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|
(**************************************************************************)
(* *)
(* SMTCoq *)
(* Copyright (C) 2011 - 2022 *)
(* *)
(* See file "AUTHORS" for the list of authors *)
(* *)
(* This file is distributed under the terms of the CeCILL-C licence *)
(* *)
(**************************************************************************)
Add Rec LoadPath "../src" as SMTCoq.
Require Import SMTCoq.
Require Import Bool PArray Int63 List ZArith Logic.
Local Open Scope Z_scope.
Infix "-->" := implb (at level 60, right associativity) : bool_scope.
Section BV.
Import BVList.BITVECTOR_LIST.
Local Open Scope bv_scope.
Goal forall (a b c: bitvector 4),
(c = (bv_and a b)) ->
((bv_and (bv_and c a) b) = c).
Proof using.
smt.
Qed.
Goal forall (bv1 bv2 bv3: bitvector 4),
bv1 = #b|0|0|0|0| /\
bv2 = #b|1|0|0|0| /\
bv3 = #b|1|1|0|0| ->
bv_ultP bv1 bv2 /\ bv_ultP bv2 bv3.
Proof using.
smt.
Qed.
Goal forall (a: bitvector 32), a = a.
Proof using.
smt.
Qed.
Goal forall (bv1 bv2: bitvector 4),
bv1 = bv2 <-> bv2 = bv1.
Proof using.
smt.
Qed.
End BV.
Section Arrays.
Import FArray.
Local Open Scope farray_scope.
Goal forall (a:farray Z Z) i j, i = j -> a[i] = a[j].
Proof using.
smt.
Qed.
Goal forall (a b: farray Z Z)
(v w x y: Z)
(g: farray Z Z -> Z)
(f: Z -> Z),
(a[x <- v] = b) /\ a[y <- w] = b ->
(f x) = (f y) \/ (g a) = (g b).
Proof using.
smt.
Qed.
(* TODO *)
(* Goal forall (a b c d: farray Z Z), *)
(* b[0%Z <- 4] = c -> *)
(* d = b[0%Z <- 4][1%Z <- 4] -> *)
(* a = d[1%Z <- b[1%Z]] -> *)
(* a = c. *)
(* Proof using. *)
(* smt. *)
(* Qed. *)
(*
Goal forall (a b: farray Z Z) i,
(select (store (store (store a i 3%Z) 1%Z (select (store b i 4) i)) 2%Z 2%Z) 1%Z) = 4.
Proof using.
smt.
rewrite read_over_other_write; try easy.
rewrite read_over_same_write; try easy; try apply Z_compdec.
rewrite read_over_same_write; try easy; try apply Z_compdec.
Qed.
*)
End Arrays.
Section EUF.
Goal forall
(x y: Z)
(f: Z -> Z),
y = x -> (f x) = (f y).
Proof using.
smt.
Qed.
Goal forall
(x y: Z)
(f: Z -> Z),
(f x) = (f y) -> (f y) = (f x).
Proof using.
smt.
Qed.
Goal forall
(x y: Z)
(f: Z -> Z),
x + 1 = (y + 1) -> (f y) = (f x).
Proof using.
smt.
Qed.
Goal forall
(x y: Z)
(f: Z -> Z),
x = (y + 1) -> (f y) = (f (x - 1)).
Proof using.
smt.
Qed.
Goal forall
(x y: Z)
(f: Z -> Z),
x = (y + 1) -> (f (y + 1)) = (f x).
Proof using.
smt.
Qed.
End EUF.
Section LIA.
Goal forall (a b: Z), a = b <-> b = a.
Proof using.
smt.
Qed.
Goal forall (x y: Z), (x >= y) -> (y < x) \/ (x = y).
Proof using.
smt.
Qed.
Goal forall (f : Z -> Z) (a:Z), ((f a) > 1) -> ((f a) + 1) >= 2 \/((f a) = 30) .
Proof using.
smt.
Qed.
Goal forall x: Z, (x = 0%Z) -> (8 >= 0).
