Module mathcomp.classical.contra

From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq.
From mathcomp Require Import boolp.

Unset SsrOldRewriteGoalsOrder.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

# Contraposition This file provides tactics to reason by contraposition and contradiction. ## Tactics ``` assume_not == add a goal negation assumption. The tactic also works for goals in Type, simplifies the added assumption, and exposes its top-level constructive content. absurd_not == proof by contradiction. Same as assume_not, but the goal is erased and replaced by False. Caveat: absurd_not cannot be used as a move/ view because its conclusion is indeterminate. The more general notP can be used instead. contra == proof by contraposition. Change a goal of the form assumption -> conclusion to ~ conclusion -> ~ assumption. As with assume_not, contra allows both assumption and conclusion to be in Type, simplifies the negation of both assumption and conclusion, and exposes the constructive contents of the negated conclusion. The contra tactic also supports a limited form of the ':' discharge pseudo tactical, whereby contra: <d-items> means move: <d-items>; contra. The only <d-items> allowed are one term, possibly preceded by a clear switch. absurd == proof by contradiction. The defective form of the tactic simply replaces the entire goal with False (just as the Ltac exfalso), leaving the user to derive a contradiction from the assumptions. The ':' form absurd: <d-items> replaces the goal with the negation of the (single) <d-item> (as with contra:, a clear switch is also allowed. Finally the Ltac absurd term form is also supported. ```

Hiding module for the internal definitions and lemmas used by the tactics defined here.

Module .

A wrapper for view lemmas with an indeterminate conclusion (of the form forall ... T ..., pattern -> T), and for which the intended view pattern may fail to match some assumptions. This wrapper ensures that such views are only used in the forward direction (as in move/), and only with the appropriate move_viewP hint, preventing its application to an arbitrary assumption A by the instatiation to A -> T' of its indeterminate conclusion T. This is similar to the implies wrapper, except move_viewP is *NOT* declared as a coercion---it must be used explicitly to apply the view manually to an assumption (as in, move_viewP my_view some_assumption).

Variant := of S -> T.
Definition { } : S -> T := let: MoveView v := mv in v.
Hint View for move/ move_viewP|2.

## Type-level equivalence

Variant := of S -> T & T -> S.

Definition : equivT S S := EquivT id id.
Definition { } : equivT S T -> equivT S U -> equivT T U :=
  fun ( : equivT S T) ( : equivT S U) =>
    let: EquivT S_T T_S := st in
    let: EquivT S_U U_S := su in
    EquivT (S_U \o T_S) (S_T \o U_S).
Definition { } : equivT S T -> equivT T S :=
   equivT_transl^~ (equivT_refl S).
Definition { } : equivT S T -> equivT T U -> equivT S U :=
   equivT_transl \o equivT_sym.
Definition { } : equivT U S -> equivT U T :=
   equivT_trans^~ eqST.
Definition ( : Prop) : (equivT P Q) <-> (equivT P Q).

Proof.

split; destruct 1; split; assumption. Defined.

Definition { } ( : equivT S T) : S -> T :=
let: EquivT S_T _ := eq in S_T.
Definition { } ( : equivT S T) : T -> S :=
let: EquivT _ T_S := eq in T_S.

Hint View for move/ equivT_LR|2 equivT_RL|2.
Hint View for apply/ equivT_RL|2 equivT_LR|2.

A generic Forall "constructor" for the Gallina forall quantifier, i.e., ``` \Forall x, P := Forall (fun x => P) := forall x, P. ``` The main use of Forall is to apply congruence to a forall equality: ``` congr1 Forall : forall P Q, P = Q -> Forall P = Forall Q. ``` in particular in a classical setting with function extensionality, where we can have (forall x, P x = Q x) -> (forall x, P x) = (forall x, Q x). We use a forallSort structure to factor the ad hoc PTS product formation rules; forallSort is keyed on the type of the entire forall expression, or (up to subsumption) the type of the forall body---this is always a sort. This implementation has two important limitations: 1. It cannot handle the SProp sort and its typing rules. However, its main application is extensionality, which is not compatible with SProp because an (A : SProp) -> B "function" is not a generic (A : Type) -> B function as SProp is not included in Type. 2. The Forall constructor can't be inserted by a straightforward unfold (as in, rewrite -[forall x, _]/(Forall _)) because of the way Coq unification handles Type constraints. The ForallI tactic mitigates this issue, but there are additional issues with its implementation---see below.

Structure :=
  ForallSort { :> Type; _ : (A -> forall_sort) -> forall_sort}.

Notation A S := (@ForallSort A S (fun => forall , T x)).
Polymorphic Definition ( := Type) ( : S) := mkForallSort A S.
Canonical TypeForall.

Canonical := mkForallSort A Prop.

Canonical ( : Set) := mkForallSort A Set.

