Library Ssreflect.ssreflect
Require Import Bool.
Declare ML Module "ssreflect".
Global Set Automatic Coercions Import.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Module SsrSyntax.
Reserved Notation "(* x 'is' y 'by' z 'of' // /= //= *)" (at level 8).
Reserved Notation "(* 69 *)" (at level 69).
End SsrSyntax.
Export SsrSyntax.
Delimit Scope general_if_scope with GEN_IF.
Notation "'if' c 'then' v1 'else' v2" :=
(if c then v1 else v2)
(at level 200, c, v1, v2 at level 200, only parsing) : general_if_scope.
Notation "'if' c 'return' t 'then' v1 'else' v2" :=
(if c return t then v1 else v2)
(at level 200, c, t, v1, v2 at level 200, only parsing) : general_if_scope.
Notation "'if' c 'as' x 'return' t 'then' v1 'else' v2" :=
(if c as x return t then v1 else v2)
(at level 200, c, t, v1, v2 at level 200, x ident, only parsing)
: general_if_scope.
Delimit Scope boolean_if_scope with BOOL_IF.
Notation "'if' c 'return' t 'then' v1 'else' v2" :=
(if c%bool is true in bool return t then v1 else v2) : boolean_if_scope.
Notation "'if' c 'then' v1 'else' v2" :=
(if c%bool is true in bool return _ then v1 else v2) : boolean_if_scope.
Notation "'if' c 'as' x 'return' t 'then' v1 'else' v2" :=
(if c%bool is true as x in bool return t then v1 else v2) : boolean_if_scope.
Open Scope boolean_if_scope.
Delimit Scope form_scope with FORM.
Open Scope form_scope.
Notation "x : T" := (x : T)
(at level 100, right associativity,
format "'[hv' x '/ ' : T ']'") : core_scope.
Notation "T : 'Type'" := (T%type : Type)
(at level 100, only parsing) : core_scope.
Notation "P : 'Prop'" := (P%type : Prop)
(at level 100, only parsing) : core_scope.
Module TheCanonical.
CoInductive put vT sT (v1 v2 : vT) (s : sT) : Type := Put.
Definition get vT sT v s (p : @put vT sT v v s) := let: Put := p in s.
Definition get_by vT sT of sT -> vT := @get vT sT.
End TheCanonical.
Import TheCanonical.
Notation "[ 'the' sT 'of' v 'by' f ]" :=
(@get_by _ sT f _ _ ((fun v' (s : sT) => Put v' (f s) s) v _))
(at level 0, only parsing) : form_scope.
Notation "[ 'the' sT 'of' v ]" := (get ((fun s : sT => Put v s s) _))
(at level 0, only parsing) : form_scope.
Notation "[ 'th' 'e' sT 'of' v 'by' f ]" := (@get_by _ sT f v _ _)
(at level 0, format "[ 'th' 'e' sT 'of' v 'by' f ]") : form_scope.
Notation "[ 'th' 'e' sT 'of' v ]" := (@get _ sT v _ _)
(at level 0, format "[ 'th' 'e' sT 'of' v ]") : form_scope.
Definition argumentType T P & forall x : T, P x := T.
Definition dependentReturnType T P & forall x : T, P x := P.
Definition returnType aT rT & aT -> rT := rT.
Notation "{ 'type' 'of' c 'for' s }" := (dependentReturnType c s)
(at level 0, format "{ 'type' 'of' c 'for' s }") : type_scope.
CoInductive phantom (T : Type) (p : T) : Type := Phantom.
Implicit Arguments phantom [].
Implicit Arguments Phantom [].
CoInductive phant (p : Type) : Type := Phant.
Definition protect_term (A : Type) (x : A) : A := x.
Declare ML Module "ssreflect".
Global Set Automatic Coercions Import.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Module SsrSyntax.
Reserved Notation "(* x 'is' y 'by' z 'of' // /= //= *)" (at level 8).
Reserved Notation "(* 69 *)" (at level 69).
End SsrSyntax.
Export SsrSyntax.
Delimit Scope general_if_scope with GEN_IF.
Notation "'if' c 'then' v1 'else' v2" :=
(if c then v1 else v2)
(at level 200, c, v1, v2 at level 200, only parsing) : general_if_scope.
Notation "'if' c 'return' t 'then' v1 'else' v2" :=
(if c return t then v1 else v2)
(at level 200, c, t, v1, v2 at level 200, only parsing) : general_if_scope.
Notation "'if' c 'as' x 'return' t 'then' v1 'else' v2" :=
(if c as x return t then v1 else v2)
(at level 200, c, t, v1, v2 at level 200, x ident, only parsing)
: general_if_scope.
Delimit Scope boolean_if_scope with BOOL_IF.
Notation "'if' c 'return' t 'then' v1 'else' v2" :=
(if c%bool is true in bool return t then v1 else v2) : boolean_if_scope.
Notation "'if' c 'then' v1 'else' v2" :=
(if c%bool is true in bool return _ then v1 else v2) : boolean_if_scope.
Notation "'if' c 'as' x 'return' t 'then' v1 'else' v2" :=
(if c%bool is true as x in bool return t then v1 else v2) : boolean_if_scope.
Open Scope boolean_if_scope.
Delimit Scope form_scope with FORM.
Open Scope form_scope.