Proof using.
smt.
Qed.
Goal forall x: Z, ~ (x < 0%Z) -> (8 >= 0).
Proof using.
smt.
Qed.
Goal forall (a b: Z), a < b -> a < (b + 1).
Proof using.
smt.
Qed.
Goal forall (a b: Z), (a < b) -> (a + 2) < (b + 3).
Proof using.
smt.
Qed.
Goal forall a b, a < b -> a < (b + 10).
Proof using.
smt.
Qed.
End LIA.
Section PR.
Local Open Scope int63_scope.
(* Simple connectors *)
Goal forall (a:bool), a || negb a.
Proof using.
smt.
Qed.
Goal forall a, negb (a || negb a) = false.
Proof using.
smt.
Qed.
Goal forall a, (a && negb a) = false.
Proof using.
smt.
Qed.
Goal forall a, negb (a && negb a).
Proof using.
smt.
Qed.
Goal forall a, a --> a.
Proof using.
smt.
Qed.
Goal forall a, negb (a --> a) = false.
Proof using.
smt.
Qed.
Goal forall a , (xorb a a) || negb (xorb a a).
Proof using.
smt.
Qed.
Goal forall a, (a||negb a) || negb (a||negb a).
Proof using.
smt.
Qed.
(* Polarities *)
Goal forall a b, andb (orb (negb (negb a)) b) (negb (orb a b)) = false.
Proof using.
smt.
Qed.
Goal forall a b, andb (orb a b) (andb (negb a) (negb b)) = false.
Proof using.
smt.
Qed.
(* Multiple negations *)
Goal forall a, orb a (negb (negb (negb a))) = true.
Proof using.
smt.
Qed.
Goal forall a b c,
(a || b || c) && ((negb a) || (negb b) || (negb c)) && ((negb a) || b) && ((negb b) || c) && ((negb c) || a) = false.
Proof using.
smt.
Qed.
Goal true.
Proof using.
smt.
Qed.
Goal negb false.
Proof using.
smt.
Qed.
Goal forall a, Bool.eqb a a.
Proof using.
smt.
Qed.
Goal forall (a:bool), a = a.
Proof using.
smt.
Qed.
Goal (false || true) && false = false.
Proof using.
smt.
Qed.
Goal negb true = false.
Proof using.
smt.
Qed.
Goal false = false.
Proof using.
smt.
Qed.
Goal forall x y, Bool.eqb (xorb x y) ((x && (negb y)) || ((negb x) && y)).
Proof using.
smt.
Qed.
Goal forall x y, Bool.eqb (x --> y) ((x && y) || (negb x)).
Proof using.
smt.
Qed.
Goal forall x y z, Bool.eqb (ifb x y z) ((x && y) || ((negb x) && z)).
Proof using.
smt.
Qed.
Goal forall a b c, (((a && b) || (b && c)) && (negb b)) = false.
Proof using.
smt.
Qed.
Goal forall a, ((a || a) && (negb a)) = false.
Proof using.
smt.
Qed.
Goal forall a, (negb (a || (negb a))) = false.
Proof using.
smt.
Qed.
Goal forall a b c,
(a || b || c) && ((negb a) || (negb b) || (negb c)) && ((negb a) || b) && ((negb b) || c) && ((negb c) || a) = false.
Proof using.
smt.
Qed.
(** The same goal above with a, b and c are concrete terms *)
Goal forall i j k,
let a := i == j in
let b := j == k in
let c := k == i in
(a || b || c) && ((negb a) || (negb b) || (negb c)) && ((negb a) || b) && ((negb b) || c) && ((negb c) || a) = false.
Proof using.
smt.
Qed.
Goal forall a b c d, ((a && b) && (c || d) && (negb (c || (a && b && d)))) = false.
Proof using.
smt.
Qed.
Goal forall a b c d, (a && b && c && ((negb a) || (negb b) || d) && ((negb d) || (negb c))) = false.
Proof using.
smt.
Qed.