Definition {} { : forallSort A} :=
  let: ForallSort _ F := S return (A -> S) -> S in F.

Notation :=
   (Forall (fun => .. (Forall (fun => T)) ..))
  (at level 200, x binder, z binder, T at level 200,
   format "'[hv' '\Forall' '[' x .. z , ']' '/ ' T ']'") : type_scope.

Tactic Notation "ForallI" ssrpatternarg(pat) :=
  let F := fresh "F" in ssrmatching.ssrpattern pat => F;
  case: F / (@erefl _ F : Forall _ = _).
Tactic Notation "ForallI" := ForallI (forall , _).

We define specialized copies of the wrapped structure of ssrfun for Prop and Type, as we need more than two alternative rules (indeed, 3 for Prop and 4 for Type). We need separate copies for Prop and Type as universe polymorphism cannot instantiate Type with Prop.

Structure := WrapProp { :> Prop}.
Definition := WrapProp.
Definition := wrap4Prop.
Definition := wrap3Prop.
Canonical := wrap2Prop P.

Polymorphic Structure @{i} := WrapType { :> Type@{i}}.
Polymorphic Definition @{i} := WrapType@{i}.
Polymorphic Definition @{i} := wrap4Type@{i}.
Polymorphic Definition @{i} := wrap3Type@{i}.
Polymorphic Definition @{i} ( : Type@{i}) := wrap2Type T.
Canonical wrap1Type.

Lemma {} { : forallSort A} { : A -> S} :
P =1 Q -> Forall P = Forall Q.

Proof.

by move/funext->. Qed.

A set of tools (tactics, views, and rewrite rules) to facilitate the handling of classical negation. The core functionality of these tools is implemented by three sets of canonical structures that provide for the simplification of negation statements (e.g., using de Morgan laws), the conversion from constructive statements in Type to purely logical ones in Prop (equivalently, expansion rules for the statement inhabited T), and conversely extraction of constructive contents from logical statements. Except for bool predicates and operators, all definitions are treated transparently when matching statements for either simplification or conversion; this is achieved by using the wrapper telescope pattern, first delegating the matching of specific logical connectives, predicates, or type constructors to an auxiliary structure that *FAILS* to match unknown operators, thus triggers the expansion of defined constants. If this ultimately fails then the wrapper is expanded, and the primary structure instance for the expanded wrapper provides an alternative default rule: not simplifying ~ P, not expanding inhabited T, or not extracting any contents from a proposition P, respectively. Additional rules, for intermediate wrapper instances, are used to handle forall statements (for which canonical instances are not yet supported), as well as addiitonal simplifications, such as inhabited P = P :> Prop. Finally various tertiary structures are used to match deeper patterns, such as bounded forall statements of the form forall x, P x -> Q x, or inequalites x != y (i.e., is_true (~~ (x == y))). As mentioned above, tertiary rules for bool subexpressions do not try to expand definitions, as this would lead to the undesirable expansion of some standard definitions. This is simply achieved by *NOT* using the wrapper telescope pattern, and just having a default instance alongside those for specific predicates and connectives.

The negatedProp structure provides simplification of the Prop negation (~ _) for standard connectives and predicates. The instances below cover the pervasive and ssrbool Prop connectives, decidable equality, as well as bool propositions (i.e., the is_true predicate), together with a few bool connectives and predicates: negation ~~, equality ==, and nat <= and <. Others can be added (e.g., Order.le/lt) by declaring appropriate instances of bool_negation and bool_affirmation, while other Prop connectives and predicates can be added by declaring instances of proper_negatedProp.

The implementation follows the wrapper telescope pattern outlined above: negatedProp instances match on the wrappedProp wrapper to try three generic matching rules, in succession: - Rule 1: match a specific connective or predicate with an instance of the properNegatedProp secondary structure, expanding definitions if needed, but failing if no proper match is found. - Rule 2: match a forall statement (including (T : Type) -> P statements). - Rule 3: match any Prop but return the trivial simplification. The simplified proposition is returned as a projection parameter nP rather than a Structure member, so that applying the corresponding views or rewrite rules doesn't expose the inferred structures; properNegatedProp does similarly. Also, negatedProp similarly returns a 'trivial' bool flag that is set when Rule 3 is used, but this is actually used in the reverse direction: views notP and rewrite rule notE force trivial := false, thus excluding trivial instances.

Structure ( : bool) :=
NegatedProp { :> wrappedProp; _ : (~ negated_Prop) = nP}.

Structure := ProperNegatedProp {
:> Prop; _ : (~ proper_negated_Prop) = nP}.

Local Notation t nP P := (unwrap_Prop (@negated_Prop t nP P)).
Local Notation t nP P x := (nProp t (nP x) (P x)).
Local Notation nP P := (@proper_negated_Prop nP P).

