Nuprl Lemma : dM-to-FL-unique
∀[I:fset(ℕ)]. ∀[g:Hom(free-DeMorgan-lattice(names(I);NamesDeq);face_lattice(I))].
∀[x:Point(dM(I))]. (dM-to-FL(I;x) = (g x) ∈ Point(face_lattice(I)))
supposing ∀i:names(I). (((g <i>) = (i=1) ∈ Point(face_lattice(I))) ∧ ((g <1-i>) = (i=0) ∈ Point(face_lattice(I))))
Proof
Definitions occuring in Statement :
dM-to-FL: dM-to-FL(I;z),
fl1: (x=1),
fl0: (x=0),
face_lattice: face_lattice(I),
dM_opp: <1-x>,
dM_inc: <x>,
dM: dM(I),
names-deq: NamesDeq,
names: names(I),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
bounded-lattice-hom: Hom(l1;l2),
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
guard: {T},
so_apply: x[s],
bdd-distributive-lattice: BoundedDistributiveLattice,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
lattice-point: Point(l),
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,
dM-to-FL: dM-to-FL(I;z),
all: ∀x:A. B[x],
implies: P ⇒ Q,
compose: f o g,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
dminc: <i>,
dM_inc: <x>,
dmopp: <1-i>,
dM_opp: <1-x>,
top: Top
Lemmas referenced :
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,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
all_wf,
names_wf,
face_lattice_wf,
dM_inc_wf,
subtype_rel-equal,
free-DeMorgan-lattice_wf,
names-deq_wf,
fl1_wf,
dM_opp_wf,
fl0_wf,
bounded-lattice-hom_wf,
bdd-distributive-lattice_wf,
fset_wf,
nat_wf,
lattice-extend-is-hom,
union-deq_wf,
face_lattice-deq_wf,
free-dist-lattice-hom-unique,
squash_wf,
true_wf,
lattice-extend-dl-inc,
subtype_rel_self,
iff_weakening_equal,
dM-point,
free-dl-point
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
because_Cache,
independent_isectElimination,
isect_memberEquality,
axiomEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
unionEquality,
lambdaFormation,
unionElimination,
dependent_functionElimination,
independent_functionElimination,
functionExtensionality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
applyLambdaEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[g:Hom(free-DeMorgan-lattice(names(I);NamesDeq);face\_lattice(I))].
\mforall{}[x:Point(dM(I))]. (dM-to-FL(I;x) = (g x))
supposing \mforall{}i:names(I). (((g <i>) = (i=1)) \mwedge{} ((g ə-i>) = (i=0)))
Date html generated:
2019_11_04-PM-05_33_49
Last ObjectModification:
2018_08_21-PM-02_03_15
Theory : cubical!type!theory
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