diff options
Diffstat (limited to 'theories/Relations/Relation_Operators.v')
| -rw-r--r-- | theories/Relations/Relation_Operators.v | 74 |
1 files changed, 37 insertions, 37 deletions
diff --git a/theories/Relations/Relation_Operators.v b/theories/Relations/Relation_Operators.v index b538620d1..e4918d40e 100644 --- a/theories/Relations/Relation_Operators.v +++ b/theories/Relations/Relation_Operators.v @@ -24,7 +24,7 @@ Require Import List. Section Transitive_Closure. Variable A : Type. Variable R : relation A. - + Inductive clos_trans (x: A) : A -> Prop := | t_step : forall y:A, R x y -> clos_trans x y | t_trans : @@ -48,16 +48,16 @@ End Reflexive_Transitive_Closure. Section Reflexive_Symetric_Transitive_Closure. Variable A : Type. Variable R : relation A. - + Inductive clos_refl_sym_trans : relation A := | rst_step : forall x y:A, R x y -> clos_refl_sym_trans x y | rst_refl : forall x:A, clos_refl_sym_trans x x | rst_sym : - forall x y:A, clos_refl_sym_trans x y -> clos_refl_sym_trans y x + forall x y:A, clos_refl_sym_trans x y -> clos_refl_sym_trans y x | rst_trans : - forall x y z:A, - clos_refl_sym_trans x y -> - clos_refl_sym_trans y z -> clos_refl_sym_trans x z. + forall x y z:A, + clos_refl_sym_trans x y -> + clos_refl_sym_trans y z -> clos_refl_sym_trans x z. End Reflexive_Symetric_Transitive_Closure. @@ -92,18 +92,18 @@ End Disjoint_Union. Section Lexicographic_Product. -(* Lexicographic order on dependent pairs *) + (* Lexicographic order on dependent pairs *) -Variable A : Set. -Variable B : A -> Set. -Variable leA : A -> A -> Prop. -Variable leB : forall x:A, B x -> B x -> Prop. + Variable A : Set. + Variable B : A -> Set. + Variable leA : A -> A -> Prop. + Variable leB : forall x:A, B x -> B x -> Prop. -Inductive lexprod : sigS B -> sigS B -> Prop := - | left_lex : + Inductive lexprod : sigS B -> sigS B -> Prop := + | left_lex : forall (x x':A) (y:B x) (y':B x'), leA x x' -> lexprod (existS B x y) (existS B x' y') - | right_lex : + | right_lex : forall (x:A) (y y':B x), leB x y y' -> lexprod (existS B x y) (existS B x y'). End Lexicographic_Product. @@ -117,9 +117,9 @@ Section Symmetric_Product. Inductive symprod : A * B -> A * B -> Prop := | left_sym : - forall x x':A, leA x x' -> forall y:B, symprod (x, y) (x', y) + forall x x':A, leA x x' -> forall y:B, symprod (x, y) (x', y) | right_sym : - forall y y':B, leB y y' -> forall x:A, symprod (x, y) (x, y'). + forall y y':B, leB y y' -> forall x:A, symprod (x, y) (x, y'). End Symmetric_Product. @@ -131,34 +131,34 @@ Section Swap. Inductive swapprod : A * A -> A * A -> Prop := | sp_noswap : forall x x':A * A, symprod A A R R x x' -> swapprod x x' | sp_swap : - forall (x y:A) (p:A * A), - symprod A A R R (x, y) p -> swapprod (y, x) p. + forall (x y:A) (p:A * A), + symprod A A R R (x, y) p -> swapprod (y, x) p. End Swap. Section Lexicographic_Exponentiation. - -Variable A : Set. -Variable leA : A -> A -> Prop. -Let Nil := nil (A:=A). -Let List := list A. - -Inductive Ltl : List -> List -> Prop := - | Lt_nil : forall (a:A) (x:List), Ltl Nil (a :: x) - | Lt_hd : forall a b:A, leA a b -> forall x y:list A, Ltl (a :: x) (b :: y) - | Lt_tl : forall (a:A) (x y:List), Ltl x y -> Ltl (a :: x) (a :: y). - - -Inductive Desc : List -> Prop := - | d_nil : Desc Nil - | d_one : forall x:A, Desc (x :: Nil) - | d_conc : + + Variable A : Set. + Variable leA : A -> A -> Prop. + Let Nil := nil (A:=A). + Let List := list A. + + Inductive Ltl : List -> List -> Prop := + | Lt_nil : forall (a:A) (x:List), Ltl Nil (a :: x) + | Lt_hd : forall a b:A, leA a b -> forall x y:list A, Ltl (a :: x) (b :: y) + | Lt_tl : forall (a:A) (x y:List), Ltl x y -> Ltl (a :: x) (a :: y). + + + Inductive Desc : List -> Prop := + | d_nil : Desc Nil + | d_one : forall x:A, Desc (x :: Nil) + | d_conc : forall (x y:A) (l:List), leA x y -> Desc (l ++ y :: Nil) -> Desc ((l ++ y :: Nil) ++ x :: Nil). -Definition Pow : Set := sig Desc. - -Definition lex_exp (a b:Pow) : Prop := Ltl (proj1_sig a) (proj1_sig b). + Definition Pow : Set := sig Desc. + + Definition lex_exp (a b:Pow) : Prop := Ltl (proj1_sig a) (proj1_sig b). End Lexicographic_Exponentiation. |