Nuprl Lemma : select-last-in-nat-missing_wf
∀[max:ℕ]. ∀[missing:{l:ℕ List| l-ordered(ℕ;x,y.x < y;l)} ].
(select-last-in-nat-missing(max;missing) ∈ {x:ℤ|
((-1) ≤ x)
∧ (x ≤ max)
∧ ((0 ≤ x) ⇒ (¬(x ∈ missing)))
∧ (∀y:ℕ. ((¬(y ∈ missing)) ⇒ y < max ⇒ (y ≤ x)))
∧ (∀y:ℕ. (y < max ⇒ x < y ⇒ (y ∈ missing)))} )
Proof
Definitions occuring in Statement :
select-last-in-nat-missing: select-last-in-nat-missing(max;missing),
l_member: (x ∈ l),
list: T List,
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n,
int: ℤ,
l-ordered: l-ordered(T;x,y.R[x; y];L)
Lemmas :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
primrec0_lemma,
in-missing_wf,
subtype_rel_list,
nat_wf,
bool_wf,
eqtt_to_assert,
false_wf,
le_wf,
not_wf,
l_member_wf,
all_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
assert-in-missing-nat-iff,
l-ordered_wf,
decidable__le,
subtract_wf,
not-ge-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,
set_wf,
list_wf,
primrec-unroll-1,
not-le-2,
le-add-cancel2,
add-mul-special,
zero-mul,
equal-wf-T-base,
int_subtype_base,
decidable__lt,
not-equal-2,
le-add-cancel-alt,
decidable__equal_int,
squash_wf,
true_wf,
iff_weakening_equal,
subtype_rel_sets
\mforall{}[max:\mBbbN{}]. \mforall{}[missing:\{l:\mBbbN{} List| l-ordered(\mBbbN{};x,y.x < y;l)\} ].
(select-last-in-nat-missing(max;missing) \mmember{} \{x:\mBbbZ{}|
((-1) \mleq{} x)
\mwedge{} (x \mleq{} max)
\mwedge{} ((0 \mleq{} x) {}\mRightarrow{} (\mneg{}(x \mmember{} missing)))
\mwedge{} (\mforall{}y:\mBbbN{}. ((\mneg{}(y \mmember{} missing)) {}\mRightarrow{} y < max {}\mRightarrow{} (y \mleq{} x)))
\mwedge{} (\mforall{}y:\mBbbN{}. (y < max {}\mRightarrow{} x < y {}\mRightarrow{} (y \mmember{} missing)))\} )
Date html generated:
2015_07_17-AM-08_21_43
Last ObjectModification:
2015_04_02-PM-05_43_25
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