SfLibSoftware Foundations Library
(* $Date: 2013-07-17 16:19:11 -0400 (Wed, 17 Jul 2013) $ *)
Here we collect together several useful definitions and theorems
from Basics.v, List.v, Poly.v, Ind.v, and Logic.v that are not
already in the Coq standard library. From now on we can Import
or Export this file, instead of cluttering our environment with
all the examples and false starts in those files.
From the Coq Standard Library
Require Omega. (* needed for using the omega tactic *)
Require Export Bool.
Require Export List.
Export ListNotations.
Require Export Arith.
Require Export Arith.EqNat. (* Contains beq_nat, among other things *)
Definition admit {T: Type} : T. Admitted.
Require String. Open Scope string_scope.
Ltac move_to_top x :=
match reverse goal with
| H : _ ⊢ _ => try move x after H
end.
Tactic Notation "assert_eq" ident(x) constr(v) :=
let H := fresh in
assert (x = v) as H by reflexivity;
clear H.
Tactic Notation "Case_aux" ident(x) constr(name) :=
first [
set (x := name); move_to_top x
| assert_eq x name; move_to_top x
| fail 1 "because we are working on a different case" ].
Tactic Notation "Case" constr(name) := Case_aux Case name.
Tactic Notation "SCase" constr(name) := Case_aux SCase name.
Tactic Notation "SSCase" constr(name) := Case_aux SSCase name.
Tactic Notation "SSSCase" constr(name) := Case_aux SSSCase name.
Tactic Notation "SSSSCase" constr(name) := Case_aux SSSSCase name.
Tactic Notation "SSSSSCase" constr(name) := Case_aux SSSSSCase name.
Tactic Notation "SSSSSSCase" constr(name) := Case_aux SSSSSSCase name.
Tactic Notation "SSSSSSSCase" constr(name) := Case_aux SSSSSSSCase name.
Fixpoint ble_nat (n m : nat) : bool :=
match n with
| O => true
| S n' =>
match m with
| O => false
| S m' => ble_nat n' m'
end
end.
Theorem andb_true_elim1 : ∀b c,
andb b c = true → b = true.
Proof.
intros b c H.
destruct b.
Case "b = true".
reflexivity.
Case "b = false".
rewrite ← H. reflexivity. Qed.
Theorem andb_true_elim2 : ∀b c,
andb b c = true → c = true.
Proof.
(* An exercise in Basics.v *)
Admitted.
Theorem beq_nat_sym : ∀(n m : nat),
beq_nat n m = beq_nat m n.
(* An exercise in Lists.v *)
Admitted.
Inductive ev : nat → Prop :=
| ev_0 : ev O
| ev_SS : ∀n:nat, ev n → ev (S (S n)).
Theorem andb_true : ∀b c,
andb b c = true → b = true ∧ c = true.
Proof.
intros b c H.
destruct b.
destruct c.
apply conj. reflexivity. reflexivity.
inversion H.
inversion H. Qed.
Theorem false_beq_nat: ∀n n' : nat,
n <> n' →
beq_nat n n' = false.
Proof.
(* An exercise in Logic.v *)
Admitted.
Theorem ex_falso_quodlibet : ∀(P:Prop),
False → P.
Proof.
intros P contra.
inversion contra. Qed.
Theorem ev_not_ev_S : ∀n,
ev n → ~ ev (S n).
Proof.
(* An exercise in Logic.v *)
Admitted.
Theorem ble_nat_true : ∀n m,
ble_nat n m = true → n <= m.
(* An exercise in Logic.v *)
Admitted.
Theorem ble_nat_false : ∀n m,
ble_nat n m = false → ~(n <= m).
(* An exercise in Logic.v *)
Admitted.
Inductive appears_in (n : nat) : list nat → Prop :=
| ai_here : ∀l, appears_in n (n::l)
| ai_later : ∀m l, appears_in n l → appears_in n (m::l).
