Nuprl Lemma : member-eclass-simple-comb-2-iff
∀[Info,A,B,C:Type]. ∀[es:EO+(Info)]. ∀[e:E]. ∀[F:bag(A) ⟶ bag(B) ⟶ bag(C)]. ∀[X:EClass(A)]. ∀[Y:EClass(B)].
(uiff(↑e ∈b F|X, Y|;(↑e ∈b X) ∧ (↑e ∈b Y) ∧ (¬↑bag-null(F {X@e} {Y@e})))) supposing
(single-valued-classrel(es;Y;B) and
single-valued-classrel(es;X;A) and
lifting2-like(A;B;F))
Proof
Definitions occuring in Statement :
lifting2-like: lifting2-like(A;B;f),
simple-comb-2: F|X, Y|,
classfun-res: X@e,
single-valued-classrel: single-valued-classrel(es;X;T),
member-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
assert: ↑b,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
bag-null: bag-null(bs),
single-bag: {x},
bag: bag(T)
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
prop: ℙ,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
lifting2-like: lifting2-like(A;B;f),
simple-comb-2: F|X, Y|,
member-eclass: e ∈b X,
simple-comb: simple-comb(F;Xs),
select: L[n],
cons: [a / b],
subtract: n - m,
eclass: EClass(A[eo; e]),
nat: ℕ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
single-valued-classrel: single-valued-classrel(es;X;T),
classrel: v ∈ X(e),
single-valued-bag: single-valued-bag(b;T),
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
nequal: a ≠ b ∈ T
Latex:
\mforall{}[Info,A,B,C:Type]. \mforall{}[es:EO+(Info)]. \mforall{}[e:E]. \mforall{}[F:bag(A) {}\mrightarrow{} bag(B) {}\mrightarrow{} bag(C)]. \mforall{}[X:EClass(A)].
\mforall{}[Y:EClass(B)].
(uiff(\muparrow{}e \mmember{}\msubb{} F|X, Y|;(\muparrow{}e \mmember{}\msubb{} X) \mwedge{} (\muparrow{}e \mmember{}\msubb{} Y) \mwedge{} (\mneg{}\muparrow{}bag-null(F \{X@e\} \{Y@e\})))) supposing
(single-valued-classrel(es;Y;B) and
single-valued-classrel(es;X;A) and
lifting2-like(A;B;F))
Date html generated:
2016_05_17-AM-11_13_30
Last ObjectModification:
2016_01_18-AM-00_10_56
Theory : process-model
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