Theory: window

Parents


Type constants


Term constants


Axioms


Definitions

PMI_DEF
|- !a b. a <== b = b ==> a

Theorems

IMP_REFL_THM
|- !x. x ==> x
IMP_TRANS_THM
|- !x y z. (x ==> y) /\ (y ==> z) ==> x ==> z
PMI_REFL_THM
|- !x. x <== x
PMI_TRANS_THM
|- !x y z. (x <== y) /\ (y <== z) ==> (x <== z)
COND1_THM
|- !R A B C D. (!x. R x x) ==> (A ==> R B D) ==> R (A => B | C) (A => D | C)
COND2_THM
|- !R A B C D. (!x. R x x) ==> (~A ==> R C D) ==> R (A => B | C) (A => B | D)
BODY2_THM
|- !c f g r. (!v. (v = c) ==> r (f v) (g v)) ==> r (f c) (g c)
LET_THM
|- !c f g r. (!v. (v = c) ==> r (f v) (g v)) ==> r (LET f c) (LET g c)
CONJ1_THM
|- !A B C. (B ==> (A = C)) ==> (A /\ B = C /\ B)
CONJ2_THM
|- !A B C. (A ==> (B = C)) ==> (A /\ B = A /\ C)
IMP1_THM
|- !A B C. (~B ==> (A = C)) ==> (A ==> B = C ==> B)
IMP2_THM
|- !A B C. (A ==> (B = C)) ==> (A ==> B = A ==> C)
PM1_THM
|- !A B C. (B ==> (A = C)) ==> (A <== B = C <== B)
PMI2_THM
|- !A B C. (~A ==> (B = C)) ==> (A <== B = A <== C)
DISJ1_THM
|- !A B C. (~B ==> (A = C)) ==> (A \/ B = C \/ B)
DISJ2_THM
|- !A B C. (~A ==> (B = C)) ==> (A \/ B = A \/ C)
DNEG_THM
|- !t. ~~t = t
NOT_DISJ_THM
|- !t1 t2. ~(A \/ B) = ~A /\ ~B
NOT_IMP_THM
|- !t1 t2. ~(t1 ==> t2) = t1 /\ ~t2
NOT_PMI_THM
|- !t1 t2. ~(t1 <== t2) = ~t1 /\ t2
COND_ABF_THM
|- !t1 t2. (t1 => t2 | F) = t1 /\ t2
COND_AFC_THM
|- !t1 t3. (t1 => F | t3) = ~t1 /\ t3
IMP_CONJ1_THM
|- !A B C. (B ==> A ==> C) ==> A /\ B ==> C /\ B
IMP_CONJ2_THM
|- !A B C. (A ==> B ==> C) ==> A /\ B ==> A /\ C
IMP_IMP1_THM
|- !A B C. (~B ==> (A <== C)) ==> (A ==> B) ==> C ==> B
IMP_IMP2_THM
|- !A B C. (A ==> B ==> C) ==> (A ==> B) ==> A ==> C
IMP_PMI1_THM
|- !A B C. (B ==> A ==> C) ==> (A <== B) ==> (C <== B)
IMP_PMI2_THM
|- !A B C. (~A ==> (B <== C)) ==> (A <== B) ==> (A <== C)
IMP_DISJ1_THM
|- !A B C. (~B ==> A ==> C) ==> A \/ B ==> C \/ B
IMP_DISJ2_THM
|- !A B C. (~A ==> B ==> C) ==> A \/ B ==> A \/ C
IMP_NEG_THM
|- !A B. (A <== B) ==> ~A ==> ~B
PMI_CONJ1_THM
|- !A B C. (B ==> (A <== C)) ==> (A /\ B <== C /\ B)
PMI_CONJ2_THM
|- !A B C. (A ==> (B <== C)) ==> (A /\ B <== A /\ C)
PMI_IMP1_THM
|- !A B C. (~B ==> A ==> C) ==> ((A ==> B) <== (C ==> B))
PMI_IMP2_THM
|- !A B C. (A ==> (B <== C)) ==> ((A ==> B) <== (A ==> C))
PMI_PMI1_THM
|- !A B C. (B ==> (A <== C)) ==> ((A <== B) <== C <== B)
PMI_PMI2_THM
|- !A B C. (~A ==> B ==> C) ==> ((A <== B) <== A <== C)
PMI_DISJ1_THM
|- !A B C. (~B ==> (A <== C)) ==> (A \/ B <== C \/ B)
PMI_DISJ2_THM
|- !A B C. (~A ==> (B <== C)) ==> (A \/ B <== A \/ C)
PMI_NEG_THM
|- !A B. (A ==> B) ==> (~A <== ~B)