Fixing tauto "special" behavior on singleton types w/ 2 parameters (bug #2680).

- tauto/intuition now works uniformly on and, prod, or, sum, False,
  Empty_set, unit, True (and isomorphic copies of them), iff, ->, and
  on all inhabited singleton types with a no-arguments constructor
  such as "eq t t" (even though the last case goes out of
  propositional logic: this features is so often used that it is
  difficult to come back on it).
- New dtauto and dintuition works on all inductive types with one
  constructors and no real arguments (for instance, they work on
  records such as "Equivalence"), in addition to -> and eq-like types.
- Moreover, both of them no longer unfold inner negations (this is a
  souce of incompatibility for intuition and evaluation of the level
  of incompatibility on contribs still needs to be done).

Incidentally, and amazingly, fixing bug #2680 made that constants
InfA_compat and InfA_eqA in SetoidList.v lost one argument: old tauto
had indeed destructed a section hypothesis "@StrictOrder A ltA@
thinking it was a conjunction, making this section hypothesis
artificially necessary while it was not.

Renouncing to the unfolding of inner negations made auto/eauto
sometimes succeeding more, sometimes succeeding less. There is by the
way a (standard) problem with not in auto/eauto: even when given as an
"unfold hint", it works only in goals, not in hypotheses, so that auto
is not able to solve something like "forall P, (forall x, ~ P x) -> P
0 -> False". Should we automatically add a lemma of type "HYPS -> A ->
False" in the hint database everytime a lemma ""HYPS -> ~A" is
declared (and "unfold not" is a hint), and similarly for all unfold
hints?

At this occasion, also re-did some proofs of Znumtheory.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@15180 85f007b7-540e-0410-9357-904b9bb8a0f7

3 files changed, 159 insertions, 95 deletions

diff --git a/tactics/hipattern.ml4 b/tactics/hipattern.ml4
index 6f07eed703..4f1174916b 100644
--- a/tactics/hipattern.ml4
+++ b/tactics/hipattern.ml4
@@ -81,9 +81,9 @@ let has_nodep_prod = has_nodep_prod_after 0
 
 (* style: None = record; Some false = conjunction; Some true = strict conj *)
 
-let match_with_one_constructor style allow_rec t =
+let match_with_one_constructor style onlybinary allow_rec t =
   let (hdapp,args) = decompose_app t in
-  match kind_of_term hdapp with
+  let res = match kind_of_term hdapp with
   | Ind ind ->
       let (mib,mip) = Global.lookup_inductive ind in
       if (Array.length mip.mind_consnames = 1)
@@ -110,22 +110,25 @@ let match_with_one_constructor style allow_rec t =
 	      None
       else
 	None
+  | _ -> None in
+  match res with
+  | Some (hdapp,args) when not onlybinary || List.length args = 2 -> res
   | _ -> None
 
-let match_with_conjunction ?(strict=false) t =
-  match_with_one_constructor (Some strict) false t
+let match_with_conjunction ?(strict=false) ?(onlybinary=false) t =
+  match_with_one_constructor (Some strict) onlybinary false t
 
 let match_with_record t =
-  match_with_one_constructor None false t
+  match_with_one_constructor None false false t
 
-let is_conjunction ?(strict=false) t =
-  op2bool (match_with_conjunction ~strict t)
+let is_conjunction ?(strict=false) ?(onlybinary=false) t =
+  op2bool (match_with_conjunction ~strict ~onlybinary t)
 
 let is_record t =
   op2bool (match_with_record t)
 
 let match_with_tuple t =
-  let t = match_with_one_constructor None true t in
+  let t = match_with_one_constructor None false true t in
   Option.map (fun (hd,l) ->
     let ind = destInd hd in
     let (mib,mip) = Global.lookup_inductive ind in
@@ -146,14 +149,15 @@ let test_strict_disjunction n lc =
     | [_,None,c] -> isRel c && destRel c = (n - i)
     | _ -> false) 0 lc
 
