Nuprl Lemma : es-interface-extensionality

{

[Info,A:Type].

[X,Y:EClass(A)].
(X = Y) supposing
((

es:EO+(Info).

e:E. ((

e

X)

(

e

Y)

(X(e) = Y(e)))) and
(

es:EO+(Info).

e:E. (

e

X

e

Y)) and
Singlevalued(X) and
Singlevalued(Y)) }

{ Proof }

Definitions occuring in Statement : sv-class: Singlevalued(X), eclass-val: X(e), in-eclass: e

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
Definitions : atom: Atom, es-base-E: es-base-E(es), token: "$token", fpf-cap: f(x)?z, apply: f a, fpf-dom: x

dom(f), bag: bag(T), bool:

, record-select: r.x, eq_atom: x =a y, eq_atom: eq_atom$n(x;y), set: {x:A| B[x]} , dep-isect: Error :dep-isect, record+: record+, lambda:

x.A[x], eclass-val: X(e), limited-type: LimitedType, top: Top, in-eclass: e

X, subtype: S

T, pair: <a, b>, fpf: a:A fp-> B[a], rev_implies: P

Q, decide: case b of inl(x) => s[x] | inr(y) => t[y], ifthenelse: if b then t else f fi , void: Void, false: False, strong-subtype: strong-subtype(A;B), le: A

B, ge: i

j , not:

A, less_than: a < b, product: x:A

B[x], and: P

Q, uiff: uiff(P;Q), subtype_rel: A

r B, axiom: Ax, implies: P

Q, event-ordering+: EO+(Info), event_ordering: EO, es-E: E, assert:

b, iff: P

Q, function: x:A

B[x], all:

x:A. B[x], equal: s = t, universe: Type, uall:

[x:A]. B[x], eclass: EClass(A[eo; e]), so_lambda:

x y.t[x; y], prop:

, uimplies: b supposing a, isect:

x:A. B[x], member: t

T, sv-class: Singlevalued(X), Auto: Error :Auto, CollapseTHEN: Error :CollapseTHEN, MaAuto: Error :MaAuto, tactic: Error :tactic, natural_number: $n, bag-size: bag-size(bs), AssertBY: Error :AssertBY, RepeatFor: Error :RepeatFor, RepUR: Error :RepUR, permutation: permutation(T;L1;L2), true: True, bag_only_single: bag_only_single{bag_only_single_compseq_tag_def:o}(x), null: null(as), set_blt: a <

b, grp_blt: a <

b, dcdr-to-bool: [d]

, bl-all: (

x

L.P[x])_b, bl-exists: (

x

L.P[x])_b, b-exists: (

i<n.P[i])_b, eq_type: eq_type(T;T'), qeq: qeq(r;s), q_less: q_less(r;s), q_le: q_le(r;s), deq-member: deq-member(eq;x;L), deq-disjoint: deq-disjoint(eq;as;bs), deq-all-disjoint: deq-all-disjoint(eq;ass;bs), name_eq: name_eq(x;y), eq_id: a = b, eq_lnk: a = b, es-eq-E: e = e', es-bless: e <loc e', es-ble: e

loc e', bnot:

b, bimplies: p

q, band: p

q, bor: p

q, bag_size_single: bag_size_single{bag_size_single_compseq_tag_def:o}(x), quotient: x,y:A//B[x; y], single-bag: {x}, eq_int: (i =

j), bag-only: only(bs), sqequal: s ~ t, union: left + right, or: P

Q, list: type List, nat_plus:

, l_contains: A

B, cmp-le: cmp-le(cmp;x;y), inject: Inj(A;B;f), reducible: reducible(a), prime: prime(a), squash:

T, l_exists: (

x

L. P[x]), l_all: (

x

L.P[x]), fun-connected: y is f*(x), rationals:

, qle: r

s, qless: r < s, q-rel: q-rel(r;x), sq_exists:

x:{A| B[x]}, atom: Atom$n, i-finite: i-finite(I), i-closed: i-closed(I), p-outcome: Outcome, dstype: dstype(TypeNames; d; a), fset-member: a

s, f-subset: xs

ys, fset: FSet{T}, fset-closed: (s closed under fs), Id: Id, IdLnk: IdLnk, Knd: Knd, MaName: MaName, l_disjoint: l_disjoint(T;l1;l2), consensus-state3: consensus-state3(T), cs-not-completed: in state s, a has not completed inning i, cs-archived: by state s, a archived v in inning i, cs-passed: by state s, a passed inning i without archiving a value, cs-inning-committed: in state s, inning i has committed v, cs-inning-committable: in state s, inning i could commit v , cs-archive-blocked: in state s, ws' blocks ws from archiving v in inning i, cs-precondition: state s may consider v in inning i, consensus-rcv: consensus-rcv(V;A), infix_ap: x f y, es-causl: (e < e'), es-locl: (e <loc e'), es-le: e

loc e' , es-causle: e c

e', existse-before:

e<e'.P[e], existse-le:

e

e'.P[e], alle-lt:

e<e'.P[e], alle-le:

e

e'.P[e], alle-between1:

e

[e1,e2).P[e], existse-between1:

e

[e1,e2).P[e], alle-between2:

e

[e1,e2].P[e], existse-between2:

e

[e1,e2].P[e], existse-between3:

e

(e1,e2].P[e], es-fset-loc: i

locs(s), exists:

x:A. B[x], es-r-immediate-pred: es-r-immediate-pred(es;R;e';e), same-thread: same-thread(es;p;e;e'), decidable: Dec(P), real:

, grp_car: |g|, nat:

, Complete: Error :Complete, AllHyps: Error :AllHyps, THENM: Error :THENM, CollapseTHENA: Error :CollapseTHENA, int:

, Decide: Error :Decide, D: Error :D, empty-bag: {}
Lemmas : bag-size-zero, empty-bag_wf, bag-size_wf, single-bag_wf, bag-only_wf, nat_wf, eq_int_wf, bag-size-one, decidable__equal_int, le_wf, iff_functionality_wrt_iff, iff_weakening_uiff, assert_of_eq_int, false_wf, ifthenelse_wf, true_wf, permutation_wf, es-E_wf, assert_wf, event-ordering+_wf, eclass_wf, iff_wf, sv-class_wf, event-ordering+_inc, in-eclass_wf, eclass-val_wf, member_wf, subtype_rel_wf, bag_wf, top_wf, dep-eclass_subtype_rel, es-base-E_wf, subtype_rel_self

\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: 2011_08_16-AM-11_38_50 Last ObjectModification: 2011_06_20-AM-00_30_23 Home Index