Nuprl Lemma : utrans_imp_sp_utrans_b
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
((∀[x,y:T]. R[x;y] supposing R[x;y])
⇒ UniformlyTrans(T;a,b.R[a;b])
⇒ {∀[a,b,c:T]. (strict_part(x,y.R[x;y];a;c)) supposing (strict_part(x,y.R[x;y];a;b) and R[b;c])})
Proof
Definitions occuring in Statement :
strict_part: strict_part(x,y.R[x; y];a;b),
utrans: UniformlyTrans(T;x,y.E[x; y]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
strict_part: strict_part(x,y.R[x; y];a;b),
guard: {T},
utrans: UniformlyTrans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
and: P ∧ Q,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
not_wf,
uall_wf,
isect_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
thin,
sqequalHypSubstitution,
productElimination,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
hypothesis,
independent_functionElimination,
voidElimination,
productEquality,
lambdaEquality,
universeEquality,
introduction,
extract_by_obid,
isectElimination,
functionEquality,
because_Cache,
independent_isectElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}[x,y:T]. R[x;y] supposing R[x;y])
{}\mRightarrow{} UniformlyTrans(T;a,b.R[a;b])
{}\mRightarrow{} \{\mforall{}[a,b,c:T]. (strict\_part(x,y.R[x;y];a;c)) supposing (strict\_part(x,y.R[x;y];a;b) and R[b;c])\}\000C)
Date html generated:
2016_10_21-AM-09_43_02
Last ObjectModification:
2016_08_01-PM-09_48_59
Theory : rel_1
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