diff options
37 files changed, 1131 insertions, 998 deletions
diff --git a/Makefile.build b/Makefile.build index 814305a2fe..3c32e5bcc2 100644 --- a/Makefile.build +++ b/Makefile.build @@ -865,9 +865,11 @@ endif # Dependencies of .v files +PLUGININCLUDES=$(addprefix -I plugins/, $(PLUGINDIRS)) + $(VDFILE).d: $(D_DEPEND_BEFORE_SRC) $(VFILES) $(D_DEPEND_AFTER_SRC) $(COQDEPBOOT) $(SHOW)'COQDEP VFILES' - $(HIDE)$(COQDEPBOOT) -vos -boot $(DYNDEP) -Q user-contrib "" $(USERCONTRIBINCLUDES) $(VFILES) $(TOTARGET) + $(HIDE)$(COQDEPBOOT) -vos -boot $(DYNDEP) -R theories Coq -R plugins Coq -Q user-contrib "" $(PLUGININCLUDES) $(USERCONTRIBINCLUDES) $(VFILES) $(TOTARGET) ########################################################################### diff --git a/checker/checkInductive.ml b/checker/checkInductive.ml index a2cf44389e..051f51bbb3 100644 --- a/checker/checkInductive.ml +++ b/checker/checkInductive.ml @@ -20,7 +20,7 @@ exception InductiveMismatch of MutInd.t * string let check mind field b = if not b then raise (InductiveMismatch (mind,field)) -let to_entry (mb:mutual_inductive_body) : Entries.mutual_inductive_entry = +let to_entry mind (mb:mutual_inductive_body) : Entries.mutual_inductive_entry = let open Entries in let nparams = List.length mb.mind_params_ctxt in (* include letins *) let mind_entry_record = match mb.mind_record with @@ -28,7 +28,27 @@ let to_entry (mb:mutual_inductive_body) : Entries.mutual_inductive_entry = | PrimRecord data -> Some (Some (Array.map (fun (x,_,_,_) -> x) data)) in let mind_entry_universes = match mb.mind_universes with - | Monomorphic univs -> Monomorphic_entry univs + | Monomorphic _ -> + (* We only need to rebuild the set of constraints for template polymorphic + inductive types. The set of monomorphic constraints is already part of + the graph at that point, but we need to emulate a broken bound variable + mechanism for template inductive types. *) + let fold accu ind = match ind.mind_arity with + | RegularArity _ -> accu + | TemplateArity ar -> + match accu with + | None -> Some ar.template_context + | Some ctx -> + (* Ensure that all template contexts agree. This is enforced by the + kernel. *) + let () = check mind "mind_arity" (ContextSet.equal ctx ar.template_context) in + Some ctx + in + let univs = match Array.fold_left fold None mb.mind_packets with + | None -> ContextSet.empty + | Some ctx -> ctx + in + Monomorphic_entry univs | Polymorphic auctx -> Polymorphic_entry (AUContext.names auctx, AUContext.repr auctx) in let mind_entry_inds = Array.map_to_list (fun ind -> @@ -69,8 +89,9 @@ let check_arity env ar1 ar2 = match ar1, ar2 with | RegularArity ar, RegularArity {mind_user_arity;mind_sort} -> Constr.equal ar.mind_user_arity mind_user_arity && Sorts.equal ar.mind_sort mind_sort - | TemplateArity ar, TemplateArity {template_param_levels;template_level} -> + | TemplateArity ar, TemplateArity {template_param_levels;template_level;template_context} -> List.equal (Option.equal Univ.Level.equal) ar.template_param_levels template_param_levels && + ContextSet.equal template_context ar.template_context && UGraph.check_leq (universes env) template_level ar.template_level (* template_level is inferred by indtypes, so functor application can produce a smaller one *) | (RegularArity _ | TemplateArity _), _ -> assert false @@ -136,7 +157,7 @@ let check_same_record r1 r2 = match r1, r2 with | (NotRecord | FakeRecord | PrimRecord _), _ -> false let check_inductive env mind mb = - let entry = to_entry mb in + let entry = to_entry mind mb in let { mind_packets; mind_record; mind_finite; mind_ntypes; mind_hyps; mind_nparams; mind_nparams_rec; mind_params_ctxt; mind_universes; mind_variance; mind_sec_variance; diff --git a/checker/values.ml b/checker/values.ml index fff166f27b..c8bbc092b4 100644 --- a/checker/values.ml +++ b/checker/values.ml @@ -228,7 +228,7 @@ let v_oracle = |] let v_pol_arity = - v_tuple "polymorphic_arity" [|List(Opt v_level);v_univ|] + v_tuple "polymorphic_arity" [|List(Opt v_level);v_univ;v_context_set|] let v_primitive = v_enum "primitive" 44 (* Number of "Primitive" in Int63.v and PrimFloat.v *) diff --git a/dev/ci/ci-basic-overlay.sh b/dev/ci/ci-basic-overlay.sh index 7342bc72e7..608cc127a0 100755 --- a/dev/ci/ci-basic-overlay.sh +++ b/dev/ci/ci-basic-overlay.sh @@ -97,11 +97,8 @@ ######################################################################## # Coquelicot ######################################################################## -# Modified until https://gitlab.inria.fr/coquelicot/coquelicot/merge_requests/2 is merged -: "${coquelicot_CI_REF:=fix-rlist-import}" -: "${coquelicot_CI_GITURL:=https://gitlab.inria.fr/pedrot/coquelicot}" -# : "${coquelicot_CI_REF:=master}" -# : "${coquelicot_CI_GITURL:=https://gitlab.inria.fr/coquelicot/coquelicot}" +: "${coquelicot_CI_REF:=master}" +: "${coquelicot_CI_GITURL:=https://gitlab.inria.fr/coquelicot/coquelicot}" : "${coquelicot_CI_ARCHIVEURL:=${coquelicot_CI_GITURL}/-/archive}" ######################################################################## diff --git a/doc/changelog/03-notations/11240-rew-dependent.rst b/doc/changelog/03-notations/11240-rew-dependent.rst new file mode 100644 index 0000000000..e9daab0c2c --- /dev/null +++ b/doc/changelog/03-notations/11240-rew-dependent.rst @@ -0,0 +1,5 @@ +- **Added** + Added :g:`rew dependent` notations for the dependent version of + :g:`rew` in :g:`Coq.Init.Logic.EqNotations` to improve the display + and parsing of :g:`match` statements on :g:`Logic.eq` (`#11240 + <https://github.com/coq/coq/pull/11240>`_, by Jason Gross). diff --git a/doc/changelog/05-tactic-language/10343-issue-10342-ltac2-standard-library.rst b/doc/changelog/05-tactic-language/10343-issue-10342-ltac2-standard-library.rst new file mode 100644 index 0000000000..4acc423d10 --- /dev/null +++ b/doc/changelog/05-tactic-language/10343-issue-10342-ltac2-standard-library.rst @@ -0,0 +1,4 @@ +- **Added:** + An array library for ltac2 (OCaml standard library compatible where possible). + (`#10343 <https://github.com/coq/coq/pull/10343>`_, + by Michael Soegtrop). diff --git a/doc/changelog/08-tools/11523-coqdep+refactor2.rst b/doc/changelog/08-tools/11523-coqdep+refactor2.rst new file mode 100644 index 0000000000..90c23d8b76 --- /dev/null +++ b/doc/changelog/08-tools/11523-coqdep+refactor2.rst @@ -0,0 +1,7 @@ +- **Changed:** + Internal options and behavior of ``coqdep`` have changed, in particular + options ``-w``, ``-D``, ``-mldep``, and ``-dumpbox`` have been removed, + and ``-boot`` will not load any path by default, ``-R/-Q`` should be + used instead + (`#11523 <https://github.com/coq/coq/pull/11523>`_, + by Emilio Jesus Gallego Arias). diff --git a/doc/changelog/10-standard-library/11404-removeRList.rst b/doc/changelog/10-standard-library/11404-removeRList.rst new file mode 100644 index 0000000000..88e22d128c --- /dev/null +++ b/doc/changelog/10-standard-library/11404-removeRList.rst @@ -0,0 +1,15 @@ +- **Removed:** + Type `RList` has been removed. All uses have been replaced by `list R`. + Functions from `RList` named `In`, `Rlength`, `cons_Rlist`, `app_Rlist` + have also been removed as they are essentially the same as `In`, `length`, + `app`, and `map` from `List`, modulo the following changes: + + - `RList.In x (RList.cons a l)` used to be convertible to + `(x = a) \\/ RList.In x l`, + but `List.In x (a :: l)` is convertible to + `(a = x) \\/ List.In l`. + The equality is reversed. + - `app_Rlist` and `List.map` take arguments in different order. + + (`#11404 <https://github.com/coq/coq/pull/11404>`_, + by Yves Bertot). diff --git a/doc/stdlib/index-list.html.template b/doc/stdlib/index-list.html.template index 5e13214a1a..b2ddf36b65 100644 --- a/doc/stdlib/index-list.html.template +++ b/doc/stdlib/index-list.html.template @@ -664,7 +664,6 @@ through the <tt>Require Import</tt> command.</p> </dt> <dd> theories/Compat/AdmitAxiom.v - theories/Compat/Coq89.v theories/Compat/Coq810.v theories/Compat/Coq811.v theories/Compat/Coq812.v @@ -25,7 +25,11 @@ (source_tree theories) (source_tree plugins) (source_tree user-contrib)) - (action (with-stdout-to .vfiles.d (bash "%{bin:coqdep} -dyndep both -noglob -boot `find theories plugins user-contrib -type f -name *.v`")))) + (action + (with-stdout-to .vfiles.d + (bash "%{bin:coqdep} -dyndep both -noglob -boot -R theories Coq -R plugins Coq -Q user-contrib/Ltac2 Ltac2 -I user-contrib/Ltac2 \ + `find plugins/ -maxdepth 1 -mindepth 1 -type d -printf '-I %p '` \ + `find theories plugins user-contrib -type f -name *.v`")))) (alias (name vodeps) diff --git a/kernel/context.ml b/kernel/context.ml index 7e394da2ed..500ed20343 100644 --- a/kernel/context.ml +++ b/kernel/context.ml @@ -196,12 +196,10 @@ struct (** Return a new rel-context enriched by with a given inner-most declaration. *) let add d ctx = d :: ctx - (** Return the number of {e local declarations} in a given context. *) + (** Return the number of {e local declarations} in a given rel-context. *) let length = List.length - (** [extended_rel_list n Γ] builds an instance [args] such that [Γ,Δ ⊢ args:Γ] - with n = |Δ| and with the local definitions of [Γ] skipped in - [args]. Example: for [x:T,y:=c,z:U] and [n]=2, it gives [Rel 5, Rel 3]. *) + (** Return the number of {e local assumptions} in a given rel-context. *) let nhyps ctx = let open Declaration in let rec nhyps acc = function @@ -413,7 +411,7 @@ struct (** empty named-context *) let empty = [] - (** empty named-context *) + (** Return a new named-context enriched by with a given inner-most declaration. *) let add d ctx = d :: ctx (** Return the number of {e local declarations} in a given named-context. *) diff --git a/kernel/context.mli b/kernel/context.mli index 8f233613da..04aa039a01 100644 --- a/kernel/context.mli +++ b/kernel/context.mli @@ -129,7 +129,7 @@ sig (** Return a new rel-context enriched by with a given inner-most declaration. *) val add : ('c, 't) Declaration.pt -> ('c, 't) pt -> ('c, 't) pt - (** Return the number of {e local declarations} in a given context. *) + (** Return the number of {e local declarations} in a given rel-context. *) val length : ('c, 't) pt -> int (** Check whether given two rel-contexts are equal. *) diff --git a/kernel/cooking.ml b/kernel/cooking.ml index cebbfe4986..f1eb000c88 100644 --- a/kernel/cooking.ml +++ b/kernel/cooking.ml @@ -312,14 +312,14 @@ let cook_one_ind ~template_check ~ntypes let arity = abstract_as_type (expmod arity) hyps in let sort = destSort (expmod (mkSort sort)) in RegularArity {mind_user_arity=arity; mind_sort=sort} - | TemplateArity {template_param_levels=levels;template_level} -> + | TemplateArity {template_param_levels=levels;template_level;template_context} -> let sec_levels = CList.map_filter (fun d -> if RelDecl.is_local_assum d then Some (template_level_of_var ~template_check d) else None) section_decls in let levels = List.rev_append sec_levels levels in - TemplateArity {template_param_levels=levels;template_level} + TemplateArity {template_param_levels=levels;template_level;template_context} in let mind_arity_ctxt = let ctx = Context.Rel.map expmod mip.mind_arity_ctxt in diff --git a/kernel/declarations.ml b/kernel/declarations.ml index 0b6e59bd5e..c550b0d432 100644 --- a/kernel/declarations.ml +++ b/kernel/declarations.ml @@ -32,6 +32,7 @@ type engagement = set_predicativity type template_arity = { template_param_levels : Univ.Level.t option list; template_level : Univ.Universe.t; + template_context : Univ.ContextSet.t; } type ('a, 'b) declaration_arity = diff --git a/kernel/declareops.ml b/kernel/declareops.ml index 27e3f84464..047027984d 100644 --- a/kernel/declareops.ml +++ b/kernel/declareops.ml @@ -49,7 +49,8 @@ let map_decl_arity f g = function let hcons_template_arity ar = { template_param_levels = ar.template_param_levels; (* List.Smart.map (Option.Smart.map Univ.hcons_univ_level) ar.template_param_levels; *) - template_level = Univ.hcons_univ ar.template_level } + template_level = Univ.hcons_univ ar.template_level; + template_context = Univ.hcons_universe_context_set ar.template_context } let universes_context = function | Monomorphic _ -> Univ.AUContext.empty diff --git a/kernel/indTyping.ml b/kernel/indTyping.ml index 719eb276df..113ee787f2 100644 --- a/kernel/indTyping.ml +++ b/kernel/indTyping.ml @@ -274,7 +274,7 @@ let abstract_packets ~template_check univs usubst params ((arity,lc),(indices,sp CErrors.user_err Pp.(strbrk "Ill-formed template inductive declaration: not polymorphic on any universe.") else - TemplateArity {template_param_levels = param_levels; template_level = min_univ} + TemplateArity {template_param_levels = param_levels; template_level = min_univ; template_context = ctx } in let kelim = allowed_sorts univ_info in diff --git a/man/coqdep.1 b/man/coqdep.1 index 02c9d4390c..4223482c99 100644 --- a/man/coqdep.1 +++ b/man/coqdep.1 @@ -6,9 +6,6 @@ coqdep \- Compute inter-module dependencies for Coq and Caml programs .SH SYNOPSIS .B coqdep [ -.BI \-w -] -[ .BI \-I \ directory ] [ @@ -21,9 +18,6 @@ coqdep \- Compute inter-module dependencies for Coq and Caml programs .BI \-i ] [ -.BI \-D -] -[ .BI \-slash ] .I filename ... @@ -61,25 +55,6 @@ directives and the dot notation .BI \-c Prints the dependencies of Caml modules. (On Caml modules, the behaviour is exactly the same as ocamldep). -\" THESE OPTIONS ARE BROKEN CURRENTLY -\" .TP -\" .BI \-w -\" Prints a warning if a Coq command -\" .IR Declare \& -\" .IR ML \& -\" .IR Module \& -\" is incorrect. (For instance, you wrote `Declare ML Module "A".', -\" but the module A contains #open "B"). The correct command is printed -\" (see option \-D). The warning is printed on standard error. -\" .TP -\" .BI \-D -\" This commands looks for every command -\" .IR Declare \& -\" .IR ML \& -\" .IR Module \& -\" of each Coq file given as argument and complete (if needed) -\" the list of Caml modules. The new command is printed on -\" the standard output. No dependency is computed with this option. .TP .BI \-f \ file Read filenames and options -I, -R and -Q from a _CoqProject FILE. @@ -93,10 +68,6 @@ Indicates where is the Coq library. The default value has been determined at installation time, and therefore this option should not be used under normal circumstances. .TP -.BI \-dumpgraph[box] \ file -Dumps a dot dependency graph in file -.IR file \&. -.TP .BI \-exclude-dir \ dir Skips subdirectory .IR dir \ during @@ -169,7 +140,7 @@ example% coqdep \-I . *.v With a warning: .IP .B -example% coqdep \-w \-I . *.v +example% coqdep \-I . *.v .RS .sp .5 .nf diff --git a/test-suite/bugs/closed/bug_11515.v b/test-suite/bugs/closed/bug_11515.v new file mode 100644 index 0000000000..fe4ba87447 --- /dev/null +++ b/test-suite/bugs/closed/bug_11515.v @@ -0,0 +1,7 @@ +Require Import Ltac2.Ltac2. + +Lemma foo : + True. +Proof. + Fail rewrite _. +Abort. diff --git a/test-suite/ltac2/array_lib.v b/test-suite/ltac2/array_lib.v new file mode 100644 index 0000000000..31227eaddb --- /dev/null +++ b/test-suite/ltac2/array_lib.v @@ -0,0 +1,181 @@ +Require Import Ltac2.Ltac2. +Import Ltac2.Message. +Import Ltac2.Array. +Require Ltac2.List. +Require Ltac2.Int. + +(* Array/List comparison functions which throw an exception on unequal *) + +Ltac2 Type exn ::= [ Regression_Test_Failure ]. + +Ltac2 check_eq_int a l := + List.iter2 + (fun a b => match Int.equal a b with true => () | false => Control.throw Regression_Test_Failure end) + (to_list a) l. + +Ltac2 check_eq_bool a l := + List.iter2 + (fun a b => match Bool.eq a b with true => () | false => Control.throw Regression_Test_Failure end) + (to_list a) l. + +Ltac2 check_eq_int_matrix m ll := + List.iter2 (fun a b => check_eq_int a b) (to_list m) ll. + +Ltac2 check_eq_bool_matrix m ll := + List.iter2 (fun a b => check_eq_bool a b) (to_list m) ll. + +(* The below printing functions are mostly for debugging below test cases *) + +Ltac2 print2 m1 m2 := print (Message.concat m1 m2). +Ltac2 print3 m1 m2 m3 := print2 m1 (Message.concat m2 m3). + +Ltac2 print_int_array a := + iteri (fun i x => print3 (of_int i) (of_string "=") (of_int x)) a. + +Ltac2 of_bool b := match b with true=>of_string "true" | false=>of_string "false" end. + +Ltac2 print_bool_array a := + iteri (fun i x => print3 (of_int i) (of_string "=") (of_bool x)) a. + +Ltac2 print_int_list a := + List.iteri (fun i x => print3 (of_int i) (of_string "=") (of_int x)) a. + +Goal True. + (* Test failure *) + Fail check_eq_int ((init 3 (fun i => (Int.add i 10)))) [10;11;13]. + + (* test empty with int *) + check_eq_int (empty ()) []. + check_eq_int (append (empty ()) (init 3 (fun i => (Int.add i 10)))) [10;11;12]. + check_eq_int (append (init 3 (fun i => (Int.add i 10))) (empty ())) [10;11;12]. + + (* test empty with bool *) + check_eq_bool (empty ()) []. + check_eq_bool (append (empty ()) (init 3 (fun i => (Int.ge i 2)))) [false;false;true]. + check_eq_bool (append (init 3 (fun i => (Int.ge i 2))) (empty ())) [false;false;true]. + + (* test init with int *) + check_eq_int (init 0 (fun i => (Int.add i 10))) []. + check_eq_int (init 4 (fun i => (Int.add i 10))) [10;11;12;13]. + + (* test init with bool *) + check_eq_bool (init 0 (fun i => (Int.ge i 2))) []. + check_eq_bool (init 4 (fun i => (Int.ge i 2))) [false;false;true;true]. + + (* test make_matrix, set, get *) + let a := make_matrix 4 3 1 in + Array.set (Array.get a 1) 2 0; + check_eq_int_matrix a [[1;1;1];[1;1;0];[1;1;1];[1;1;1]]. + + let a := make_matrix 3 4 false in + Array.set (Array.get a 2) 1 true; + check_eq_bool_matrix a [[false;false;false;false];[false;false;false;false];[false;true;false;false]]. + + (* test copy *) + let a := init 4 (fun i => (Int.add i 10)) in + let b := copy a in + check_eq_int b [10;11;12;13]. + + (* test append *) + let a := init 3 (fun i => (Int.add i 10)) in + let b := init 4 (fun i => (Int.add i 20)) in + check_eq_int (append a b) [10;11;12;20;21;22;23]. + + (* test concat *) + let a := init 3 (fun i => (Int.add i 10)) in + let b := init 4 (fun i => (Int.add i 20)) in + let c := init 5 (fun i => (Int.add i 30)) in + check_eq_int (concat (a::b::c::[])) [10;11;12;20;21;22;23;30;31;32;33;34]. + + (* test sub *) + let a := init 10 (fun i => (Int.add i 10)) in + let b := (sub a 3 0) in + let c := (append b (init 3 (fun i => (Int.add i 10)))) in + check_eq_int b []; + check_eq_int c [10;11;12]. + + let a := init 10 (fun i => (Int.add i 10)) in + let b := (sub a 3 4) in + check_eq_int b [13;14;15;16]. + + (* test fill *) + let a := init 10 (fun i => (Int.add i 10)) in + fill a 3 4 0; + check_eq_int a [10;11;12;0;0;0;0;17;18;19]. + + (* test blit *) + let a := init 10 (fun i => (Int.add i 10)) in + let b := init 10 (fun i => (Int.add i 20)) in + blit a 5 b 3 4; + check_eq_int b [20;21;22;15;16;17;18;27;28;29]. + + (* test iter *) + let a := init 4 (fun i => (Int.add i 3)) in + let b := init 10 (fun i => 10) in + iter (fun x => Array.set b x x) a; + check_eq_int b [10;10;10;3;4;5;6;10;10;10]. + + (* test iter2 *) + let a := init 4 (fun i => (Int.add i 2)) in + let b := init 4 (fun i => (Int.add i 4)) in + let c := init 8 (fun i => 10) in + iter2 (fun x y => Array.set c x y) a b; + check_eq_int c [10;10;4;5;6;7;10;10]. + + (* test map *) + let a := init 4 (fun i => (Int.add i 10)) in + check_eq_bool (map (fun i => (Int.ge i 12)) a) [false;false;true;true]. + + (* test map2 *) + let a := init 4 (fun i => (Int.add 10 i)) in + let b := init 4 (fun i => (Int.sub 13 i)) in + check_eq_bool (map2 (fun x y => (Int.ge x y)) a b) [false;false;true;true]. + + (* test iteri *) + let a := init 4 (fun i => (Int.add i 10)) in + let m := make_matrix 4 2 100 in + iteri (fun i x => Array.set (Array.get m i) 0 i; Array.set (Array.get m i) 1 x) a; + check_eq_int_matrix m [[0;10];[1;11];[2;12];[3;13]]. + + (* test mapi *) + let a := init 4 (fun i => (Int.sub 3 i)) in + check_eq_bool (mapi (fun i x => (Int.ge i x)) a) [false;false;true;true]. + + (* to_list is already tested in check_eq_... *) + + (* test of_list *) + check_eq_int (of_list ([0;1;2;3])) [0;1;2;3]. + + (* test fold_left *) + let a := init 4 (fun i => (Int.add 10 i)) in + check_eq_int (of_list (fold_left (fun a b => b::a) [] a)) [13;12;11;10]. + + (* test fold_right *) + let a := init 4 (fun i => (Int.add 10 i)) in + check_eq_int (of_list (fold_right (fun a b => b::a) [] a)) [10;11;12;13]. + + (* test exist *) + let a := init 4 (fun i => (Int.add 10 i)) in + let l := [ + exist (fun x => Int.equal x 10) a; + exist (fun x => Int.equal x 13) a; + exist (fun x => Int.equal x 14) a] in + check_eq_bool (of_list l) [true;true;false]. + + (* test for_all *) + let a := init 4 (fun i => (Int.add 10 i)) in + let l := [ + for_all (fun x => Int.lt x 14) a; + for_all (fun x => Int.lt x 13) a] in + check_eq_bool (of_list l) [true;false]. + + (* test mem *) + let a := init 4 (fun i => (Int.add 10 i)) in + let l := [ + mem Int.equal 10 a; + mem Int.equal 13 a; + mem Int.equal 14 a] in + check_eq_bool (of_list l) [true;true;false]. + +exact I. +Qed. diff --git a/test-suite/output/Notations.out b/test-suite/output/Notations.out index 94b86fc222..b870fa6f6f 100644 --- a/test-suite/output/Notations.out +++ b/test-suite/output/Notations.out @@ -137,3 +137,71 @@ end = p : forall x : nat, x = x -> Prop bar 0 : nat +let k := rew [P] p in v in k + : P y +let k := rew [P] p in v in k + : P y +let k := rew <- [P] p in v' in k + : P x +let k := rew [P] p in v in k + : P y +let k := rew [P] p in v in k + : P y +let k := rew <- [P] p in v' in k + : P x +let k := rew [fun y : A => P y] p in v in k + : P y +let k := rew [fun y : A => P y] p in v in k + : P y +let k := rew <- [fun y : A => P y] p in v' in k + : P x +let k := rew [fun y : A => P y] p in v in k + : P y +let k := rew [fun y : A => P y] p in v in k + : P y +let k := rew <- [fun y : A => P y] p in v' in k + : P x +let k := rew dependent [P] p in v in k + : P y p +let k := rew dependent [P] p in v in k + : P y p +let k := rew dependent <- [P'] p in v' in k + : P' x (eq_sym p) +let k := rew dependent [P] p in v in k + : P y p +let k := rew dependent [P] p in v in k + : P y p +let k := rew dependent <- [P'] p in v' in k + : P' x (eq_sym p) +let k := rew dependent [P] p in v in k + : P y p +let k := rew dependent [P] p in v in k + : P y p +let k := rew dependent <- [P'] p in v' in k + : P' x (eq_sym p) +let k := rew dependent [fun y p => id (P y p)] p in v in k + : P y p +let k := rew dependent [fun y p => id (P y p)] p in v in k + : P y p +let k := rew dependent <- [fun y0 p => id (P' y0 p)] p in v' in k + : P' x (eq_sym p) +let k := rew dependent [P] p in v in k + : P y p +let k := rew dependent [P] p in v in k + : P y p +let k := rew dependent <- [P'] p in v' in k + : P' x (eq_sym p) +let k := rew dependent [fun y p0 => id (P y p0)] p in v in k + : P y p +let k := rew dependent [fun y p0 => id (P y p0)] p in v in k + : P y p +let k := rew dependent <- [fun y0 p0 => id (P' y0 p0)] p in v' in k + : P' x (eq_sym p) +rew dependent [P] p in v + : P y p +rew dependent <- [P'] p in v' + : P' x (eq_sym p) +rew dependent [fun a x => id (P a x)] p in v + : id (P y p) +rew dependent <- [fun a p' => id (P' a p')] p in v' + : id (P' x (eq_sym p)) diff --git a/test-suite/output/Notations.v b/test-suite/output/Notations.v index adab324cf0..7d2f1e9ba8 100644 --- a/test-suite/output/Notations.v +++ b/test-suite/output/Notations.v @@ -251,11 +251,11 @@ Notation NONE := None. Check (fun x => match x with SOME x => x | NONE => 0 end). Notation NONE2 := (@None _). -Notation SOME2 := (@Some _). +Notation SOME2 := (@Some _). Check (fun x => match x with SOME2 x => x | NONE2 => 0 end). Notation NONE3 := @None. -Notation SOME3 := @Some. +Notation SOME3 := @Some. Check (fun x => match x with SOME3 _ x => x | NONE3 _ => 0 end). Notation "a :'" := (cons a) (at level 12). @@ -300,3 +300,61 @@ Definition bar (a b : nat) := plus a b. Notation "" := A (format "", only printing). Check (bar A 0). End M. + +(* Check eq notations *) +Module EqNotationsCheck. + Import EqNotations. + Section nd. + Context (A : Type) (x : A) (P : A -> Type) + (y : A) (p : x = y) (v : P x) (v' : P y). + + Check let k : P y := rew p in v in k. + Check let k : P y := rew -> p in v in k. + Check let k : P x := rew <- p in v' in k. + Check let k : P y := rew [P] p in v in k. + Check let k : P y := rew -> [P] p in v in k. + Check let k : P x := rew <- [P] p in v' in k. + Check let k : P y := rew [fun y => P y] p in v in k. + Check let k : P y := rew -> [fun y => P y] p in v in k. + Check let k : P x := rew <- [fun y => P y] p in v' in k. + Check let k : P y := rew [fun (y : A) => P y] p in v in k. + Check let k : P y := rew -> [fun (y : A) => P y] p in v in k. + Check let k : P x := rew <- [fun (y : A) => P y] p in v' in k. + End nd. + Section dep. + Context (A : Type) (x : A) (P : forall y, x = y -> Type) + (y : A) (p : x = y) (P' : forall x, y = x -> Type) + (v : P x eq_refl) (v' : P' y eq_refl). + + Check let k : P y p := rew dependent p in v in k. + Check let k : P y p := rew dependent -> p in v in k. + Check let k : P' x (eq_sym p) := rew dependent <- p in v' in k. + Check let k : P y p := rew dependent [P] p in v in k. + Check let k : P y p := rew dependent -> [P] p in v in k. + Check let k : P' x (eq_sym p) := rew dependent <- [P'] p in v' in k. + Check let k : P y p := rew dependent [fun y p => P y p] p in v in k. + Check let k : P y p := rew dependent -> [fun y p => P y p] p in v in k. + Check let k : P' x (eq_sym p) := rew dependent <- [fun y p => P' y p] p in v' in k. + Check let k : P y p := rew dependent [fun y p => id (P y p)] p in v in k. + Check let k : P y p := rew dependent -> [fun y p => id (P y p)] p in v in k. + Check let k : P' x (eq_sym p) := rew dependent <- [fun y p => id (P' y p)] p in v' in k. + Check let k : P y p := rew dependent [(fun (y : A) (p : x = y) => P y p)] p in v in k. + Check let k : P y p := rew dependent -> [(fun (y : A) (p : x = y) => P y p)] p in v in k. + Check let k : P' x (eq_sym p) := rew dependent <- [(fun (x : A) (p : y = x) => P' x p)] p in v' in k. + Check let k : P y p := rew dependent [(fun (y : A) (p : x = y) => id (P y p))] p in v in k. + Check let k : P y p := rew dependent -> [(fun (y : A) (p : x = y) => id (P y p))] p in v in k. + Check let k : P' x (eq_sym p) := rew dependent <- [(fun (x : A) (p : y = x) => id (P' x p))] p in v' in k. + Check match p as x in _ = a return P a x with + | eq_refl => v + end. + Check match eq_sym p as p' in _ = a return P' a p' with + | eq_refl => v' + end. + Check match p as x in _ = a return id (P a x) with + | eq_refl => v + end. + Check match eq_sym p as p' in _ = a return id (P' a p') with + | eq_refl => v' + end. + End dep. +End EqNotationsCheck. diff --git a/test-suite/success/CompatOldOldFlag.v b/test-suite/success/CompatOldOldFlag.v deleted file mode 100644 index dd259988ac..0000000000 --- a/test-suite/success/CompatOldOldFlag.v +++ /dev/null @@ -1,6 +0,0 @@ -(* -*- coq-prog-args: ("-compat" "8.9") -*- *) -(** Check that the current-minus-three compatibility flag actually requires the relevant modules. *) -Import Coq.Compat.Coq812. -Import Coq.Compat.Coq811. -Import Coq.Compat.Coq810. -Import Coq.Compat.Coq89. diff --git a/test-suite/tools/update-compat/run.sh b/test-suite/tools/update-compat/run.sh index 61273c4f37..7ff5571ffb 100755 --- a/test-suite/tools/update-compat/run.sh +++ b/test-suite/tools/update-compat/run.sh @@ -6,4 +6,4 @@ SCRIPT_DIR="$( cd "$( dirname "${BASH_SOURCE[0]}" )" >/dev/null && pwd )" # we assume that the script lives in test-suite/tools/update-compat/, # and that update-compat.py lives in dev/tools/ cd "${SCRIPT_DIR}/../../.." -dev/tools/update-compat.py --assert-unchanged --master || exit $? +dev/tools/update-compat.py --assert-unchanged --release || exit $? diff --git a/theories/Compat/Coq89.v b/theories/Compat/Coq89.v deleted file mode 100644 index 274cb4afd3..0000000000 --- a/theories/Compat/Coq89.v +++ /dev/null @@ -1,19 +0,0 @@ -(************************************************************************) -(* * The Coq Proof Assistant / The Coq Development Team *) -(* v * INRIA, CNRS and contributors - Copyright 1999-2019 *) -(* <O___,, * (see CREDITS file for the list of authors) *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(* * (see LICENSE file for the text of the license) *) -(************************************************************************) - -(** Compatibility file for making Coq act similar to Coq v8.9 *) -Local Set Warnings "-deprecated". - -Require Export Coq.Compat.Coq810. - -Unset Private Polymorphic Universes. - -(** Unsafe flag, can hide inconsistencies. *) -Global Unset Template Check. diff --git a/theories/Init/Logic.v b/theories/Init/Logic.v index 4d84d61f9f..8ba17e38c8 100644 --- a/theories/Init/Logic.v +++ b/theories/Init/Logic.v @@ -460,6 +460,58 @@ Module EqNotations. Notation "'rew' -> [ P ] H 'in' H'" := (eq_rect _ P H' _ H) (at level 10, H' at level 10, only parsing). + Notation "'rew' 'dependent' H 'in' H'" + := (match H with + | eq_refl => H' + end) + (at level 10, H' at level 10, + format "'[' 'rew' 'dependent' H in '/' H' ']'"). + Notation "'rew' 'dependent' -> H 'in' H'" + := (match H with + | eq_refl => H' + end) + (at level 10, H' at level 10, only parsing). + Notation "'rew' 'dependent' <- H 'in' H'" + := (match eq_sym H with + | eq_refl => H' + end) + (at level 10, H' at level 10, + format "'[' 'rew' 'dependent' <- H in '/' H' ']'"). + Notation "'rew' 'dependent' [ 'fun' y p => P ] H 'in' H'" + := (match H as p in (_ = y) return P with + | eq_refl => H' + end) + (at level 10, H' at level 10, y ident, p ident, + format "'[' 'rew' 'dependent' [ 'fun' y p => P ] '/ ' H in '/' H' ']'"). + Notation "'rew' 'dependent' -> [ 'fun' y p => P ] H 'in' H'" + := (match H as p in (_ = y) return P with + | eq_refl => H' + end) + (at level 10, H' at level 10, y ident, p ident, only parsing). + Notation "'rew' 'dependent' <- [ 'fun' y p => P ] H 'in' H'" + := (match eq_sym H as p in (_ = y) return P with + | eq_refl => H' + end) + (at level 10, H' at level 10, y ident, p ident, + format "'[' 'rew' 'dependent' <- [ 'fun' y p => P ] '/ ' H in '/' H' ']'"). + Notation "'rew' 'dependent' [ P ] H 'in' H'" + := (match H as p in (_ = y) return P y p with + | eq_refl => H' + end) + (at level 10, H' at level 10, + format "'[' 'rew' 'dependent' [ P ] '/ ' H in '/' H' ']'"). + Notation "'rew' 'dependent' -> [ P ] H 'in' H'" + := (match H as p in (_ = y) return P y p with + | eq_refl => H' + end) + (at level 10, H' at level 10, + only parsing). + Notation "'rew' 'dependent' <- [ P ] H 'in' H'" + := (match eq_sym H as p in (_ = y) return P y p with + | eq_refl => H' + end) + (at level 10, H' at level 10, + format "'[' 'rew' 'dependent' <- [ P ] '/ ' H in '/' H' ']'"). End EqNotations. Import EqNotations. @@ -793,13 +845,6 @@ Qed. Declare Left Step iff_stepl. Declare Right Step iff_trans. -Local Notation "'rew' 'dependent' H 'in' H'" - := (match H with - | eq_refl => H' - end) - (at level 10, H' at level 10, - format "'[' 'rew' 'dependent' '/ ' H in '/' H' ']'"). - (** Equality for [ex] *) Section ex. Local Unset Implicit Arguments. diff --git a/theories/Reals/RList.v b/theories/Reals/RList.v index 128543d8ab..18cc3aa034 100644 --- a/theories/Reals/RList.v +++ b/theories/Reals/RList.v @@ -8,98 +8,90 @@ (* * (see LICENSE file for the text of the license) *) (************************************************************************) +Require Import List. Require Import Rbase. Require Import Rfunctions. Local Open Scope R_scope. -Inductive Rlist : Type := -| nil : Rlist -| cons : R -> Rlist -> Rlist. -Fixpoint In (x:R) (l:Rlist) : Prop := - match l with - | nil => False - | cons a l' => x = a \/ In x l' - end. +#[deprecated(since="8.12",note="use (list R) instead")] +Notation Rlist := (list R). -Fixpoint Rlength (l:Rlist) : nat := - match l with - | nil => 0%nat - | cons a l' => S (Rlength l') - end. +#[deprecated(since="8.12",note="use List.length instead")] +Notation Rlength := List.length. -Fixpoint MaxRlist (l:Rlist) : R := +Fixpoint MaxRlist (l:list R) : R := match l with | nil => 0 - | cons a l1 => + | a :: l1 => match l1 with | nil => a - | cons a' l2 => Rmax a (MaxRlist l1) + | a' :: l2 => Rmax a (MaxRlist l1) end end. -Fixpoint MinRlist (l:Rlist) : R := +Fixpoint MinRlist (l:list R) : R := match l with | nil => 1 - | cons a l1 => + | a :: l1 => match l1 with | nil => a - | cons a' l2 => Rmin a (MinRlist l1) + | a' :: l2 => Rmin a (MinRlist l1) end end. -Lemma MaxRlist_P1 : forall (l:Rlist) (x:R), In x l -> x <= MaxRlist l. +Lemma MaxRlist_P1 : forall (l:list R) (x:R), In x l -> x <= MaxRlist l. Proof. intros; induction l as [| r l Hrecl]. simpl in H; elim H. induction l as [| r0 l Hrecl0]. simpl in H; elim H; intro. - simpl; right; assumption. + simpl; right; symmetry; assumption. elim H0. - replace (MaxRlist (cons r (cons r0 l))) with (Rmax r (MaxRlist (cons r0 l))). + replace (MaxRlist (r :: r0 :: l)) with (Rmax r (MaxRlist (r0 :: l))). simpl in H; decompose [or] H. rewrite H0; apply RmaxLess1. - unfold Rmax; case (Rle_dec r (MaxRlist (cons r0 l))); intro. + unfold Rmax; case (Rle_dec r (MaxRlist (r0 :: l))); intro. apply Hrecl; simpl; tauto. - apply Rle_trans with (MaxRlist (cons r0 l)); + apply Rle_trans with (MaxRlist (r0 :: l)); [ apply Hrecl; simpl; tauto | left; auto with real ]. - unfold Rmax; case (Rle_dec r (MaxRlist (cons r0 l))); intro. + unfold Rmax; case (Rle_dec r (MaxRlist (r0 :: l))); intro. apply Hrecl; simpl; tauto. - apply Rle_trans with (MaxRlist (cons r0 l)); + apply Rle_trans with (MaxRlist (r0 :: l)); [ apply Hrecl; simpl; tauto | left; auto with real ]. reflexivity. Qed. -Fixpoint AbsList (l:Rlist) (x:R) : Rlist := +Fixpoint AbsList (l:list R) (x:R) : list R := match l with | nil => nil - | cons a l' => cons (Rabs (a - x) / 2) (AbsList l' x) + | a :: l' => (Rabs (a - x) / 2) :: (AbsList l' x) end. -Lemma MinRlist_P1 : forall (l:Rlist) (x:R), In x l -> MinRlist l <= x. +Lemma MinRlist_P1 : forall (l:list R) (x:R), In x l -> MinRlist l <= x. Proof. intros; induction l as [| r l Hrecl]. simpl in H; elim H. induction l as [| r0 l Hrecl0]. simpl in H; elim H; intro. - simpl; right; symmetry ; assumption. + simpl; right; assumption. elim H0. - replace (MinRlist (cons r (cons r0 l))) with (Rmin r (MinRlist (cons r0 l))). + replace (MinRlist (r :: r0 :: l)) with (Rmin r (MinRlist (r0 :: l))). simpl in H; decompose [or] H. rewrite H0; apply Rmin_l. - unfold Rmin; case (Rle_dec r (MinRlist (cons r0 l))); intro. - apply Rle_trans with (MinRlist (cons r0 l)). + unfold Rmin; case (Rle_dec r (MinRlist (r0 :: l))); intro. + apply Rle_trans with (MinRlist (r0 :: l)). assumption. apply Hrecl; simpl; tauto. apply Hrecl; simpl; tauto. - apply Rle_trans with (MinRlist (cons r0 l)). + apply Rle_trans with (MinRlist (r0 :: l)). apply Rmin_r. apply Hrecl; simpl; tauto. reflexivity. Qed. Lemma AbsList_P1 : - forall (l:Rlist) (x y:R), In y l -> In (Rabs (y - x) / 2) (AbsList l x). + forall (l:list R) (x y:R), In y l -> In (Rabs (y - x) / 2) (AbsList l x). Proof. intros; induction l as [| r l Hrecl]. elim H. @@ -109,21 +101,21 @@ Proof. Qed. Lemma MinRlist_P2 : - forall l:Rlist, (forall y:R, In y l -> 0 < y) -> 0 < MinRlist l. + forall l:list R, (forall y:R, In y l -> 0 < y) -> 0 < MinRlist l. Proof. intros; induction l as [| r l Hrecl]. apply Rlt_0_1. induction l as [| r0 l Hrecl0]. simpl; apply H; simpl; tauto. - replace (MinRlist (cons r (cons r0 l))) with (Rmin r (MinRlist (cons r0 l))). - unfold Rmin; case (Rle_dec r (MinRlist (cons r0 l))); intro. + replace (MinRlist (r :: r0 :: l)) with (Rmin r (MinRlist (r0 :: l))). + unfold Rmin; case (Rle_dec r (MinRlist (r0 :: l))); intro. apply H; simpl; tauto. apply Hrecl; intros; apply H; simpl; simpl in H0; tauto. reflexivity. Qed. Lemma AbsList_P2 : - forall (l:Rlist) (x y:R), + forall (l:list R) (x y:R), In y (AbsList l x) -> exists z : R, In z l /\ y = Rabs (z - x) / 2. Proof. intros; induction l as [| r l Hrecl]. @@ -131,47 +123,48 @@ Proof. elim H; intro. exists r; split. simpl; tauto. + symmetry. assumption. assert (H1 := Hrecl H0); elim H1; intros; elim H2; clear H2; intros; exists x0; simpl; simpl in H2; tauto. Qed. Lemma MaxRlist_P2 : - forall l:Rlist, (exists y : R, In y l) -> In (MaxRlist l) l. + forall l:list R, (exists y : R, In y l) -> In (MaxRlist l) l. Proof. intros; induction l as [| r l Hrecl]. simpl in