Nuprl Lemma : ancestral-logic-lemma1
∀x,y:Dom.  (TC(λa,b.R a b)(x,y) 
⇒ ((R x y) ∨ (∃z:Dom. ((R x z) ∧ TC(λa,b.R a b)(z,y)))))
Proof
Definitions occuring in Statement : 
TC: TC(λx,y.F[x; y])(a,b)
, 
language1: language1{i:l}(D,A,B,C,P,Q,R,H.fml[D;A;B;C;P;Q;R;H])
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
implies: P 
⇒ Q
, 
or: P ∨ Q
, 
and: P ∧ Q
, 
apply: f a
Definitions unfolded in proof : 
language1: language1{i:l}(D,A,B,C,P,Q,R,H.fml[D;A;B;C;P;Q;R;H])
, 
uall: ∀[x:A]. B[x]
, 
!hyp_hide: x
, 
member: t ∈ T
, 
prop: ℙ
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
or: P ∨ Q
, 
so_lambda: λ2x.t[x]
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
so_apply: x[s]
, 
guard: {T}
, 
exists: ∃x:A. B[x]
, 
and: P ∧ Q
Lemmas referenced : 
exists_wf, 
and_wf, 
TC_wf, 
TC-min, 
or_wf, 
TC-base, 
TC-trans
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
sqequalHypSubstitution, 
functionEquality, 
cumulativity, 
hypothesisEquality, 
universeEquality, 
because_Cache, 
cut, 
lambdaFormation, 
inlFormation, 
lemma_by_obid, 
isectElimination, 
thin, 
sqequalRule, 
lambdaEquality, 
applyEquality, 
hypothesis, 
independent_functionElimination, 
inrFormation, 
unionElimination, 
dependent_pairFormation, 
independent_pairFormation, 
dependent_functionElimination, 
productElimination
Latex:
\mforall{}x,y:Dom.    (TC(\mlambda{}a,b.R  a  b)(x,y)  {}\mRightarrow{}  ((R  x  y)  \mvee{}  (\mexists{}z:Dom.  ((R  x  z)  \mwedge{}  TC(\mlambda{}a,b.R  a  b)(z,y)))))
Date html generated:
2016_05_16-AM-09_08_18
Last ObjectModification:
2015_12_28-PM-07_03_33
Theory : first-order!and!ancestral!logic
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