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]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
union-nat-missing-pos: union-nat-missing-pos(s1;max;missing),
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
cand: A c∧ B,
nat-missing-type: nat-missing-type(),
pi1: fst(t),
pi2: snd(t),
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
guard: {T},
member-nat-missing: member-nat-missing(i;s),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Latex:
\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:
2016_05_17-PM-01_45_32
Last ObjectModification:
2016_01_17-AM-11_37_54
Theory : datatype-signatures
Home
Index