(* Pigeon hole: 4 holes, 5 pigeons *)
Goal forall x11 x12 x13 x14 x15 x21 x22 x23 x24 x25 x31 x32 x33 x34 x35 x41 x42 x43 x44 x45, (
(orb (negb x11) (negb x21)) &&
(orb (negb x11) (negb x31)) &&
(orb (negb x11) (negb x41)) &&
(orb (negb x21) (negb x31)) &&
(orb (negb x21) (negb x41)) &&
(orb (negb x31) (negb x41)) &&
(orb (negb x12) (negb x22)) &&
(orb (negb x12) (negb x32)) &&
(orb (negb x12) (negb x42)) &&
(orb (negb x22) (negb x32)) &&
(orb (negb x22) (negb x42)) &&
(orb (negb x32) (negb x42)) &&
(orb (negb x13) (negb x23)) &&
(orb (negb x13) (negb x33)) &&
(orb (negb x13) (negb x43)) &&
(orb (negb x23) (negb x33)) &&
(orb (negb x23) (negb x43)) &&
(orb (negb x33) (negb x43)) &&
(orb (negb x14) (negb x24)) &&
(orb (negb x14) (negb x34)) &&
(orb (negb x14) (negb x44)) &&
(orb (negb x24) (negb x34)) &&
(orb (negb x24) (negb x44)) &&
(orb (negb x34) (negb x44)) &&
(orb (negb x15) (negb x25)) &&
(orb (negb x15) (negb x35)) &&
(orb (negb x15) (negb x45)) &&
(orb (negb x25) (negb x35)) &&
(orb (negb x25) (negb x45)) &&
(orb (negb x35) (negb x45)) &&
(orb (negb x11) (negb x12)) &&
(orb (negb x11) (negb x13)) &&
(orb (negb x11) (negb x14)) &&
(orb (negb x11) (negb x15)) &&
(orb (negb x12) (negb x13)) &&
(orb (negb x12) (negb x14)) &&
(orb (negb x12) (negb x15)) &&
(orb (negb x13) (negb x14)) &&
(orb (negb x13) (negb x15)) &&
(orb (negb x14) (negb x15)) &&
(orb (negb x21) (negb x22)) &&
(orb (negb x21) (negb x23)) &&
(orb (negb x21) (negb x24)) &&
(orb (negb x21) (negb x25)) &&
(orb (negb x22) (negb x23)) &&
(orb (negb x22) (negb x24)) &&
(orb (negb x22) (negb x25)) &&
(orb (negb x23) (negb x24)) &&
(orb (negb x23) (negb x25)) &&
(orb (negb x24) (negb x25)) &&
(orb (negb x31) (negb x32)) &&
(orb (negb x31) (negb x33)) &&
(orb (negb x31) (negb x34)) &&
(orb (negb x31) (negb x35)) &&
(orb (negb x32) (negb x33)) &&
(orb (negb x32) (negb x34)) &&
(orb (negb x32) (negb x35)) &&
(orb (negb x33) (negb x34)) &&
(orb (negb x33) (negb x35)) &&
(orb (negb x34) (negb x35)) &&
(orb (negb x41) (negb x42)) &&
(orb (negb x41) (negb x43)) &&
(orb (negb x41) (negb x44)) &&
(orb (negb x41) (negb x45)) &&
(orb (negb x42) (negb x43)) &&
(orb (negb x42) (negb x44)) &&
(orb (negb x42) (negb x45)) &&
(orb (negb x43) (negb x44)) &&
(orb (negb x43) (negb x45)) &&
(orb (negb x44) (negb x45)) &&
(orb (orb (orb x11 x21) x31) x41) &&
(orb (orb (orb x12 x22) x32) x42) &&
(orb (orb (orb x13 x23) x33) x43) &&
(orb (orb (orb x14 x24) x34) x44) &&
(orb (orb (orb x15 x25) x35) x45)) = false.
Proof using.
smt.
Qed.
Goal forall a b c f p, ((Z.eqb a c) && (Z.eqb b c) && ((negb (Z.eqb (f a) (f b))) || ((p a) && (negb (p b))))) = false.