H; elim H; trivial. induction l as [| r0 l Hrecl0]. simpl; left; reflexivity. - change (In (Rmax r (MaxRlist (cons r0 l))) (cons r (cons r0 l))); - unfold Rmax; case (Rle_dec r (MaxRlist (cons r0 l))); + change (In (Rmax r (MaxRlist (r0 :: l))) (r :: r0 :: l)); + unfold Rmax; case (Rle_dec r (MaxRlist (r0 :: l))); intro. right; apply Hrecl; exists r0; left; reflexivity. left; reflexivity. Qed. -Fixpoint pos_Rl (l:Rlist) (i:nat) : R := +Fixpoint pos_Rl (l:list R) (i:nat) : R := match l with | nil => 0 - | cons a l' => match i with + | a :: l' => match i with | O => a | S i' => pos_Rl l' i' end end. Lemma pos_Rl_P1 : - forall (l:Rlist) (a:R), - (0 < Rlength l)%nat -> - pos_Rl (cons a l) (Rlength l) = pos_Rl l (pred (Rlength l)). + forall (l:list R) (a:R), + (0 < length l)%nat -> + pos_Rl (a :: l) (length l) = pos_Rl l (pred (length l)). Proof. intros; induction l as [| r l Hrecl]; [ elim (lt_n_O _ H) - | simpl; case (Rlength l); [ reflexivity | intro; reflexivity ] ]. + | simpl; case (length l); [ reflexivity | intro; reflexivity ] ]. Qed. Lemma pos_Rl_P2 : - forall (l:Rlist) (x:R), - In x l <-> (exists i : nat, (i < Rlength l)%nat /\ x = pos_Rl l i). + forall (l:list R) (x:R), + In x l <-> (exists i : nat, (i < length l)%nat /\ x = pos_Rl l i). Proof. intros; induction l as [| r l Hrecl]. split; intro; @@ -179,12 +172,12 @@ Proof. split; intro. elim H; intro. exists 0%nat; split; - [ simpl; apply lt_O_Sn | simpl; apply H0 ]. + [ simpl; apply lt_O_Sn | simpl; symmetry; apply H0 ]. elim Hrecl; intros; assert (H3 := H1 H0); elim H3; intros; elim H4; intros; exists (S x0); split; [ simpl; apply lt_n_S; assumption | simpl; assumption ]. elim H; intros; elim H0; intros; destruct (zerop x0) as [->|]. - simpl in H2; left; assumption. + simpl in H2; left; symmetry; assumption. right; elim Hrecl; intros H4 H5; apply H5; assert (H6 : S (pred x0) = x0). symmetry ; apply S_pred with 0%nat; assumption. exists (pred x0); split; @@ -193,21 +186,21 @@ Proof. Qed. Lemma Rlist_P1 : - forall (l:Rlist) (P:R -> R -> Prop), + forall (l:list R) (P:R -> R -> Prop), (forall x:R, In x l -> exists y : R, P x y) -> - exists l' : Rlist, - Rlength l = Rlength l' /\ - (forall i:nat, (i < Rlength l)%nat -> P (pos_Rl l i) (pos_Rl l' i)). + exists l' : list R, + length l = length l' /\ + (forall i:nat, (i < length l)%nat -> P (pos_Rl l i) (pos_Rl l' i)). Proof. intros; induction l as [| r l Hrecl]. exists nil; intros; split; [ reflexivity | intros; simpl in H0; elim (lt_n_O _ H0) ]. - assert (H0 : In r (cons r l)). + assert (H0 : In r (r :: l)). simpl; left; reflexivity. assert (H1 := H _ H0); assert (H2 : forall x:R, In x l -> exists y : R, P x y). intros; apply H; simpl; right; assumption. - assert (H3 := Hrecl H2); elim H1; intros; elim H3; intros; exists (cons x x0); + assert (H3 := Hrecl H2); elim H1; intros; elim H3; intros; exists (x :: x0); intros; elim H5; clear H5; intros; split. simpl; rewrite H5; reflexivity. intros; destruct (zerop i) as [->|]. @@ -218,57 +211,45 @@ Proof. assumption. Qed. -Definition ordered_Rlist (l:Rlist) : Prop := - forall i:nat, (i < pred (Rlength l))%nat -> pos_Rl l i <= pos_Rl l (S i). +Definition ordered_Rlist (l:list R) : Prop := + forall i:nat, (i < pred (length l))%nat -> pos_Rl l i <= pos_Rl l (S i). -Fixpoint insert (l:Rlist) (x:R) : Rlist := +Fixpoint insert (l:list R) (x:R) : list R := match l with - | nil => cons x nil - | cons a l' => + | nil => x :: nil + | a :: l' => match Rle_dec a x with - | left _ => cons a (insert l' x) - | right _ => cons x l + | left _ => a :: (insert l' x) + | right _ => x :: l end end. -Fixpoint cons_Rlist (l k:Rlist) : Rlist := - match l with - | nil => k - | cons a l' => cons a (cons_Rlist l' k) - end. - -Fixpoint cons_ORlist (k l:Rlist) : Rlist := +Fixpoint cons_ORlist (k l:list R) : list R := match k with | nil => l - | cons a k' => cons_ORlist k' (insert l a) + | a :: k' => cons_ORlist k' (insert l a) end. -Fixpoint app_Rlist (l:Rlist) (f:R -> R) : Rlist := +Fixpoint mid_Rlist (l:list R) (x:R) : list R := match l with | nil => nil - | cons a l' => cons (f a) (app_Rlist l' f) + | a :: l' => ((x + a) / 2) :: (mid_Rlist l' a) end. -Fixpoint mid_Rlist (l:Rlist) (x:R) : Rlist := +Definition Rtail (l:list R) : list R := match l with | nil => nil - | cons a l' => cons ((x + a) / 2) (mid_Rlist l' a) + | a :: l' => l' end. -Definition Rtail (l:Rlist) : Rlist := +Definition FF (l:list R) (f:R -> R) : list R := match l with | nil => nil - | cons a l' => l' - end. - -Definition FF (l:Rlist) (f:R -> R) : Rlist := - match l with - | nil => nil - | cons a l' => app_Rlist (mid_Rlist l' a) f + | a :: l' => map f (mid_Rlist l' a) end. Lemma RList_P0 : - forall (l:Rlist) (a:R), + forall (l:list R) (a:R), pos_Rl (insert l a) 0 = a \/ pos_Rl (insert l a) 0 = pos_Rl l 0. Proof. intros; induction l as [| r l Hrecl]; @@ -278,7 +259,7 @@ Proof. Qed. Lemma RList_P1 : - forall (l:Rlist) (a:R), ordered_Rlist l -> ordered_Rlist (insert l a). + forall (l:list R) (a:R), ordered_Rlist l -> ordered_Rlist (insert l a). Proof. intros; induction l as [| r l Hrecl]. simpl; unfold ordered_Rlist; intros; simpl in H0; @@ -286,8 +267,8 @@ Proof. simpl; case (Rle_dec r a); intro. assert (H1 : ordered_Rlist l). unfold ordered_Rlist; unfold ordered_Rlist in H; intros; - assert (H1 : (S i < pred (Rlength (cons r l)))%nat); - [ simpl; replace (Rlength l) with (S (pred (Rlength l))); + assert (H1 : (S i < pred (length (r :: l)))%nat); + [ simpl; replace (length l) with (S (pred (length l))); [ apply lt_n_S; assumption | symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red; intro; rewrite <- H1 in H0; simpl in H0; elim (lt_n_O _ H0) ] @@ -300,18 +281,18 @@ Proof. [ simpl; assumption | rewrite H4; apply (H 0%nat); simpl; apply lt_O_Sn ]. simpl; apply H2; simpl in H0; apply lt_S_n; - replace (S (pred (Rlength (insert l a)))) with (Rlength (insert l a)); + replace (S (pred (length (insert l a)))) with (length (insert l a)); [ assumption | apply S_pred with 0%nat; apply neq_O_lt; red; intro; rewrite <- H3 in H0; elim (lt_n_O _ H0) ]. unfold ordered_Rlist; intros; induction i as [| i Hreci]; [ simpl; auto with real - | change (pos_Rl (cons r l) i <= pos_Rl (cons r l) (S i)); apply H; + | change (pos_Rl (r :: l) i <= pos_Rl (r :: l) (S i)); apply H; simpl in H0; simpl; apply (lt_S_n _ _ H0) ]. Qed. Lemma RList_P2 : - forall l1 l2:Rlist, ordered_Rlist l2 -> ordered_Rlist (cons_ORlist l1 l2). + forall l1 l2:list R, ordered_Rlist l2 -> ordered_Rlist (cons_ORlist l1 l2). Proof. simple induction l1; [ intros; simpl; apply H @@ -319,36 +300,36 @@ Proof. Qed. Lemma RList_P3 : - forall (l:Rlist) (x:R), - In x l <-> (exists i : nat, x = pos_Rl l i /\ (i < Rlength l)%nat). + forall (l:list R) (x:R), + In x l <-> (exists i : nat, x = pos_Rl l i /\ (i < length l)%nat). Proof. intros; split; intro; [ induction l as [| r l Hrecl] | induction l as [| r l Hrecl] ]. elim H. elim H; intro; - [ exists 0%nat; split; [ apply H0 | simpl; apply lt_O_Sn ] + [ exists 0%nat; split; [ symmetry; apply H0 | simpl; apply lt_O_Sn ] | elim (Hrecl H0); intros; elim H1; clear H1; intros; exists (S x0); split; [ apply H1 | simpl; apply lt_n_S; assumption ] ]. elim H; intros; elim H0; intros; elim (lt_n_O _ H2). simpl; elim H; intros; elim H0; clear H0; intros; induction x0 as [| x0 Hrecx0]; - [ left; apply H0 + [ left; symmetry; apply H0 | right; apply Hrecl; exists x0; split; [ apply H0 | simpl in H1; apply lt_S_n; assumption ] ]. Qed. Lemma RList_P4 : - forall (l1:Rlist) (a:R), ordered_Rlist (cons a l1) -> ordered_Rlist l1. + forall (l1:list R) (a:R), ordered_Rlist (a :: l1) -> ordered_Rlist l1. Proof. intros; unfold ordered_Rlist; intros; apply (H (S i)); simpl; - replace (Rlength l1) with (S (pred (Rlength l1))); + replace (length l1) with (S (pred (length l1))); [ apply lt_n_S; assumption | symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red; intro; rewrite <- H1 in H0; elim (lt_n_O _ H0) ]. Qed. Lemma RList_P5 : - forall (l:Rlist) (x:R), ordered_Rlist l -> In x l -> pos_Rl l 0 <= x. + forall (l:list R) (x:R), ordered_Rlist l -> In x l -> pos_Rl l 0 <= x. Proof. intros; induction l as [| r l Hrecl]; [ elim H0 @@ -361,14 +342,14 @@ Proof. Qed. Lemma RList_P6 : - forall l:Rlist, + forall l:list R, ordered_Rlist l <-> (forall i j:nat, - (i <= j)%nat -> (j < Rlength l)%nat -> pos_Rl l i <= pos_Rl l j). + (i <= j)%nat -> (j < length l)%nat -> pos_Rl l i <= pos_Rl l j). Proof. - simple induction l; split; intro. + induction l as [ | r r0 H]; split; intro. intros; right; reflexivity. - unfold ordered_Rlist; intros; simpl in H0; elim (lt_n_O _ H0). + unfold ordered_Rlist;intros; simpl in H0; elim (lt_n_O _ H0). intros; induction i as [| i Hreci]; [ induction j as [| j Hrecj]; [ right; reflexivity @@ -391,14 +372,14 @@ Proof. Qed. Lemma RList_P7 : - forall (l:Rlist) (x:R), - ordered_Rlist l -> In x l -> x <= pos_Rl l (pred (Rlength l)). + forall (l:list R) (x:R), + ordered_Rlist l -> In x l -> x <= pos_Rl l (pred (length l)). Proof. intros; assert (H1 := RList_P6 l); elim H1; intros H2 _; assert (H3 := H2 H); clear H1 H2; assert (H1 := RList_P3 l x); elim H1; clear H1; intros; assert (H4 := H1 H0); elim H4; clear H4; intros; elim H4; clear H4; intros; rewrite H4; - assert (H6 : Rlength l = S (pred (Rlength l))). + assert (H6 : length l = S (pred (length l))). apply S_pred with 0%nat; apply neq_O_lt; red; intro; rewrite <- H6 in H5; elim (lt_n_O _ H5). apply H3; @@ -408,52 +389,55 @@ Proof. Qed. Lemma RList_P8 : - forall (l:Rlist) (a x:R), In x (insert l a) <-> x = a \/ In x l. -Proof. - simple induction l. - intros; split; intro; simpl in H; apply H. - intros; split; intro; - [ simpl in H0; generalize H0; case (Rle_dec r a); intros; - [ simpl in H1; elim H1; intro; - [ right; left; assumption - | elim (H a x); intros; elim (H3 H2); intro; - [ left; assumption | right; right; assumption ] ] - | simpl in H1; decompose [or] H1; - [ left; assumption - | right; left; assumption - | right; right; assumption ] ] - | simpl; case (Rle_dec r a); intro; - [ simpl in H0; decompose [or] H0; - [ right; elim (H a x); intros; apply H3; left - | left - | right; elim (H a x); intros; apply H3; right ] - | simpl in H0; decompose [or] H0; [ left | right; left | right; right ] ]; - assumption ]. + forall (l:list R) (a x:R), In x (insert l a) <-> x = a \/ In x l. +Proof. + induction l as [ | r r0 H]. + intros; split; intro; destruct H as [ax | []]; left; symmetry; exact ax. + intros; split; intro. + simpl in H0; generalize H0; case (Rle_dec r a); intros. + simpl in H1; elim H1; intro. + right; left; assumption. + elim (H a x); intros; elim (H3 H2); intro. + left; assumption. + right; right; assumption. + simpl in H1; decompose [or] H1. + left; symmetry; assumption. + right; left; assumption. + right; right; assumption. + simpl; case (Rle_dec r a); intro. + simpl in H0; decompose [or] H0. + right; elim (H a x); intros; apply H3; left. assumption. + left. assumption. + right; elim (H a x); intros; apply H3; right; assumption. + simpl in H0; decompose [or] H0; [ left | right; left | right; right]; + trivial; symmetry; assumption. Qed. Lemma RList_P9 : - forall (l1 l2:Rlist) (x:R), In x (cons_ORlist l1 l2) <-> In x l1 \/ In x l2. + forall (l1 l2:list R) (x:R), In x (cons_ORlist l1 l2) <-> In x l1 \/ In x l2. Proof. - simple induction l1. + induction l1 as [ | r r0 H]. intros; split; intro; [ simpl in H; right; assumption | simpl; elim H; intro; [ elim H0 | assumption ] ]. intros; split. simpl; intros; elim (H (insert l2 r) x); intros; assert (H3 := H1 H0); - elim H3; intro; - [ left; right; assumption - | elim (RList_P8 l2 r x); intros H5 _; assert (H6 := H5 H4); elim H6; intro; - [ left; left; assumption | right; assumption ] ]. + elim H3; intro. + left; right; assumption. + elim (RList_P8 l2 r x); intros H5 _; assert (H6 := H5 H4); elim H6; intro. + left; left; symmetry; assumption. + right; assumption. intro; simpl; elim (H (insert l2 r) x); intros _ H1; apply H1; - elim H0; intro; - [ elim H2; intro; - [ right; elim (RList_P8 l2 r x); intros _ H4; apply H4; left; assumption - | left; assumption ] - | right; elim (RList_P8 l2 r x); intros _ H3; apply H3; right; assumption ]. + elim H0; intro. + elim H2; intro. + right; elim (RList_P8 l2 r x); intros _ H4; apply H4; left. + symmetry; assumption. + left; assumption. + right; elim (RList_P8 l2 r x); intros _ H3; apply H3; right; assumption. Qed. Lemma RList_P10 : - forall (l:Rlist) (a:R), Rlength (insert l a) = S (Rlength l). + forall (l:list R) (a:R), length (insert l a) = S (length l). Proof. intros; induction l as [| r l Hrecl]; [ reflexivity @@ -462,10 +446,10 @@ Proof. Qed. Lemma RList_P11 : - forall l1 l2:Rlist, - Rlength (cons_ORlist l1 l2) = (Rlength l1 + Rlength l2)%nat. + forall l1 l2:list R, + length (cons_ORlist l1 l2) = (length l1 + length l2)%nat. Proof. - simple induction l1; + induction l1 as [ | r r0 H]; [ intro; reflexivity | intros; simpl; rewrite (H (insert l2 r)); rewrite RList_P10; apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; @@ -473,8 +457,8 @@ Proof. Qed. Lemma RList_P12 : - forall (l:Rlist) (i:nat) (f:R -> R), - (i < Rlength l)%nat -> pos_Rl (app_Rlist l f) i = f (pos_Rl l i). + forall (l:list R) (i:nat) (f:R -> R), + (i < length l)%nat -> pos_Rl (map f l) i = f (pos_Rl l i). Proof. simple induction l; [ intros; elim (lt_n_O _ H) @@ -483,30 +467,30 @@ Proof. Qed. Lemma RList_P13 : - forall (l:Rlist) (i:nat) (a:R), - (i < pred (Rlength l))%nat -> + forall (l:list R) (i:nat) (a:R), + (i < pred (length l))%nat -> pos_Rl (mid_Rlist l a) (S i) = (pos_Rl l i + pos_Rl l (S i)) / 2. Proof. - simple induction l. + induction l as [ | r r0 H]. intros; simpl in H; elim (lt_n_O _ H). - simple induction r0. + induction r0 as [ | r1 r2 H0]. intros; simpl in H0; elim (lt_n_O _ H0). intros; simpl in H1; induction i as [| i Hreci]. reflexivity. change - (pos_Rl (mid_Rlist (cons r1 r2) r) (S i) = - (pos_Rl (cons r1 r2) i + pos_Rl (cons r1 r2) (S i)) / 2) - ; apply H0; simpl; apply lt_S_n; assumption. + (pos_Rl (mid_Rlist (r1 :: r2) r) (S i) = + (pos_Rl (r1 :: r2) i + pos_Rl (r1 :: r2) (S i)) / 2). + apply H; simpl; apply lt_S_n; assumption. Qed. -Lemma RList_P14 : forall (l:Rlist) (a:R), Rlength (mid_Rlist l a) = Rlength l. +Lemma RList_P14 : forall (l:list R) (a:R), length (mid_Rlist l a) = length l. Proof. - simple induction l; intros; + induction l as [ | r r0 H]; intros; [ reflexivity | simpl; rewrite (H r); reflexivity ]. Qed. Lemma RList_P15 : - forall l1 l2:Rlist, + forall l1 l2:list R, ordered_Rlist l1 -> ordered_Rlist l2 -> pos_Rl l1 0 = pos_Rl l2 0 -> pos_Rl (cons_ORlist l1 l2) 0 = pos_Rl l1 0. @@ -514,10 +498,10 @@ Proof. intros; apply Rle_antisym. induction l1 as [| r l1 Hrecl1]; [ simpl; simpl in H1; right; symmetry ; assumption - | elim (RList_P9 (cons r l1) l2 (pos_Rl (cons r l1) 0)); intros; + | elim (RList_P9 (r :: l1) l2 (pos_Rl (r :: l1) 0)); intros; assert (H4 : - In (pos_Rl (cons r l1) 0) (cons r l1) \/ In (pos_Rl (cons r l1) 0) l2); + In (pos_Rl (r :: l1) 0) (r :: l1) \/ In (pos_Rl (r :: l1) 0) l2); [ left; left; reflexivity | assert (H5 := H3 H4); apply RList_P5; [ apply RList_P2; assumption | assumption ] ] ]. @@ -525,25 +509,25 @@ Proof. [ simpl; simpl in H1; right; assumption | assert (H2 : - In (pos_Rl (cons_ORlist (cons r l1) l2) 0) (cons_ORlist (cons r l1) l2)); + In (pos_Rl (cons_ORlist (r :: l1) l2) 0) (cons_ORlist (r :: l1) l2)); [ elim - (RList_P3 (cons_ORlist (cons r l1) l2) - (pos_Rl (cons_ORlist (cons r l1) l2) 0)); + (RList_P3 (cons_ORlist (r :: l1) l2) + (pos_Rl (cons_ORlist (r :: l1) l2) 0)); intros; apply H3; exists 0%nat; split; [ reflexivity | rewrite RList_P11; simpl; apply lt_O_Sn ] - | elim (RList_P9 (cons r l1) l2 (pos_Rl (cons_ORlist (cons r l1) l2) 0)); + | elim (RList_P9 (r :: l1) l2 (pos_Rl (cons_ORlist (r :: l1) l2) 0)); intros; assert (H5 := H3 H2); elim H5; intro; [ apply RList_P5; assumption | rewrite H1; apply RList_P5; assumption ] ] ]. Qed. Lemma RList_P16 : - forall l1 l2:Rlist, + forall l1 l2:list R, ordered_Rlist l1 -> ordered_Rlist l2 -> - pos_Rl l1 (pred (Rlength l1)) = pos_Rl l2 (pred (Rlength l2)) -> - pos_Rl (cons_ORlist l1 l2) (pred (Rlength (cons_ORlist l1 l2))) = - pos_Rl l1 (pred (Rlength l1)). + pos_Rl l1 (pred (length l1)) = pos_Rl l2 (pred (length l2)) -> + pos_Rl (cons_ORlist l1 l2) (pred (length (cons_ORlist l1 l2))) = + pos_Rl l1 (pred (length l1)). Proof. intros; apply Rle_antisym. induction l1 as [| r l1 Hrecl1]. @@ -551,99 +535,99 @@ Proof. assert (H2 : In - (pos_Rl (cons_ORlist (cons r l1) l2) - (pred (Rlength (cons_ORlist (cons r l1) l2)))) - (cons_ORlist (cons r l1) l2)); + (pos_Rl (cons_ORlist (r :: l1) l2) + (pred (length (cons_ORlist (r :: l1) l2)))) + (cons_ORlist (r :: l1) l2)); [ elim - (RList_P3 (cons_ORlist (cons r l1) l2) - (pos_Rl (cons_ORlist (cons r l1) l2) - (pred (Rlength (cons_ORlist (cons r l1) l2))))); - intros; apply H3; exists (pred (Rlength (cons_ORlist (cons r l1) l2))); + (RList_P3 (cons_ORlist (r :: l1) l2) + (pos_Rl (cons_ORlist (r :: l1) l2) + (pred (length (cons_ORlist (r :: l1) l2))))); + intros; apply H3; exists (pred (length (cons_ORlist (r :: l1) l2))); split; [ reflexivity | rewrite RList_P11; simpl; apply lt_n_Sn ] | elim - (RList_P9 (cons r l1) l2 - (pos_Rl (cons_ORlist (cons r l1) l2) - (pred (Rlength (cons_ORlist (cons r l1) l2))))); + (RList_P9 (r :: l1) l2 + (pos_Rl (cons_ORlist (r :: l1) l2) + (pred (length (cons_ORlist (r :: l1) l2))))); intros; assert (H5 := H3 H2); elim H5; intro; [ apply RList_P7; assumption | rewrite H1; apply RList_P7; assumption ] ]. induction l1 as [| r l1 Hrecl1]. simpl; simpl in H1; right; assumption. elim - (RList_P9 (cons r l1) l2 (pos_Rl (cons r l1) (pred (Rlength (cons r l1))))); + (RList_P9 (r :: l1) l2 (pos_Rl (r :: l1) (pred (length (r :: l1))))). intros; assert (H4 : - In (pos_Rl (cons r l1) (pred (Rlength (cons r l1)))) (cons r l1) \/ - In (pos_Rl (cons r l1) (pred (Rlength (cons r l1)))) l2); - [ left; change (In (pos_Rl (cons r l1) (Rlength l1)) (cons r l1)); - elim (RList_P3 (cons r l1) (pos_Rl (cons r l1) (Rlength l1))); - intros; apply H5; exists (Rlength l1); split; + In (pos_Rl (r :: l1) (pred (length (r :: l1)))) (r :: l1) \/ + In (pos_Rl (r :: l1) (pred (length (r :: l1)))) l2); + [ left; change (In (pos_Rl (r :: l1) (length l1)) (r :: l1)); + elim (RList_P3 (r :: l1) (pos_Rl (r :: l1) (length l1))); + intros; apply H5; exists (length l1); split; [ reflexivity | simpl; apply lt_n_Sn ] | assert (H5 := H3 H4); apply RList_P7; [ apply RList_P2; assumption | elim - (RList_P9 (cons r l1) l2 - (pos_Rl (cons r l1) (pred (Rlength (cons r l1))))); + (RList_P9 (r :: l1) l2 + (pos_Rl (r :: l1) (pred (length (r :: l1))))); intros; apply H7; left; elim - (RList_P3 (cons r l1) - (pos_Rl (cons r l1) (pred (Rlength (cons r l1))))); - intros; apply H9; exists (pred (Rlength (cons r l1))); + (RList_P3 (r :: l1) + (pos_Rl (r :: l1) (pred (length (r :: l1))))); + intros; apply H9; exists (pred (length (r :: l1))); split; [ reflexivity | simpl; apply lt_n_Sn ] ] ]. Qed. Lemma RList_P17 : - forall (l1:Rlist) (x:R) (i:nat), + forall (l1:list R) (x:R) (i:nat), ordered_Rlist l1 -> In x l1 -> - pos_Rl l1 i < x -> (i < pred (Rlength l1))%nat -> pos_Rl l1 (S i) <= x. + pos_Rl l1 i < x -> (i < pred (length l1))%nat -> pos_Rl l1 (S i) <= x. Proof. - simple induction l1. + induction l1 as [ | r r0 H]. intros; elim H0. intros; induction i as [| i Hreci]. simpl; elim H1; intro; [ simpl in H2; rewrite H4 in H2; elim (Rlt_irrefl _ H2) | apply RList_P5; [ apply RList_P4 with r; assumption | assumption ] ]. simpl; simpl in H2; elim H1; intro. - rewrite H4 in H2; assert (H5 : r <= pos_Rl r0 i); + rewrite <- H4 in H2; assert (H5 : r <= pos_Rl r0 i); [ apply Rle_trans with (pos_Rl r0 0); [ apply (H0 0%nat); simpl; simpl in H3; apply neq_O_lt; red; intro; rewrite <- H5 in H3; elim (lt_n_O _ H3) | elim (RList_P6 r0); intros; apply H5; [ apply RList_P4 with r; assumption | apply le_O_n - | simpl in H3; apply lt_S_n; apply lt_trans with (Rlength r0); + | simpl in H3; apply lt_S_n; apply lt_trans with (length r0); [ apply H3 | apply lt_n_Sn ] ] ] | elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H5 H2)) ]. apply H; try assumption; [ apply RList_P4 with r; assumption | simpl in H3; apply lt_S_n; - replace (S (pred (Rlength r0))) with (Rlength r0); + replace (S (pred (length r0))) with (length r0); [ apply H3 | apply S_pred with 0%nat; apply neq_O_lt; red; intro; rewrite <- H5 in H3; elim (lt_n_O _ H3) ] ]. Qed. Lemma RList_P18 : - forall (l:Rlist) (f:R -> R), Rlength (app_Rlist l f) = Rlength l. + forall (l:list R) (f:R -> R), length (map f l) = length l. Proof. simple induction l; intros; [ reflexivity | simpl; rewrite H; reflexivity ]. Qed. Lemma RList_P19 : - forall l:Rlist, - l <> nil -> exists r : R, (exists r0 : Rlist, l = cons r r0). + forall l:list R, + l <> nil -> exists r : R, (exists r0 : list R, l = r :: r0). Proof. intros; induction l as [| r l Hrecl]; [ elim H; reflexivity | exists r; exists l; reflexivity ]. Qed. Lemma RList_P20 : - forall l:Rlist, - (2 <= Rlength l)%nat -> + forall l:list R, + (2 <= length l)%nat -> exists r : R, - (exists r1 : R, (exists l' : Rlist, l = cons r (cons r1 l'))). + (exists r1 : R, (exists l' : list R, l = r :: r1 :: l')). Proof. intros; induction l as [| r l Hrecl]; [ simpl in H; elim (le_Sn_O _ H) @@ -652,40 +636,32 @@ Proof. | exists r; exists r0; exists l; reflexivity ] ]. Qed. -Lemma RList_P21 : forall l l':Rlist, l = l' -> Rtail l = Rtail l'. +Lemma RList_P21 : forall l l':list R, l = l' -> Rtail l = Rtail l'. Proof. intros; rewrite H; reflexivity. Qed. Lemma RList_P22 : - forall l1 l2:Rlist, l1 <> nil -> pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0. + forall l1 l2:list R, l1 <> nil -> pos_Rl (app l1 l2) 0 = pos_Rl l1 0. Proof. simple induction l1; [ intros; elim H; reflexivity | intros; reflexivity ]. Qed. -Lemma RList_P23 : - forall l1 l2:Rlist, - Rlength (cons_Rlist l1 l2) = (Rlength l1 + Rlength l2)%nat. -Proof. - simple induction l1; - [ intro; reflexivity | intros; simpl; rewrite H; reflexivity ]. -Qed. - Lemma RList_P24 : - forall l1 l2:Rlist, + forall l1 l2:list R, l2 <> nil -> - pos_Rl (cons_Rlist l1 l2) (pred (Rlength (cons_Rlist l1 l2))) = - pos_Rl l2 (pred (Rlength l2)). + pos_Rl (app l1 l2) (pred (length (app l1 l2))) = + pos_Rl l2 (pred (length l2)). Proof. - simple induction l1. + induction l1 as [ | r r0 H]. intros; reflexivity. intros; rewrite <- (H l2 H0); induction l2 as [| r1 l2 Hrecl2]. elim H0; reflexivity. - do 2 rewrite RList_P23; - replace (Rlength (cons r r0) + Rlength (cons r1 l2))%nat with - (S (S (Rlength r0 + Rlength l2))); - [ replace (Rlength r0 + Rlength (cons r1 l2))%nat with - (S (Rlength r0 + Rlength l2)); + do 2 rewrite app_length; + replace (length (r :: r0) + length (r1 :: l2))%nat with + (S (S (length r0 + length l2))); + [ replace (length r0 + length (r1 :: l2))%nat with + (S (length r0 + length l2)); [ reflexivity | simpl; apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; rewrite S_INR; ring ] @@ -694,39 +670,39 @@ Proof. Qed. Lemma RList_P25 : - forall l1 l2:Rlist, + forall l1 l2:list R, ordered_Rlist l1 -> ordered_Rlist l2 -> - pos_Rl l1 (pred (Rlength l1)) <= pos_Rl l2 0 -> - ordered_Rlist (cons_Rlist l1 l2). + pos_Rl l1 (pred (length l1)) <= pos_Rl l2 0 -> + ordered_Rlist (app l1 l2). Proof. - simple induction l1. + induction l1 as [ | r r0 H]. intros; simpl; assumption. - simple induction r0. + induction r0 as [ | r1 r2 H0]. intros; simpl; simpl in H2; unfold ordered_Rlist; intros; simpl in H3. induction i as [| i Hreci]. simpl; assumption. change (pos_Rl l2 i <= pos_Rl l2 (S i)); apply (H1 i); apply lt_S_n; - replace (S (pred (Rlength l2))) with (Rlength l2); + replace (S (pred (length l2))) with (length l2); [ assumption | apply S_pred with 0%nat; apply neq_O_lt; red; intro; rewrite <- H4 in H3; elim (lt_n_O _ H3) ]. - intros; clear H; assert (H : ordered_Rlist (cons_Rlist (cons r1 r2) l2)). - apply H0; try assumption. + intros; assert (H4 : ordered_Rlist (app (r1 :: r2) l2)). + apply H; try assumption. apply RList_P4 with r; assumption. - unfold ordered_Rlist; intros; simpl in H4; + unfold ordered_Rlist; intros i H5; simpl in H5. induction i as [| i Hreci]. simpl; apply (H1 0%nat); simpl; apply lt_O_Sn. change - (pos_Rl (cons_Rlist (cons r1 r2) l2) i <= - pos_Rl (cons_Rlist (cons r1 r2) l2) (S i)); - apply (H i); simpl; apply lt_S_n; assumption. + (pos_Rl (app (r1 :: r2) l2) i <= + pos_Rl (app (r1 :: r2) l2) (S i)); + apply (H4 i); simpl; apply lt_S_n; assumption. Qed. Lemma RList_P26 : - forall (l1 l2:Rlist) (i:nat), - (i < Rlength l1)%nat -> pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i. + forall (l1 l2:list R) (i:nat), + (i < length l1)%nat -> pos_Rl (app l1 l2) i = pos_Rl l1 i. Proof. simple induction l1. intros; elim (lt_n_O _ H). @@ -735,49 +711,41 @@ Proof. apply (H l2 i); simpl in H0; apply lt_S_n; assumption. Qed. -Lemma RList_P27 : - forall l1 l2 l3:Rlist, - cons_Rlist l1 (cons_Rlist l2 l3) = cons_Rlist (cons_Rlist l1 l2) l3. -Proof. - simple induction l1; intros; - [ reflexivity | simpl; rewrite (H l2 l3); reflexivity ]. -Qed. - -Lemma RList_P28 : forall l:Rlist, cons_Rlist l nil = l. -Proof. - simple induction l; - [ reflexivity | intros; simpl; rewrite H; reflexivity ]. -Qed. - Lemma RList_P29 : - forall (l2 l1:Rlist) (i:nat), - (Rlength l1 <= i)%nat -> - (i < Rlength (cons_Rlist l1 l2))%nat -> - pos_Rl (cons_Rlist l1 l2) i = pos_Rl l2 (i - Rlength l1). + forall (l2 l1:list R) (i:nat), + (length l1 <= i)%nat -> + (i < length (app l1 l2))%nat -> + pos_Rl (app l1 l2) i = pos_Rl l2 (i - length l1). Proof. - simple induction l2. - intros; rewrite RList_P28 in H0; elim (lt_irrefl _ (le_lt_trans _ _ _ H H0)). + induction l2 as [ | r r0 H]. + intros; rewrite app_nil_r in H0; elim (lt_irrefl _ (le_lt_trans _ _ _ H H0)). intros; - replace (cons_Rlist l1 (cons r r0)) with - (cons_Rlist (cons_Rlist l1 (cons r nil)) r0). + replace (app l1 (r :: r0)) with + (app (app l1 (r :: nil)) r0). inversion H0. rewrite <- minus_n_n; simpl; rewrite RList_P26. - clear l2 r0 H i H0 H1 H2; induction l1 as [| r0 l1 Hrecl1]. + clear r0 H i H0 H1 H2; induction l1 as [| r0 l1 Hrecl1]. reflexivity. simpl; assumption. - rewrite RList_P23; rewrite plus_comm; simpl; apply lt_n_Sn. - replace (S m - Rlength l1)%nat with (S (S m - S (Rlength l1))). + rewrite app_length; rewrite plus_comm; simpl; apply lt_n_Sn. + replace (S m - length l1)%nat with (S (S m - S (length l1))). rewrite H3; simpl; - replace (S (Rlength l1)) with (Rlength (cons_Rlist l1 (cons r nil))). - apply (H (cons_Rlist l1 (cons r nil)) i). - rewrite RList_P23; rewrite plus_comm; simpl; rewrite <- H3; + replace (S (length l1)) with (length (app l1 (r :: nil))). + apply (H (app l1 (r :: nil)) i). + rewrite app_length; rewrite plus_comm; simpl; rewrite <- H3; apply le_n_S; assumption. - repeat rewrite RList_P23; simpl; rewrite RList_P23 in H1; - rewrite plus_comm in H1; simpl in H1; rewrite (plus_comm (Rlength l1)); + repeat rewrite app_length; simpl; rewrite app_length in H1; + rewrite plus_comm in H1; simpl in H1; rewrite (plus_comm (length l1)); simpl; rewrite plus_comm; apply H1. - rewrite RList_P23; rewrite plus_comm; reflexivity. - change (S (m - Rlength l1) = (S m - Rlength l1)%nat); + rewrite app_length; rewrite plus_comm; reflexivity. + change (S (m - length l1) = (S m - length l1)%nat); apply minus_Sn_m; assumption. - replace (cons r r0) with (cons_Rlist (cons r nil) r0); - [ symmetry ; apply RList_P27 | reflexivity ]. + replace (r :: r0) with (app (r :: nil) r0); + [ symmetry ; apply app_assoc | reflexivity ]. Qed. + +#[deprecated(since="8.12",note="use List.cons instead")] +Notation cons := List.cons. + +#[deprecated(since="8.12",note="use List.nil instead")] +Notation nil := List.nil. diff --git a/theories/Reals/RiemannInt.v b/theories/Reals/RiemannInt.v index 0337b12cad..23094c6b93 100644 --- a/theories/Reals/RiemannInt.v +++ b/theories/Reals/RiemannInt.v @@ -464,7 +464,7 @@ Proof. elim (Rlt_irrefl _ H7) ] ]. Qed. -Fixpoint SubEquiN (N:nat) (x y:R) (del:posreal) : Rlist := +Fixpoint SubEquiN (N:nat) (x y:R) (del:posreal) : list R := match N with | O => cons y nil | S p => cons x (SubEquiN p (x + del) y del) @@ -473,7 +473,7 @@ Fixpoint SubEquiN (N:nat) (x y:R) (del:posreal) : Rlist := Definition max_N (a b:R) (del:posreal) (h:a < b) : nat := let (N,_) := maxN del h in N. -Definition SubEqui (a b:R) (del:posreal) (h:a < b) : Rlist := +Definition SubEqui (a b:R) (del:posreal) (h:a < b) : list R := SubEquiN (S (max_N del h)) a b del. Lemma Heine_cor1 : @@ -566,25 +566,25 @@ Qed. Lemma SubEqui_P2 : forall (a b:R) (del:posreal) (h:a < b), - pos_Rl (SubEqui del h) (pred (Rlength (SubEqui del h))) = b. + pos_Rl (SubEqui del h) (pred (length (SubEqui del h))) = b. Proof. intros; unfold SubEqui; destruct (maxN del h)as (x,_). cut (forall (x:nat) (a:R) (del:posreal), pos_Rl (SubEquiN (S x) a b del) - (pred (Rlength (SubEquiN (S x) a b del))) = b); + (pred (length (SubEquiN (S x) a b del))) = b); [ intro; apply H | simple induction x0; [ intros; reflexivity | intros; change (pos_Rl (SubEquiN (S n) (a0 + del0) b del0) - (pred (Rlength (SubEquiN (S n) (a0 + del0) b del0))) = b) + (pred (length (SubEquiN (S n) (a0 + del0) b del0))) = b) ; apply H ] ]. Qed. Lemma SubEqui_P3 : - forall (N:nat) (a b:R) (del:posreal), Rlength (SubEquiN N a b del) = S N. + forall (N:nat) (a b:R) (del:posreal), length (SubEquiN N a b del) = S N. Proof. simple induction N; intros; [ reflexivity | simpl; rewrite H; reflexivity ]. @@ -605,7 +605,7 @@ Qed. Lemma SubEqui_P5 : forall (a b:R) (del:posreal) (h:a < b), - Rlength (SubEqui del h) = S (S (max_N del h)). + length (SubEqui del h) = S (S (max_N del h)). Proof. intros; unfold SubEqui; apply SubEqui_P3. Qed. @@ -623,7 +623,7 @@ Proof. intros; unfold ordered_Rlist; intros; rewrite SubEqui_P5 in H; simpl in H; inversion H. rewrite (SubEqui_P6 del h (i:=(max_N del h))). - replace (S (max_N del h)) with (pred (Rlength (SubEqui del h))). + replace (S (max_N del h)) with (pred (length (SubEqui del h))). rewrite SubEqui_P2; unfold max_N; case (maxN del h) as (?&?&?); left; assumption. rewrite SubEqui_P5; reflexivity. @@ -639,7 +639,7 @@ Qed. Lemma SubEqui_P8 : forall (a b:R) (del:posreal) (h:a < b) (i:nat), - (i < Rlength (SubEqui del h))%nat -> a <= pos_Rl (SubEqui del h) i <= b. + (i < length (SubEqui del h))%nat -> a <= pos_Rl (SubEqui del h) i <= b. Proof. intros; split. pattern a at 1; rewrite <- (SubEqui_P1 del h); apply RList_P5. @@ -657,7 +657,7 @@ Lemma SubEqui_P9 : { g:StepFun a b | g b = f b /\ (forall i:nat, - (i < pred (Rlength (SubEqui del h)))%nat -> + (i < pred (length (SubEqui del h)))%nat -> constant_D_eq g (co_interval (pos_Rl (SubEqui del h) i) (pos_Rl (SubEqui del h) (S i))) @@ -713,7 +713,7 @@ Proof. a <= t <= b -> t = b \/ (exists i : nat, - (i < pred (Rlength (SubEqui del H)))%nat /\ + (i < pred (length (SubEqui del H)))%nat /\ co_interval (pos_Rl (SubEqui del H) i) (pos_Rl (SubEqui del H) (S i)) t)). intro; elim (H8 _ H7); intro. @@ -722,7 +722,7 @@ Proof. elim H9; clear H9; intros I [H9 H10]; assert (H11 := H6 I H9 t H10); rewrite H11; left; apply H4. assumption. - apply SubEqui_P8; apply lt_trans with (pred (Rlength (SubEqui del H))). + apply SubEqui_P8; apply lt_trans with (pred (length (SubEqui del H))). assumption. apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H12 in H9; elim (lt_n_O _ H9). @@ -734,7 +734,7 @@ Proof. (t - pos_Rl (SubEqui del H) (max_N del H))) with t; [ idtac | ring ]; apply Rlt_le_trans with b. rewrite H14 in H12; - assert (H13 : S (max_N del H) = pred (Rlength (SubEqui del H))). + assert (H13 : S (max_N del H) = pred (length (SubEqui del H))). rewrite SubEqui_P5; reflexivity. rewrite H13 in H12; rewrite SubEqui_P2 in H12; apply H12. rewrite SubEqui_P6. @@ -785,7 +785,7 @@ Proof. apply H5. assumption. inversion H7. - replace (S (max_N del H)) with (pred (Rlength (SubEqui del H))). + replace (S (max_N del H)) with (pred (length (SubEqui del H))). rewrite (SubEqui_P2 del H); elim H8; intros. elim H11; intro. assumption. @@ -1753,7 +1753,7 @@ Proof. rewrite <- H5; elim (RList_P6 l); intros; apply H10. assumption. apply le_O_n. - apply lt_trans with (pred (Rlength l)); [ assumption | apply lt_pred_n_n ]. + apply lt_trans with (pred (length l)); [ assumption | apply lt_pred_n_n ]. apply neq_O_lt; intro; rewrite <- H12 in H6; discriminate. unfold Rmin; decide (Rle_dec a b) with H; reflexivity. assert (H11 : pos_Rl l (S i) <= b). @@ -1960,7 +1960,7 @@ Proof. replace b with (Rmin b c). rewrite <- H5; elim (RList_P6 l1); intros; apply H10; try assumption. apply le_O_n. - apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n; + apply lt_trans with (pred (length l1)); try assumption; apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H12 in H6; discriminate. unfold Rmin; decide (Rle_dec b c) with Hyp2; @@ -1991,7 +1991,7 @@ Proof. replace a with (Rmin a b). rewrite <- H5; elim (RList_P6 l1); intros; apply H11; try assumption. apply le_O_n. - apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n; + apply lt_trans with (pred (length l1)); try assumption; apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H13 in H6; discriminate. unfold Rmin; decide (Rle_dec a b) with Hyp1; reflexivity. @@ -2018,7 +2018,7 @@ Proof. replace a with (Rmin a b). rewrite <- H5; elim (RList_P6 l1); intros; apply H11; try assumption. apply le_O_n. - apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n; + apply lt_trans with (pred (length l1)); try assumption; apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H13 in H6; discriminate. unfold Rmin; decide (Rle_dec a b) with Hyp1; reflexivity. @@ -2037,7 +2037,7 @@ Proof. replace b with (Rmin b c). rewrite <- H5; elim (RList_P6 l1); intros; apply H10; try assumption. apply le_O_n. - apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n; + apply lt_trans with (pred (length l1)); try assumption; apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H12 in H6; discriminate. unfold Rmin; decide (Rle_dec b c) with Hyp2; reflexivity. diff --git a/theories/Reals/RiemannInt_SF.v b/theories/Reals/RiemannInt_SF.v index c8ec4782d9..65221c67d2 100644 --- a/theories/Reals/RiemannInt_SF.v +++ b/theories/Reals/RiemannInt_SF.v @@ -12,6 +12,7 @@ Require Import Rbase. Require Import Rfunctions. Require Import Ranalysis_reg. Require Import Classical_Prop. +Require Import List. Require Import RList. Local Open Scope R_scope. @@ -114,41 +115,41 @@ Qed. Definition open_interval (a b x:R) : Prop := a < x < b. Definition co_interval (a b x:R) : Prop := a <= x < b. -Definition adapted_couple (f:R -> R) (a b:R) (l lf:Rlist) : Prop := +Definition adapted_couple (f:R -> R) (a b:R) (l lf:list R) : Prop := ordered_Rlist l /\ pos_Rl l 0 = Rmin a b /\ - pos_Rl l (pred (Rlength l)) = Rmax a b /\ - Rlength l = S (Rlength lf) /\ + pos_Rl l (pred (length l)) = Rmax a b /\ + length l = S (length lf) /\ (forall i:nat, - (i < pred (Rlength l))%nat -> + (i < pred (length l))%nat -> constant_D_eq f (open_interval (pos_Rl l i) (pos_Rl l (S i))) (pos_Rl lf i)). -Definition adapted_couple_opt (f:R -> R) (a b:R) (l lf:Rlist) := +Definition adapted_couple_opt (f:R -> R) (a b:R) (l lf:list R) := adapted_couple f a b l lf /\ (forall i:nat, - (i < pred (Rlength lf))%nat -> + (i < pred (length lf))%nat -> pos_Rl lf i <> pos_Rl lf (S i) \/ f (pos_Rl l (S i)) <> pos_Rl lf i) /\ - (forall i:nat, (i < pred (Rlength l))%nat -> pos_Rl l i <> pos_Rl l (S i)). + (forall i:nat, (i < pred (length l))%nat -> pos_Rl l i <> pos_Rl l (S i)). -Definition is_subdivision (f:R -> R) (a b:R) (l:Rlist) : Type := - { l0:Rlist & adapted_couple f a b l l0 }. +Definition is_subdivision (f:R -> R) (a b:R) (l:list R) : Type := + { l0:list R & adapted_couple f a b l l0 }. Definition IsStepFun (f:R -> R) (a b:R) : Type := - { l:Rlist & is_subdivision f a b l }. + { l:list R & is_subdivision f a b l }. (** ** Class of step functions *) Record StepFun (a b:R) : Type := mkStepFun {fe :> R -> R; pre : IsStepFun fe a b}. -Definition subdivision (a b:R) (f:StepFun a b) : Rlist := projT1 (pre f). +Definition subdivision (a b:R) (f:StepFun a b) : list R := projT1 (pre f). -Definition subdivision_val (a b:R) (f:StepFun a b) : Rlist := +Definition subdivision_val (a b:R) (f:StepFun a b) : list R := match projT2 (pre f) with | existT _ a b => a