Nuprl Lemma : TC-trans
∀[Dom:Type]. ∀[R:Dom ⟶ Dom ⟶ ℙ]. ∀[x,y,z:Dom].  (TC(λa,b.R a b)(x,y) 
⇒ TC(λa,b.R a b)(y,z) 
⇒ TC(λa,b.R a b)(x,z))
Proof
Definitions occuring in Statement : 
TC: TC(λx,y.F[x; y])(a,b)
, 
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
, 
TC: TC(λx,y.F[x; y])(a,b)
, 
member: t ∈ T
, 
so_lambda: λ2x y.t[x; y]
, 
infix_ap: x f y
, 
prop: ℙ
, 
so_apply: x[s1;s2]
, 
guard: {T}
, 
utrans: UniformlyTrans(T;x,y.E[x; y])
Lemmas referenced : 
transitive-closure-transitive, 
TC_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
lambdaFormation, 
sqequalHypSubstitution, 
cut, 
lemma_by_obid, 
isectElimination, 
thin, 
because_Cache, 
sqequalRule, 
lambdaEquality, 
applyEquality, 
hypothesisEquality, 
hypothesis, 
functionEquality, 
cumulativity, 
universeEquality, 
independent_functionElimination
Latex:
\mforall{}[Dom:Type].  \mforall{}[R:Dom  {}\mrightarrow{}  Dom  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[x,y,z:Dom].
    (TC(\mlambda{}a,b.R  a  b)(x,y)  {}\mRightarrow{}  TC(\mlambda{}a,b.R  a  b)(y,z)  {}\mRightarrow{}  TC(\mlambda{}a,b.R  a  b)(x,z))
Date html generated:
2016_05_16-AM-09_07_48
Last ObjectModification:
2015_12_28-PM-07_03_22
Theory : first-order!and!ancestral!logic
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