Nuprl Lemma : rel-preserving-composes
∀[T1,T2,T3:Type]. ∀[R1:T1 ⟶ T1 ⟶ Type]. ∀[R2:T2 ⟶ T2 ⟶ Type]. ∀[R3:T3 ⟶ T3 ⟶ ℙ].
∀f:T2 ⟶ T1. ∀g:T3 ⟶ T2.
(λx.f[x]:T2->T1 takes R2 into R1*) ⇒ λx.g[x]:T3->T2 takes R3 into R2*) ⇒ λx.f[g[x]]:T3->T1 takes R3 into R1*))
Proof
Definitions occuring in Statement :
rel-preserving: λx.f[x]:T2->T1 takes R2 into R1*),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
rel-preserving: λx.f[x]:T2->T1 takes R2 into R1*),
infix_ap: x f y,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
rel-preserving-star,
rel-preserving_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
independent_functionElimination,
hypothesis,
addLevel,
levelHypothesis,
functionEquality,
cumulativity,
universeEquality,
because_Cache
Latex:
\mforall{}[T1,T2,T3:Type]. \mforall{}[R1:T1 {}\mrightarrow{} T1 {}\mrightarrow{} Type]. \mforall{}[R2:T2 {}\mrightarrow{} T2 {}\mrightarrow{} Type]. \mforall{}[R3:T3 {}\mrightarrow{} T3 {}\mrightarrow{} \mBbbP{}].
\mforall{}f:T2 {}\mrightarrow{} T1. \mforall{}g:T3 {}\mrightarrow{} T2.
(\mlambda{}x.f[x]:T2->T1 takes R2 into R1*)
{}\mRightarrow{} \mlambda{}x.g[x]:T3->T2 takes R3 into R2*)
{}\mRightarrow{} \mlambda{}x.f[g[x]]:T3->T1 takes R3 into R1*))
Date html generated:
2016_05_15-PM-05_40_59
Last ObjectModification:
2015_12_27-PM-00_32_38
Theory : general
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