end. -Fixpoint Int_SF (l k:Rlist) : R := +Fixpoint Int_SF (l k:list R) : R := match l with | nil => 0 | cons a l' => @@ -179,7 +180,7 @@ Proof. Qed. Lemma StepFun_P2 : - forall (a b:R) (f:R -> R) (l lf:Rlist), + forall (a b:R) (f:R -> R) (l lf:list R), adapted_couple f a b l lf -> adapted_couple f b a l lf. Proof. unfold adapted_couple; intros; decompose [and] H; clear H; @@ -219,7 +220,7 @@ Proof. Qed. Lemma StepFun_P5 : - forall (a b:R) (f:R -> R) (l:Rlist), + forall (a b:R) (f:R -> R) (l:list R), is_subdivision f a b l -> is_subdivision f b a l. Proof. destruct 1 as (x,(H0,(H1,(H2,(H3,H4))))); exists x; @@ -236,7 +237,7 @@ Proof. Qed. Lemma StepFun_P7 : - forall (a b r1 r2 r3:R) (f:R -> R) (l lf:Rlist), + forall (a b r1 r2 r3:R) (f:R -> R) (l lf:list R), a <= b -> adapted_couple f a b (cons r1 (cons r2 l)) (cons r3 lf) -> adapted_couple f r2 b (cons r2 l) lf. @@ -257,31 +258,36 @@ Proof. rewrite H4; reflexivity. intros; unfold constant_D_eq, open_interval; intros; unfold constant_D_eq, open_interval in H6; - assert (H9 : (S i < pred (Rlength (cons r1 (cons r2 l))))%nat). + assert (H9 : (S i < pred (length (cons r1 (cons r2 l))))%nat). simpl; simpl in H0; apply lt_n_S; assumption. assert (H10 := H6 _ H9); apply H10; assumption. Qed. Lemma StepFun_P8 : - forall (f:R -> R) (l1 lf1:Rlist) (a b:R), + forall (f:R -> R) (l1 lf1:list R) (a b:R), adapted_couple f a b l1 lf1 -> a = b -> Int_SF lf1 l1 = 0. Proof. simple induction l1. intros; induction lf1 as [| r lf1 Hreclf1]; reflexivity. - simple induction r0. + intros r r0. + induction r0 as [ | r1 r2 H0]. intros; induction lf1 as [| r1 lf1 Hreclf1]. reflexivity. unfold adapted_couple in H0; decompose [and] H0; clear H0; simpl in H5; discriminate. - intros; induction lf1 as [| r3 lf1 Hreclf1]. + intros H. + induction lf1 as [| r3 lf1 Hreclf1]; intros a b H1 H2. reflexivity. simpl; cut (r = r1). - intro; rewrite H3; rewrite (H0 lf1 r b). + intros H3. + rewrite H3; rewrite (H lf1 r b). ring. rewrite H3; apply StepFun_P7 with a r r3; [ right; assumption | assumption ]. - clear H H0 Hreclf1 r0; unfold adapted_couple in H1; decompose [and] H1; + clear H H0 Hreclf1; unfold adapted_couple in H1. + decompose [and] H1. intros; simpl in H4; rewrite H4; unfold Rmin; case (Rle_dec a b); intro; [ assumption | reflexivity ]. + unfold adapted_couple in H1; decompose [and] H1; intros; apply Rle_antisym. apply (H3 0%nat); simpl; apply lt_O_Sn. simpl in H5; rewrite H2 in H5; rewrite H5; replace (Rmin b b) with (Rmax a b); @@ -292,8 +298,8 @@ Proof. Qed. Lemma StepFun_P9 : - forall (a b:R) (f:R -> R) (l lf:Rlist), - adapted_couple f a b l lf -> a <> b -> (2 <= Rlength l)%nat. + forall (a b:R) (f:R -> R) (l lf:list R), + adapted_couple f a b l lf -> a <> b -> (2 <= length l)%nat. Proof. intros; unfold adapted_couple in H; decompose [and] H; clear H; induction l as [| r l Hrecl]; @@ -307,13 +313,13 @@ Proof. Qed. Lemma StepFun_P10 : - forall (f:R -> R) (l lf:Rlist) (a b:R), + forall (f:R -> R) (l lf:list R) (a b:R), a <= b -> adapted_couple f a b l lf -> - exists l' : Rlist, - (exists lf' : Rlist, adapted_couple_opt f a b l' lf'). + exists l' : list R, + (exists lf' : list R, adapted_couple_opt f a b l' lf'). Proof. - simple induction l. + induction l as [ | r r0 H]. intros; unfold adapted_couple in H0; decompose [and] H0; simpl in H4; discriminate. intros; case (Req_dec a b); intro. @@ -503,7 +509,7 @@ Proof. Qed. Lemma StepFun_P11 : - forall (a b r r1 r3 s1 s2 r4:R) (r2 lf1 s3 lf2:Rlist) + forall (a b r r1 r3 s1 s2 r4:R) (r2 lf1 s3 lf2:list R) (f:R -> R), a < b -> adapted_couple f a b (cons r (cons r1 r2)) (cons r3 lf1) -> @@ -627,7 +633,7 @@ Proof. Qed. Lemma StepFun_P12 : - forall (a b:R) (f:R -> R) (l lf:Rlist), + forall (a b:R) (f:R -> R) (l lf:list R), adapted_couple_opt f a b l lf -> adapted_couple_opt f b a l lf. Proof. unfold adapted_couple_opt; unfold adapted_couple; intros; @@ -643,7 +649,7 @@ Proof. Qed. Lemma StepFun_P13 : - forall (a b r r1 r3 s1 s2 r4:R) (r2 lf1 s3 lf2:Rlist) + forall (a b r r1 r3 s1 s2 r4:R) (r2 lf1 s3 lf2:list R) (f:R -> R), a <> b -> adapted_couple f a b (cons r (cons r1 r2)) (cons r3 lf1) -> @@ -657,15 +663,15 @@ Proof. Qed. Lemma StepFun_P14 : - forall (f:R -> R) (l1 l2 lf1 lf2:Rlist) (a b:R), + forall (f:R -> R) (l1 l2 lf1 lf2:list R) (a b:R), a <= b -> adapted_couple f a b l1 lf1 -> adapted_couple_opt f a b l2 lf2 -> Int_SF lf1 l1 = Int_SF lf2 l2. Proof. - simple induction l1. + induction l1 as [ | r r0 H0]. intros l2 lf1 lf2 a b Hyp H H0; unfold adapted_couple in H; decompose [and] H; clear H H0 H2 H3 H1 H6; simpl in H4; discriminate. - simple induction r0. + induction r0 as [|r1 r2 H]. intros; case (Req_dec a b); intro. unfold adapted_couple_opt in H2; elim H2; intros; rewrite (StepFun_P8 H4 H3); rewrite (StepFun_P8 H1 H3); reflexivity. @@ -798,7 +804,7 @@ Proof. rewrite H9; change (forall i:nat, - (i < pred (Rlength (cons r4 lf2)))%nat -> + (i < pred (length (cons r4 lf2)))%nat -> pos_Rl (cons r4 lf2) i <> pos_Rl (cons r4 lf2) (S i) \/ f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <> pos_Rl (cons r4 lf2) i) ; rewrite <- H5; apply H3. @@ -840,7 +846,7 @@ Proof. rewrite <- H10; unfold open_interval; apply H2. elim H3; clear H3; intros; split. rewrite H5 in H3; intros; apply (H3 (S i)). - simpl; replace (Rlength lf2) with (S (pred (Rlength lf2))). + simpl; replace (length lf2) with (S (pred (length lf2))). apply lt_n_S; apply H12. symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red; intro; rewrite <- H13 in H12; elim (lt_n_O _ H12). @@ -863,7 +869,7 @@ Proof. Qed. Lemma StepFun_P15 : - forall (f:R -> R) (l1 l2 lf1 lf2:Rlist) (a b:R), + forall (f:R -> R) (l1 l2 lf1 lf2:list R) (a b:R), adapted_couple f a b l1 lf1 -> adapted_couple_opt f a b l2 lf2 -> Int_SF lf1 l1 = Int_SF lf2 l2. Proof. @@ -876,10 +882,10 @@ Proof. Qed. Lemma StepFun_P16 : - forall (f:R -> R) (l lf:Rlist) (a b:R), + forall (f:R -> R) (l lf:list R) (a b:R), adapted_couple f a b l lf -> - exists l' : Rlist, - (exists lf' : Rlist, adapted_couple_opt f a b l' lf'). + exists l' : list R, + (exists lf' : list R, adapted_couple_opt f a b l' lf'). Proof. intros; destruct (Rle_dec a b) as [Hle|Hnle]; [ apply (StepFun_P10 Hle H) @@ -891,7 +897,7 @@ Proof. Qed. Lemma StepFun_P17 : - forall (f:R -> R) (l1 l2 lf1 lf2:Rlist) (a b:R), + forall (f:R -> R) (l1 l2 lf1 lf2:list R) (a b:R), adapted_couple f a b l1 lf1 -> adapted_couple f a b l2 lf2 -> Int_SF lf1 l1 = Int_SF lf2 l2. Proof. @@ -922,7 +928,7 @@ Proof. Qed. Lemma StepFun_P19 : - forall (l1:Rlist) (f g:R -> R) (l:R), + forall (l1:list R) (f g:R -> R) (l:R), Int_SF (FF l1 (fun x:R => f x + l * g x)) l1 = Int_SF (FF l1 f) l1 + l * Int_SF (FF l1 g) l1. Proof. @@ -933,8 +939,8 @@ Proof. Qed. Lemma StepFun_P20 : - forall (l:Rlist) (f:R -> R), - (0 < Rlength l)%nat -> Rlength l = S (Rlength (FF l f)). + forall (l:list R) (f:R -> R), + (0 < length l)%nat -> length l = S (length (FF l f)). Proof. intros l f H; induction l; [ elim (lt_irrefl _ H) @@ -942,7 +948,7 @@ Proof. Qed. Lemma StepFun_P21 : - forall (a b:R) (f:R -> R) (l:Rlist), + forall (a b:R) (f:R -> R) (l:list R), is_subdivision f a b l -> adapted_couple f a b l (FF l f). Proof. intros * (x & H & H1 & H0 & H2 & H4). @@ -979,7 +985,7 @@ Proof. Qed. Lemma StepFun_P22 : - forall (a b:R) (f g:R -> R) (lf lg:Rlist), + forall (a b:R) (f g:R -> R) (lf lg:list R), a <= b -> is_subdivision f a b lf -> is_subdivision g a b lg -> is_subdivision f a b (cons_ORlist lf lg). @@ -1032,25 +1038,25 @@ Proof. (H8 : In (pos_Rl (cons_ORlist (cons r lf) lg) - (pred (Rlength (cons_ORlist (cons r lf) lg)))) + (pred (length (cons_ORlist (cons r lf) lg)))) (cons_ORlist (cons r lf) lg)). elim (RList_P3 (cons_ORlist (cons r lf) lg) (pos_Rl (cons_ORlist (cons r lf) lg) - (pred (Rlength (cons_ORlist (cons r lf) lg))))); + (pred (length (cons_ORlist (cons r lf) lg))))); intros _ H10; apply H10; - exists (pred (Rlength (cons_ORlist (cons r lf) lg))); + exists (pred (length (cons_ORlist (cons r lf) lg))); split; [ reflexivity | rewrite RList_P11; simpl; apply lt_n_Sn ]. elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) - (pred (Rlength (cons_ORlist (cons r lf) lg))))); + (pred (length (cons_ORlist (cons r lf) lg))))); intros H10 _. assert (H11 := H10 H8); elim H11; intro. elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) - (pred (Rlength (cons_ORlist (cons r lf) lg))))); + (pred (length (cons_ORlist (cons r lf) lg))))); intros H13 _; assert (H14 := H13 H12); elim H14; intros; elim H15; clear H15; intros; rewrite H15; rewrite <- H5; elim (RList_P6 (cons r lf)); intros; apply H17; @@ -1060,10 +1066,10 @@ Proof. elim (RList_P3 lg (pos_Rl (cons_ORlist (cons r lf) lg) - (pred (Rlength (cons_ORlist (cons r lf) lg))))); + (pred (length (cons_ORlist (cons r lf) lg))))); intros H13 _; assert (H14 := H13 H12); elim H14; intros; elim H15; clear H15; intros. - rewrite H15; assert (H17 : Rlength lg = S (pred (Rlength lg))). + rewrite H15; assert (H17 : length lg = S (pred (length lg))). apply S_pred with 0%nat; apply neq_O_lt; red; intro; rewrite <- H17 in H16; elim (lt_n_O _ H16). rewrite <- H0; elim (RList_P6 lg); intros; apply H18; @@ -1075,7 +1081,7 @@ Proof. assert (H8 : In b (cons_ORlist (cons r lf) lg)). elim (RList_P9 (cons r lf) lg b); intros; apply H10; left; elim (RList_P3 (cons r lf) b); intros; apply H12; - exists (pred (Rlength (cons r lf))); split; + exists (pred (length (cons r lf))); split; [ symmetry ; assumption | simpl; apply lt_n_Sn ]. apply RList_P7; [ apply RList_P2; assumption | assumption ]. apply StepFun_P20; rewrite RList_P11; rewrite H2; rewrite H7; simpl; @@ -1089,7 +1095,7 @@ Proof. intros; elim H11; clear H11; intros; assert (H12 := H11); assert (Hyp_cons : - exists r : R, (exists r0 : Rlist, cons_ORlist lf lg = cons r r0)). + exists r : R, (exists r0 : list R, cons_ORlist lf lg = cons r r0)). apply RList_P19; red; intro; rewrite H13 in H8; elim (lt_n_O _ H8). elim Hyp_cons; clear Hyp_cons; intros r [r0 Hyp_cons]; rewrite Hyp_cons; unfold FF; rewrite RList_P12. @@ -1128,7 +1134,7 @@ Proof. elim (RList_P6 (cons_ORlist lf lg)); intros; apply H11. apply RList_P2; assumption. apply le_O_n. - apply lt_trans with (pred (Rlength (cons_ORlist lf lg))); + apply lt_trans with (pred (length (cons_ORlist lf lg))); [ assumption | apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H13 in H8; elim (lt_n_O _ H8) ]. @@ -1147,9 +1153,9 @@ Proof. set (I := fun j:nat => - pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\ (j < Rlength lf)%nat); + pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\ (j < length lf)%nat); assert (H12 : Nbound I). - unfold Nbound; exists (Rlength lf); intros; unfold I in H12; elim H12; + unfold Nbound; exists (length lf); intros; unfold I in H12; elim H12; intros; apply lt_le_weak; assumption. assert (H13 : exists n : nat, I n). exists 0%nat; unfold I; split. @@ -1159,7 +1165,7 @@ Proof. elim (RList_P6 (cons_ORlist lf lg)); intros; apply H13. apply RList_P2; assumption. apply le_O_n. - apply lt_trans with (pred (Rlength (cons_ORlist lf lg))). + apply lt_trans with (pred (length (cons_ORlist lf lg))). assumption. apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H15 in H8; elim (lt_n_O _ H8). @@ -1167,12 +1173,12 @@ Proof. rewrite <- H6 in H11; rewrite <- H5 in H11; elim (Rlt_irrefl _ H11). assert (H14 := Nzorn H13 H12); elim H14; clear H14; intros x0 H14; exists (pos_Rl lf0 x0); unfold constant_D_eq, open_interval; - intros; assert (H16 := H9 x0); assert (H17 : (x0 < pred (Rlength lf))%nat). + intros; assert (H16 := H9 x0); assert (H17 : (x0 < pred (length lf))%nat). elim H14; clear H14; intros; unfold I in H14; elim H14; clear H14; intros; - apply lt_S_n; replace (S (pred (Rlength lf))) with (Rlength lf). + apply lt_S_n; replace (S (pred (length lf))) with (length lf). inversion H18. 2: apply lt_n_S; assumption. - cut (x0 = pred (Rlength lf)). + cut (x0 = pred (length lf)). intro; rewrite H19 in H14; rewrite H5 in H14; cut (pos_Rl (cons_ORlist lf lg) i < b). intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H14 H21)). @@ -1180,7 +1186,7 @@ Proof. elim H10; intros; apply Rlt_trans with x; assumption. rewrite <- H5; apply Rle_trans with - (pos_Rl (cons_ORlist lf lg) (pred (Rlength (cons_ORlist lf lg)))). + (pos_Rl (cons_ORlist lf lg) (pred (length (cons_ORlist lf lg)))). elim (RList_P6 (cons_ORlist lf lg)); intros; apply H21. apply RList_P2; assumption. apply lt_n_Sm_le; apply lt_n_S; assumption. @@ -1197,8 +1203,8 @@ Proof. elim H14; clear H14; intros; split. apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i); assumption. apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)); try assumption. - assert (H22 : (S x0 < Rlength lf)%nat). - replace (Rlength lf) with (S (pred (Rlength lf))); + assert (H22 : (S x0 < length lf)%nat). + replace (length lf) with (S (pred (length lf))); [ apply lt_n_S; assumption | symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red; intro; rewrite <- H22 in H21; elim (lt_n_O _ H21) ]. @@ -1216,7 +1222,7 @@ Proof. Qed. Lemma StepFun_P23 : - forall (a b:R) (f g:R -> R) (lf lg:Rlist), + forall (a b:R) (f g:R -> R) (lf lg:list R), is_subdivision f a b lf -> is_subdivision g a b lg -> is_subdivision f a b (cons_ORlist lf lg). Proof. @@ -1229,7 +1235,7 @@ Proof. Qed. Lemma StepFun_P24 : - forall (a b:R) (f g:R -> R) (lf lg:Rlist), + forall (a b:R) (f g:R -> R) (lf lg:list R), a <= b -> is_subdivision f a b lf -> is_subdivision g a b lg -> is_subdivision g a b (cons_ORlist lf lg). @@ -1282,24 +1288,24 @@ Proof. (H8 : In (pos_Rl (cons_ORlist (cons r lf) lg) - (pred (Rlength (cons_ORlist (cons r lf) lg)))) + (pred (length (cons_ORlist (cons r lf) lg)))) (cons_ORlist (cons r lf) lg)). elim (RList_P3 (cons_ORlist (cons r lf) lg) (pos_Rl (cons_ORlist (cons r lf) lg) - (pred (Rlength (cons_ORlist (cons r lf) lg))))); + (pred (length (cons_ORlist (cons r lf) lg))))); intros _ H10; apply H10; - exists (pred (Rlength (cons_ORlist (cons r lf) lg))); + exists (pred (length (cons_ORlist (cons r lf) lg))); split; [ reflexivity | rewrite RList_P11; simpl; apply lt_n_Sn ]. elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) - (pred (Rlength (cons_ORlist (cons r lf) lg))))); + (pred (length (cons_ORlist (cons r lf) lg))))); intros H10 _; assert (H11 := H10 H8); elim H11; intro. elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) - (pred (Rlength (cons_ORlist (cons r lf) lg))))); + (pred (length (cons_ORlist (cons r lf) lg))))); intros H13 _; assert (H14 := H13 H12); elim H14; intros; elim H15; clear H15; intros; rewrite H15; rewrite <- H5; elim (RList_P6 (cons r lf)); intros; apply H17; @@ -1309,10 +1315,10 @@ Proof. elim (RList_P3 lg (pos_Rl (cons_ORlist (cons r lf) lg) - (pred (Rlength (cons_ORlist (cons r lf) lg))))); + (pred (length (cons_ORlist (cons r lf) lg))))); intros H13 _; assert (H14 := H13 H12); elim H14; intros; elim H15; clear H15; intros; rewrite H15; - assert (H17 : Rlength lg = S (pred (Rlength lg))). + assert (H17 : length lg = S (pred (length lg))). apply S_pred with 0%nat; apply neq_O_lt; red; intro; rewrite <- H17 in H16; elim (lt_n_O _ H16). rewrite <- H0; elim (RList_P6 lg); intros; apply H18; @@ -1324,7 +1330,7 @@ Proof. assert (H8 : In b (cons_ORlist (cons r lf) lg)). elim (RList_P9 (cons r lf) lg b); intros; apply H10; left; elim (RList_P3 (cons r lf) b); intros; apply H12; - exists (pred (Rlength (cons r lf))); split; + exists (pred (length (cons r lf))); split; [ symmetry ; assumption | simpl; apply lt_n_Sn ]. apply RList_P7; [ apply RList_P2; assumption | assumption ]. apply StepFun_P20; rewrite RList_P11; rewrite H7; rewrite H2; simpl; @@ -1338,7 +1344,7 @@ Proof. intros; elim H11; clear H11; intros; assert (H12 := H11); assert (Hyp_cons : - exists r : R, (exists r0 : Rlist, cons_ORlist lf lg = cons r r0)). + exists r : R, (exists r0 : list R, cons_ORlist lf lg = cons r r0)). apply RList_P19; red; intro; rewrite H13 in H8; elim (lt_n_O _ H8). elim Hyp_cons; clear Hyp_cons; intros r [r0 Hyp_cons]; rewrite Hyp_cons; unfold FF; rewrite RList_P12. @@ -1377,7 +1383,7 @@ Proof. elim (RList_P6 (cons_ORlist lf lg)); intros; apply H11. apply RList_P2; assumption. apply le_O_n. - apply lt_trans with (pred (Rlength (cons_ORlist lf lg))); + apply lt_trans with (pred (length (cons_ORlist lf lg))); [ assumption | apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H13 in H8; elim (lt_n_O _ H8) ]. @@ -1394,9 +1400,9 @@ Proof. set (I := fun j:nat => - pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\ (j < Rlength lg)%nat); + pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\ (j < length lg)%nat); assert (H12 : Nbound I). - unfold Nbound; exists (Rlength lg); intros; unfold I in H12; elim H12; + unfold Nbound; exists (length lg); intros; unfold I in H12; elim H12; intros; apply lt_le_weak; assumption. assert (H13 : exists n : nat, I n). exists 0%nat; unfold I; split. @@ -1406,7 +1412,7 @@ Proof. elim (RList_P6 (cons_ORlist lf lg)); intros; apply H13; [ apply RList_P2; assumption | apply le_O_n - | apply lt_trans with (pred (Rlength (cons_ORlist lf lg))); + | apply lt_trans with (pred (length (cons_ORlist lf lg))); [ assumption | apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H15 in H8; elim (lt_n_O _ H8) ] ]. @@ -1414,12 +1420,12 @@ Proof. rewrite <- H1 in H11; rewrite <- H0 in H11; elim (Rlt_irrefl _ H11). assert (H14 := Nzorn H13 H12); elim H14; clear H14; intros x0 H14; exists (pos_Rl lg0 x0); unfold constant_D_eq, open_interval; - intros; assert (H16 := H4 x0); assert (H17 : (x0 < pred (Rlength lg))%nat). + intros; assert (H16 := H4 x0); assert (H17 : (x0 < pred (length lg))%nat). elim H14; clear H14; intros; unfold I in H14; elim H14; clear H14; intros; - apply lt_S_n; replace (S (pred (Rlength lg))) with (Rlength lg). + apply lt_S_n; replace (S (pred (length lg))) with (length lg). inversion H18. 