Nuprl Lemma : can-find-first-ext
∀[T:Type]. ∀L:T List. ∀P:{x:T| (x ∈ L)} ⟶ 𝔹. ((∃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]),
l_member: (x ∈ l),
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,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
ifthenelse: if b then t else f fi ,
subtract: n - m,
can-find-first2,
can-find-first1-ext
Lemmas referenced :
can-find-first2,
can-find-first1-ext
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}. ((\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_45
Last ObjectModification:
2018_05_19-PM-04_41_05
Theory : general
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