Nuprl Lemma : transitive-closure-induction-ext
∀[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 :
member: t ∈ T,
spreadn: spread3,
transitive-closure-induction,
transitive-closure-minimal,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
top: Top,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
transitive-closure-induction,
lifting-strict-spread,
istype-void,
strict4-apply,
transitive-closure-minimal
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
Error :isect_memberEquality_alt,
voidElimination,
independent_isectElimination
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_32
Last ObjectModification:
2019_01_11-PM-04_20_45
Theory : relations2
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