Nuprl Lemma : restriction-of-transitive
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[P:T ⟶ ℙ]. (Trans(T;x,y.R x y) ⇒ Trans(T;x,y.R|P x y))
Proof
Definitions occuring in Statement :
rel-restriction: R|P,
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
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,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
trans: Trans(T;x,y.E[x; y]),
rel-restriction: R|P,
all: ∀x:A. B[x],
and: P ∧ Q
Lemmas referenced :
trans_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
productElimination,
independent_pairFormation,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. (Trans(T;x,y.R x y) {}\mRightarrow{} Trans(T;x,y.R|P x y))
Date html generated:
2016_05_14-AM-06_06_05
Last ObjectModification:
2015_12_26-AM-11_32_26
Theory : relations
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