-let match_with_disjunction ?(strict=false) t =
+let match_with_disjunction ?(strict=false) ?(onlybinary=false) t =
   let (hdapp,args) = decompose_app t in
-  match kind_of_term hdapp with
+  let res = match kind_of_term hdapp with
   | Ind ind  ->
       let car = mis_constr_nargs ind in
       let (mib,mip) = Global.lookup_inductive ind in
-      if array_for_all (fun ar -> ar = 1) car &&
-	not (mis_is_recursive (ind,mib,mip))
+      if array_for_all (fun ar -> ar = 1) car
+	&& not (mis_is_recursive (ind,mib,mip))
+        && (mip.mind_nrealargs = 0)
       then
 	if strict then
 	  if test_strict_disjunction mib.mind_nparams mip.mind_nf_lc then
@@ -167,10 +171,13 @@ let match_with_disjunction ?(strict=false) t =
 	  Some (hdapp,Array.to_list cargs)
       else
 	None
+  | _ -> None in
+  match res with
+  | Some (hdapp,args) when not onlybinary || List.length args = 2 -> res
   | _ -> None
 
-let is_disjunction ?(strict=false) t =
-  op2bool (match_with_disjunction ~strict t)
+let is_disjunction ?(strict=false) ?(onlybinary=false) t =
+  op2bool (match_with_disjunction ~strict ~onlybinary t)
 
 (* An empty type is an inductive type, possible with indices, that has no
    constructors *)
diff --git a/tactics/hipattern.mli b/tactics/hipattern.mli
index 82e9e3b8eb..e4ae554091 100644
--- a/tactics/hipattern.mli
+++ b/tactics/hipattern.mli
@@ -53,13 +53,13 @@ val is_non_recursive_type         : testing_function
 
 (** Non recursive type with no indices and exactly one argument for each
    constructor; canonical definition of n-ary disjunction if strict *)
-val match_with_disjunction : ?strict:bool -> (constr * constr list) matching_function
-val is_disjunction         : ?strict:bool -> testing_function
+val match_with_disjunction : ?strict:bool -> ?onlybinary:bool -> (constr * constr list) matching_function
+val is_disjunction         : ?strict:bool -> ?onlybinary:bool -> testing_function
 
 (** Non recursive tuple (one constructor and no indices) with no inner
    dependencies; canonical definition of n-ary conjunction if strict *)
-val match_with_conjunction : ?strict:bool -> (constr * constr list) matching_function
-val is_conjunction         : ?strict:bool -> testing_function
+val match_with_conjunction : ?strict:bool -> ?onlybinary:bool -> (constr * constr list) matching_function
+val is_conjunction         : ?strict:bool -> ?onlybinary:bool -> testing_function
 
 (** Non recursive tuple, possibly with inner dependencies *)
 val match_with_record      : (constr * constr list) matching_function
diff --git a/tactics/tauto.ml4 b/tactics/tauto.ml4
index 036c4f0ea8..12bb36eb41 100644
--- a/tactics/tauto.ml4
+++ b/tactics/tauto.ml4
@@ -28,22 +28,30 @@ let assoc_var s ist =
 
 (** Parametrization of tauto *)
 
+type tauto_flags = {
+
 (* Whether conjunction and disjunction are restricted to binary connectives *)
-(* (this is the compatibility mode) *)
-let binary_mode = true
+  binary_mode : bool;
+
+(* Whether compatibility for buggy detection of binary connective is on *)
+  binary_mode_bugged_detection : bool;
 
 (* Whether conjunction and disjunction are restricted to the connectives *)
 (* having the structure of "and" and "or" (up to the choice of sorts) in *)
-(* contravariant position in an hypothesis (this is the compatibility mode) *)
-let strict_in_contravariant_hyp = true
+(* contravariant position in an hypothesis *)
+  strict_in_contravariant_hyp : bool;
 
 (* Whether conjunction and disjunction are restricted to the connectives *)
 (* having the structure of "and" and "or" (up to the choice of sorts) in *)
 (* an hypothesis and in the conclusion *)
-let strict_in_hyp_and_ccl = false
+  strict_in_hyp_and_ccl : bool;
 
 (* Whether unit type includes equality types *)
-let strict_unit = false
+  strict_unit : bool;
+
+(* Whether inner unfolding is allowed *)
+  inner_unfolding : bool
+}
 
 (* Whether inner iff are unfolded *)
 let iff_unfolding = ref false
@@ -70,8 +78,8 @@ let is_empty ist =
 
