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
Home
Index