Nuprl Lemma : pv11_p1_add_if_new_iff2
∀A:Type. ∀p,x:A. ∀L:A List. ∀test:A ─→ A ─→ 𝔹.
((p ∈ pv11_p1_add_if_new() test x L) ⇐⇒ (p ∈ L) ∨ if (∃z∈L.test x z)_b then False else p = x ∈ A fi )
Proof
Definitions occuring in Statement :
pv11_p1_add_if_new: pv11_p1_add_if_new(),
bl-exists: (∃x∈L.P[x])_b,
l_member: (x ∈ l),
list: T List,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
false: False,
apply: f a,
function: x:A ─→ B[x],
universe: Type,
equal: s = t ∈ T
Lemmas :
l_member_wf,
pv11_p1_add_if_new_wf,
bool_wf,
list_wf,
or_wf,
bl-exists_wf,
eqtt_to_assert,
assert-bl-exists,
false_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
l_exists_wf,
assert_wf,
bool_cases,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
member_append,
cons_wf,
nil_wf,
member_singleton,
equal-wf-T-base,
cons_member
Latex:
\mforall{}A:Type. \mforall{}p,x:A. \mforall{}L:A List. \mforall{}test:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbB{}.
((p \mmember{} pv11\_p1\_add\_if\_new() test x L) \mLeftarrow{}{}\mRightarrow{} (p \mmember{} L) \mvee{} if (\mexists{}z\mmember{}L.test x z)\_b then False else p = x fi )
Date html generated:
2015_07_23-PM-04_44_30
Last ObjectModification:
2015_01_29-AM-11_20_46
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