Nuprl Lemma : union-nat-missing-pos-prop
∀s1:nat-missing-type(). ∀max:ℕ. ∀missing:{l:ℕ List| l-ordered(ℕ;x,y.x < y;l)} . ∀x:ℕ.
(↑member-nat-missing(x;union-nat-missing-pos(s1;max;missing))
⇐⇒ (↑member-nat-missing(x;s1)) ∨ ((x ≤ max) ∧ (¬(x ∈ missing))))
Proof
Definitions occuring in Statement :
union-nat-missing-pos: union-nat-missing-pos(s1;max;missing),
member-nat-missing: member-nat-missing(i;s),
nat-missing-type: nat-missing-type(),
l_member: (x ∈ l),
list: T List,
nat: ℕ,
assert: ↑b,
less_than: a < b,
le: A ≤ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
l-ordered: l-ordered(T;x,y.R[x; y];L)
Lemmas :
iff_wf,
assert_wf,
member-nat-missing_wf,
union-nat-missing-pos_wf,
decidable__le,
subtract_wf,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_wf,
or_wf,
not_wf,
l_member_wf,
set_wf,
less_than_wf,
primrec-wf2,
nat_wf,
list_wf,
l-ordered_wf,
nat-missing-type_wf,
primrec1_lemma,
in-missing_wf,
subtype_rel_list,
bool_wf,
eqtt_to_assert,
assert-in-missing-nat-iff,
le_antisymmetry,
squash_wf,
true_wf,
iff_weakening_equal,
assert-member-nat-missing,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
sq_stable__le,
le_weakening,
equal-wf-T-base,
add-nat-missing-prop,
add-nat-missing_wf,
primrec-unroll-1,
decidable__lt,
general_arith_equation1,
trivial-int-eq1,
le_weakening2,
le-add-cancel2,
decidable__equal_int,
int_subtype_base,
not-equal-2,
and_wf
\mforall{}s1:nat-missing-type(). \mforall{}max:\mBbbN{}. \mforall{}missing:\{l:\mBbbN{} List| l-ordered(\mBbbN{};x,y.x < y;l)\} . \mforall{}x:\mBbbN{}.
(\muparrow{}member-nat-missing(x;union-nat-missing-pos(s1;max;missing))
\mLeftarrow{}{}\mRightarrow{} (\muparrow{}member-nat-missing(x;s1)) \mvee{} ((x \mleq{} max) \mwedge{} (\mneg{}(x \mmember{} missing))))
Date html generated:
2015_07_17-AM-08_21_35
Last ObjectModification:
2015_04_02-PM-05_43_20
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