Nuprl Lemma : TC-min
∀[Dom:Type]. ∀[R,Q:Dom ⟶ Dom ⟶ ℙ].
((∀x,y,z:Dom. ((Q x y) ⇒ (Q y z) ⇒ (Q x z)))
⇒ (∀x,y:Dom. ((R x y) ⇒ (Q x y)))
⇒ (∀x,y:Dom. (TC(λa,b.R a b)(x,y) ⇒ (Q x y))))
Proof
Definitions occuring in Statement :
TC: TC(λx,y.F[x; y])(a,b),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
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,
all: ∀x:A. B[x],
TC: TC(λx,y.F[x; y])(a,b),
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
rel_implies: R1 => R2,
prop: ℙ,
trans: Trans(T;x,y.E[x; y]),
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
transitive-closure-minimal,
TC_wf,
all_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,
independent_functionElimination,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination
Latex:
\mforall{}[Dom:Type]. \mforall{}[R,Q:Dom {}\mrightarrow{} Dom {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y,z:Dom. ((Q x y) {}\mRightarrow{} (Q y z) {}\mRightarrow{} (Q x z)))
{}\mRightarrow{} (\mforall{}x,y:Dom. ((R x y) {}\mRightarrow{} (Q x y)))
{}\mRightarrow{} (\mforall{}x,y:Dom. (TC(\mlambda{}a,b.R a b)(x,y) {}\mRightarrow{} (Q x y))))
Date html generated:
2016_05_16-AM-09_07_44
Last ObjectModification:
2015_12_28-PM-07_03_26
Theory : first-order!and!ancestral!logic
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