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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