Nuprl Lemma : dMpair-eq-meet
∀[I:fset(ℕ)]. ∀[i,j:ℕ]. (dMpair(i;j) = <i> ∧ <j> ∈ Point(dM(I))) supposing (j ∈ I and i ∈ I)
Proof
Definitions occuring in Statement :
dMpair: dMpair(i;j),
dM_inc: <x>,
dM: dM(I),
lattice-meet: a ∧ b,
lattice-point: Point(l),
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
dM_inc: <x>,
dM: dM(I),
lattice-meet: a ∧ b,
dMpair: dMpair(i;j),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
all: ∀x:A. B[x],
top: Top,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
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,
lattice-point: Point(l),
names: names(I),
dminc: <i>,
free-dl-inc: free-dl-inc(x),
implies: P ⇒ Q,
fset-ac-glb: fset-ac-glb(eq;ac1;ac2),
f-union: f-union(domeq;rngeq;s;x.g[x]),
fset-singleton: {x},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
empty-fset: {},
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
fset_wf,
rec_select_update_lemma,
fset-antichain-singleton,
names_wf,
union-deq_wf,
names-deq_wf,
fset-pair_wf,
assert_wf,
fset-antichain_wf,
equal_wf,
list_accum_cons_lemma,
list_accum_nil_lemma,
fset-singleton_wf,
f-proper-subset-dec_wf,
deq-fset_wf,
fset-union_wf,
fset-image-singleton,
squash_wf,
true_wf,
fset-minimals_wf,
bool_wf,
empty-fset-union,
iff_weakening_equal,
fset-minimals-singleton,
fset-pair-is-union
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
applyEquality,
intEquality,
independent_isectElimination,
sqequalRule,
lambdaEquality,
natural_numberEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
unionEquality,
inlEquality,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageElimination,
universeEquality,
functionEquality,
cumulativity,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i,j:\mBbbN{}]. (dMpair(i;j) = <i> \mwedge{} <j>) supposing (j \mmember{} I and i \mmember{} I)
Date html generated:
2017_10_05-AM-00_59_26
Last ObjectModification:
2017_07_28-AM-09_25_19
Theory : cubical!type!theory
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