Nuprl Lemma : least-equiv-induction2
∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ].
∀x,y:A. ((x least-equiv(A;R) y) ⇒ {∀[P:A ⟶ ℙ]. ((∀x,y:A. ((x R y) ⇒ (P[x] ⇐⇒ P[y]))) ⇒ P[x] ⇒ P[y])})
Proof
Definitions occuring in Statement :
least-equiv: least-equiv(A;R),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
infix_ap: x f y
Lemmas referenced :
least-equiv-induction,
all_wf,
iff_wf,
least-equiv_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
independent_functionElimination,
hypothesis,
applyEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
\mforall{}x,y:A.
((x least-equiv(A;R) y)
{}\mRightarrow{} \{\mforall{}[P:A {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x,y:A. ((x R y) {}\mRightarrow{} (P[x] \mLeftarrow{}{}\mRightarrow{} P[y]))) {}\mRightarrow{} P[x] {}\mRightarrow{} P[y])\})
Date html generated:
2018_05_21-PM-00_52_07
Last ObjectModification:
2018_05_04-AM-10_25_19
Theory : relations2
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