diff options
Diffstat (limited to 'theories/Logic/Diaconescu.v')
| -rw-r--r-- | theories/Logic/Diaconescu.v | 28 |
1 files changed, 14 insertions, 14 deletions
diff --git a/theories/Logic/Diaconescu.v b/theories/Logic/Diaconescu.v index 55eed096..2b982963 100644 --- a/theories/Logic/Diaconescu.v +++ b/theories/Logic/Diaconescu.v @@ -6,9 +6,9 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Diaconescu.v,v 1.5.2.1 2004/07/16 19:31:06 herbelin Exp $ i*) +(*i $Id: Diaconescu.v,v 1.5.2.3 2004/08/01 09:36:44 herbelin Exp $ i*) -(* R. Diaconescu [Diaconescu] showed that the Axiom of Choice in Set Theory +(** R. Diaconescu [Diaconescu] showed that the Axiom of Choice in Set Theory entails Excluded-Middle; S. Lacas and B. Werner [LacasWerner] adapted the proof to show that the axiom of choice in equivalence classes entails Excluded-Middle in Type Theory. @@ -27,12 +27,12 @@ Section PredExt_GuardRelChoice_imp_EM. -(* The axiom of extensionality for predicates *) +(** The axiom of extensionality for predicates *) Definition PredicateExtensionality := forall P Q:bool -> Prop, (forall b:bool, P b <-> Q b) -> P = Q. -(* From predicate extensionality we get propositional extensionality +(** From predicate extensionality we get propositional extensionality hence proof-irrelevance *) Require Import ClassicalFacts. @@ -54,7 +54,7 @@ Proof. apply (ext_prop_dep_proof_irrel_cic prop_ext). Qed. -(* From proof-irrelevance and relational choice, we get guarded +(** From proof-irrelevance and relational choice, we get guarded relational choice *) Require Import ChoiceFacts. @@ -73,8 +73,8 @@ Proof. (rel_choice_and_proof_irrel_imp_guarded_rel_choice rel_choice proof_irrel). Qed. -(* The form of choice we need: there is a functional relation which chooses - an element in any non empty subset of bool *) +(** The form of choice we need: there is a functional relation which chooses + an element in any non empty subset of bool *) Require Import Bool. @@ -90,30 +90,30 @@ Proof. exact (fun _ H => H). Qed. -(* The proof of the excluded middle *) -(* Remark: P could have been in Set or Type *) +(** The proof of the excluded middle *) +(** Remark: P could have been in Set or Type *) Theorem pred_ext_and_rel_choice_imp_EM : forall P:Prop, P \/ ~ P. Proof. intro P. -(* first we exhibit the choice functional relation R *) +(** first we exhibit the choice functional relation R *) destruct AC as [R H]. set (class_of_true := fun b => b = true \/ P). set (class_of_false := fun b => b = false \/ P). -(* the actual "decision": is (R class_of_true) = true or false? *) +(** the actual "decision": is (R class_of_true) = true or false? *) destruct (H class_of_true) as [b0 [H0 [H0' H0'']]]. exists true; left; reflexivity. destruct H0. -(* the actual "decision": is (R class_of_false) = true or false? *) +(** the actual "decision": is (R class_of_false) = true or false? *) destruct (H class_of_false) as [b1 [H1 [H1' H1'']]]. exists false; left; reflexivity. destruct H1. -(* case where P is false: (R class_of_true)=true /\ (R class_of_false)=false *) +(** case where P is false: (R class_of_true)=true /\ (R class_of_false)=false *) right. intro HP. assert (Hequiv : forall b:bool, class_of_true b <-> class_of_false b). @@ -129,7 +129,7 @@ rewrite <- H0''. reflexivity. rewrite Heq. assumption. -(* cases where P is true *) +(** cases where P is true *) left; assumption. left; assumption. |