Nuprl Lemma : transitive-closure-induction
∀[A:Type]. ∀[P:A ⟶ ℙ]. ∀[R:A ⟶ A ⟶ ℙ].
((∀x,y:A. ((x R y) ⇒ P[x] ⇒ P[y])) ⇒ (∀x,y:A. ((x TC(R) y) ⇒ P[x] ⇒ P[y])))
Proof
Definitions occuring in Statement :
transitive-closure: TC(R),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
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_apply: x[s],
subtype_rel: A ⊆r B,
rel_implies: R1 => R2,
infix_ap: x f y,
trans: Trans(T;x,y.E[x; y]),
guard: {T}
Lemmas referenced :
transitive-closure-minimal,
subtype_rel_self,
transitive-closure_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
Error :lambdaEquality_alt,
functionEquality,
applyEquality,
hypothesis,
because_Cache,
sqequalRule,
Error :inhabitedIsType,
independent_functionElimination,
Error :universeIsType,
instantiate,
universeEquality,
Error :functionIsType,
dependent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbP{}]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:A. ((x R y) {}\mRightarrow{} P[x] {}\mRightarrow{} P[y])) {}\mRightarrow{} (\mforall{}x,y:A. ((x TC(R) y) {}\mRightarrow{} P[x] {}\mRightarrow{} P[y])))
Date html generated:
2019_06_20-PM-02_01_30
Last ObjectModification:
2018_10_06-AM-11_23_55
Theory : relations2
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