Proof using.
smt.
Qed.
Goal forall a b c (p : Z -> bool), ((((p a) && (p b)) || ((p b) && (p c))) && (negb (p b))) = false.
Proof using.
smt.
Qed.
Goal forall x y z f, ((Z.eqb x y) && (Z.eqb y z) && (negb (Z.eqb (f x) (f z)))) = false.
Proof using.
smt.
Qed.
Goal forall x y z f, ((negb (Z.eqb (f x) (f y))) && (Z.eqb y z) && (Z.eqb (f x) (f (f z))) && (Z.eqb x y)) = false.
Proof using.
smt.
Qed.
Goal forall a b c d e f, ((Z.eqb a b) && (Z.eqb b c) && (Z.eqb c d) && (Z.eqb c e) && (Z.eqb e f) && (negb (Z.eqb a f))) = false.
Proof using.
smt.
Qed.
Goal forall x y z, ((x <=? 3) && ((y <=? 7) || (z <=? 9)))%Z -->
((x + y <=? 10) || (x + z <=? 12))%Z = true.
Proof using.
smt.
Qed.
Goal forall x, (Z.eqb (x - 3) 7) --> (x >=? 10) = true.
Proof using.
smt.
Qed.
Goal forall x y, (x >? y)%Z --> (y + 1 <=? x)%Z = true.
Proof using.
smt.
Qed.
Goal forall x y, Bool.eqb (x <? y)%Z (x <=? y - 1)%Z = true.
Proof using.
smt.
Qed.
Goal forall x y, ((x + y <=? - (3)) && (y >=? 0)
|| (x <=? - (3)))%Z && (x >=? 0) = false.
Proof using.
smt.
Qed.
Goal forall x, (andb ((x - 3) <=? 7) (7 <=? (x - 3)))%Z --> (x >=? 10) = true.
Proof using.
smt.
Qed.
Goal forall x, (Z.eqb (x - 3) 7)%Z --> (10 <=? x)%Z = true.
Proof using.
smt.
Qed.
(* More general examples *)
Goal forall a b c, ((a || b || c) && ((negb a) || (negb b) || (negb c)) && ((negb a) || b) && ((negb b) || c) && ((negb c) || a)) = false.
Proof using.
smt.
Qed.
Goal forall (a b : Z) (P : Z -> bool) (f : Z -> Z),
(negb (Z.eqb (f a) b)) || (negb (P (f a))) || (P b).
Proof using.
smt.
Qed.
Goal forall b1 b2 x1 x2,
(ifb b1
(ifb b2 (Z.eqb (2*x1+1) (2*x2+1)) (Z.eqb (2*x1+1) (2*x2)))
(ifb b2 (Z.eqb (2*x1) (2*x2+1)) (Z.eqb (2*x1) (2*x2)))) -->
((b1 --> b2) && (b2 --> b1) && (Z.eqb x1 x2)).
Proof using.
smt.
Qed.
(* With let ... in ... *)
Goal forall b,
let a := b in
a && (negb a) = false.
Proof using.
smt.
Qed.
Goal forall b,
let a := b in
a || (negb a) = true.
Proof using.
smt.
Qed.
(* With concrete terms *)
Goal forall i j,
let a := i == j in
a && (negb a) = false.
Proof using.
smt.
Qed.
Goal forall i j,
let a := i == j in
a || (negb a) = true.
Proof using.
smt.
Qed.
(* Congruence in which some premises are REFL *)
Goal forall (f:Z -> Z -> Z) x y z,
(Z.eqb x y) --> (Z.eqb (f z x) (f z y)).
Proof using.
smt.
Qed.
Goal forall (f:Z -> Z -> Z) x y z,
(x = y) -> (f z x) = (f z y).
Proof using.
smt.
Qed.
Goal forall (P:Z -> Z -> bool) x y z,
(Z.eqb x y) --> ((P z x) --> (P z y)).
Proof using.
smt.
