Nuprl Lemma : set-sig-add-prop
∀[Item:Type]
∀s:set-sig{i:l}(Item). ∀set:set-sig-set(s). ∀x,y:Item.
(↑(set-sig-member(s) x (set-sig-add(s) y set)) ⇐⇒ (x = y ∈ Item) ∨ (↑(set-sig-member(s) x set)))
Proof
Definitions occuring in Statement :
set-sig-add: set-sig-add(s),
set-sig-member: set-sig-member(s),
set-sig-set: set-sig-set(s),
set-sig: set-sig{i:l}(Item),
assert: ↑b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
apply: f a,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
set-sig: set-sig{i:l}(Item),
record+: record+,
record-select: r.x,
subtype_rel: A ⊆r B,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
btrue: tt,
implies: P ⇒ Q,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
or: P ∨ Q,
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
uimplies: b supposing a,
assert: ↑b,
sq_stable: SqStable(P),
true: True,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
false: False,
set-sig-member: set-sig-member(s),
set-sig-set: set-sig-set(s),
set-sig-add: set-sig-add(s)
Latex:
\mforall{}[Item:Type]
\mforall{}s:set-sig\{i:l\}(Item). \mforall{}set:set-sig-set(s). \mforall{}x,y:Item.
(\muparrow{}(set-sig-member(s) x (set-sig-add(s) y set)) \mLeftarrow{}{}\mRightarrow{} (x = y) \mvee{} (\muparrow{}(set-sig-member(s) x set)))
Date html generated:
2016_05_17-PM-01_44_31
Last ObjectModification:
2016_01_17-AM-11_37_40
Theory : datatype-signatures
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