User views and rewrite rules. The plain versions (notP, notE and notI) do not match trivial instances; lax_XXX versions allow them. In addition, the negation introduction rewrite rule notI does not match forall or -> statements---lax_notI must be used for these.

Lemma { } : (~ nProp t nP P) = nP

Proof.

by case: P. Qed.

Lemma { } : ~ nProp t nP P -> nP

Proof.

by rewrite lax_notE. Qed.

Definition { } : nProp t nP P = (~ nP) := canRL notK (lax_notE P).

#[warn(note="A different `notE` used to be provided by `boolp.v` before MathComp-Analysis 1.15.0.")]
Definition {} : (~ nProp false nP P) = nP := lax_notE P.
#[warn(note="A different `notP` used to be provided by `boolp.v` before MathComp-Analysis 1.15.0.")]
Definition { } := MoveView (@lax_notP false nP P).

Fact : (~ pnProp nP P) = nP

Proof.

by case: P. Qed.

Definition {} : pnProp nP P = (~ nP) := canRL notK (proper_nPropP P).

Rule 1: proper negation simplification, delegated to properNegatedProp.

Canonical :=
@NegatedProp false nP (wrap1Prop (pnProp nP P)) (proper_nPropP P).

Rule 2: forall_nProp is defined below as it uses exists_nProp.

Rule 3: trivial negation.

Canonical := @NegatedProp true (~ P) (wrap3Prop P) erefl.

properNegatedProp instances.

Canonical := @ProperNegatedProp False True notB.1.
Canonical := @ProperNegatedProp True False notB.2.
Canonical := @ProperNegatedProp P (~ P) (notK P).

Fact : (~ (P /\ nProp tQ nQ Q)) = (P -> nQ).

Proof.

by rewrite -implypN lax_notE. Qed.

Canonical :=
ProperNegatedProp (@and_nPropP P tQ nQ Q).

Fact : (~ [/\ P, Q & nProp tR nR R]) = (P -> Q -> nR).

Proof.

#[warnings="-user-warn"] by hnf; rewrite and3E notE. Qed.

Canonical :=
ProperNegatedProp (@and3_nPropP P Q tR nR R).

Fact :
(~ [/\ P, Q, R & nProp tS nS S]) = (P -> Q -> R -> nS).

Proof.

#[warnings="-user-warn"] by hnf; rewrite and4E notE. Qed.

Canonical :=
ProperNegatedProp (@and4_nPropP P Q R tS nS S).

Fact :
(~ [/\ P, Q, R, S & nProp tT nT T]) = (P -> Q -> R -> S -> nT).

Proof.

#[warnings="-user-warn"] by hnf; rewrite and5E notE. Qed.

Canonical :=
ProperNegatedProp (@and5_nPropP P Q R S tT nT T).

Fact :
(~ (nProp tP nP P \/ nProp tQ nQ Q)) = (nP /\ nQ).

Proof.

by rewrite not_orE !lax_notE. Qed.

Canonical :=
ProperNegatedProp (@or_nPropP tP nP P tQ nQ Q).

Fact :
(~ [\/ nProp tP nP P, nProp tQ nQ Q | nProp tR nR R]) = [/\ nP, nQ & nR].

Proof.

#[warnings="-user-warn"] by rewrite or3E notE and3E. Qed.

Canonical :=
  ProperNegatedProp (@or3_nPropP tP nP P tQ nQ Q tR nR R).

Fact :
  (~ [\/ nProp tP nP P, nProp tQ nQ Q, nProp tR nR R | nProp tS nS S])
     = [/\ nP, nQ, nR & nS].

Proof.

#[warnings="-user-warn"] by rewrite or4E notE and4E. Qed.

Canonical :=
ProperNegatedProp (@or4_nPropP tP nP P tQ nQ Q tR nR R tS nS S).

The andRHS tertiary structure used to simplify (~ (P -> False)) to P, both here for the imply_nProp instance and for bounded_forall_nProp below. Because the andRHS instances match the Prop RETURNED by negatedProp they do not need to expand definitions, hence do not need to use the wrapper telescope pattern.

Notation binary P Q PQ := (PQ = if binary then P /\ Q else Q)%type.
Structure :=
AndRHS { :> Prop; _ : (P /\ and_RHS) = PQ; _ : and_def binary P Q PQ}.
Canonical := @AndRHS false P P P True (andB.1.2 P) erefl.
Canonical := @AndRHS true P Q (P /\ Q) Q erefl erefl.

Fact ( : andRHS b P nQ PnQ) :
(~ (P -> nProp tR nR R)) = PnQ.

Proof.

by rewrite -orNp {R}lax_notE; case: nR. Qed.

Canonical :=
ProperNegatedProp (@imply_nPropP b P nQ PnQ tR nR R).

