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diff --git a/test-suite/misc/berardi_test.v b/test-suite/misc/berardi_test.v deleted file mode 100644 index a64db4da..00000000 --- a/test-suite/misc/berardi_test.v +++ /dev/null @@ -1,153 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(** This file formalizes Berardi's paradox which says that in - the calculus of constructions, excluded middle (EM) and axiom of - choice (AC) imply proof irrelevance (PI). - Here, the axiom of choice is not necessary because of the use - of inductive types. -<< -@article{Barbanera-Berardi:JFP96, - author = {F. Barbanera and S. Berardi}, - title = {Proof-irrelevance out of Excluded-middle and Choice - in the Calculus of Constructions}, - journal = {Journal of Functional Programming}, - year = {1996}, - volume = {6}, - number = {3}, - pages = {519-525} -} ->> *) - -Set Implicit Arguments. - -Section Berardis_paradox. - -(** Excluded middle *) -Hypothesis EM : forall P:Prop, P \/ ~ P. - -(** Conditional on any proposition. *) -Definition IFProp (P B:Prop) (e1 e2:P) := - match EM B with - | or_introl _ => e1 - | or_intror _ => e2 - end. - -(** Axiom of choice applied to disjunction. - Provable in Coq because of dependent elimination. *) -Lemma AC_IF : - forall (P B:Prop) (e1 e2:P) (Q:P -> Prop), - (B -> Q e1) -> (~ B -> Q e2) -> Q (IFProp B e1 e2). -Proof. -intros P B e1 e2 Q p1 p2. -unfold IFProp. -case (EM B); assumption. -Qed. - - -(** We assume a type with two elements. They play the role of booleans. - The main theorem under the current assumptions is that [T=F] *) -Variable Bool : Prop. -Variable T : Bool. -Variable F : Bool. - -(** The powerset operator *) -Definition pow (P:Prop) := P -> Bool. - - -(** A piece of theory about retracts *) -Section Retracts. - -Variables A B : Prop. - -Record retract : Prop := - {i : A -> B; j : B -> A; inv : forall a:A, j (i a) = a}. - -Record retract_cond : Prop := - {i2 : A -> B; j2 : B -> A; inv2 : retract -> forall a:A, j2 (i2 a) = a}. - - -(** The dependent elimination above implies the axiom of choice: *) -Lemma AC : forall r:retract_cond, retract -> forall a:A, j2 r (i2 r a) = a. -Proof. -intros r. -case r; simpl. -trivial. -Qed. - -End Retracts. - -(** This lemma is basically a commutation of implication and existential - quantification: (EX x | A -> P(x)) <=> (A -> EX x | P(x)) - which is provable in classical logic ( => is already provable in - intuitionnistic logic). *) - -Lemma L1 : forall A B:Prop, retract_cond (pow A) (pow B). -Proof. -intros A B. -destruct (EM (retract (pow A) (pow B))) as [(f0,g0,e) | hf]. - exists f0 g0; trivial. - exists (fun (x:pow A) (y:B) => F) (fun (x:pow B) (y:A) => F); intros; - destruct hf; auto. -Qed. - - -(** The paradoxical set *) -Definition U := forall P:Prop, pow P. - -(** Bijection between [U] and [(pow U)] *) -Definition f (u:U) : pow U := u U. - -Definition g (h:pow U) : U := - fun X => let lX := j2 (L1 X U) in let rU := i2 (L1 U U) in lX (rU h). - -(** We deduce that the powerset of [U] is a retract of [U]. - This lemma is stated in Berardi's article, but is not used - afterwards. *) -Lemma retract_pow_U_U : retract (pow U) U. -Proof. -exists g f. -intro a. -unfold f, g; simpl. -apply AC. -exists (fun x:pow U => x) (fun x:pow U => x). -trivial. -Qed. - -(** Encoding of Russel's paradox *) - -(** The boolean negation. *) -Definition Not_b (b:Bool) := IFProp (b = T) F T. - -(** the set of elements not belonging to itself *) -Definition R : U := g (fun u:U => Not_b (u U u)). - - -Lemma not_has_fixpoint : R R = Not_b (R R). -Proof. -unfold R at 1. -unfold g. -rewrite AC with (r := L1 U U) (a := fun u:U => Not_b (u U u)). -trivial. -exists (fun x:pow U => x) (fun x:pow U => x); trivial. -Qed. - - -Theorem classical_proof_irrelevence : T = F. -Proof. -generalize not_has_fixpoint. -unfold Not_b. -apply AC_IF. -intros is_true is_false. -elim is_true; elim is_false; trivial. - -intros not_true is_true. -elim not_true; trivial. -Qed. - -End Berardis_paradox. |