Nuprl Lemma : least-equiv-induction
∀[A:Type]. ∀[P:A ⟶ ℙ]. ∀[R:A ⟶ A ⟶ ℙ].
((∀x,y:A. ((x R y) ⇒ (P[x] ⇐⇒ P[y]))) ⇒ (∀x,y:A. ((x least-equiv(A;R) y) ⇒ P[x] ⇒ P[y])))
Proof
Definitions occuring in Statement :
least-equiv: least-equiv(A;R),
uall: ∀[x:A]. B[x],
prop: ℙ,
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],
implies: P ⇒ Q,
all: ∀x:A. B[x],
infix_ap: x f y,
least-equiv: least-equiv(A;R),
transitive-reflexive-closure: R^*,
or: P ∨ Q,
prop: ℙ,
and: P ∧ Q,
member: t ∈ T,
so_apply: x[s],
so_lambda: λ2x.t[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
and_wf,
equal_wf,
transitive-closure-induction,
or_wf,
infix_ap_wf,
least-equiv_wf,
all_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
unionElimination,
thin,
cut,
hypothesis,
addLevel,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
applyLambdaEquality,
setElimination,
rename,
productElimination,
applyEquality,
levelHypothesis,
lambdaEquality,
independent_functionElimination,
instantiate,
cumulativity,
universeEquality,
because_Cache,
dependent_functionElimination,
functionEquality
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] \mLeftarrow{}{}\mRightarrow{} P[y]))) {}\mRightarrow{} (\mforall{}x,y:A. ((x least-equiv(A;R) y) {}\mRightarrow{} P[x] {}\mRightarrow{} P[y])))
Date html generated:
2018_05_21-PM-00_52_06
Last ObjectModification:
2018_05_04-AM-10_21_29
Theory : relations2
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