- TRIV_CHAINF_LEMMA
-
|- !p1 p2. p1 <<< p2 ==> CHAINF (TRIV_CHAINF p1 p2)
- CHAINF_FUNPOW
-
|- !fun. MONOTONE fun ==> CHAINF (\n. FUNPOW n fun BOT)
- SUP_CHAINF_LEMMA
-
|- !p1 p2. p1 <<< p2 = SUP (TRIV_CHAINF p1 p2) = p2
- LAMB_TRIV_CHAINF
-
|- !fun p1 p2. (\n. fun (TRIV_CHAINF p1 p2 n)) = TRIV_CHAINF (fun p1) (fun p2)
- CONT_MONOTONE
-
|- !fun. CONTINUOUSF fun ==> MONOTONE fun
- FIX_LEMMA
-
|- !fun. CONTINUOUSF fun ==> (fun (LIM_FUNPOW fun) = LIM_FUNPOW fun)
- LEAST_FIX_LEMMA
-
|- !fun.
CONTINUOUSF fun ==> (!f. (fun f = f) ==> (!n. FUNPOW n fun BOT <<< f))
- LEAST_FIX_THM
-
|- !fun. CONTINUOUSF fun ==> (!f. (fun f = f) ==> LIM_FUNPOW fun <<< f)
- LIM_FUNPOW_THM
-
|- !fun.
CONTINUOUSF fun ==>
(fun (LIM_FUNPOW fun) = LIM_FUNPOW fun) /\
(!f. (fun f = f) ==> LIM_FUNPOW fun <<< f)
- FIX_EXISTS
-
|- !fun.
CONTINUOUSF fun ==>
(?f. (fun f = f) /\ (!f'. (fun f' = f') ==> f <<< f'))
- FIX_THM
-
|- !fun.
CONTINUOUSF fun ==>
(fun (FIX fun) = FIX fun) /\ (!f. (fun f = f) ==> FIX fun <<< f)
- ANTISYM_LEMMA
-
|- !f g. f <<< g /\ g <<< f ==> (f = g)
- FIX_LIM_FUNPOW_THM
-
|- !fun. CONTINUOUSF fun ==> (FIX fun = LIM_FUNPOW fun)
- HOARE_ADMITS_LEMMA
-
|- !p q. ADMITS_INDUCTION (\f. !s1 s2. p s1 /\ f (s1,s2) ==> q s2)
- SCOTT_INDUCTION_LEMMA
-
|- !p fun.
CONTINUOUSF fun /\ p BOT /\ (!f. p f ==> p (fun f)) ==>
(!n. p (FUNPOW n fun BOT))
- SCOTT_INDUCTION_THM
-
|- !p fun.
ADMITS_INDUCTION p /\
CONTINUOUSF fun /\
p BOT /\
(!f. p f ==> p (fun f)) ==>
p (FIX fun)