Nuprl Lemma : dma-neg-dM_opp
∀[I:fset(ℕ)]. ∀[x:names(I)]. (¬(<1-x>) = <x> ∈ Point(dM(I)))
Proof
Definitions occuring in Statement :
dM_opp: <1-x>,
dM_inc: <x>,
dM: dM(I),
names: names(I),
dma-neg: ¬(x),
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
squash: ↓T,
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
dma-neg: ¬(x),
record-select: r.x,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
btrue: tt,
dm-neg: ¬(x),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
dM_opp: <1-x>,
dmopp: <1-i>,
free-dl-inc: free-dl-inc(x),
fset-singleton: {x},
cons: [a / b],
nil: [],
it: ⋅,
fset-union: x ⋃ y,
l-union: as ⋃ bs,
insert: insert(a;L),
eval_list: eval_list(t),
deq-member: x ∈b L,
bfalse: ff,
lattice-join: a ∨ b,
opposite-lattice: opposite-lattice(L),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
so_lambda: λ2x y.t[x; y],
lattice-meet: a ∧ b,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
fset-ac-glb: fset-ac-glb(eq;ac1;ac2),
fset-minimals: fset-minimals(x,y.less[x; y]; s),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
lattice-fset-meet: /\(s),
fset-ac-lub: fset-ac-lub(eq;ac1;ac2),
lattice-1: 1,
lattice-0: 0,
empty-fset: {},
fset-minimal: fset-minimal(x,y.less[x; y];s;a),
fset-null: fset-null(s),
null: null(as),
f-proper-subset-dec: f-proper-subset-dec(eq;xs;ys),
band: p ∧b q,
deq-f-subset: deq-f-subset(eq),
isl: isl(x),
decidable__f-subset,
decidable__all_fset,
decidable_functionality,
iff_preserves_decidability,
iff_weakening_uiff,
fset-all-iff,
decidable__assert,
bnot: ¬bb,
decidable__fset-member,
assert-deq-fset-member,
deq-fset-member: a ∈b s,
bor: p ∨bq,
union-deq: union-deq(A;B;a;b),
sumdeq: sumdeq(a;b),
names-deq: NamesDeq,
int-deq: IntDeq,
eq_int: (i =z j),
lattice-point: Point(l),
bdd-distributive-lattice: BoundedDistributiveLattice,
true: True
Lemmas referenced :
neg-dM_opp,
equal_wf,
squash_wf,
true_wf,
istype-universe,
lattice-point_wf,
dM_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
dm-neg_wf,
names_wf,
names-deq_wf,
dM_opp_wf,
subtype_rel-equal,
free-DeMorgan-lattice_wf,
fset_wf,
nat_wf,
decidable__f-subset,
decidable__all_fset,
decidable_functionality,
iff_preserves_decidability,
iff_weakening_uiff,
fset-all-iff,
decidable__assert,
decidable__fset-member,
assert-deq-fset-member
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hyp_replacement,
equalitySymmetry,
sqequalRule,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
universeIsType,
inhabitedIsType,
instantiate,
universeEquality,
productEquality,
independent_isectElimination,
cumulativity,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[x:names(I)]. (\mneg{}(ə-x>) = <x>)
Date html generated:
2019_11_04-PM-05_30_33
Last ObjectModification:
2018_11_08-AM-10_18_17
Theory : cubical!type!theory
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