 (* Strictly speaking, this exceeds the propositional fragment as it
    matches also equality types (and solves them if a reflexivity) *)
-let is_unit_or_eq ist =
-  let test = if strict_unit then is_unit_type else is_unit_or_eq_type in
+let is_unit_or_eq flags ist =
+  let test = if flags.strict_unit then is_unit_type else is_unit_or_eq_type in
   if test (assoc_var "X1" ist) then
     <:tactic<idtac>>
   else
@@ -85,13 +93,13 @@ let is_record t =
 	    mib.Declarations.mind_record
       | _ -> false
 
-let is_binary t =
+let bugged_is_binary t =
   isApp t &&
   let (hdapp,args) = decompose_app t in
     match (kind_of_term hdapp) with
     | Ind ind  ->
         let (mib,mip) = Global.lookup_inductive ind in
-	  mib.Declarations.mind_nparams = 2
+         mib.Declarations.mind_nparams = 2
     | _ -> false
 
 let iter_tac tacl =
@@ -99,70 +107,72 @@ let iter_tac tacl =
 
 (** Dealing with conjunction *)
 
-let is_conj ist =
+let is_conj flags ist =
   let ind = assoc_var "X1" ist in
-    if (not binary_mode || is_binary ind) (* && not (is_record ind) *)
-      && is_conjunction ~strict:strict_in_hyp_and_ccl ind
+    if (not flags.binary_mode_bugged_detection || bugged_is_binary ind) &&
+       is_conjunction
+         ~strict:flags.strict_in_hyp_and_ccl
+         ~onlybinary:flags.binary_mode ind
     then
       <:tactic<idtac>>
     else
       <:tactic<fail>>
 
-let flatten_contravariant_conj ist =
+let flatten_contravariant_conj flags ist =
   let typ = assoc_var "X1" ist in
   let c = assoc_var "X2" ist in
   let hyp = assoc_var "id" ist in
-  match match_with_conjunction ~strict:strict_in_contravariant_hyp typ with
+  match match_with_conjunction
+          ~strict:flags.strict_in_contravariant_hyp
+          ~onlybinary:flags.binary_mode typ
+  with
   | Some (_,args) ->
-      let i = List.length args in
-      if not binary_mode || i = 2 then
-	let newtyp = valueIn (VConstr ([],List.fold_right mkArrow args c)) in
-	let hyp = valueIn (VConstr ([],hyp)) in
-	let intros =
-	  iter_tac (List.map (fun _ -> <:tactic< intro >>) args)
-	  <:tactic< idtac >> in
-	<:tactic<
-          let newtyp := $newtyp in
-          let hyp := $hyp in
-	  assert newtyp by ($intros; apply hyp; split; assumption);
-	  clear hyp
-	>>
-      else
-	<:tactic<fail>>
+      let newtyp = valueIn (VConstr ([],List.fold_right mkArrow args c)) in
+      let hyp = valueIn (VConstr ([],hyp)) in
+      let intros =
+	iter_tac (List.map (fun _ -> <:tactic< intro >>) args)
+	<:tactic< idtac >> in
+      <:tactic<
+        let newtyp := $newtyp in
+        let hyp := $hyp in
+	assert newtyp by ($intros; apply hyp; split; assumption);
+	clear hyp
+      >>
   | _ ->
       <:tactic<fail>>
 
 (** Dealing with disjunction *)
 
-let is_disj ist =
+let is_disj flags ist =
   let t = assoc_var "X1" ist in
-  if (not binary_mode || is_binary t) &&
-    is_disjunction ~strict:strict_in_hyp_and_ccl t
+  if (not flags.binary_mode_bugged_detection || bugged_is_binary t) &&
+     is_disjunction
+       ~strict:flags.strict_in_hyp_and_ccl
+       ~onlybinary:flags.binary_mode t
   then
     <:tactic<idtac>>
   else
     <:tactic<fail>>
 
