77From HB Require Import structures.
88From mathcomp Require Import all_ssreflect.
99
10- (***** ************************************************************************ *)
11- (* Classical Logic *)
10+ (***md ************************************************************************ *)
11+ (* # Classical Logic *)
1212(* *)
1313(* This file provides the axioms of classical logic and tools to perform *)
1414(* classical reasoning in the Mathematical Compnent framework. The three *)
1515(* axioms are taken from the standard library of Coq, more details can be *)
1616(* found in Section 5 of *)
17- (* Reynald Affeldt, Cyril Cohen, Damien Rouhling: *)
18- (* Formalization Techniques for Asymptotic Reasoning in Classical Analysis. *)
19- (* Journal of Formalized Reasoning, 2018 *)
17+ (* - R. Affeldt, C. Cohen, D. Rouhling. Formalization Techniques for *)
18+ (* Asymptotic Reasoning in Classical Analysis. JFR 2018 *)
2019(* *)
21- (* * Axioms *)
20+ (* ## Axioms *)
21+ (* ``` *)
2222(* functional_extensionality_dep == functional extensionality (on dependently *)
2323(* typed functions), i.e., functions that are pointwise *)
2424(* equal are equal *)
@@ -27,14 +27,19 @@ From mathcomp Require Import all_ssreflect.
2727(* constructive_indefinite_description == existential in Prop (ex) implies *)
2828(* existential in Type (sig) *)
2929(* cid := constructive_indefinite_description (shortcut) *)
30- (* --> A number of properties are derived below from these axioms and are *)
30+ (* ``` *)
31+ (* *)
32+ (* A number of properties are derived below from these axioms and are *)
3133(* often more pratical to use than directly using the axioms. For instance *)
3234(* propext, funext, the excluded middle (EM),... *)
3335(* *)
34- (* * Boolean View of Prop *)
36+ (* ## Boolean View of Prop *)
37+ (* ``` *)
3538(* `[< P >] == boolean view of P : Prop, see all lemmas about asbool *)
39+ (* ``` *)
3640(* *)
37- (* * Mathematical Components Structures *)
41+ (* ## Mathematical Components Structures *)
42+ (* ``` *)
3843(* {classic T} == Endow T : Type with a canonical eqType/choiceType. *)
3944(* This is intended for local use. *)
4045(* E.g., T : Type |- A : {fset {classic T}} *)
@@ -43,8 +48,9 @@ From mathcomp Require Import all_ssreflect.
4348(* {eclassic T} == Endow T : eqType with a canonical choiceType. *)
4449(* On the model of {classic _}. *)
4550(* See also the lemmas Peq and eqPchoice. *)
51+ (* ``` *)
4652(* *)
47- (* --> Functions into a porderType (resp. latticeType) are equipped with *)
53+ (* Functions into a porderType (resp. latticeType) are equipped with *)
4854(* a porderType (resp. latticeType), (f <= g)%O when f x <= g x for all x, *)
4955(* see lemma lefP. *)
5056(***************************************************************************** *)
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