Nuprl Lemma : es-interface-extensionality
∀[Info,A:Type]. ∀[X,Y:EClass(A)].
(X = Y ∈ EClass(A)) supposing
((∀es:EO+(Info). ∀e:E. ((↑e ∈b X) ⇒ (↑e ∈b Y) ⇒ (X(e) = Y(e) ∈ A))) and
(∀es:EO+(Info). ∀e:E. (↑e ∈b X ⇐⇒ ↑e ∈b Y)) and
Singlevalued(X) and
Singlevalued(Y))
Proof
Definitions occuring in Statement :
sv-class: Singlevalued(X),
eclass-val: X(e),
in-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
eclass: EClass(A[eo; e]),
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
top: Top,
so_apply: x[s],
guard: {T},
sv-class: Singlevalued(X),
in-eclass: e ∈b X,
eclass-val: X(e),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
iff: P ⇐⇒ Q,
true: True,
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
nequal: a ≠ b ∈ T ,
nat: ℕ,
rev_implies: P ⇐ Q,
squash: ↓T,
le: A ≤ B
Latex:
\mforall{}[Info,A:Type]. \mforall{}[X,Y:EClass(A)].
(X = Y) supposing
((\mforall{}es:EO+(Info). \mforall{}e:E. ((\muparrow{}e \mmember{}\msubb{} X) {}\mRightarrow{} (\muparrow{}e \mmember{}\msubb{} Y) {}\mRightarrow{} (X(e) = Y(e)))) and
(\mforall{}es:EO+(Info). \mforall{}e:E. (\muparrow{}e \mmember{}\msubb{} X \mLeftarrow{}{}\mRightarrow{} \muparrow{}e \mmember{}\msubb{} Y)) and
Singlevalued(X) and
Singlevalued(Y))
Date html generated:
2016_05_16-PM-02_32_03
Last ObjectModification:
2016_01_17-PM-07_37_44
Theory : event-ordering
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