Nuprl Lemma : TC-ind
∀[Dom:Type]. ∀[B:Dom ⟶ ℙ]. ∀[R:Dom ⟶ Dom ⟶ ℙ].
((∀x,y:Dom. ((R x y) ⇒ (B x) ⇒ (B y))) ⇒ (∀x,y:Dom. (TC(λa,b.R a b)(x,y) ⇒ (B x) ⇒ (B 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],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
TC: TC(λx,y.F[x; y])(a,b),
infix_ap: x f y
Lemmas referenced :
TC_wf,
all_wf,
transitive-closure-induction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
applyEquality,
hypothesisEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}[Dom:Type]. \mforall{}[B:Dom {}\mrightarrow{} \mBbbP{}]. \mforall{}[R:Dom {}\mrightarrow{} Dom {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:Dom. ((R x y) {}\mRightarrow{} (B x) {}\mRightarrow{} (B y))) {}\mRightarrow{} (\mforall{}x,y:Dom. (TC(\mlambda{}a,b.R a b)(x,y) {}\mRightarrow{} (B x) {}\mRightarrow{} (B y))))
Date html generated:
2016_05_16-AM-09_07_42
Last ObjectModification:
2015_12_28-PM-07_03_19
Theory : first-order!and!ancestral!logic
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