Nuprl Lemma : Wzero-leq
∀[A:Type]. ∀[B:A ⟶ Type]. ∀w:W(A;a.B[a]). (isZero(w) ⇐⇒ ∀w2:W(A;a.B[a]). (w ≤ w2))
Proof
Definitions occuring in Statement :
Wzero: isZero(w),
Wcmp: Wcmp(A;a.B[a];leq),
W: W(A;a.B[a]),
btrue: tt,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
infix_ap: x f y,
implies: P ⇒ Q,
all: ∀x:A. B[x],
Wsup: Wsup(a;b),
Wcmp: Wcmp(A;a.B[a];leq),
ifthenelse: if b then t else f fi ,
btrue: tt,
Wzero: isZero(w),
pi1: fst(t),
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q
Lemmas referenced :
W-induction,
iff_wf,
Wzero_wf,
all_wf,
W_wf,
Wcmp_wf,
btrue_wf,
not_wf,
bfalse_wf,
infix_ap_wf,
Wless_antireflexive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
independent_functionElimination,
lambdaFormation,
independent_pairFormation,
voidElimination,
because_Cache,
cumulativity,
instantiate,
universeEquality,
functionEquality,
rename,
dependent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}w:W(A;a.B[a]). (isZero(w) \mLeftarrow{}{}\mRightarrow{} \mforall{}w2:W(A;a.B[a]). (w \mleq{} w2))
Date html generated:
2016_05_14-AM-06_16_39
Last ObjectModification:
2015_12_26-PM-00_04_22
Theory : co-recursion
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