Qed.
End PR.
Section A_BV_EUF_LIA_PR.
Import BVList.BITVECTOR_LIST FArray.
Local Open Scope farray_scope.
Local Open Scope bv_scope.
(* TODO *)
(* Goal forall (bv1 bv2 : bitvector 4) *)
(* (a b c d : farray (bitvector 4) Z), *)
(* bv1 = #b|0|0|0|0| -> *)
(* bv2 = #b|1|0|0|0| -> *)
(* c = b[bv1 <- 4] -> *)
(* d = b[bv1 <- 4][bv2 <- 4] -> *)
(* a = d[bv2 <- b[bv2]] -> *)
(* a = c. *)
(* Proof using. *)
(* smt. *)
(* Qed. *)
(* TODO *)
(* Goal forall (bv1 bv2 : bitvector 4) *)
(* (a b c d : farray (bitvector 4) Z), *)
(* bv1 = #b|0|0|0|0| /\ *)
(* bv2 = #b|1|0|0|0| /\ *)
(* c = b[bv1 <- 4] /\ *)
(* d = b[bv1 <- 4][bv2 <- 4] /\ *)
(* a = d[bv2 <- b[bv2]] -> *)
(* a = c. *)
(* Proof using. *)
(* smt. *)
(* Qed. *)
(** the example in the CAV paper *)
Goal forall (a b: farray Z Z) (v w x y: Z)
(r s: bitvector 4)
(f: Z -> Z)
(g: farray Z Z -> Z)
(h: bitvector 4 -> Z),
a[x <- v] = b /\ a[y <- w] = b ->
r = s /\ h r = v /\ h s = y ->
v < x + 1 /\ v > x - 1 ->
f (h r) = f (h s) \/ g a = g b.
Proof using.
smt. (** "cvc4. verit." also solves the goal *)
Qed.
(** the example in the FroCoS paper *)
Goal forall (a b: farray Z Z) (v w x y z t: Z)
(r s: bitvector 4)
(f: Z -> Z)
(g: farray Z Z -> Z)
(h: bitvector 4 -> Z),
a[x <- v] = b /\ a[y <- w] = b ->
a[z <- w] = b /\ a[t <- v] = b ->
r = s -> v < x + 10 /\ v > x - 5 ->
~ (g a = g b) \/ f (h r) = f (h s).
Proof using.
smt. (** "cvc4. verit." also solves the goal *)
Qed.
Goal forall (a b: farray (bitvector 4) Z)
(x y: bitvector 4)
(v: Z),
b[bv_add y x <- v] = a /\
b = a[bv_add x y <- v] ->
a = b.
Proof using.
smt.
(* CVC4 preprocessing hole on BV *)
replace (bv_add x y) with (bv_add y x)
by apply bv_eq_reflect, bv_add_comm.
verit.
Qed.
Goal forall (a:farray Z Z), a = a.
Proof using.
smt.
Qed.
Goal forall (a b: farray Z Z), a = b <-> b = a.
Proof using.
smt.
Qed.
(* Testing bit-vectors substraction: this is a syntactic check that it
is correctly exported to SMT-LIB, CVC4-1.6 does not provide useful
proofs anyway. *)
Goal forall (x: bitvector 1), bv_subt x #b|0| = x.
Proof using.
smt.
Abort.
(* The original issue (invalid) *)
Goal forall (x: bitvector 1), bv_subt (bv_shl #b|0| x) #b|0| = #b|0|.
Proof using.
smt.
Abort.
End A_BV_EUF_LIA_PR.
(* Example of the webpage *)
Section Group.
Variable G : Type.
(* We suppose that G has a decidable equality *)
Variable HG : CompDec G.
Variable op : G -> G -> G.
Variable inv : G -> G.
Variable e : G.
Local Notation "a ==? b" := (@eqb_of_compdec G HG a b) (at level 60).
(* We can prove automatically that we have a group if we only have the
"left" versions of the axioms of a group *)
Hypothesis associative :
forall a b c : G, op a (op b c) ==? op (op a b) c.
