Nuprl Lemma : equipollent-nat-subset-ext
∀[T:Type]. ∀P:T ⟶ ℙ. ((∀x:T. Dec(P[x])) ⇒ (∀L:T List. ∃x:T. (P[x] ∧ (¬(x ∈ L)))) ⇒ ℕ ~ T ⇒ ℕ ~ {x:T| P[x]} )
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
l_member: (x ∈ l),
list: T List,
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
isl: isl(x),
btrue: tt,
it: ⋅,
bfalse: ff,
enumerate: enumerate(P;n),
compose: f o g,
bool-size: 𝔹size(k;f),
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
equipollent-nat-subset,
equipollent_transitivity,
equipollent-nat-decidable-subset,
iff_weakening_equal,
decidable__le,
any: any x,
decidable__assert,
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
guard: {T},
or: P ∨ Q,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
equipollent-nat-subset,
spread-axiom-sqequal-bottom,
lifting-strict-spread,
has-value_wf_base,
base_wf,
is-exception_wf,
strict4-spread,
lifting-strict-decide,
strict4-decide,
value-type-has-value,
int-value-type,
top_wf,
equal_wf,
equipollent_transitivity,
equipollent-nat-decidable-subset,
iff_weakening_equal,
decidable__le,
decidable__assert
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
baseClosed,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
callbyvalueApply,
baseApply,
closedConclusion,
hypothesisEquality,
applyExceptionCases,
inrFormation,
imageMemberEquality,
imageElimination,
inlFormation,
callbyvalueAdd,
productElimination,
because_Cache,
addExceptionCases,
exceptionSqequal,
intEquality,
sqequalSqle,
divergentSqle,
callbyvalueSpread,
productEquality,
sqleReflexivity,
dependent_functionElimination,
independent_functionElimination,
spreadExceptionCases,
axiomSqleEquality
Latex:
\mforall{}[T:Type]
\mforall{}P:T {}\mrightarrow{} \mBbbP{}
((\mforall{}x:T. Dec(P[x])) {}\mRightarrow{} (\mforall{}L:T List. \mexists{}x:T. (P[x] \mwedge{} (\mneg{}(x \mmember{} L)))) {}\mRightarrow{} \mBbbN{} \msim{} T {}\mRightarrow{} \mBbbN{} \msim{} \{x:T| P[x]\} )
Date html generated:
2018_05_21-PM-07_59_31
Last ObjectModification:
2018_05_19-PM-04_54_00
Theory : general
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