Nuprl Lemma : dM-join-inc-opp
∀[i:ℕ]. (<i> ∨ <1-i> ~ {{inr i },{inl i}})
Proof
Definitions occuring in Statement :
dM_opp: <1-x>,
dM_inc: <x>,
dM: dM(I),
lattice-join: a ∨ b,
fset-pair: {a,b},
fset-singleton: {x},
nat: ℕ,
uall: ∀[x:A]. B[x],
inr: inr x ,
inl: inl x,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
lattice-join: a ∨ b,
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,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
btrue: tt,
fset-ac-lub: fset-ac-lub(eq;ac1;ac2),
fset-minimals: fset-minimals(x,y.less[x; y]; s),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-union: x ⋃ y,
l-union: as ⋃ bs,
dM_opp: <1-x>,
dmopp: <1-i>,
free-dl-inc: free-dl-inc(x),
fset-singleton: {x},
cons: [a / b],
insert: insert(a;L),
eval_list: eval_list(t),
nil: [],
it: ⋅,
dM_inc: <x>,
dminc: <i>,
deq-member: x ∈b L,
bor: p ∨bq,
deq-fset: deq-fset(eq),
isl: isl(x),
decidable__equal_fset,
decidable_functionality,
iff_preserves_decidability,
decidable__and2,
decidable__f-subset,
decidable__all_fset,
decidable__assert,
fset-null: fset-null(s),
null: null(as),
bnot: ¬bb,
decidable__fset-member,
deq-fset-member: a ∈b s,
mk_deq: mk_deq(p),
union-deq: union-deq(A;B;a;b),
sumdeq: sumdeq(a;b),
fset-minimal: fset-minimal(x,y.less[x; y];s;a),
f-proper-subset-dec: f-proper-subset-dec(eq;xs;ys),
band: p ∧b q,
deq-f-subset: deq-f-subset(eq),
names-deq: NamesDeq,
int-deq: IntDeq,
eq_int: (i =z j),
fset-pair: {a,b},
nat: ℕ,
iff_weakening_uiff,
fset-all-iff,
decidable__and
Lemmas referenced :
nat_wf,
decidable__equal_fset,
decidable_functionality,
iff_preserves_decidability,
decidable__and2,
decidable__f-subset,
decidable__all_fset,
decidable__assert,
decidable__fset-member,
iff_weakening_uiff,
fset-all-iff,
decidable__and
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
int_eqReduceTrueSq,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalAxiom,
extract_by_obid
Latex:
\mforall{}[i:\mBbbN{}]. (<i> \mvee{} ə-i> \msim{} \{\{inr i \},\{inl i\}\})
Date html generated:
2018_05_23-AM-08_28_47
Last ObjectModification:
2018_05_20-PM-05_39_01
Theory : cubical!type!theory
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