Hypothesis inverse :
forall a : G, op (inv a) a ==? e.
Hypothesis identity :
forall a : G, op e a ==? a.
Add_lemmas associative inverse identity.
(* The "right" version of inverse *)
Lemma inverse' :
forall a : G, op a (inv a) ==? e.
Proof. smt. Qed.
(* The "right" version of identity *)
Lemma identity' :
forall a : G, op a e ==? a.
Proof. smt inverse'. Qed.
(* Some other interesting facts about groups *)
Lemma unique_identity e':
(forall z, op e' z ==? z) -> e' ==? e.
Proof. smt. Qed.
Lemma simplification_right x1 x2 y:
op x1 y ==? op x2 y -> x1 ==? x2.
Proof. smt_no_check inverse'. Qed.
Lemma simplification_left x1 x2 y:
op y x1 ==? op y x2 -> x1 ==? x2.
Proof. smt_no_check inverse'. Qed.
Clear_lemmas.
End Group.
Section EqualityOnUninterpretedType1.
Variable A : Type.
Hypothesis HA : CompDec A.
Goal forall (f : A -> Z) (a b : A), a = b -> f a = f b.
Proof. cvc4. Qed.
Goal forall (f : A -> Z) (a b : A), a = b -> f a = f b.
Proof. smt. Qed.
End EqualityOnUninterpretedType1.
Section EqualityOnUninterpretedType2.
Variable A B : Type.
Hypothesis HA : CompDec A.
Hypothesis HB : CompDec B.
Goal forall (f : A -> Z) (a b : A), a = b -> f a = f b.
Proof. cvc4. Qed.
Goal forall (f : A -> Z) (a b : A), a = b -> f a = f b.
Proof. smt. Qed.
Goal forall (f : Z -> B) (a b : Z), a = b -> f a = f b.
Proof. cvc4. Qed.
Goal forall (f : Z -> B) (a b : Z), a = b -> f a = f b.
Proof. smt. Qed.
Goal forall (f : A -> B) (a b : A), a = b -> f a = f b.
Proof. cvc4. Qed.
Goal forall (f : A -> B) (a b : A), a = b -> f a = f b.
Proof. smt. Qed.
End EqualityOnUninterpretedType2.
Section EqualityOnUninterpretedType3.
Variable A B : Type.
Goal forall (f : A -> Z) (a b : A), a = b -> f a = f b.
Proof. cvc4. Abort.
Goal forall (f : A -> Z) (a b : A), a = b -> f a = f b.
Proof. smt. Abort.
Goal forall (f : Z -> B) (a b : Z), a = b -> f a = f b.
Proof. cvc4. Abort.
Goal forall (f : Z -> B) (a b : Z), a = b -> f a = f b.
Proof. smt. Abort.
Goal forall (f : A -> B) (a b : A), a = b -> f a = f b.
Proof. cvc4. Abort.
Goal forall (f : A -> B) (a b : A), a = b -> f a = f b.
Proof. smt. Abort.
Goal forall (f : A -> A -> B) (a b c d : A), a = b -> c = d -> f a c = f b d.
Proof. cvc4. Abort.
Goal forall (f : A -> A -> B) (a b c d : A), a = b -> c = d -> f a c = f b d.
Proof. smt. Abort.
End EqualityOnUninterpretedType3.
Section Issue17.
Variable A : Type.
Variable cd : CompDec A.
Goal forall (a:A), a = a.
Proof. smt. Qed.
End Issue17.
(* TODO *)
(* From cvc4_bool : Uncaught exception Not_found *)
(* Goal forall (a b c d: farray Z Z), *)
(* b[0 <- 4] = c -> *)
(* d = b[0 <- 4][1 <- 4] -> *)
(* a = d[1 <- b[1]] -> *)
(* a = c. *)
(* Proof. *)
(* smt. *)
(* Qed. *)
(*
Local Variables:
coq-load-path: ((rec "../src" "SMTCoq"))
End:
*)
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