Fact :
(~ exists : A, nPred tP nP P x) = (forall : A, nP x).

Proof.

eqProp=> [nEP x | AnP [x]]; last by rewrite -/(~ _) lax_notE.
by rewrite -(lax_notE (P x)) => Px; case: nEP; exists x.
Qed.

Canonical :=
ProperNegatedProp (@exists_nPropP A tP nP P).

Fact :
(~ exists2 : A, P x & nPred tQ nQ Q x) = (forall : A, P x -> nQ x).

Proof.

#[warnings="-user-warn"] by rewrite exists2E notE. Qed.

Canonical :=
ProperNegatedProp (@exists2_nPropP A P tQ nQ Q).

Fact : (~ inhabited T) = (T -> False).

Proof.

#[warnings="-user-warn"] by rewrite inhabitedE notE. Qed.

Canonical := ProperNegatedProp (inhabited_nPropP T).

Rule 2: forall negation, including (T : Type) -> P statements. We use tertiary structures to recognize bounded foralls and simplify, e.g., ~ forall x, P -> Q to exists2 x, P & ~ Q, or even exists x, P when Q := False (as above for imply). As forall_body_nProp and forall_body_proper_nProp are telescopes over negatedProp and properNegatedProp, respectively, their instances match instances declared above without the need to expand definitions, hence do not need to use the wrapper telescope idiom.

Structure := NegatedForallBody {
:> negatedProp tR nR; _ : and_def bounded P nQ nR}.
Structure := ProperNegatedForallBody {
:> properNegatedProp nR; _ : and_def b P nQ nR}.
Notation b P nQ t nR x := (negatedForallBody b (P x) (nQ x) t (nR x)).

The explicit argument to fun_if is a workaround for a bug in the Coq unification code that prevents default instances from ever matching match constructs. Furthermore rewriting with ifE would not work here, because the if_expr definition would be expanded by the eta expansion needed to match the exists_nProp rule.

Fact ( : forall , nBody b P nQ tR nR x) :
(~ forall : A, R x) = if b then exists2 , P x & nQ x else exists , nQ x.

Proof.

rewrite exists2E -(fun_if (fun => exists , idfun P x)) notI /=; congr not.
apply/generic_forall_extensionality=> x; rewrite if_arg lax_notI.
by case: (R x) => _ <-.
Qed.

Canonical ( : forall , nBody b P nQ tR nR x) :=
@NegatedProp false _ (wrap2Prop (forall : A, R x)) (forall_nPropP R).

Fact :
properNegatedForallBody b P nQ nR -> and_def b P nQ nR.

Proof.

by case. Qed.

Canonical :=
  let := @proper_nBodyP b P nQ nR R in
  @NegatedForallBody b P nQ false nR (proper_nProp R) def_nR.
Canonical :=
  @NegatedForallBody false True nP tP nP P erefl.

Fact : andRHS b P Q PQ -> and_def b P Q PQ.

Proof.

by case. Qed.

Canonical :=
ProperNegatedForallBody (@imply_nProp b P nQ PnQ tR nR R) (andRHS_def nR).
Canonical :=
@ProperNegatedForallBody false True nQ nQ Q erefl.

The properNegatedProp instance that handles boolean statements. We use two tertiary structures to handle positive and negative boolean statements so that the contra tactic below will mostly subsume the collection of contraXX lemmas in ssrbool and eqtype. We only match manifest ~~ connectives, the true and false constants, and the ==, <=%N, and <%N predicates. In particular we do not use de Morgan laws to push boolean negation into connectives, as we did above for Prop connectives. It will be up to the user to use rewriting to put the negated statement in its desired shape.

Structure :=
NegatedBool { :> bool; _ : (~ negated_bool) = nP}.
Structure :=
PositedBool { :> bool; _ : is_true posited_bool = P}.

Local Fact ( : negatedBool nP) : (~ b) = nP.

Proof.

by case: b. Qed.

Canonical := ProperNegatedProp (@is_true_nPropP nP b).

Local Fact : (~ true) = False

Proof.

by eqProp. Qed.

Local Fact : (true : Prop) = True

Proof.

by eqProp. Qed.

Local Fact : (~ false) = True

Proof.

by eqProp. Qed.

Local Fact : (false : Prop) = False

Proof.

by eqProp. Qed.

Canonical := NegatedBool true_negP.
Canonical := PositedBool true_posP.
Canonical := NegatedBool false_negP.
Canonical := PositedBool false_posP.

Local Fact ( : bool) : (~ b) = ~~ b

Proof.

exact/reflect_eq/negP. Qed.

Canonical := NegatedBool (id_negP b).
Canonical ( : bool) := @PositedBool b b erefl.

Local Fact ( : positedBool P) : (~ ~~ b) = P.

Proof.

by rewrite (reflect_eq negP) negbK; case: b. Qed.

