Theory "res_quan"

Parents     pred_set

Theorems

RES_SELECT_UNIV
|- !p. RES_SELECT UNIV p = $@ p
RES_SELECT_EMPTY
|- !p. RES_SELECT {} p = @x. F
RES_EXISTS_UNIQUE_ALT
|- !p m. RES_EXISTS_UNIQUE p m = ?x::p. m x /\ !y::p. m y ==> (y = x)
RES_EXISTS_UNIQUE_NULL
|- !p m. (?!x::p. m) = (?x. p = {x}) /\ m
RES_EXISTS_UNIQUE_UNIV
|- !p. RES_EXISTS_UNIQUE UNIV p = $?! p
RES_EXISTS_UNIQUE_EMPTY
|- !p. ~RES_EXISTS_UNIQUE {} p
RES_EXISTS_ALT
|- !p m. RES_EXISTS p m = RES_SELECT p m IN p /\ m (RES_SELECT p m)
RES_EXISTS_NULL
|- !p m. (?x::p. m) = ~(p = {}) /\ m
RES_EXISTS_UNIV
|- !p. RES_EXISTS UNIV p = $? p
RES_EXISTS_EMPTY
|- !p. ~RES_EXISTS {} p
RES_EXISTS_REORDER
|- !P Q R. (?(i::P) (j::Q). R i j) = ?(j::Q) (i::P). R i j
RES_EXISTS_EQUAL
|- !P j. (?i::$= j. P i) = P j
RES_DISJ_EXISTS_DIST
|- !P Q R. (?i::(\i. P i \/ Q i). R i) = (?i::P. R i) \/ ?i::Q. R i
RES_EXISTS_DISJ_DIST
|- !P Q R. (?i::P. Q i \/ R i) = (?i::P. Q i) \/ ?i::P. R i
RES_FORALL_NULL
|- !p m. (!x::p. m) = (p = {}) \/ m
RES_FORALL_UNIV
|- !p. RES_FORALL UNIV p = $! p
RES_FORALL_EMPTY
|- !p. RES_FORALL {} p
RES_FORALL_REORDER
|- !P Q R. (!(i::P) (j::Q). R i j) = !(j::Q) (i::P). R i j
RES_FORALL_FORALL
|- !P R x. (!x (i::P). R i x) = !(i::P) x. R i x
RES_FORALL_UNIQUE
|- !P j. (!i::$= j. P i) = P j
RES_FORALL_DISJ_DIST
|- !P Q R. (!i::(\j. P j \/ Q j). R i) = (!i::P. R i) /\ !i::Q. R i
RES_FORALL_CONJ_DIST
|- !P Q R. (!i::P. Q i /\ R i) = (!i::P. Q i) /\ !i::P. R i
RES_FORALL
|- !p m. RES_FORALL p m = !x. x IN p ==> m x
RES_EXISTS
|- !p m. RES_EXISTS p m = ?x. x IN p /\ m x
RES_EXISTS_UNIQUE
|- !p m.
     RES_EXISTS_UNIQUE p m = (?x::p. m x) /\ !x y::p. m x /\ m y ==> (x = y)
RES_SELECT
|- !p m. RES_SELECT p m = @x. x IN p /\ m x
RES_ABSTRACT
|- !p m x. x IN p ==> (RES_ABSTRACT p m x = m x)
RES_ABSTRACT_EQUAL
|- !p m1 m2.
     (!x. x IN p ==> (m1 x = m2 x)) ==>
     (RES_ABSTRACT p m1 = RES_ABSTRACT p m2)
RES_ABSTRACT_IDEMPOT
|- !p m. RES_ABSTRACT p (RES_ABSTRACT p m) = RES_ABSTRACT p m
RES_ABSTRACT_EQUAL_EQ
|- !p m1 m2.
     (RES_ABSTRACT p m1 = RES_ABSTRACT p m2) = !x. x IN p ==> (m1 x = m2 x)