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: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
select-last-in-nat-missing: select-last-in-nat-missing(max;missing),
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_apply: x[s],
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
cand: A c∧ B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
label: ...$L... t
Latex:
\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:
2016_05_17-PM-01_45_48
Last ObjectModification:
2016_01_17-AM-11_37_33
Theory : datatype-signatures
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