Canonical := NegatedBool (@negb_negP P b).
Local Fact ( : negatedBool nP) : (~~ b = nP :> Prop).

Proof.

#[warnings="-user-warn"] by rewrite -(reflect_eq negP) notE. Qed.

Canonical := PositedBool (@negb_posP nP b).

We use a tertiary structure to handle the negation of nat comparisons, and simplify ~ m <= n to n < m, and ~ m < n to n <= m. As m < n is merely notation for m.+1 <= n, we need to dispatch on the left hand side of the comparison to perform the latter simplification.

Structure :=
NegatedLeqLHS { :> nat; _ : (n < negated_leq_LHS) = lt_nm}.
Canonical := @NegatedLeqLHS n (n <= m) m.+1 erefl.
Canonical := @NegatedLeqLHS n (n < m) m erefl.

Local Fact ( : negatedLeqLHS n lt_nm) : (~ m <= n) = lt_nm.

Proof.

#[warnings="-user-warn"] by rewrite notE -ltnNge; case: m => /= m ->. Qed.

Canonical := NegatedBool (@leq_negP n lt_nm m).

We use two tertiary structures to simplify negation of boolean constant and decidable equalities, simplifying b <> true to ~~ b, b <> false to b, x <> y to x != y, and ~ x != y to x = y. We do need to use the wrapper telescope pattern here, as we want to simplify instances of x <> y when y evaluates to true or false. Since we only need two rules (true/false RHS or generic eqType RHS) we can use the generic wrapped type from ssrfun. The actual matching of the true and false RHS is delegated to a fourth level bool_eq_negation_rhs structure. Finally observe that the ~ x != y to x = y simplification can be handled by a bool_affirmation instance.

Structure :=
NeqRHS { :> wrapped T; _ : (x <> unwrap neq_RHS) = nP}.
Structure ( : bool) :=
BoolNeqRHS {; _ : (x <> bool_neq_RHS) = nP}.

Local Fact ( : neqRHS nP x) : (x <> unwrap y :> T) = nP.

Proof.

by case: y. Qed.

Canonical := ProperNegatedProp (@eq_nPropP nP T x y).

Local Fact : (x <> @bool_neq_RHS nP x y) = nP.

Proof.

by case: y. Qed.

Canonical := @NeqRHS nP bool x (wrap _) (@bool_neqP nP x y).
Canonical := BoolNeqRHS (@is_true_nPropP nP b).
Local Fact ( : positedBool P) : (b <> false :> bool) = P.

Proof.

by move: b => [] [] /= <-; exact/propext. Qed.

Canonical := BoolNeqRHS (@false_neqP P b).

Local Fact ( : eqType) ( : T) : (x <> y) = (x != y).

Proof.

by rewrite [in LHS](reflect_eq eqP) (reflect_eq negP). Qed.

Canonical ( : eqType) :=
@NeqRHS (x != y) T x (Wrap y) (eqType_neqP x y).
Local Fact ( : eqType) : (x == y :> T : Prop) = (x = y).

Proof.

exact/esym/reflect_eq/eqP. Qed.

Canonical := PositedBool (@eq_op_posP T x y).

The witnessedType structure provides conversion between Type and Prop in goals; the conversion is mostly used in the Type-to-Prop direction, e.g., as a preprocessing step preceding proof by contradiction (see absurd_not below), but the Prop-to-Type direction is required for contraposition. Thus witnessedType associates to a type T a "witness" proposition P equivalent to the existence of an x of type T. As in a `{classical_logic} context inhabited T is such a proposition, witnessedType can be understood as providing simplification for inhabited T, much like negatedProp provides simplification for ~ P for standard connectives and predicates.

Similarly to negatedProp, witnessedType returns the witness proposition via a projection argument P, but does not need to signal "trivial" instances as the default value for P is nontrivial (namely, inhabited T), while the "trivial" case where P = T is actually desireable and handled by an extra top-priority rule.

Structure := WitnessedType {
:> wrappedType; _ : inhabited witnessed_Type = P}.
Structure := ProperWitnessedType {
:> Type; _ : inhabited proper_witnessed_Type = P}.
Local Notation P T := (unwrap_Type (@witnessed_Type P T)).
Local Notation P T x := (wType (P x) (T x)).

Lemma { : Prop} : P -> wType P T.

Proof.

by case: T => /= T <- /inhabited_witness. Qed.

Local Coercion witnessedType_intro : witnessedType >-> Funclass.

Lemma {} : wType P T -> P.

Proof.

by case: T => /= T <-. Qed.

Local Notation := witnessedType_elim.

Local Fact ( : Prop) : (T -> P) -> (P -> T) -> inhabited T = P.

Proof.

by move=> T_P P_T; eqProp=> [[/T_P] | /P_T]. Qed.