-let flatten_contravariant_disj ist =
+let flatten_contravariant_disj flags ist =
   let typ = assoc_var "X1" ist in
   let c = assoc_var "X2" ist in
   let hyp = assoc_var "id" ist in
-  match match_with_disjunction ~strict:strict_in_contravariant_hyp typ with
+  match match_with_disjunction
+          ~strict:flags.strict_in_contravariant_hyp
+          ~onlybinary:flags.binary_mode
+          typ with
   | Some (_,args) ->
-      let i = List.length args in
-      if not binary_mode || i = 2 then
-	let hyp = valueIn (VConstr ([],hyp)) in
-	iter_tac (list_map_i (fun i arg ->
-	  let typ = valueIn (VConstr ([],mkArrow arg c)) in
-	  let i = Tacexpr.Integer i in
-	  <:tactic<
-            let typ := $typ in
-            let hyp := $hyp in
-	    let i := $i in
-	    assert typ by (intro; apply hyp; constructor i; assumption)
-	  >>) 1 args) <:tactic< let hyp := $hyp in clear hyp >>
-      else
-	<:tactic<fail>>
+      let hyp = valueIn (VConstr ([],hyp)) in
+      iter_tac (list_map_i (fun i arg ->
+	let typ = valueIn (VConstr ([],mkArrow arg c)) in
+	let i = Tacexpr.Integer i in
+	<:tactic<
+          let typ := $typ in
+          let hyp := $hyp in
+	  let i := $i in
+	  assert typ by (intro; apply hyp; constructor i; assumption)
+	>>) 1 args) <:tactic< let hyp := $hyp in clear hyp >>
   | _ ->
       <:tactic<fail>>
 
@@ -173,13 +183,11 @@ let not_dep_intros ist =
   <:tactic<
   repeat match goal with
   | |- (forall (_: ?X1), ?X2) => intro
-  | |- (Coq.Init.Logic.not _)   => unfold Coq.Init.Logic.not at 1
-  | H:(Coq.Init.Logic.not _)|-_    => unfold Coq.Init.Logic.not at 1 in H
-  | H:forall (_: Coq.Init.Logic.not _), _|-_    => unfold Coq.Init.Logic.not at 1 in H
+  | |- (Coq.Init.Logic.not _) => unfold Coq.Init.Logic.not at 1; intro
   end >>
 
-let axioms ist =
-  let t_is_unit_or_eq = tacticIn is_unit_or_eq
+let axioms flags ist =
+  let t_is_unit_or_eq = tacticIn (is_unit_or_eq flags)
   and t_is_empty = tacticIn is_empty in
     <:tactic<
     match reverse goal with
@@ -189,12 +197,12 @@ let axioms ist =
     end >>
 
 
-let simplif ist =
-  let t_is_unit_or_eq = tacticIn is_unit_or_eq
-  and t_is_conj = tacticIn is_conj
-  and t_flatten_contravariant_conj = tacticIn flatten_contravariant_conj
-  and t_flatten_contravariant_disj = tacticIn flatten_contravariant_disj
-  and t_is_disj = tacticIn is_disj
+let simplif flags ist =
+  let t_is_unit_or_eq = tacticIn (is_unit_or_eq flags)
+  and t_is_conj = tacticIn (is_conj flags)
+  and t_flatten_contravariant_conj = tacticIn (flatten_contravariant_conj flags)
+  and t_flatten_contravariant_disj = tacticIn (flatten_contravariant_disj flags)
+  and t_is_disj = tacticIn (is_disj flags)
   and t_not_dep_intros = tacticIn not_dep_intros in
   <:tactic<
     $t_not_dep_intros;
@@ -231,11 +239,11 @@ let simplif ist =
         end;
         $t_not_dep_intros) >>
 
-let rec tauto_intuit t_reduce solver ist =
-  let t_axioms = tacticIn axioms
-  and t_simplif = tacticIn simplif
-  and t_is_disj = tacticIn is_disj
-  and t_tauto_intuit = tacticIn (tauto_intuit t_reduce solver) in
+let rec tauto_intuit flags t_reduce solver ist =
+  let t_axioms = tacticIn (axioms flags)
+  and t_simplif = tacticIn (simplif flags)
+  and t_is_disj = tacticIn (is_disj flags)
+  and t_tauto_intuit = tacticIn (tauto_intuit flags t_reduce solver) in
   let t_solver = globTacticIn (fun _ist -> solver) in
   <:tactic<
    ($t_simplif;$t_axioms
@@ -247,6 +255,10 @@ let rec tauto_intuit t_reduce solver ist =
 		[ exact id
 		| generalize (fun y:X2 => id (fun x:X1 => y)); intro; clear id;
 		  solve [ $t_tauto_intuit ]]]
+      | id:forall (_:not ?X1), ?X3|- _ =>
+	  cut X3;
+	    [ intro; clear id; $t_tauto_intuit
+	    | cut (not X1); [ exact id | clear id; intro; solve [$t_tauto_intuit ]]]
       | |- ?X1 =>
           $t_is_disj; solve [left;$t_tauto_intuit | right;$t_tauto_intuit]
       end
@@ -267,11 +279,13 @@ let reduction_not _ist =
 
 let t_reduction_not = tacticIn reduction_not
 
-let intuition_gen tac =
-  interp (tacticIn (tauto_intuit t_reduction_not tac))
+let intuition_gen flags tac =
+  let t_reduce =
+    if flags.inner_unfolding then t_reduction_not else  <:tactic<idtac>> in
+  interp (tacticIn (tauto_intuit flags t_reduce tac))
 