2: apply lt_n_S; assumption. - cut (x0 = pred (Rlength lg)). + cut (x0 = pred (length lg)). intro; rewrite H19 in H14; rewrite H0 in H14; cut (pos_Rl (cons_ORlist lf lg) i < b). intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H14 H21)). @@ -1427,7 +1433,7 @@ Proof. elim H10; intros; apply Rlt_trans with x; assumption. rewrite <- H0; apply Rle_trans with - (pos_Rl (cons_ORlist lf lg) (pred (Rlength (cons_ORlist lf lg)))). + (pos_Rl (cons_ORlist lf lg) (pred (length (cons_ORlist lf lg)))). elim (RList_P6 (cons_ORlist lf lg)); intros; apply H21. apply RList_P2; assumption. apply lt_n_Sm_le; apply lt_n_S; assumption. @@ -1445,8 +1451,8 @@ Proof. elim H14; clear H14; intros; split. apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i); assumption. apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)); try assumption. - assert (H22 : (S x0 < Rlength lg)%nat). - replace (Rlength lg) with (S (pred (Rlength lg))). + assert (H22 : (S x0 < length lg)%nat). + replace (length lg) with (S (pred (length lg))). apply lt_n_S; assumption. symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red; intro; rewrite <- H22 in H21; elim (lt_n_O _ H21). @@ -1463,7 +1469,7 @@ Proof. Qed. Lemma StepFun_P25 : - forall (a b:R) (f g:R -> R) (lf lg:Rlist), + forall (a b:R) (f g:R -> R) (lf lg:list R), is_subdivision f a b lf -> is_subdivision g a b lg -> is_subdivision g a b (cons_ORlist lf lg). Proof. @@ -1476,7 +1482,7 @@ Proof. Qed. Lemma StepFun_P26 : - forall (a b l:R) (f g:R -> R) (l1:Rlist), + forall (a b l:R) (f g:R -> R) (l1:list R), is_subdivision f a b l1 -> is_subdivision g a b l1 -> is_subdivision (fun x:R => f x + l * g x) a b l1. @@ -1494,7 +1500,7 @@ Proof. change (pos_Rl x0 i + l * pos_Rl x i = pos_Rl - (app_Rlist (mid_Rlist (cons r r0) r) (fun x2:R => f x2 + l * g x2)) + (map (fun x2:R => f x2 + l * g x2) (mid_Rlist (cons r r0) r)) (S i)); rewrite RList_P12. rewrite RList_P13. rewrite <- H12; rewrite (H9 _ H8); try rewrite (H4 _ H8); @@ -1521,7 +1527,7 @@ Proof. Qed. Lemma StepFun_P27 : - forall (a b l:R) (f g:R -> R) (lf lg:Rlist), + forall (a b l:R) (f g:R -> R) (lf lg:list R), is_subdivision f a b lf -> is_subdivision g a b lg -> is_subdivision (fun x:R => f x + l * g x) a b (cons_ORlist lf lg). @@ -1586,9 +1592,9 @@ Proof. Qed. Lemma StepFun_P31 : - forall (a b:R) (f:R -> R) (l lf:Rlist), + forall (a b:R) (f:R -> R) (l lf:list R), adapted_couple f a b l lf -> - adapted_couple (fun x:R => Rabs (f x)) a b l (app_Rlist lf Rabs). + adapted_couple (fun x:R => Rabs (f x)) a b l (map Rabs lf). Proof. unfold adapted_couple; intros; decompose [and] H; clear H; repeat split; try assumption. @@ -1604,15 +1610,15 @@ Lemma StepFun_P32 : Proof. intros a b f; unfold IsStepFun; apply existT with (subdivision f); unfold is_subdivision; - apply existT with (app_Rlist (subdivision_val f) Rabs); + apply existT with (map Rabs (subdivision_val f)); apply StepFun_P31; apply StepFun_P1. Qed. Lemma StepFun_P33 : - forall l2 l1:Rlist, - ordered_Rlist l1 -> Rabs (Int_SF l2 l1) <= Int_SF (app_Rlist l2 Rabs) l1. + forall l2 l1:list R, + ordered_Rlist l1 -> Rabs (Int_SF l2 l1) <= Int_SF (map Rabs l2) l1. Proof. - simple induction l2; intros. + induction l2 as [ | r r0 H]; intros. simpl; rewrite Rabs_R0; right; reflexivity. simpl; induction l1 as [| r1 l1 Hrecl1]. rewrite Rabs_R0; right; reflexivity. @@ -1635,7 +1641,7 @@ Proof. replace (Int_SF (subdivision_val (mkStepFun (StepFun_P32 f))) (subdivision (mkStepFun (StepFun_P32 f)))) with - (Int_SF (app_Rlist (subdivision_val f) Rabs) (subdivision f)). + (Int_SF (map Rabs (subdivision_val f)) (subdivision f)). apply StepFun_P33; assert (H0 := StepFun_P29 f); unfold is_subdivision in H0; elim H0; intros; unfold adapted_couple in p; decompose [and] p; assumption. @@ -1645,14 +1651,14 @@ Proof. Qed. Lemma StepFun_P35 : - forall (l:Rlist) (a b:R) (f g:R -> R), + forall (l:list R) (a b:R) (f g:R -> R), ordered_Rlist l -> pos_Rl l 0 = a -> - pos_Rl l (pred (Rlength l)) = b -> + pos_Rl l (pred (length l)) = b -> (forall x:R, a < x < b -> f x <= g x) -> Int_SF (FF l f) l <= Int_SF (FF l g) l. Proof. - simple induction l; intros. + induction l as [ | r r0 H]; intros. right; reflexivity. simpl; induction r0 as [| r0 r1 Hrecr0]. right; reflexivity. @@ -1682,7 +1688,7 @@ Proof. rewrite <- Rinv_r_sym. rewrite Rmult_1_l; rewrite double; assert (H5 : r0 <= b). replace b with - (pos_Rl (cons r (cons r0 r1)) (pred (Rlength (cons r (cons r0 r1))))). + (pos_Rl (cons r (cons r0 r1)) (pred (length (cons r (cons r0 r1))))). replace r0 with (pos_Rl (cons r (cons r0 r1)) 1). elim (RList_P6 (cons r (cons r0 r1))); intros; apply H5. assumption. @@ -1712,7 +1718,7 @@ Proof. Qed. Lemma StepFun_P36 : - forall (a b:R) (f g:StepFun a b) (l:Rlist), + forall (a b:R) (f g:StepFun a b) (l:list R), a <= b -> is_subdivision f a b l -> is_subdivision g a b l -> @@ -1748,18 +1754,18 @@ Proof. Qed. Lemma StepFun_P38 : - forall (l:Rlist) (a b:R) (f:R -> R), + forall (l:list R) (a b:R) (f:R -> R), ordered_Rlist l -> pos_Rl l 0 = a -> - pos_Rl l (pred (Rlength l)) = b -> + pos_Rl l (pred (length l)) = b -> { g:StepFun a b | g b = f b /\ (forall i:nat, - (i < pred (Rlength l))%nat -> + (i < pred (length l))%nat -> constant_D_eq g (co_interval (pos_Rl l i) (pos_Rl l (S i))) (f (pos_Rl l i))) }. Proof. - intros l a b f; generalize a; clear a; induction l. + intros l a b f; generalize a; clear a; induction l as [|r l IHl]. intros a H H0 H1; simpl in H0; simpl in H1; exists (mkStepFun (StepFun_P4 a b (f b))); split. reflexivity. @@ -1772,7 +1778,7 @@ Proof. apply RList_P4 with r; assumption. assert (H3 : pos_Rl (cons r1 l) 0 = r1). reflexivity. - assert (H4 : pos_Rl (cons r1 l) (pred (Rlength (cons r1 l))) = b). + assert (H4 : pos_Rl (cons r1 l) (pred (length (cons r1 l))) = b). rewrite <- H1; reflexivity. elim (IHl r1 H2 H3 H4); intros g [H5 H6]. set @@ -1796,7 +1802,7 @@ Proof. simpl in H0; rewrite <- H0; apply (H 0%nat); simpl; apply lt_O_Sn. unfold Rmin; decide (Rle_dec r1 b) with H7; reflexivity. apply (H10 i); apply lt_S_n. - replace (S (pred (Rlength lg))) with (Rlength lg). + replace (S (pred (length lg))) with (length lg). apply H9. apply S_pred with 0%nat; apply neq_O_lt; intro; rewrite <- H14 in H9; elim (lt_n_O _ H9). @@ -1825,9 +1831,9 @@ Proof. change (constant_D_eq g' (open_interval (pos_Rl lg i) (pos_Rl lg (S i))) (pos_Rl lg2 i)); clear Hreci; assert (H16 := H15 i); - assert (H17 : (i < pred (Rlength lg))%nat). + assert (H17 : (i < pred (length lg))%nat). apply lt_S_n. - replace (S (pred (Rlength lg))) with (Rlength lg). + replace (S (pred (length lg))) with (length lg). assumption. apply S_pred with 0%nat; apply neq_O_lt; red; intro; rewrite <- H14 in H9; elim (lt_n_O _ H9). @@ -1843,7 +1849,7 @@ Proof. assumption. elim (RList_P3 lg (pos_Rl lg i)); intros; apply H21; exists i; split. reflexivity. - apply lt_trans with (pred (Rlength lg)); try assumption. + apply lt_trans with (pred (length lg)); try assumption. apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H22 in H17; elim (lt_n_O _ H17). unfold Rmin; decide (Rle_dec r1 b) with H7; reflexivity. @@ -1860,7 +1866,7 @@ Proof. (constant_D_eq (mkStepFun H8) (co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i))) (f (pos_Rl (cons r1 l) i))); assert (H10 := H6 i); - assert (H11 : (i < pred (Rlength (cons r1 l)))%nat). + assert (H11 : (i < pred (length (cons r1 l)))%nat). simpl; apply lt_S_n; assumption. assert (H12 := H10 H11); unfold constant_D_eq, co_interval in H12; unfold constant_D_eq, co_interval; intros; @@ -1873,7 +1879,7 @@ Proof. elim (RList_P6 (cons r1 l)); intros; apply H15; [ assumption | apply le_O_n - | simpl; apply lt_trans with (Rlength l); + | simpl; apply lt_trans with (length l); [ apply lt_S_n; assumption | apply lt_n_Sn ] ]. Qed. @@ -1912,12 +1918,12 @@ Proof. Qed. Lemma StepFun_P40 : - forall (f:R -> R) (a b c:R) (l1 l2 lf1 lf2:Rlist), + forall (f:R -> R) (a b c:R) (l1 l2 lf1 lf2:list R), a < b -> b < c -> adapted_couple f a b l1 lf1 -> adapted_couple f b c l2 lf2 -> - adapted_couple f a c (cons_Rlist l1 l2) (FF (cons_Rlist l1 l2) f). + adapted_couple f a c (app l1 l2) (FF (app l1 l2) f). Proof. intros f a b c l1 l2 lf1 lf2 H H0 H1 H2; unfold adapted_couple in H1, H2; unfold adapted_couple; decompose [and] H1; @@ -1941,28 +1947,28 @@ Proof. | left; assumption ]. red; intro; rewrite H1 in H11; discriminate. apply StepFun_P20. - rewrite RList_P23; apply neq_O_lt; red; intro. - assert (H2 : (Rlength l1 + Rlength l2)%nat = 0%nat). + rewrite app_length; apply neq_O_lt; red; intro. + assert (H2 : (length l1 + length l2)%nat = 0%nat). symmetry ; apply H1. elim (plus_is_O _ _ H2); intros; rewrite H12 in H6; discriminate. unfold constant_D_eq, open_interval; intros; - elim (le_or_lt (S (S i)) (Rlength l1)); intro. - assert (H14 : pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i). + elim (le_or_lt (S (S i)) (length l1)); intro. + assert (H14 : pos_Rl (app l1 l2) i = pos_Rl l1 i). apply RList_P26; apply lt_S_n; apply le_lt_n_Sm; apply le_S_n; - apply le_trans with (Rlength l1); [ assumption | apply le_n_Sn ]. - assert (H15 : pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l1 (S i)). + apply le_trans with (length l1); [ assumption | apply le_n_Sn ]. + assert (H15 : pos_Rl (app l1 l2) (S i) = pos_Rl l1 (S i)). apply RList_P26; apply lt_S_n; apply le_lt_n_Sm; assumption. - rewrite H14 in H2; rewrite H15 in H2; assert (H16 : (2 <= Rlength l1)%nat). + rewrite H14 in H2; rewrite H15 in H2; assert (H16 : (2 <= length l1)%nat). apply le_trans with (S (S i)); [ repeat apply le_n_S; apply le_O_n | assumption ]. elim (RList_P20 _ H16); intros r1 [r2 [r3 H17]]; rewrite H17; change - (f x = pos_Rl (app_Rlist (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1) f) i) + (f x = pos_Rl (map f (mid_Rlist (app (cons r2 r3) l2) r1)) i) ; rewrite RList_P12. induction i as [| i Hreci]. simpl; assert (H18 := H8 0%nat); unfold constant_D_eq, open_interval in H18; - assert (H19 : (0 < pred (Rlength l1))%nat). + assert (H19 : (0 < pred (length l1))%nat). rewrite H17; simpl; apply lt_O_Sn. assert (H20 := H18 H19); repeat rewrite H20. reflexivity. @@ -1991,14 +1997,14 @@ Proof. clear Hreci; rewrite RList_P13. rewrite H17 in H14; rewrite H17 in H15; change - (pos_Rl (cons_Rlist (cons r2 r3) l2) i = + (pos_Rl (app (cons r2 r3) l2) i = pos_Rl (cons r1 (cons r2 r3)) (S i)) in H14; rewrite H14; change - (pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) = + (pos_Rl (app (cons r2 r3) l2) (S i) = pos_Rl (cons r1 (cons r2 r3)) (S (S i))) in H15; rewrite H15; assert (H18 := H8 (S i)); unfold constant_D_eq, open_interval in H18; - assert (H19 : (S i < pred (Rlength l1))%nat). + assert (H19 : (S i < pred (length l1))%nat). apply lt_pred; apply lt_S_n; apply le_lt_n_Sm; assumption. assert (H20 := H18 H19); repeat rewrite H20. reflexivity. @@ -2025,7 +2031,7 @@ Proof. simpl; rewrite H17 in H1; simpl in H1; apply lt_S_n; assumption. rewrite RList_P14; rewrite H17 in H1; simpl in H1; apply H1. inversion H12. - assert (H16 : pos_Rl (cons_Rlist l1 l2) (S i) = b). + assert (H16 : pos_Rl (app l1 l2) (S i) = b). rewrite RList_P29. rewrite H15; rewrite <- minus_n_n; rewrite H10; unfold Rmin; case (Rle_dec b c) as [|[]]; [ reflexivity | left; assumption ]. @@ -2033,30 +2039,30 @@ Proof. induction l1 as [| r l1 Hrecl1]. simpl in H15; discriminate. clear Hrecl1; simpl in H1; simpl; apply lt_n_S; assumption. - assert (H17 : pos_Rl (cons_Rlist l1 l2) i = b). + assert (H17 : pos_Rl (app l1 l2) i = b). rewrite RList_P26. - replace i with (pred (Rlength l1)); + replace i with (pred (length l1)); [ rewrite H4; unfold Rmax; case (Rle_dec a b) as [|[]]; [ reflexivity | left; assumption ] | rewrite H15; reflexivity ]. rewrite H15; apply lt_n_Sn. rewrite H16 in H2; rewrite H17 in H2; elim H2; intros; elim (Rlt_irrefl _ (Rlt_trans _ _ _ H14 H18)). - assert (H16 : pos_Rl (cons_Rlist l1 l2) i = pos_Rl l2 (i - Rlength l1)). + assert (H16 : pos_Rl (app l1 l2) i = pos_Rl l2 (i - length l1)). apply RList_P29. apply le_S_n; assumption. - apply lt_le_trans with (pred (Rlength (cons_Rlist l1 l2))); + apply lt_le_trans with (pred (length (app l1 l2))); [ assumption | apply le_pred_n ]. assert - (H17 : pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l2 (S (i - Rlength l1))). - replace (S (i - Rlength l1)) with (S i - Rlength l1)%nat. + (H17 : pos_Rl (app l1 l2) (S i) = pos_Rl l2 (S (i - length l1))). + replace (S (i - length l1)) with (S i - length l1)%nat. apply RList_P29. apply le_S_n; apply le_trans with (S i); [ assumption | apply le_n_Sn ]. induction l1 as [| r l1 Hrecl1]. simpl in H6; discriminate. clear Hrecl1; simpl in H1; simpl; apply lt_n_S; assumption. symmetry ; apply minus_Sn_m; apply le_S_n; assumption. - assert (H18 : (2 <= Rlength l1)%nat). + assert (H18 : (2 <= length l1)%nat). clear f c l2 lf2 H0 H3 H8 H7 H10 H9 H11 H13 i H1 x H2 H12 m H14 H15 H16 H17; induction l1 as [| r l1 Hrecl1]. discriminate. @@ -2068,7 +2074,7 @@ Proof. clear Hrecl1; simpl; repeat apply le_n_S; apply le_O_n. elim (RList_P20 _ H18); intros r1 [r2 [r3 H19]]; rewrite H19; change - (f x = pos_Rl (app_Rlist (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1) f) i) + (f x = pos_Rl (map f (mid_Rlist (app (cons r2 r3) l2) r1)) i) ; rewrite RList_P12. induction i as [| i Hreci]. assert (H20 := le_S_n _ _ H15); assert (H21 := le_trans _ _ _ H18 H20); @@ -2076,31 +2082,31 @@ Proof. clear Hreci; rewrite RList_P13. rewrite H19 in H16; rewrite H19 in H17; change - (pos_Rl (cons_Rlist (cons r2 r3) l2) i = - pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))) + (pos_Rl (app (cons r2 r3) l2) i = + pos_Rl l2 (S i - length (cons r1 (cons r2 r3)))) in H16; rewrite H16; change - (pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) = - pos_Rl l2 (S (S i - Rlength (cons r1 (cons r2 r3))))) - in H17; rewrite H17; assert (H20 := H13 (S i - Rlength l1)%nat); + (pos_Rl (app (cons r2 r3) l2) (S i) = + pos_Rl l2 (S (S i - length (cons r1 (cons r2 r3))))) + in H17; rewrite H17; assert (H20 := H13 (S i - length l1)%nat); unfold constant_D_eq, open_interval in H20; - assert (H21 : (S i - Rlength l1 < pred (Rlength l2))%nat). + assert (H21 : (S i - length l1 < pred (length l2))%nat). apply lt_pred; rewrite minus_Sn_m. - apply plus_lt_reg_l with (Rlength l1); rewrite <- le_plus_minus. + apply plus_lt_reg_l with (length l1); rewrite <- le_plus_minus. rewrite H19 in H1; simpl in H1; rewrite H19; simpl; - rewrite RList_P23 in H1; apply lt_n_S; assumption. + rewrite app_length in H1; apply lt_n_S; assumption. apply le_trans with (S i); [ apply le_S_n; assumption | apply le_n_Sn ]. apply le_S_n; assumption. assert (H22 := H20 H21); repeat rewrite H22. reflexivity. rewrite <- H19; assert - (H23 : pos_Rl l2 (S i - Rlength l1) <= pos_Rl l2 (S (S i - Rlength l1))). + (H23 : pos_Rl l2 (S i - length l1) <= pos_Rl l2 (S (S i - length l1))). apply H7; apply lt_pred. rewrite minus_Sn_m. - apply plus_lt_reg_l with (Rlength l1); rewrite <- le_plus_minus. + apply plus_lt_reg_l with (length l1); rewrite <- le_plus_minus. rewrite H19 in H1; simpl in H1; rewrite H19; simpl; - rewrite RList_P23 in H1; apply lt_n_S; assumption. + rewrite app_length in H1; apply lt_n_S; assumption. apply le_trans with (S i); [ apply le_S_n; assumption | apply le_n_Sn ]. apply le_S_n; assumption. elim H23; intro. @@ -2115,7 +2121,7 @@ Proof. [ prove_sup0 | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; - [ rewrite Rmult_1_l; rewrite (Rplus_comm (pos_Rl l2 (S i - Rlength l1))); + [ rewrite Rmult_1_l; rewrite (Rplus_comm (pos_Rl l2 (S i - length l1))); rewrite double; apply Rplus_lt_compat_l; assumption | discrR ] ]. rewrite <- H19 in H16; rewrite <- H19 in H17; elim H2; intros; @@ -2123,11 +2129,11 @@ Proof. simpl in H16; rewrite H16 in H25; simpl in H26; simpl in H17; rewrite H17 in H26; simpl in H24; rewrite H24 in H25; elim (Rlt_irrefl _ (Rlt_trans _ _ _ H25 H26)). - assert (H23 : pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l2 (S i - Rlength l1)). + assert (H23 : pos_Rl (app l1 l2) (S i) = pos_Rl l2 (S i - length l1)). rewrite H19; simpl; simpl in H16; apply H16. assert (H24 : - pos_Rl (cons_Rlist l1 l2) (S (S i)) = pos_Rl l2 (S (S i - Rlength l1))). + pos_Rl (app l1 l2) (S (S i)) = pos_Rl l2 (S (S i - length l1))). rewrite H19; simpl; simpl in H17; apply H17. rewrite <- H23; rewrite <- H24; assumption. simpl; rewrite H19 in H1; simpl in H1; apply lt_S_n; assumption. @@ -2141,7 +2147,7 @@ Proof. intros f a b c H H0 (l1,(lf1,H1)) (l2,(lf2,H2)); destruct (total_order_T a b) as [[Hltab|Hab]|Hgtab]. destruct (total_order_T b c) as [[Hltbc|Hbc]|Hgtbc]. - exists (cons_Rlist l1 l2); exists (FF (cons_Rlist l1 l2) f); + exists (app l1 l2); exists (FF (app l1 l2) f); apply StepFun_P40 with b lf1 lf2; assumption. exists l1; exists lf1; rewrite Hbc in H1; assumption. