Nuprl Lemma : preserved_by_monotone
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀[R1,R2:T ⟶ T ⟶ ℙ]. ((∀x,y:T. ((x R1 y) ⇒ (x R2 y))) ⇒ R2 preserves P ⇒ R1 preserves P)
Proof
Definitions occuring in Statement :
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,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
preserved_by: R preserves P,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
infix_ap: x f y,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
lambdaFormation,
applyEquality,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality,
functionEquality,
hypothesis,
Error :inhabitedIsType,
Error :functionIsType,
Error :universeIsType,
universeEquality,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[R1,R2:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:T. ((x R1 y) {}\mRightarrow{} (x R2 y))) {}\mRightarrow{} R2 preserves P {}\mRightarrow{} R1 preserves P)
Date html generated:
2019_06_20-PM-00_30_29
Last ObjectModification:
2018_09_26-PM-00_40_01
Theory : relations
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