -> aexp.
(** Again, extending [aeval] would be tricky, since now evaluation is
_not_ a deterministic function from expressions to number... | Inductive | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | aexp | |
aevalR: aexp -> nat -> Prop :=
| E_Any : forall (n:nat),
AAny \\ n (* <--- new *)
| E_ANum : forall (n:nat),
(ANum n) \\ n
| E_APlus : forall (a1 a2: aexp) (n1 n2 : nat),
(a1 \\ n1) -> (a2 \\ n2) -> (APlus a1 a2) \\ (n1 + n2)
| E_AMinus : forall (a1 a2: aexp) (n1 n2 : nat),
... | Inductive | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | aevalR | |
state:= total_map nat. | Definition | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | state | |
empty_state: state :=
t_empty 0.
(** We can add variables to the arithmetic expressions we had before by
simply adding one more constructor: *) | Definition | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | empty_state | |
aexp: Type :=
| ANum : nat -> aexp
| AId : id -> aexp (* <----- NEW *)
| APlus : aexp -> aexp -> aexp
| AMinus : aexp -> aexp -> aexp
| AMult : aexp -> aexp -> aexp.
(** Defining a few variable names as notational shorthands will make
examples easier to read: *) | Inductive | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | aexp | |
W: id := Id "W". | Definition | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | W | |
X: id := Id "X". | Definition | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | X | |
Y: id := Id "Y". | Definition | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | Y | |
Z: id := Id "Z".
(** (This convention for naming program variables ([X], [Y],
[Z]) clashes a bit with our earlier use of uppercase letters for
types. Since we're not using polymorphism heavily in the chapters
devoped to Imp, this overloading should not cause confusion.) *)
(** The definition of [bexp]s i... | Definition | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | Z | |
bexp: Type :=
| BTrue : bexp
| BFalse : bexp
| BEq : aexp -> aexp -> bexp
| BLe : aexp -> aexp -> bexp
| BNot : bexp -> bexp
| BAnd : bexp -> bexp -> bexp.
(** The arith and boolean evaluators are extended to handle
variables in the obvious way, taking a state as an extra
argument: *) | Inductive | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | bexp | |
aeval(st : state) (a : aexp) : nat :=
match a with
| ANum n => n
| AId x => st x (* <----- NEW *)
| APlus a1 a2 => (aeval st a1) + (aeval st a2)
| AMinus a1 a2 => (aeval st a1) - (aeval st a2)
| AMult a1 a2 => (aeval st a1) * (aeval st a2)
end. | Fixpoint | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | aeval | |
beval(st : state) (b : bexp) : bool :=
match b with
| BTrue => true
| BFalse => false
| BEq a1 a2 => beq_nat (aeval st a1) (aeval st a2)
| BLe a1 a2 => leb (aeval st a1) (aeval st a2)
| BNot b1 => negb (beval st b1)
| BAnd b1 b2 => andb (beval st b1) (beval st b2)
end. | Fixpoint | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | beval | |
aexp1:
aeval (t_update empty_state X 5)
(APlus (ANum 3) (AMult (AId X) (ANum 2)))
= 13.
Proof. reflexivity. Qed. | Example | sf-experiment | [
"Require Import Coq.Bool.Bool",
"Require Import Coq.Arith.Arith",
"Require Import Coq.Arith.EqNat",
"Require Import Coq.omega.Omega",
"Require Import Coq.Lists.List",
"Require Import Maps"
] | sf-experiment/Imp.v | aexp1 |
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