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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