Nuprl Lemma : trans_rel_self_functionality
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. (Trans(T;x,y.R[x;y]) ⇒ {∀a,a',b,b':T. (R[b;a] ⇒ R[a';b'] ⇒ R[a;a'] ⇒ R[b;b'])})
Proof
Definitions occuring in Statement :
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality,
functionEquality,
hypothesis,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
because_Cache
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(Trans(T;x,y.R[x;y]) {}\mRightarrow{} \{\mforall{}a,a',b,b':T. (R[b;a] {}\mRightarrow{} R[a';b'] {}\mRightarrow{} R[a;a'] {}\mRightarrow{} R[b;b'])\})
Date html generated:
2016_10_21-AM-09_41_55
Last ObjectModification:
2016_08_01-PM-09_49_27
Theory : rel_1
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