Ltac eqInh := apply: eq_inhabited.

Rule 1: Prop goals are left as is.

Canonical :=
@WitnessedType P (wrap1Type P) (eq_inhabited (@id P) id).

Rule 2: Specific type constructors (sigs, sums, and pairs) are delegated to the secondary properWitnessedType structure.

Lemma ( : properWitnessedType P) : inhabited T = P.

Proof.

by case: T. Qed.

Canonical :=
@WitnessedType P (wrap2Type _) (@proper_wTypeP P T).

Rule 3: Forall (and -> as a special case).

Local Fact :
inhabited (forall : A, wTycon P T x) = (forall : A, P x) .

Proof.

by do [eqInh=> allP x; have:= allP x] => [/wTypeP | /T]. Qed.

Canonical :=
@WitnessedType _ (wrap3Type _) (@forall_wTypeP A P T).

Rule 4: Default to inhabited if all else fails.

Canonical := @WitnessedType (inhabited T) (wrap4Type T) erefl.

Specific proper_witnessedType instances.

Local Fact : inhabited void = False

Proof.

by eqInh. Qed.

Canonical := ProperWitnessedType void_wTypeP.

Local Fact : inhabited unit = True

Proof.

by eqInh. Qed.

Canonical := ProperWitnessedType unit_wTypeP.

Local Fact : inhabited (wType P S * wType Q T) = (P /\ Q).

Proof.

by eqInh=> [[/wTypeP-isP /wTypeP] | [/S-x /T]]. Qed.

Canonical := ProperWitnessedType (@pair_wTypeP P Q S T).

Local Fact : inhabited (wType P S + wType Q T) = (P \/ Q).

Proof.

by eqInh=> [[] /wTypeP | /decide_or[/S | /T]]; by [left | right]. Qed.

Canonical := ProperWitnessedType (@sum_wTypeP P Q S T).

Local Fact : inhabited ({P} + {Q}) = (P \/ Q).

Proof.

by eqInh=> [[] | /decide_or[]]; by [left | right]. Qed.

Canonical := ProperWitnessedType (@sumbool_wTypeP P Q).

Local Fact : inhabited (wType P T + {Q}) = (P \/ Q).

Proof.

by eqInh=> [[/wTypeP|] | /decide_or[/T|]]; by [left | right]. Qed.

Canonical := ProperWitnessedType (@sumor_wTypeP P Q T).

Local Fact : inhabited { : T | P x} = (exists : T, P x).

Proof.

by eqInh=> [[x Px] | /cid//]; exists x. Qed.

Canonical := ProperWitnessedType (@sig1_wTypeP T P).

Local Fact :
inhabited { : T | P x & Q x} = exists2 : T, P x & Q x.

Proof.

by eqInh=> [[x Px Qx] | /cid2//]; exists x. Qed.

Canonical := ProperWitnessedType (@sig2_wTypeP T P Q).

Local Fact :
inhabited { : A & wTycon P T x} = (exists : A, P x).

Proof.

by eqInh=> [[x /wTypeP] | /cid[x /T]]; exists x. Qed.

Canonical := ProperWitnessedType (@sigT_wTypeP A P T).

Local Fact :
inhabited { : A & wTycon P S x & wTycon Q T x} = (exists2 : A, P x & Q x).

Proof.

by eqInh=> [[x /wTypeP-Px /wTypeP] | /cid2[x /S-y /T]]; exists x. Qed.

Canonical :=
ProperWitnessedType (@sigT2_wTypeP A P Q S T).

The witnessProp structure provides for conversion of some Prop assumptions to Type values with some constructive contents, e.g., convert a P \/ Q assumption to a {P} + {Q} sumbool value. This is not the same as the forward direction of witnessedType, because instances here match the Prop statement: witness_Prop find a T such that P -> T while witnessedType finds a P such that P -> T (and T -> P for the converse direction).

