Nuprl Lemma : dM-to-FL-properties
∀[I:fset(ℕ)]
((∀x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
(dM-to-FL(I;x ∨ y) = dM-to-FL(I;x) ∨ dM-to-FL(I;y) ∈ Point(face_lattice(I))))
∧ (∀x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
(dM-to-FL(I;x ∧ y) = dM-to-FL(I;x) ∧ dM-to-FL(I;y) ∈ Point(face_lattice(I))))
∧ (dM-to-FL(I;0) = 0 ∈ Point(face_lattice(I)))
∧ (dM-to-FL(I;1) = 1 ∈ Point(face_lattice(I))))
Proof
Definitions occuring in Statement :
dM-to-FL: dM-to-FL(I;z),
face_lattice: face_lattice(I),
names-deq: NamesDeq,
names: names(I),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
lattice-0: 0,
lattice-1: 1,
lattice-join: a ∨ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
dM-to-FL: dM-to-FL(I;z),
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,
lattice-0: 0,
record-select: r.x,
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),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
empty-fset: {},
nil: [],
it: ⋅,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
implies: P ⇒ Q,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
dM-to-FL-is-hom,
lattice-point_wf,
free-DeMorgan-lattice_wf,
names_wf,
names-deq_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
lattice-0_wf,
face_lattice_wf,
bdd-distributive-lattice_wf,
fset_wf,
nat_wf,
bounded-lattice-hom_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
because_Cache,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
universeEquality,
independent_isectElimination,
independent_pairFormation,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
productElimination,
equalityUniverse,
levelHypothesis,
functionExtensionality,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[I:fset(\mBbbN{})]
((\mforall{}x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
(dM-to-FL(I;x \mvee{} y) = dM-to-FL(I;x) \mvee{} dM-to-FL(I;y)))
\mwedge{} (\mforall{}x,y:Point(free-DeMorgan-lattice(names(I);NamesDeq)).
(dM-to-FL(I;x \mwedge{} y) = dM-to-FL(I;x) \mwedge{} dM-to-FL(I;y)))
\mwedge{} (dM-to-FL(I;0) = 0)
\mwedge{} (dM-to-FL(I;1) = 1))
Date html generated:
2017_10_05-AM-01_12_03
Last ObjectModification:
2017_07_28-AM-09_30_18
Theory : cubical!type!theory
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