Nuprl Lemma : cond_rel_star_monotonic
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀[R1,R2:T ⟶ T ⟶ ℙ].
  (when P, R1 => R2 
⇒ R1 preserves P 
⇒ (∀x,y:T.  ((P x) 
⇒ (x (R1^*) y) 
⇒ (x (R2^*) y))))
Proof
Definitions occuring in Statement : 
rel_star: R^*
, 
cond_rel_implies: when P, R1 => R2
, 
preserved_by: R preserves P
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
infix_ap: x f y
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
cond_rel_implies: when P, R1 => R2
, 
prop: ℙ
, 
infix_ap: x f y
Lemmas referenced : 
cond_rel_star_monotone, 
rel_star_wf, 
preserved_by_wf, 
cond_rel_implies_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :isect_memberFormation_alt, 
lambdaFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
because_Cache, 
hypothesisEquality, 
independent_functionElimination, 
hypothesis, 
dependent_functionElimination, 
applyEquality, 
Error :inhabitedIsType, 
Error :functionIsType, 
Error :universeIsType, 
universeEquality
Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[R1,R2:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    (when  P,  R1  =>  R2
    {}\mRightarrow{}  R1  preserves  P
    {}\mRightarrow{}  (\mforall{}x,y:T.    ((P  x)  {}\mRightarrow{}  (x  (R1\^{}*)  y)  {}\mRightarrow{}  (x  (R2\^{}*)  y))))
Date html generated:
2019_06_20-PM-00_30_42
Last ObjectModification:
2018_09_26-PM-00_48_05
Theory : relations
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