The implementation follows the wrapper telescope pattern similarly to negatedProp, with three rules, one for Prop constructors with proper constructive contents, one for forall propositions (also with proper constructive contents) and one default rule that just returns P : Prop as is (thus, with no other contents except the provability of P). The witnessProp structure also uses projection parameters to return the inferred Type T together with a bool 'trivial' flag that is set when the trivial rule is used. Here, however, this flag is used in both directions: the 'witness' view forces it to false to prevent trivial instances, but the flag is also used to fine tune the choice of T, selecting between sum, sumor, and sumbool, between sig and sigT, and sig2 and sigT2. This relies on the fact that the tactic engine will eagerly iota reduce the returned type, so that the user will never see the conditionals specified in the proper_witness_Prop instances. However, it would not be possible to construct the specialised types for trivial witnesses (e.g., {P} + {Q}) using the types returned by witnessProp instances, since thes are in Type, and the information that they are actully in Prop has been lost. This is solved by returning an additional Prop P0 that is a copy of the matched Prop P when trivial = true. (We put P0 = True when trivial = false, as we only need to ensure P -> P0.) Caveat: although P0 should in principle be the last parameter of witness_Prop, and we use this order for the wProp and wPred projector local notation, it is important to put P0 _BEFORE_ T, to circumvent an incompleteness in Coq's implementation of higher-order pattern unification that would cause the trivial rule to fail for the body of an exists. In such a case the rule needs to unify (1) ?P0 x ~ ?P and (2) ?T x ~ ?P for some type A some x : A in the context of ?P, but not ?P0 nor ?T. This succeeds easily if (1) is performed before (2), setting ?P := ?P0 x and ?T := ?P0, but if (2) is attempted first Coq tries to perform ?P := ?T x, which fails Type/Prop universe constraints, and then fails outright, instead of using pattern unification to solve (2) as ?P := ?Q x, ?T := ?Q for a fresh ?Q : A -> Prop.

Structure ( : bool) ( : Prop) ( : Type) :=
WitnessProp { :> wrappedProp; _ : witness_Prop -> T * P0}.
Structure :=
ProperWitnessProp { :> Prop; _ : proper_witness_Prop -> T}.

Local Notation t T P0 P := (unwrap_Prop (@witness_Prop t P0 T P)).
Local Notation t T P0 P x := (wProp t (T x) (P0 x) (P x)).

Local Fact : wProp t T P0 P -> T * P0

Proof.

by case: P. Qed.

Lemma { } : move_view (wProp t T P0 P) T.

Proof.

by split=> /wPropP[]. Qed.

Definition { } := @lax_witness false T P0 P.

Rule 1: proper instances (except forall), delegated to an auxiliary structures.

Local Fact : wrap1Prop (@proper_witness_Prop T P) -> T * True.

Proof.

by case: P => _ P_T {}/P_T. Qed.

Canonical := WitnessProp false (@proper_wPropP T P).

Rule 2: forall types (including implication); as only proper instances are allowed, we set trivial = false for the recursive body instance.

Local Fact :
wrap2Prop (forall : A, wPred false T P0 P x) -> (forall , T x) * True.

Proof.

by move=> P_A; split=> // x; have /witness := P_A x. Qed.

Canonical := WitnessProp false (@forall_wPropP A T P0 P).

Rule 3: trivial (proof) self-witness.

Canonical :=
WitnessProp true (fun : wrap3Prop P => (p, p) : P * P).

Specific proper_witnesss_Prop instances.

Canonical := ProperWitnessProp (@inhabited_witness T).

Conjunctions P /\ Q are a little delicate to handle, as we should not produce a proper instance (and thus fail) if neither P nor Q is proper. We use a tertiary structure for this : nand_bool b, which has instances only for booleans b0 such that ~~ (b0 && b). We allow the witness_Prop instance for P to return an arbitrary 'trivial' flag s, but then force the 'trivial' flag for Q to be an instance of nand_bool s.

Structure := NandBool { :> bool; _ : ~~ (nand_bool && b)}.
Canonical := @NandBool b false isT.
Canonical := @NandBool false true isT.

Local Fact ( : nandBool s) :
wProp s S P0 P /\ wProp t T Q0 Q -> S * T.

Proof.

by case=> /lax_witness-x /lax_witness. Qed.

Canonical :=
  ProperWitnessProp (@and_wPropP s S P0 P t T Q0 Q).

Local Fact :
    wProp s S P0 P \/ wProp t T Q0 Q ->
  if t then if s then {P0} + {Q0} : Type else S + {Q0} else S + T : Type.

Proof.

by case: s t => -[] in P Q *; (case/decide_or=> /wPropP[]; [left | right]).
Qed.

Canonical :=
ProperWitnessProp (@or_wPropP s S P0 P t T Q0 Q).

Local Fact :
(exists : A, wPred t T P0 P x) -> if t then { | P0 x} else { & T x}.

Proof.

by case/cid => x /wPropP[]; case t; exists x. Qed.

Canonical :=
  ProperWitnessProp (@exists_wPropP A t T P0 P).

Local Fact ( := if s then t else false) :
    (exists2 : A, wPred s S P0 P x & wPred t T Q0 Q x) ->
  if st then { | P0 x & Q0 x} else { : A & S x & T x}.

Proof.

by case/cid2=> x /wPropP[P0x y] /wPropP[]; case: ifP; exists x. Qed.

Canonical :=
ProperWitnessProp (@exists2_wPropP A s S P0 P t T Q0 Q).

