Nuprl Lemma : trans_imp_sp_trans_b
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(Trans(T;a,b.R[a;b]) ⇒ {∀a,b,c:T. (strict_part(x,y.R[x;y];a;b) ⇒ R[b;c] ⇒ strict_part(x,y.R[x;y];a;c))})
Proof
Definitions occuring in Statement :
strict_part: strict_part(x,y.R[x; y];a;b),
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 :
strict_part: strict_part(x,y.R[x; y];a;b),
guard: {T},
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
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,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
independent_pairFormation,
hypothesis,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
productEquality,
lambdaEquality,
universeEquality,
introduction,
extract_by_obid,
isectElimination,
functionEquality,
dependent_functionElimination,
independent_functionElimination,
voidElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(Trans(T;a,b.R[a;b])
{}\mRightarrow{} \{\mforall{}a,b,c:T. (strict\_part(x,y.R[x;y];a;b) {}\mRightarrow{} R[b;c] {}\mRightarrow{} strict\_part(x,y.R[x;y];a;c))\})
Date html generated:
2016_10_21-AM-09_42_59
Last ObjectModification:
2016_08_01-PM-09_48_50
Theory : rel_1
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