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 Hgtbc)). @@ -2150,9 +2156,9 @@ Proof. Qed. Lemma StepFun_P42 : - forall (l1 l2:Rlist) (f:R -> R), - pos_Rl l1 (pred (Rlength l1)) = pos_Rl l2 0 -> - Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2) = + forall (l1 l2:list R) (f:R -> R), + pos_Rl l1 (pred (length l1)) = pos_Rl l2 0 -> + Int_SF (FF (app l1 l2) f) (app l1 l2) = Int_SF (FF l1 f) l1 + Int_SF (FF l2 f) l2. Proof. intros l1 l2 f; induction l1 as [| r l1 IHl1]; intros H; @@ -2193,7 +2199,7 @@ Proof. elim Hle; intro. elim Hle'; intro. replace (Int_SF lf3 l3) with - (Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2)). + (Int_SF (FF (app l1 l2) f) (app l1 l2)). replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). symmetry ; apply StepFun_P42. @@ -2225,7 +2231,7 @@ Proof. elim Hle''; intro. rewrite Rplus_comm; replace (Int_SF lf1 l1) with - (Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2)). + (Int_SF (FF (app l3 l2) f) (app l3 l2)). replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). apply StepFun_P42. @@ -2249,7 +2255,7 @@ Proof. ring. elim Hle; intro. replace (Int_SF lf2 l2) with - (Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1)). + (Int_SF (FF (app l3 l1) f) (app l3 l1)). replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). symmetry ; apply StepFun_P42. @@ -2277,7 +2283,7 @@ Proof. ring. rewrite Rplus_comm; elim Hle''; intro. replace (Int_SF lf2 l2) with - (Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3)). + (Int_SF (FF (app l1 l3) f) (app l1 l3)). replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). symmetry ; apply StepFun_P42. @@ -2304,7 +2310,7 @@ Proof. ring. elim Hle'; intro. replace (Int_SF lf1 l1) with - (Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3)). + (Int_SF (FF (app l2 l3) f) (app l2 l3)). replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). symmetry ; apply StepFun_P42. @@ -2334,7 +2340,7 @@ Proof. replace (Int_SF lf3 l3) with (Int_SF lf2 l2 + Int_SF lf1 l1). ring. replace (Int_SF lf3 l3) with - (Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1)). + (Int_SF (FF (app l2 l1) f) (app l2 l1)). replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). symmetry ; apply StepFun_P42. @@ -2395,17 +2401,17 @@ Proof. elim H; clear H; intros; unfold IsStepFun in X; unfold is_subdivision in X; elim X; clear X; intros l1 [lf1 H2]; cut - (forall (l1 lf1:Rlist) (a b c:R) (f:R -> R), + (forall (l1 lf1:list R) (a b c:R) (f:R -> R), adapted_couple f a b l1 lf1 -> a <= c <= b -> - { l:Rlist & { l0:Rlist & adapted_couple f a c l l0 } }). + { l:list R & { l0:list R & adapted_couple f a c l l0 } }). intro X; unfold IsStepFun; unfold is_subdivision; eapply X. apply H2. split; assumption. clear f a b c H0 H H1 H2 l1 lf1; simple induction l1. intros; unfold adapted_couple in H; decompose [and] H; clear H; simpl in H4; discriminate. - simple induction r0. + intros r r0; elim r0. intros X lf1 a b c f H H0; assert (H1 : a = b). unfold adapted_couple in H; decompose [and] H; clear H; simpl in H3; simpl in H2; assert (H7 : a <= b). @@ -2438,7 +2444,7 @@ Proof. unfold constant_D_eq, open_interval; intros; simpl in H8; inversion H8. simpl; assert (H10 := H7 0%nat); - assert (H12 : (0 < pred (Rlength (cons r (cons r1 r2))))%nat). + assert (H12 : (0 < pred (length (cons r (cons r1 r2))))%nat). simpl; apply lt_O_Sn. apply (H10 H12); unfold open_interval; simpl; rewrite H11 in H9; simpl in H9; elim H9; clear H9; @@ -2479,7 +2485,7 @@ Proof. intros; simpl in H; unfold constant_D_eq, open_interval; intros; induction i as [| i Hreci]. simpl; assert (H17 := H10 0%nat); - assert (H18 : (0 < pred (Rlength (cons r (cons r1 r2))))%nat). + assert (H18 : (0 < pred (length (cons r (cons r1 r2))))%nat). simpl; apply lt_O_Sn. apply (H17 H18); unfold open_interval; simpl; simpl in H4; elim H4; clear H4; intros; split; try assumption; @@ -2507,16 +2513,16 @@ Proof. elim H; clear H; intros; unfold IsStepFun in X; unfold is_subdivision in X; elim X; clear X; intros l1 [lf1 H2]; cut - (forall (l1 lf1:Rlist) (a b c:R) (f:R -> R), + (forall (l1 lf1:list R) (a b c:R) (f:R -> R), adapted_couple f a b l1 lf1 -> a <= c <= b -> - { l:Rlist & { l0:Rlist & adapted_couple f c b l l0 } }). + { l:list R & { l0:list R & adapted_couple f c b l l0 } }). intro X; unfold IsStepFun; unfold is_subdivision; eapply X; [ apply H2 | split; assumption ]. clear f a b c H0 H H1 H2 l1 lf1; simple induction l1. intros; unfold adapted_couple in H; decompose [and] H; clear H; simpl in H4; discriminate. - simple induction r0. + intros r r0; elim r0. intros X lf1 a b c f H H0; assert (H1 : a = b). unfold adapted_couple in H; decompose [and] H; clear H; simpl in H3; simpl in H2; assert (H7 : a <= b). diff --git a/theories/Reals/Rtopology.v b/theories/Reals/Rtopology.v index d21042884e..fa5442e86f 100644 --- a/theories/Reals/Rtopology.v +++ b/theories/Reals/Rtopology.v @@ -12,6 +12,7 @@ Require Import Rbase. Require Import Rfunctions. Require Import Ranalysis1. Require Import RList. +Require Import List. Require Import Classical_Prop. Require Import Classical_Pred_Type. Local Open Scope R_scope. @@ -388,7 +389,7 @@ Record family : Type := mkfamily Definition family_open_set (f:family) : Prop := forall x:R, open_set (f x). Definition domain_finite (D:R -> Prop) : Prop := - exists l : Rlist, (forall x:R, D x <-> In x l). + exists l : list R, (forall x:R, D x <-> In x l). Definition family_finite (f:family) : Prop := domain_finite (ind f). @@ -669,7 +670,7 @@ Proof. intro H14; simpl in H14; unfold intersection_domain in H14; specialize H13 with x0; destruct H13 as (H13,H15); destruct (Req_dec x0 y0) as [H16|H16]. - simpl; left; apply H16. + simpl; left. symmetry; apply H16. simpl; right; apply H13. simpl; unfold intersection_domain; unfold Db in H14; decompose [and or] H14. @@ -678,8 +679,8 @@ Proof. intro H14; simpl in H14; destruct H14 as [H15|H15]; simpl; unfold intersection_domain. split. - apply (cond_fam f0); rewrite H15; exists b; apply H6. - unfold Db; right; assumption. + apply (cond_fam f0); rewrite <- H15; exists b; apply H6. + unfold Db; right; symmetry; assumption. simpl; unfold intersection_domain; elim (H13 x0). intros _ H16; assert (H17 := H16 H15); simpl in H17; unfold intersection_domain in H17; split. @@ -750,15 +751,15 @@ Proof. intro H14; simpl in H14; unfold intersection_domain in H14; specialize (H13 x0); destruct H13 as (H13,H15); destruct (Req_dec x0 y0) as [Heq|Hneq]. - simpl; left; apply Heq. + simpl; left; symmetry; apply Heq. simpl; right; apply H13; simpl; unfold intersection_domain; unfold Db in H14; decompose [and or] H14. split; assumption. elim Hneq; assumption. intros [H15|H15]. split. - apply (cond_fam f0); rewrite H15; exists m; apply H6. - unfold Db; right; assumption. + apply (cond_fam f0); rewrite <- H15; exists m; apply H6. + unfold Db; right; symmetry; assumption. elim (H13 x0); intros _ H16. assert (H17 := H16 H15). simpl in H17. @@ -810,9 +811,10 @@ Proof. unfold family_finite; unfold domain_finite; exists (cons y0 nil); intro; split. simpl; unfold intersection_domain; intros (H3,H4). - unfold D' in H4; left; apply H4. + unfold D' in H4; left; symmetry; apply H4. simpl; unfold intersection_domain; intros [H4|[]]. - split; [ rewrite H4; apply (cond_fam f0); exists a; apply H2 | apply H4 ]. + split; [ rewrite <- H4; apply (cond_fam f0); exists a; apply H2 | + symmetry; apply H4 ]. split; [ right; reflexivity | apply Hle ]. apply compact_eqDom with (fun c:R => False). apply compact_EMP. diff --git a/tools/coqdep.ml b/tools/coqdep.ml index 2140014c58..745cf950b5 100644 --- a/tools/coqdep.ml +++ b/tools/coqdep.ml @@ -20,7 +20,6 @@ open Minisys As of today, this module depends on the following Coq modules: - - Flags - Envars - CoqProject_file @@ -28,10 +27,7 @@ open Minisys coqlib handling up so this can be bootstrapped earlier. *) -let option_D = ref false -let option_w = ref false let option_sort = ref false -let option_dump = ref None let warning_mult suf iter = let tab = Hashtbl.create 151 in @@ -74,378 +70,10 @@ let sort () = in List.iter (fun (name,_) -> loop name) !vAccu -let (dep_tab : (string,string list) Hashtbl.t) = Hashtbl.create 151 - -let mL_dep_list b f = - try - Hashtbl.find dep_tab f - with Not_found -> - let deja_vu = ref ([] : string list) in - try - let chan = open_in f in - let buf = Lexing.from_channel chan in - try - while true do - let (Use_module str) = caml_action buf in - if str = b then begin - coqdep_warning "in file %s the notation %s. is useless !\n" f b - end else - if not (List.mem str !deja_vu) then addQueue deja_vu str - done; [] - with Fin_fichier -> begin - close_in chan; - let rl = List.rev !deja_vu in - Hashtbl.add dep_tab f rl; - rl - end - with Sys_error _ -> [] - -let affiche_Declare f dcl = - printf "\n*** In file %s: \n" f; - printf "Declare ML Module"; - List.iter (fun str -> printf " \"%s\"" str) dcl; - printf ".\n%!" - -let warning_Declare f dcl = - eprintf "*** Warning : in file %s, the ML modules declaration should be\n" f; - eprintf "*** Declare ML Module"; - List.iter (fun str -> eprintf " \"%s\"" str) dcl; - eprintf ".\n%!" - -let traite_Declare f = - let decl_list = ref ([] : string list) in - let rec treat = function - | s :: ll -> - let s' = basename_noext s in - (match search_ml_known s with - | Some mldir when not (List.mem s' !decl_list) -> - let fullname = file_name s' mldir in - let depl = mL_dep_list s (fullname ^ ".ml") in - treat depl; - decl_list := s :: !decl_list - | _ -> ()); - treat ll - | [] -> () - in - try - let chan = open_in f in - let buf = Lexing.from_channel chan in - begin try - while true do - let tok = coq_action buf in - (match tok with - | Declare sl -> - decl_list := []; - treat sl; - decl_list := List.rev !decl_list; - if !option_D then - affiche_Declare f !decl_list - else if !decl_list <> sl then - warning_Declare f !decl_list - | _ -> ()) - done - with Fin_fichier -> () end; - close_in chan - with Sys_error _ -> () - -let declare_dependencies () = - List.iter - (fun (name,_) -> - traite_Declare (name^".v"); - pp_print_flush std_formatter ()) - (List.rev !vAccu) - -(** DAGs guaranteed to be transitive reductions *) -module DAG (Node : Set.OrderedType) : -sig - type node = Node.t - type t - val empty : t - val add_transitive_edge : node -> node -> t -> t - val iter : (node -> node -> unit) -> t -> unit -end = -struct - type node = Node.t - module NSet = Set.Make(Node) - module NMap = Map.Make(Node) - - (** Associate to a node the set of its neighbours *) - type _t = NSet.t NMap.t - - (** Optimisation: construct the reverse graph at the same time *) - type t = { dir : _t; rev : _t; } - - - let node_equal x y = Node.compare x y = 0 - - let add_edge x y graph = - let set = try NMap.find x graph with Not_found -> NSet.empty in - NMap.add x (NSet.add y set) graph - - let remove_edge x y graph = - let set = try NMap.find x graph with Not_found -> NSet.empty in - let set = NSet.remove y set in - if NSet.is_empty set then NMap.remove x graph - else NMap.add x set graph - - let has_edge x y graph = - let set = try NMap.find x graph with Not_found -> NSet.empty in - NSet.mem y set - - let connected x y graph = - let rec aux rem seen = - if NSet.is_empty rem then false - else - let x = NSet.choose rem in - if node_equal x y then true - else - let rem = NSet.remove x rem in - if NSet.mem x seen then - aux rem seen - else - let seen = NSet.add x seen in - let next = try NMap.find x graph with Not_found -> NSet.empty in - let rem = NSet.union next rem in - aux rem seen - in - aux (NSet.singleton x) NSet.empty - - (** Check whether there is a path from a to b going through the edge - x -> y. *) - let connected_through a b x y graph = - let rec aux rem seen = - if NMap.is_empty rem then false - else - let (n, through) = NMap.choose rem in - if node_equal n b && through then true - else - let rem = NMap.remove n rem in - let is_seen = try Some (NMap.find n seen) with Not_found -> None in - match is_seen with - | None -> - let seen = NMap.add n through seen in - let next = try NMap.find n graph with Not_found -> NSet.empty in - let is_x = node_equal n x in - let push m accu = - let through = through || (is_x && node_equal m y) in - NMap.add m through accu - in - let rem = NSet.fold push next rem in - aux rem seen - | Some false -> - (* The path we took encountered x -> y but not the one in seen *) - if through then aux (NMap.add n true rem) (NMap.add n true seen) - else aux rem seen - | Some true -> aux rem seen - in - aux (NMap.singleton a false) NMap.empty - - let closure x graph = - let rec aux rem seen = - if NSet.is_empty rem then seen - else - let x = NSet.choose rem in - let rem = NSet.remove x rem in - if NSet.mem x seen then - aux rem seen - else - let seen = NSet.add x seen in - let next = try NMap.find x graph with Not_found -> NSet.empty in - let rem = NSet.union next rem in - aux rem seen - in - aux (NSet.singleton x) NSet.empty - - let empty = { dir = NMap.empty; rev = NMap.empty; } - - (** Online transitive reduction algorithm *) - let add_transitive_edge x y graph = - if connected x y graph.dir then graph - else - let dir = add_edge x y graph.dir in - let rev = add_edge y x graph.rev in - let graph = { dir; rev; } in - let ancestors = closure x rev in - let descendents = closure y dir in - let fold_ancestor a graph = - let fold_descendent b graph = - let to_remove = has_edge a b graph.dir in - let to_remove = to_remove && not (node_equal x a && node_equal y b) in - let to_remove = to_remove && connected_through a b x y graph.dir in - if to_remove then - let dir = remove_edge a b graph.dir in - let rev = remove_edge b a graph.rev in - { dir; rev; } - else graph - in - NSet.fold fold_descendent descendents graph - in - NSet.fold fold_ancestor ancestors graph - - let iter f graph = - let iter x set = NSet.iter (fun y -> f x y) set in - NMap.iter iter graph.dir - -end - -module Graph = -struct -(** Dumping a dependency graph **) - -module DAG = DAG(struct type t = string let compare = compare end) - -(** TODO: we should share this code with Coqdep_common *) -module VData = struct - type t = string list option * string list - let compare = Util.pervasives_compare -end - -module VCache = Set.Make(VData) - -let treat_coq_file chan = - let buf = Lexing.from_channel chan in - let deja_vu_v = ref VCache.empty in - let deja_vu_ml = ref StrSet.empty in - let mark_v_done from acc str = - let seen = VCache.mem (from, str) !deja_vu_v in - if not seen then - let () = deja_vu_v := VCache.add (from, str) !deja_vu_v in - match search_v_known ?from str with - | None -> acc - | Some file_str -> (canonize file_str, !suffixe) :: acc - else acc - in - let rec loop acc = - let token = try Some (coq_action buf) with Fin_fichier -> None in - match token with - | None -> acc - | Some action -> - let acc = match action with - | Require (from, strl) -> - List.fold_left (fun accu v -> mark_v_done from accu v) acc strl - | Declare sl -> - let declare suff dir s = - let base = escape (file_name s dir) in - match !option_dynlink with - | No -> [] - | Byte -> [base,suff] - | Opt -> [base,".cmxs"] - | Both -> [base,suff; base,".cmxs"] - | Variable -> - if suff=".cmo" then [base,"$(DYNOBJ)"] - else [base,"$(DYNLIB)"] - in - let decl acc str = - let s = basename_noext str in - if not (StrSet.mem s !deja_vu_ml) then - let () = deja_vu_ml := StrSet.add s !deja_vu_ml in - match search_mllib_known s with - | Some mldir -> (declare ".cma" mldir s) @ acc - | None -> - match search_ml_known s with - | Some mldir -> (declare ".cmo" mldir s) @ acc - | None -> acc - else acc - in - List.fold_left decl acc sl - | Load str -> - let str = Filename.basename str in - let seen = VCache.mem (None, [str]) !deja_vu_v in - if not seen then - let () = deja_vu_v := VCache.add (None, [str]) !deja_vu_v in - match search_v_known [str] with - | None -> acc - | Some file_str -> (canonize file_str, ".v") :: acc - else acc - | AddLoadPath _ | AddRecLoadPath _ -> acc (* TODO *) - in - loop acc - in - loop [] - -let treat_coq_file f = - let chan = try Some (open_in f) with Sys_error _ -> None in - match chan with - | None -> [] - | Some chan -> - try - let ans = treat_coq_file chan in - let () = close_in chan in - ans - with Syntax_error (i, j) -> close_in chan; error_cannot_parse f (i, j) - -type graph = - | Element of string - | Subgraph of string * graph list - -let rec insert_graph name path graphs = match path, graphs with - | [] , graphs -> (Element name) :: graphs - | (box :: boxes), (Subgraph (hd, names)) :: tl when hd = box -> - Subgraph (hd, insert_graph name boxes names) :: tl - | _, hd :: tl -> hd :: (insert_graph name path tl) - | (box :: boxes), [] -> [ Subgraph (box, insert_graph name boxes []) ] - -let print_graphs chan graph = - let rec print_aux name = function - | [] -> name - | (Element str) :: tl -> fprintf chan "\"%s\";\n" str; print_aux name tl - | Subgraph (box, names) :: tl -> - fprintf chan "subgraph cluster%n {\nlabel=\"%s\";\n" name box; - let name = print_aux (name + 1) names in - fprintf chan "}\n"; print_aux name tl - in - ignore (print_aux 0 graph) - -let rec pop_common_prefix = function - | [Subgraph (_, graphs)] -> pop_common_prefix graphs - | graphs -> graphs - -let split_path = Str.split (Str.regexp "/") - -let rec pop_last = function - | [] -> [] - | [ x ] -> [] - | x :: xs -> x :: pop_last xs - -let get_boxes path = pop_last (split_path path) - -let insert_raw_graph file = - insert_graph file (get_boxes file) - -let rec get_dependencies name args = - let vdep = treat_coq_file (name ^ ".v") in - let fold (deps, graphs, alseen) (dep, _) = - let dag = DAG.add_transitive_edge name dep deps in - if not (List.mem dep alseen) then - get_dependencies dep (dag, insert_raw_graph dep graphs, dep :: alseen) - else - (dag, graphs, alseen) - in - List.fold_left fold args vdep - -let coq_dependencies_dump chan dumpboxes = - let (deps, graphs, _) = - List.fold_left (fun ih (name, _) -> get_dependencies name ih) - (DAG.empty, List.fold_left (fun ih (file, _) -> insert_raw_graph file ih) [] !vAccu, - List.map fst !vAccu) !vAccu - in - fprintf chan "digraph dependencies {\n"; - if dumpboxes then print_graphs chan (pop_common_prefix graphs) - else List.iter (fun (name, _) -> fprintf chan "\"%s\"[label=\"%s\"]\n" name (basename_noext name)) !vAccu; - DAG.iter (fun name dep -> fprintf chan "\"%s\" -> \"%s\"\n" dep name) deps; - fprintf chan "}\n%!" - -end - let usage () = eprintf " usage: coqdep [options] <filename>+\n"; eprintf " options:\n"; eprintf " -c : Also print the dependencies of caml modules (=ocamldep).\n"; - (* Does not work anymore *) - (* eprintf " -w : Print informations on missing or wrong \"Declare - ML Module\" commands in coq files.\n"; *) - (* Does not work anymore: *) - (* eprintf " -D : Prints the missing ocmal module names. No dependency computed.\n"; *) eprintf " -boot : For coq developers, prints dependencies over coq library files (omitted by default).