Nuprl Lemma : can-find-first1-ext
∀[T:Type]. ∀P:T ⟶ 𝔹. ∀L:T List. ((∃x:T [first-member(T;x;L;P)]) ∨ (∀x∈L.¬↑(P x)))
Proof
Definitions occuring in Statement :
first-member: first-member(T;x;L;P),
l_all: (∀x∈L.P[x]),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
not: ¬A,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
can-find-first,
list_induction,
l_all_nil,
any: any x,
ifthenelse: if b then t else f fi ,
decidable__assert,
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],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
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,
l_all_cons,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
l_all_iff,
cons_member,
select_member,
eq_int: (i =z j),
btrue: tt,
bfalse: ff,
iff_weakening_equal,
subtract: n - m
Lemmas referenced :
can-find-first,
lifting-strict-decide,
top_wf,
equal_wf,
has-value_wf_base,
base_wf,
is-exception_wf,
lifting-strict-spread,
lifting-strict-int_eq,
list_induction,
l_all_nil,
decidable__assert,
l_all_cons,
l_all_iff,
cons_member,
select_member,
iff_weakening_equal
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
isectElimination,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
callbyvalueDecide,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
unionEquality,
unionElimination,
sqleReflexivity,
dependent_functionElimination,
independent_functionElimination,
baseApply,
closedConclusion,
decideExceptionCases,
inrFormation,
because_Cache,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation,
callbyvalueApply,
applyExceptionCases
Latex:
\mforall{}[T:Type]. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List. ((\mexists{}x:T [first-member(T;x;L;P)]) \mvee{} (\mforall{}x\mmember{}L.\mneg{}\muparrow{}(P x)))
Date html generated:
2018_05_21-PM-06_34_10
Last ObjectModification:
2017_07_26-PM-04_52_26
Theory : general
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