Inductive next_nat (n:nat) : nat → Prop :=
| nn : next_nat n (S n).
Inductive total_relation : nat → nat → Prop :=
tot : ∀n m : nat, total_relation n m.
Inductive empty_relation : nat → nat → Prop := .
Definition relation (X:Type) := X → X → Prop.
Definition deterministic {X: Type} (R: relation X) :=
∀x y1 y2 : X, R x y1 → R x y2 → y1 = y2.
Inductive multi (X:Type) (R: relation X)
: X → X → Prop :=
| multi_refl : ∀(x : X),
multi X R x x
| multi_step : ∀(x y z : X),
R x y →
multi X R y z →
multi X R x z.
Implicit Arguments multi [[X]].
Tactic Notation "multi_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "multi_refl" | Case_aux c "multi_step" ].
Theorem multi_R : ∀(X:Type) (R:relation X) (x y : X),
R x y → multi R x y.
Proof.
intros X R x y r.
apply multi_step with y. apply r. apply multi_refl. Qed.
Theorem multi_trans :
∀(X:Type) (R: relation X) (x y z : X),
multi R x y →
multi R y z →
multi R x z.
Proof.
(* FILL IN HERE *) Admitted.
Identifiers and polymorphic partial maps.
Inductive id : Type :=
Id : nat → id.
Theorem eq_id_dec : ∀id1 id2 : id, {id1 = id2} + {id1 <> id2}.
Proof.
intros id1 id2.
destruct id1 as [n1]. destruct id2 as [n2].
destruct (eq_nat_dec n1 n2) as [Heq | Hneq].
Case "n1 = n2".
left. rewrite Heq. reflexivity.
Case "n1 <> n2".
right. intros contra. inversion contra. apply Hneq. apply H0.
Defined.
Lemma eq_id : ∀(T:Type) x (p q:T),
(if eq_id_dec x x then p else q) = p.
Proof.
intros.
destruct (eq_id_dec x x); try reflexivity.
apply ex_falso_quodlibet; auto.
Qed.
Lemma neq_id : ∀(T:Type) x y (p q:T), x <> y →
(if eq_id_dec x y then p else q) = q.
Proof.
(* FILL IN HERE *) Admitted.
Definition partial_map (A:Type) := id → option A.
Definition empty {A:Type} : partial_map A := (fun _ => None).
Notation "'λempty'" := empty.
Definition extend {A:Type} (Γ : partial_map A) (x:id) (T : A) :=
fun x' => if eq_id_dec x x' then Some T else Γ x'.
Lemma extend_eq : ∀A (ctxt: partial_map A) x T,
(extend ctxt x T) x = Some T.
Proof.
intros. unfold extend. rewrite eq_id; auto.
Qed.
Lemma extend_neq : ∀A (ctxt: partial_map A) x1 T x2,
x2 <> x1 →
(extend ctxt x2 T) x1 = ctxt x1.
Proof.
intros. unfold extend. rewrite neq_id; auto.
Qed.
Lemma extend_shadow : ∀A (ctxt: partial_map A) t1 t2 x1 x2,
extend (extend ctxt x2 t1) x2 t2 x1 = extend ctxt x2 t2 x1.
Proof with auto.
intros. unfold extend. destruct (eq_id_dec x2 x1)...
Qed.
Tactic Notation "solve_by_inversion_step" tactic(t) :=
match goal with
| H : _ ⊢ _ => solve [ inversion H; subst; t ]
end
⇓ fail "because the goal is not solvable by inversion.".
Tactic Notation "solve" "by" "inversion" "1" :=
solve_by_inversion_step idtac.
Tactic Notation "solve" "by" "inversion" "2" :=
solve_by_inversion_step (solve by inversion 1).
Tactic Notation "solve" "by" "inversion" "3" :=
solve_by_inversion_step (solve by inversion 2).
Tactic Notation "solve" "by" "inversion" :=
solve by inversion 1.