## User lemmas and tactics for proof by contradiction and contraposition.

Helper lemmas: - push_goal_copy makes a copy of the goal that can then be matched with witnessedType and negatedProp instances to generate a contradiction assuption, without disturbing the original form of the goal. - assume_not_with turns the copy generated by push_identity into an equivalent negative assumption, which can then be simplified using the lax_notP and lax_witness views. - absurd and absurdW replace the goal with False; absurdW does this under an assumption, and is used to weaken proof-by-assuming-negation to proof-by-contradiction. - contra_Type converts an arbitrary function goal (with assumption and conclusion in Type) to an equivalent contrapositive Prop implication. - contra_notP simplifies a contrapositive ~ Q -> ~ P goal using negatedProp instances.

Local Fact {} : ((T -> T) -> T) -> T

Proof.

exact. Qed.

Local Fact { } : (~ P -> R) -> (wType P T -> R) -> R.

Proof.

by move=> nP_T T_R; have [/T|] := asboolP P. Qed.

Local Fact { } : (S -> False) -> S -> T

Proof.

by []. Qed.

Local Fact { } : (~ Q -> ~ P) -> wType P S -> wType Q T.

Proof.

by rewrite implyNN => P_Q /wTypeP/P_Q/T. Qed.

Local Fact ( : Prop) :
(nP -> nQ) -> (~ nProp tP nP P -> ~ nProp tQ nQ Q).

Proof.

by rewrite 2!lax_notE. Qed.

End Internals.
Import Internals.
#[warnings="-user-warn"] Definition := @Internals.notP.
Hint View for move/ move_viewP|2.
Hint View for move/ Internals.equivT_LR|2 Internals.equivT_RL|2.
Hint View for apply/ Internals.equivT_RL|2 Internals.equivT_LR|2.
Export (canonicals) Internals.

Lemma and tactic assume_not: add a goal negation assumption. The tactic also works for goals in Type, simplifies the added assumption, and exposes its top-level constructive content.

Lemma {} : (~ P -> P) -> P

Proof.

by rewrite implyNp orB. Qed.

Ltac assume_not :=
apply: Internals.push_goal_copy; apply: Internals.assume_not_with
=> /Internals.lax_notP-/Internals.lax_witness.

Lemma and tactic absurd_not: proof by contradiction. Same as assume_not, but the goal is erased and replaced by False. Caveat: absurd_not cannot be used as a move/ view because its conclusion is indeterminate. The more general notP defined above can be used instead.

Lemma {} : (~ P -> False) -> P

Proof.

#[warnings="-user-warn"] by move/Internals.notP. Qed.

Ltac absurd_not := assume_not; apply: Internals.absurdW.

Tactic contra: proof by contraposition. Assume the negation of the goal conclusion, and prove the negation of a given assumption. The defective form contra (which can also be written contrapose) expects the assumption to be pushed on the goal which thus has the form assumption -> conclusion. As with assume_not, contra allows both assumption and conclusion to be in Type, simplifies the negation of both assumption and conclusion, and exposes the constructive contents of the negated conclusion. The contra tactic also supports a limited form of the ':' discharge pseudo tactical, whereby contra: <d-items> means move: <d-items>; contra. The only <d-items> allowed are one term, possibly preceded by a clear switch.

Ltac contrapose :=
  apply: Internals.contra_Type;
  apply: Internals.contra_notP => /Internals.lax_witness.
Tactic Notation "contra" := contrapose.
Tactic Notation "contra" ":" constr(H) := move: (H); contra.
Tactic Notation "contra" ":" ident(H) := move: H; contra.
Tactic Notation "contra" ":" "{" hyp_list(Hs) "}" constr(H) :=
  contra: (H); clear Hs.
Tactic Notation "contra" ":" "{" hyp_list(Hs) "}" ident(H) :=
  contra: H; clear Hs.
Tactic Notation "contra" ":" "{" "-" "}" constr(H) := contra: (H).

Lemma and tactic absurd: proof by contradiction. The defective form of the lemma simply replaces the entire goal with False (just as the Ltac exfalso), leaving the user to derive a contradiction from the assumptions. The ':' form absurd: <d-items> replaces the goal with the negation of the (single) <d-item> (as with contra:, a clear switch is also allowed. Finally the Ltac absurd term form is also supported.

Lemma : False -> T

Proof.

by []. Qed.

Tactic Notation (at level 0) "absurd" := apply absurd.
Tactic Notation (at level 0) "absurd" constr(P) := have []: ~ P.
Tactic Notation "absurd" ":" constr(H) := absurd; contra: (H) => _.
Tactic Notation "absurd" ":" ident(H) := absurd; contra: H => _.
Tactic Notation "absurd" ":" "{" hyp_list(Hs) "}" constr(H) :=
absurd: (H) => _; clear Hs.
Tactic Notation "absurd" ":" "{" hyp_list(Hs) "}" ident(H) :=
absurd: H => _; clear Hs.

Files

Module mathcomp.classical.contra