Nuprl Lemma : rel-immediate-preserves-order
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(Trans(T;x,y.R x y) ⇒ sum_of_torder(T;R) ⇒ (∀x,y,x',y':T. ((R x y) ⇒ (R! x' x) ⇒ (R! y' y) ⇒ (R x' y'))))
Proof
Definitions occuring in Statement :
sum_of_torder: sum_of_torder(T;R),
rel-immediate: R!,
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
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,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
trans: Trans(T;x,y.E[x; y]),
rel-immediate: R!,
and: P ∧ Q,
cand: A c∧ B,
or: P ∨ Q
Lemmas referenced :
rel-immediate_wf,
sum_of_torder_wf,
trans_wf,
rel-immediate-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
applyEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
functionExtensionality,
hypothesis,
sqequalRule,
lambdaEquality,
functionEquality,
universeEquality,
independent_functionElimination,
dependent_functionElimination,
productElimination,
unionElimination,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(Trans(T;x,y.R x y)
{}\mRightarrow{} sum\_of\_torder(T;R)
{}\mRightarrow{} (\mforall{}x,y,x',y':T. ((R x y) {}\mRightarrow{} (R! x' x) {}\mRightarrow{} (R! y' y) {}\mRightarrow{} (R x' y'))))
Date html generated:
2016_10_25-AM-11_01_30
Last ObjectModification:
2016_07_12-AM-07_08_11
Theory : general
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