Notation "x : T" := (x : T)
(at level 100, right associativity,
format "'[hv' x '/ ' : T ']'") : core_scope.
Notation "T : 'Type'" := (T%type : Type)
(at level 100, only parsing) : core_scope.
Notation "P : 'Prop'" := (P%type : Prop)
(at level 100, only parsing) : core_scope.
Module TheCanonical.
CoInductive put vT sT (v1 v2 : vT) (s : sT) : Type := Put.
Definition get vT sT v s (p : @put vT sT v v s) := let: Put := p in s.
Definition get_by vT sT of sT -> vT := @get vT sT.
End TheCanonical.
Import TheCanonical.
Notation "[ 'the' sT 'of' v 'by' f ]" :=
(@get_by _ sT f _ _ ((fun v' (s : sT) => Put v' (f s) s) v _))
(at level 0, only parsing) : form_scope.
Notation "[ 'the' sT 'of' v ]" := (get ((fun s : sT => Put v s s) _))
(at level 0, only parsing) : form_scope.
Notation "[ 'th' 'e' sT 'of' v 'by' f ]" := (@get_by _ sT f v _ _)
(at level 0, format "[ 'th' 'e' sT 'of' v 'by' f ]") : form_scope.
Notation "[ 'th' 'e' sT 'of' v ]" := (@get _ sT v _ _)
(at level 0, format "[ 'th' 'e' sT 'of' v ]") : form_scope.
Definition argumentType T P & forall x : T, P x := T.
Definition dependentReturnType T P & forall x : T, P x := P.
Definition returnType aT rT & aT -> rT := rT.
Notation "{ 'type' 'of' c 'for' s }" := (dependentReturnType c s)
(at level 0, format "{ 'type' 'of' c 'for' s }") : type_scope.
CoInductive phantom (T : Type) (p : T) : Type := Phantom.
Implicit Arguments phantom [].
Implicit Arguments Phantom [].
CoInductive phant (p : Type) : Type := Phant.
Definition protect_term (A : Type) (x : A) : A := x.
Term tagging (user-level).
Notation "'nosimpl' t" := (let: tt := tt in t) (at level 10, t at level 8).
Structure unlockable (T : Type) (v : T) : Type :=
Unlockable {unlocked : T; _ : unlocked = v}.
Lemma unlock : forall aT rT (f : forall x : aT, rT x) (u : unlockable f) x,
unlocked u x = f x.
Proof. move=> aT rT f [u /= ->]; split. Qed.
Lemma master_key : unit. Proof. exact tt. Qed.
Definition locked A := let: tt := master_key in fun x : A => x.
Lemma lock : forall A x, x = locked x :> A.
Proof. rewrite /locked; case master_key; split. Qed.
Lemma not_locked_false_eq_true : locked false <> true.
Proof. rewrite -lock; discriminate. Qed.
Ltac done :=
trivial; hnf; intros; solve
[ do ![solve [trivial | apply: sym_equal; trivial]
| discriminate | contradiction | split]
| case not_locked_false_eq_true; assumption
| match goal with H : ~ _ |- _ => solve [case H; trivial] end ].
Definition ssr_have Plemma Pgoal (step : Plemma) rest : Pgoal := rest step.
Implicit Arguments ssr_have [Pgoal].
Definition ssr_suff Plemma Pgoal step (rest : Plemma) : Pgoal := step rest.
Implicit Arguments ssr_suff [Pgoal].
Definition ssr_wlog := ssr_suff.
Implicit Arguments ssr_wlog [Pgoal].
Fixpoint nary_congruence_statement (n : nat)
: (forall B, (B -> B -> Prop) -> Prop) -> Prop :=
match n with
| O => fun k => forall B, k B (fun x1 x2 : B => x1 = x2)
| S n' =>
let k' A B e (f1 f2 : A -> B) :=
forall x1 x2, x1 = x2 -> (e (f1 x1) (f2 x2) : Prop) in
fun k => forall A, nary_congruence_statement n' (fun B e => k _ (k' A B e))
end.
Lemma nary_congruence : forall n : nat,
let k B e := forall y : B, (e y y : Prop) in nary_congruence_statement n k.
Proof.
move=> n k; have: k _ _ := _; rewrite {1}/k.
elim: n k => [|n IHn] k Hk /= A; auto.
by apply: IHn => B e He; apply: Hk => f x1 x2 <-.
Qed.
Lemma ssr_congr_arrow : forall Plemma Pgoal, Plemma = Pgoal -> Plemma -> Pgoal.
Proof. by move=> H G ->; apply. Qed.
Implicit Arguments ssr_congr_arrow [].
Section ApplyIff.
Variables P Q : Prop.
Hypothesis eqPQ : P <-> Q.
Lemma iffLR : P -> Q. Proof. by case eqPQ. Qed.
Lemma iffRL : Q -> P. Proof. by case eqPQ. Qed.
Lemma iffLRn : ~P -> ~Q. Proof. by move=> nP tQ; case: nP; case: eqPQ tQ. Qed.
Lemma iffRLn : ~Q -> ~P. Proof. by move=> nQ tP; case: nQ; case: eqPQ tP. Qed.
End ApplyIff.
Hint View for move/ iffLRn|2 iffRLn|2 iffLR|2 iffRL|2.
Hint View for apply/ iffRLn|2 iffLRn|2 iffRL|2 iffLR|2.
Unset Implicit Arguments.