-let tauto_intuitionistic g =
-  try intuition_gen <:tactic<fail>> g
+let tauto_intuitionistic flags g =
+  try intuition_gen flags <:tactic<fail>> g
   with
     Refiner.FailError _ | UserError _ ->
       errorlabstrm "tauto" (str "tauto failed.")
@@ -280,26 +294,69 @@ let coq_nnpp_path =
   let dir = List.map id_of_string ["Classical_Prop";"Logic";"Coq"] in
   Libnames.make_path (make_dirpath dir) (id_of_string "NNPP")
 
-let tauto_classical nnpp g =
-  try tclTHEN (apply nnpp) tauto_intuitionistic g
+let tauto_classical flags nnpp g =
+  try tclTHEN (apply nnpp) (tauto_intuitionistic flags) g
   with UserError _ -> errorlabstrm "tauto" (str "Classical tauto failed.")
 
-let tauto g =
+let tauto_gen flags g =
   try
     let nnpp = constr_of_global (Nametab.global_of_path coq_nnpp_path) in
     (* try intuitionistic version first to avoid an axiom if possible *)
-    tclORELSE tauto_intuitionistic (tauto_classical nnpp) g
+    tclORELSE (tauto_intuitionistic flags) (tauto_classical flags nnpp) g
   with Not_found ->
-    tauto_intuitionistic g
-
+    tauto_intuitionistic flags g
 
 let default_intuition_tac = <:tactic< auto with * >>
 
+(* This is the uniform mode dealing with ->, not, iff and types isomorphic to
+   /\ and *, \/ and +, False and Empty_set, True and unit, _and_ eq-like types;
+   not and iff are unfolded only if necessary *)
+let tauto_uniform_unit_flags = {
+  binary_mode = true;
+  binary_mode_bugged_detection = false;
+  strict_in_contravariant_hyp = true;
+  strict_in_hyp_and_ccl = true;
+  strict_unit = false;
+  inner_unfolding = false
+}
+
+(* This is the compatibility mode (not used) *)
+let tauto_legacy_flags = {
+  binary_mode = true;
+  binary_mode_bugged_detection = true;
+  strict_in_contravariant_hyp = true;
+  strict_in_hyp_and_ccl = false;
+  strict_unit = false;
+  inner_unfolding = true
+}
+
+(* This is the improved mode *)
+let tauto_power_flags = {
+  binary_mode = false; (* support n-ary connectives *)
+  binary_mode_bugged_detection = false;
+  strict_in_contravariant_hyp = false; (* supports non-regular connectives *)
+  strict_in_hyp_and_ccl = false;
+  strict_unit = false;
+  inner_unfolding = false
+}
+
+let tauto = tauto_gen tauto_uniform_unit_flags
+let dtauto = tauto_gen tauto_power_flags
+
 TACTIC EXTEND tauto
 | [ "tauto" ] -> [ tauto ]
 END
 
+TACTIC EXTEND dtauto
+| [ "dtauto" ] -> [ dtauto ]
+END
+
 TACTIC EXTEND intuition
-| [ "intuition" ] -> [ intuition_gen default_intuition_tac ]
-| [ "intuition" tactic(t) ] -> [ intuition_gen t ]
+| [ "intuition" ] -> [ intuition_gen tauto_uniform_unit_flags default_intuition_tac ]
+| [ "intuition" tactic(t) ] -> [ intuition_gen tauto_uniform_unit_flags t ]
+END
+
+TACTIC EXTEND dintuition
+| [ "dintuition" ] -> [ intuition_gen tauto_power_flags default_intuition_tac ]
+| [ "dintuition" tactic(t) ] -> [ intuition_gen tauto_power_flags t ]
 END