\n"; eprintf " -sort : output the given file name ordered by dependencies\n"; eprintf " -noglob | -no-glob : \n"; @@ -456,8 +84,6 @@ let usage () = eprintf " -R dir logname : add and import dir recursively to coq load path under logical name logname\n"; eprintf " -Q dir logname : add (recursively) and open (non recursively) dir to coq load path under logical name logname\n"; eprintf " -vos : also output dependencies about .vos files\n"; - eprintf " -dumpgraph f : print a dot dependency graph in file 'f'\n"; - eprintf " -dumpgraphbox f : print a dot dependency graph box in file 'f'\n"; eprintf " -exclude-dir dir : skip subdirectories named 'dir' during -R/-Q search\n"; eprintf " -coqlib dir : set the coq standard library directory\n"; eprintf " -suffix s : \n"; @@ -468,7 +94,6 @@ let usage () = let split_period = Str.split (Str.regexp (Str.quote ".")) let add_q_include path l = add_rec_dir_no_import add_known path (split_period l) - let add_r_include path l = add_rec_dir_import add_known path (split_period l) let treat_coqproject f = @@ -482,9 +107,8 @@ let treat_coqproject f = iter_sourced (fun f -> treat_file None f) (all_files project) let rec parse = function + (* TODO, deprecate option -c *) | "-c" :: ll -> option_c := true; parse ll - | "-D" :: ll -> option_D := true; parse ll - | "-w" :: ll -> option_w := true; parse ll | "-boot" :: ll -> option_boot := true; parse ll | "-sort" :: ll -> option_sort := true; parse ll | "-vos" :: ll -> write_vos := true; parse ll @@ -495,17 +119,12 @@ let rec parse = function | "-R" :: r :: ln :: ll -> add_r_include r ln; parse ll | "-Q" :: r :: ln :: ll -> add_q_include r ln; parse ll | "-R" :: ([] | [_]) -> usage () - | "-dumpgraph" :: f :: ll -> option_dump := Some (false, f); parse ll - | "-dumpgraphbox" :: f :: ll -> option_dump := Some (true, f); parse ll | "-exclude-dir" :: r :: ll -> System.exclude_directory r; parse ll | "-exclude-dir" :: [] -> usage () | "-coqlib" :: r :: ll -> Envars.set_user_coqlib r; parse ll | "-coqlib" :: [] -> usage () | "-suffix" :: s :: ll -> suffixe := s ; parse ll | "-suffix" :: [] -> usage () - | "-slash" :: ll -> - coqdep_warning "warning: option -slash has no effect and is deprecated."; - parse ll | "-dyndep" :: "no" :: ll -> option_dynlink := No; parse ll | "-dyndep" :: "opt" :: ll -> option_dynlink := Opt; parse ll | "-dyndep" :: "byte" :: ll -> option_dynlink := Byte; parse ll @@ -525,19 +144,8 @@ let coqdep () = (* Add current dir with empty logical path if not set by options above. *) (try ignore (Coqdep_common.find_dir_logpath (Sys.getcwd())) with Not_found -> add_norec_dir_import add_known "." []); - (* NOTE: These directories are searched from last to first *) - if !option_boot then begin - add_rec_dir_import add_known "theories" ["Coq"]; - add_rec_dir_import add_known "plugins" ["Coq"]; - add_rec_dir_import (fun _ -> add_caml_known) "theories" ["Coq"]; - add_rec_dir_import (fun _ -> add_caml_known) "plugins" ["Coq"]; - let user = "user-contrib" in - if Sys.file_exists user then begin - add_rec_dir_no_import add_known user []; - add_rec_dir_no_import (fun _ -> add_caml_known) user []; - end; - end else begin - (* option_boot is actually always false in this branch *) + (* We don't setup any loadpath if the -boot is passed *) + if not !option_boot then begin Envars.set_coqlib ~fail:(fun msg -> raise (CoqlibError msg)); let coqlib = Envars.coqlib () in add_rec_dir_import add_coqlib_known (coqlib//"theories") ["Coq"]; @@ -554,17 +162,9 @@ let coqdep () = warning_mult ".mli" iter_mli_known; warning_mult ".ml" iter_ml_known; if !option_sort then begin sort (); exit 0 end; - if !option_c && not !option_D then mL_dependencies (); - if not !option_D then coq_dependencies (); - if !option_w || !option_D then declare_dependencies (); - begin match !option_dump with - | None -> () - | Some (box, file) -> - let chan = open_out file in - let chan_fmt = formatter_of_out_channel chan in - try Graph.coq_dependencies_dump chan_fmt box; close_out chan - with e -> close_out chan; raise e - end + if !option_c then mL_dependencies (); + coq_dependencies (); + () let _ = try diff --git a/tools/coqdep_boot.ml b/tools/coqdep_boot.ml index 1730dd3d68..1cebb3638e 100644 --- a/tools/coqdep_boot.ml +++ b/tools/coqdep_boot.ml @@ -19,6 +19,7 @@ open Coqdep_common let split_period = Str.split (Str.regexp (Str.quote ".")) let add_q_include path l = add_rec_dir_no_import add_known path (split_period l) +let add_r_include path l = add_rec_dir_import add_known path (split_period l) let rec parse = function | "-dyndep" :: "no" :: ll -> option_dynlink := No; parse ll @@ -26,16 +27,14 @@ let rec parse = function | "-dyndep" :: "byte" :: ll -> option_dynlink := Byte; parse ll | "-dyndep" :: "both" :: ll -> option_dynlink := Both; parse ll | "-dyndep" :: "var" :: ll -> option_dynlink := Variable; parse ll - | "-c" :: ll -> option_c := true; parse ll | "-boot" :: ll -> parse ll (* We're already in boot mode by default *) - | "-mldep" :: ocamldep :: ll -> - option_mldep := Some ocamldep; option_c := true; parse ll | "-I" :: r :: ll -> (* To solve conflict (e.g. same filename in kernel and checker) we allow to state an explicit order *) add_caml_dir r; norec_dirs := StrSet.add r !norec_dirs; parse ll + | "-R" :: r :: ln :: ll -> add_r_include r ln; parse ll | "-Q" :: r :: ln :: ll -> add_q_include r ln; parse ll | f :: ll -> treat_file None f; parse ll | [] -> () @@ -44,16 +43,4 @@ let _ = let () = option_boot := true in if Array.length Sys.argv < 2 then exit 1; parse (List.tl (Array.to_list Sys.argv)); - if !option_c then begin - add_rec_dir_import add_known "." []; - add_rec_dir_import (fun _ -> add_caml_known) "." ["Coq"]; - end - else begin - add_rec_dir_import add_known "theories" ["Coq"]; - add_rec_dir_import add_known "plugins" ["Coq"]; - add_caml_dir "tactics"; - add_rec_dir_import (fun _ -> add_caml_known) "theories" ["Coq"]; - add_rec_dir_import (fun _ -> add_caml_known) "plugins" ["Coq"]; - end; - if !option_c then mL_dependencies (); coq_dependencies () diff --git a/tools/coqdep_common.ml b/tools/coqdep_common.ml index 775c528176..bd72a52adf 100644 --- a/tools/coqdep_common.ml +++ b/tools/coqdep_common.ml @@ -35,7 +35,6 @@ let option_c = ref false let option_noglob = ref false let option_dynlink = ref Both let option_boot = ref false -let option_mldep = ref None let norec_dirs = ref StrSet.empty @@ -246,26 +245,7 @@ let depend_ML str = (" "^mlifile^".cmi"," "^mlifile^".cmi") | None, None -> "", "" -let soustraite_fichier_ML dep md ext = - try - let chan = open_process_in (dep^" -modules "^md^ext) in - let list = ocamldep_parse (Lexing.from_channel chan) in - let a_faire = ref "" in - let a_faire_opt = ref "" in - List.iter - (fun str -> - let byte,opt = depend_ML str in - a_faire := !a_faire ^ byte; - a_faire_opt := !a_faire_opt ^ opt) - (List.rev list); - (!a_faire, !a_faire_opt) - with - | Sys_error _ -> ("","") - | _ -> - Printf.eprintf "Coqdep: subprocess %s failed on file %s%s\n" dep md ext; - exit 1 - -let autotraite_fichier_ML md ext = +let traite_fichier_ML md ext = try let chan = open_in (md ^ ext) in let buf = Lexing.from_channel chan in @@ -290,11 +270,6 @@ let autotraite_fichier_ML md ext = (!a_faire, !a_faire_opt) with Sys_error _ -> ("","") -let traite_fichier_ML md ext = - match !option_mldep with - | Some dep -> soustraite_fichier_ML dep md ext - | None -> autotraite_fichier_ML md ext - let traite_fichier_modules md ext = try let chan = open_in (md ^ ext) in diff --git a/tools/coqdep_common.mli b/tools/coqdep_common.mli index 6d49f7e06c..96266b8e36 100644 --- a/tools/coqdep_common.mli +++ b/tools/coqdep_common.mli @@ -30,7 +30,6 @@ val write_vos : bool ref type dynlink = Opt | Byte | Both | No | Variable val option_dynlink : dynlink ref -val option_mldep : string option ref val norec_dirs : StrSet.t ref val suffixe : string ref type dir = string option diff --git a/toplevel/coqargs.ml b/toplevel/coqargs.ml index 74d9c113d6..7d919956e8 100644 --- a/toplevel/coqargs.ml +++ b/toplevel/coqargs.ml @@ -271,8 +271,7 @@ let get_compat_file = function | "8.12" -> "Coq.Compat.Coq812" | "8.11" -> "Coq.Compat.Coq811" | "8.10" -> "Coq.Compat.Coq810" - | "8.9" -> "Coq.Compat.Coq89" - | ("8.8" | "8.7" | "8.6" | "8.5" | "8.4" | "8.3" | "8.2" | "8.1" | "8.0") as s -> + | ("8.9" | "8.8" | "8.7" | "8.6" | "8.5" | "8.4" | "8.3" | "8.2" | "8.1" | "8.0") as s -> CErrors.user_err ~hdr:"get_compat_file" Pp.(str "Compatibility with version " ++ str s ++ str " not supported.") | s -> diff --git a/user-contrib/Ltac2/Array.v b/user-contrib/Ltac2/Array.v index c55e20bc88..ee3bf88647 100644 --- a/user-contrib/Ltac2/Array.v +++ b/user-contrib/Ltac2/Array.v @@ -8,9 +8,220 @@ (* * (see LICENSE file for the text of the license) *) (************************************************************************) +(* This is mostly a translation of OCaml stdlib/array.ml *) + +(**************************************************************************) +(* *) +(* OCaml *) +(* *) +(* Xavier Leroy, projet Cristal, INRIA Rocquencourt *) +(* *) +(* Copyright 1996 Institut National de Recherche en Informatique et *) +(* en Automatique. *) +(* *) +(* All rights reserved. This file is distributed under the terms of *) +(* the GNU Lesser General Public License version 2.1, with the *) +(* special exception on linking described in the file LICENSE. *) +(* *) +(**************************************************************************) + Require Import Ltac2.Init. +Require Ltac2.Int. +Require Ltac2.Control. +Require Ltac2.Bool. +Require Ltac2.Message. + +(* Question: what is returned in case of an out of range value? + Answer: Ltac2 throws a panic *) +Ltac2 @external empty : unit -> 'a array := "ltac2" "array_empty". Ltac2 @external make : int -> 'a -> 'a array := "ltac2" "array_make". Ltac2 @external length : 'a array -> int := "ltac2" "array_length". Ltac2 @external get : 'a array -> int -> 'a := "ltac2" "array_get". Ltac2 @external set : 'a array -> int -> 'a -> unit := "ltac2" "array_set". +Ltac2 @external lowlevel_blit : 'a array -> int -> 'a array -> int -> int -> unit := "ltac2" "array_blit". +Ltac2 @external lowlevel_fill : 'a array -> int -> int -> 'a -> unit := "ltac2" "array_fill". +Ltac2 @external concat : ('a array) list -> 'a array := "ltac2" "array_concat". + +(* Low level array operations *) + +Ltac2 lowlevel_sub (arr : 'a array) (start : int) (len : int) := + let l := length arr in + match Int.equal l 0 with + | true => empty () + | false => + let newarr:=make len (get arr 0) in + lowlevel_blit arr start newarr 0 len; + newarr + end. + +(* Array functions as defined in the OCaml library *) + +Ltac2 init (l : int) (f : int->'a) := + let rec init_aux (dst : 'a array) (pos : int) (len : int) (f : int->'a) := + match Int.equal len 0 with + | true => () + | false => + set dst pos (f pos); + init_aux dst (Int.add pos 1) (Int.sub len 1) f + end + in + match Int.le l 0 with + | true => empty () + | false => + let arr:=make l (f 0) in + init_aux arr 0 (length arr) f; + arr + end. + +Ltac2 make_matrix (sx : int) (sy : int) (v : 'a) := + let init1 i := v in + let initr i := init sy init1 in + init sx initr. + +Ltac2 copy a := lowlevel_sub a 0 (length a). + +Ltac2 append (a1 : 'a array) (a2 : 'a array) := + match Int.equal (length a1) 0 with + | true => copy a2 + | false => match Int.equal (length a2) 0 with + | true => copy a1 + | false => + let newarr:=make (Int.add (length a1) (length a2)) (get a1 0) in + lowlevel_blit a1 0 newarr 0 (length a1); + lowlevel_blit a2 0 newarr (length a1) (length a2); + newarr + end + end. + +Ltac2 sub (a : 'a array) (ofs : int) (len : int) := + Control.assert_valid_argument "Array.sub ofs<0" (Int.ge ofs 0); + Control.assert_valid_argument "Array.sub len<0" (Int.ge len 0); + Control.assert_bounds "Array.sub" (Int.le ofs (Int.sub (length a) len)); + lowlevel_sub a ofs len. + +Ltac2 fill (a : 'a array) (ofs : int) (len : int) (v : 'a) := + Control.assert_valid_argument "Array.fill ofs<0" (Int.ge ofs 0); + Control.assert_valid_argument "Array.fill len<0" (Int.ge len 0); + Control.assert_bounds "Array.fill" (Int.le ofs (Int.sub (length a) len)); + lowlevel_fill a ofs len v. + +Ltac2 blit (a1 : 'a array) (ofs1 : int) (a2 : 'a array) (ofs2 : int) (len : int) := + Control.assert_valid_argument "Array.blit ofs1<0" (Int.ge ofs1 0); + Control.assert_valid_argument "Array.blit ofs2<0" (Int.ge ofs2 0); + Control.assert_valid_argument "Array.blit len<0" (Int.ge len 0); + Control.assert_bounds "Array.blit ofs1+len>len a1" (Int.le ofs1 (Int.sub (length a1) len)); + Control.assert_bounds "Array.blit ofs2+len>len a2" (Int.le ofs2 (Int.sub (length a2) len)); + lowlevel_blit a1 ofs1 a2 ofs2 len. + +Ltac2 rec iter_aux (f : 'a -> unit) (a : 'a array) (pos : int) (len : int) := + match Int.equal len 0 with + | true => () + | false => f (get a pos); iter_aux f a (Int.add pos 1) (Int.sub len 1) + end. + +Ltac2 iter (f : 'a -> unit) (a : 'a array) := iter_aux f a 0 (length a). + +Ltac2 rec iter2_aux (f : 'a -> 'b -> unit) (a : 'a array) (b : 'b array) (pos : int) (len : int) := + match Int.equal len 0 with + | true => () + | false => f (get a pos) (get b pos); iter2_aux f a b (Int.add pos 1) (Int.sub len 1) + end. + +Ltac2 rec iter2 (f : 'a -> 'b -> unit) (a : 'a array) (b : 'b array) := + Control.assert_valid_argument "Array.iter2" (Int.equal (length a) (length b)); + iter2_aux f a b 0 (length a). + +Ltac2 map (f : 'a -> 'b) (a : 'a array) := + init (length a) (fun i => f (get a i)). + +Ltac2 map2 (f : 'a -> 'b -> 'c) (a : 'a array) (b : 'b array) := + Control.assert_valid_argument "Array.map2" (Int.equal (length a) (length b)); + init (length a) (fun i => f (get a i) (get b i)). + +Ltac2 rec iteri_aux (f : int -> 'a -> unit) (a : 'a array) (pos : int) (len : int) := + match Int.equal len 0 with + | true => () + | false => f pos (get a pos); iteri_aux f a (Int.add pos 1) (Int.sub len 1) + end. + +Ltac2 iteri (f : int -> 'a -> unit) (a : 'a array) := iteri_aux f a 0 (length a). + +Ltac2 mapi (f : int -> 'a -> 'b) (a : 'a array) := + init (length a) (fun i => f i (get a i)). + +Ltac2 rec to_list_aux (a : 'a array) (pos : int) (len : int) := + match Int.equal len 0 with + | true => [] + | false => get a pos :: to_list_aux a (Int.add pos 1) (Int.sub len 1) + end. + +Ltac2 to_list (a : 'a array) := to_list_aux a 0 (length a). + +Ltac2 rec of_list_aux (ls : 'a list) (dst : 'a array) (pos : int) := + match ls with + | [] => () + | hd::tl => + set dst pos hd; + of_list_aux tl dst (Int.add pos 1) + end. + +Ltac2 of_list (ls : 'a list) := + (* Don't use List.length here because the List module might depend on Array some day *) + let rec list_length (ls : 'a list) := + match ls with + | [] => 0 + | _ :: tl => Int.add 1 (list_length tl) + end in + match ls with + | [] => empty () + | hd::tl => + let anew := make (list_length ls) hd in + of_list_aux ls anew 0; + anew + end. + +Ltac2 rec fold_left_aux (f : 'a -> 'b -> 'a) (x : 'a) (a : 'b array) (pos : int) (len : int) := + match Int.equal len 0 with + | true => x + | false => fold_left_aux f (f x (get a pos)) a (Int.add pos 1) (Int.sub len 1) + end. + +Ltac2 fold_left (f : 'a -> 'b -> 'a) (x : 'a) (a : 'b array) := fold_left_aux f x a 0 (length a). + +Ltac2 rec fold_right_aux (f : 'a -> 'b -> 'a) (x : 'a) (a : 'b array) (pos : int) (len : int) := + (* Note: one could compare pos<0. + We keep an extra len parameter so that the function can be used for any sub array *) + match Int.equal len 0 with + | true => x + | false => fold_right_aux f (f x (get a pos)) a (Int.sub pos 1) (Int.sub len 1) + end. + +Ltac2 fold_right (f : 'a -> 'b -> 'a) (x : 'a) (a : 'b array) := fold_right_aux f x a (Int.sub (length a) 1) (length a). + +Ltac2 rec exist_aux (p : 'a -> bool) (a : 'a array) (pos : int) (len : int) := + match Int.equal len 0 with + | true => false + | false => match p (get a pos) with + | true => true + | false => exist_aux p a (Int.add pos 1) (Int.sub len 1) + end + end. + +(* Note: named exist (as in Coq library) rather than exists cause exists is a notation *) +Ltac2 exist (p : 'a -> bool) (a : 'a array) := exist_aux p a 0 (length a). + +Ltac2 rec for_all_aux (p : 'a -> bool) (a : 'a array) (pos : int) (len : int) := + match Int.equal len 0 with + | true => true + | false => match p (get a pos) with + | true => for_all_aux p a (Int.add pos 1) (Int.sub len 1) + | false => false + end + end. + +Ltac2 for_all (p : 'a -> bool) (a : 'a array) := for_all_aux p a 0 (length a). + +(* Note: we don't have (yet) a generic equality function in Ltac2 *) +Ltac2 mem (eq : 'a -> 'a -> bool) (x : 'a) (a : 'a array) := + exist (eq x) a. diff --git a/user-contrib/Ltac2/tac2core.ml b/user-contrib/Ltac2/tac2core.ml index 55cd7f7692..431589aa30 100644 --- a/user-contrib/Ltac2/tac2core.ml +++ b/user-contrib/Ltac2/tac2core.ml @@ -213,6 +213,14 @@ let define3 name r0 r1 r2 f = define_primitive name (arity_suc (arity_suc arity_ f (Value.repr_to r0 x) (Value.repr_to r1 y) (Value.repr_to r2 z) end +let define4 name r0 r1 r2 r3 f = define_primitive name (arity_suc (arity_suc (arity_suc arity_one))) begin fun x0 x1 x2 x3 -> + f (Value.repr_to r0 x0) (Value.repr_to r1 x1) (Value.repr_to r2 x2) (Value.repr_to r3 x3) +end + +let define5 name r0 r1 r2 r3 r4 f = define_primitive name (arity_suc (arity_suc (arity_suc (arity_suc arity_one)))) begin fun x0 x1 x2 x3 x4 -> + f (Value.repr_to r0 x0) (Value.repr_to r1 x1) (Value.repr_to r2 x2) (Value.repr_to r3 x3) (Value.repr_to r4 x4) +end + (** Printing *) let () = define1 "print" pp begin fun pp -> @@ -253,6 +261,10 @@ end (** Array *) +let () = define0 "array_empty" begin + return (v_blk 0 (Array.of_list [])) +end + let () = define2 "array_make" int valexpr begin fun n x -> if n < 0 || n > Sys.max_array_length then throw err_outofbounds else wrap (fun () -> v_blk 0 (Array.make n x)) @@ -272,6 +284,20 @@ let () = define2 "array_get" block int begin fun (_, v) n -> else wrap (fun () -> v.(n)) end +let () = define5 "array_blit" block int block int int begin fun (_, v0) s0 (_, v1) s1 l -> + if s0 < 0 || s0+l > Array.length v0 || s1 < 0 || s1+l > Array.length v1 || l<0 then throw err_outofbounds + else wrap_unit (fun () -> Array.blit v0 s0 v1 s1 l) +end + +let () = define4 "array_fill" block int int valexpr begin fun (_, d) s l v -> + if s < 0 || s+l > Array.length d || l<0 then throw err_outofbounds + else wrap_unit (fun () -> Array.fill d s l v) +end + +let () = define1 "array_concat" (list block) begin fun l -> + wrap (fun () -> v_blk 0 (Array.concat (List.map snd l))) +end + (** Ident *) let () = define2 "ident_equal" ident ident begin fun id1 id2 -> diff --git a/user-contrib/Ltac2/tac2tactics.ml b/user-contrib/Ltac2/tac2tactics.ml index 561bd9c0c5..8a14be9ca7 100644 --- a/user-contrib/Ltac2/tac2tactics.ml +++ b/user-contrib/Ltac2/tac2tactics.ml @@ -33,6 +33,7 @@ let delayed_of_tactic tac env sigma = let _, pv = Proofview.init sigma [] in let name, poly = Id.of_string "ltac2_delayed", false in let c, pv, _, _ = Proofview.apply ~name ~poly env tac pv in + let _, sigma = Proofview.proofview pv in (sigma, c) let delayed_of_thunk r tac env sigma = |