Nuprl Lemma : face_lattice-le
∀[I:fset(ℕ)]. ∀[x,y:Point(face_lattice(I))].
uiff(x ≤ y;∀f:I ⟶ I. (((x)<f> = 1 ∈ Point(face_lattice(I))) ⇒ ((y)<f> = 1 ∈ Point(face_lattice(I)))))
Proof
Definitions occuring in Statement :
fl-morph: <f>,
face_lattice: face_lattice(I),
names-hom: I ⟶ J,
lattice-1: 1,
lattice-le: a ≤ b,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
lattice-le: a ≤ b,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
face_lattice: face_lattice(I),
rev_implies: P ⇐ Q,
isl: isl(x),
outl: outl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
false: False,
isr: isr(x),
outr: outr(x),
union-deq: union-deq(A;B;a;b),
face_lattice-deq: face_lattice-deq(),
top: Top,
face-lattice0: (x=0),
fl0: (x=0),
face-lattice1: (x=1),
fl1: (x=1),
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
not: ¬A,
exists: ∃x:A. B[x],
btrue: tt,
cand: A c∧ B,
names: names(I),
nat: ℕ,
sq_type: SQType(T),
face-lattice-constraints: face-lattice-constraints(x),
f-subset: xs ⊆ ys,
or: P ∨ Q,
name-morph-satisfies: (psi f) = 1,
irr_face: irr_face(I;as;bs),
respects-equality: respects-equality(S;T),
decidable: Dec(P),
fset-singleton: {x},
cons: [a / b],
irr-face-morph: irr-face-morph(I;as;bs),
bool: 𝔹,
unit: Unit,
it: ⋅,
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,
dm-neg: ¬(x),
dM1: 1,
lattice-1: 1,
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],
eq_atom: x =a y,
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),
nil: [],
fset-union: x ⋃ y,
l-union: as ⋃ bs,
insert: insert(a;L),
eval_list: eval_list(t),
deq-member: x ∈b L,
lattice-join: a ∨ b,
opposite-lattice: opposite-lattice(L),
so_lambda: λ2x y.t[x; y],
lattice-meet: a ∧ b,
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),
empty-fset: {},
lattice-0: 0,
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)
Lemmas referenced :
fl-morph_wf,
lattice-1_wf,
face_lattice_wf,
names-hom_wf,
lattice-le_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
lattice-point_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
fset_wf,
nat_wf,
squash_wf,
true_wf,
iff_weakening_equal,
lattice-1-le-iff,
bdd-distributive-lattice-subtype-bdd-lattice,
lattice-hom-le,
face-lattice-le,
names_wf,
names-deq_wf,
fset-member_wf,
deq-fset_wf,
union-deq_wf,
irr-face-morph_wf,
fset-mapfilter_wf,
btrue_wf,
bfalse_wf,
assert_wf,
istype-assert,
isl_wf,
isr_wf,
face-lattice-basis,
istype-universe,
subtype_rel_self,
face_lattice-deq_wf,
fl-point-sq,
istype-void,
fl0_wf,
fl1_wf,
fset-image_wf,
lattice-fset-meet_wf,
decidable__equal_face_lattice,
face_lattice-fset-join-eq-1,
lattice-hom-fset-join,
irr-face-morph-satisfies,
sq_stable__assert,
fset-disjoint_wf,
fset-all-iff,
fset-contains-none_wf,
face-lattice-constraints_wf,
assert-fset-disjoint,
assert-fset-contains-none,
member-fset-mapfilter,
iff_weakening_uiff,
exists_wf,
subtype_base_sq,
subtype_rel_universe1,
set_subtype_base,
int-deq_wf,
istype-nat,
le_wf,
istype-int,
int_subtype_base,
fset-pair_wf,
member-fset-singleton,
member-fset-pair,
fset-member_witness,
deq_wf,
member-fset-image-iff,
irr_face_wf,
all_wf,
decidable_wf,
bdd-lattice_wf,
fset-extensionality,
fset-union_wf,
member-fset-union,
equal-wf-T-base,
strong-subtype-deq-subtype,
strong-subtype-set3,
strong-subtype-self,
fset-singleton_wf,
respects-equality-face-lattice-point-2,
top_wf,
istype-top,
subtype_rel_union,
decidable__fset-member,
decidable__or,
decidable__squash_exists_fset,
outl_wf,
f-subset_wf,
lattice-hom-fset-meet,
lattice-fset-meet-is-1,
fl-morph-fl0,
deq-fset-member_wf,
eqtt_to_assert,
assert-deq-fset-member,
lattice-0_wf,
face-lattice-0-not-1,
not_wf,
dM-to-FL_wf,
neg-dM_inc,
dM-to-FL-opp,
fl0-not-1,
fl-morph-fl1,
dM-to-FL-dM0,
dM-to-FL-inc,
fl1-not-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaFormation_alt,
hypothesis,
equalityIstype,
inhabitedIsType,
hypothesisEquality,
applyEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality_alt,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
universeIsType,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
instantiate,
productEquality,
cumulativity,
because_Cache,
independent_isectElimination,
functionIsType,
productElimination,
independent_pairEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
unionEquality,
unionElimination,
unionIsType,
voidElimination,
setIsType,
universeEquality,
productIsType,
promote_hyp,
intEquality,
inlEquality_alt,
inrEquality_alt,
dependent_pairFormation_alt,
baseApply,
closedConclusion,
inlFormation_alt,
inrFormation_alt,
hyp_replacement,
dependent_set_memberEquality_alt,
applyLambdaEquality,
voidEquality,
sqequalBase,
equalityElimination
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[x,y:Point(face\_lattice(I))]. uiff(x \mleq{} y;\mforall{}f:I {}\mrightarrow{} I. (((x)<f> = 1) {}\mRightarrow{} ((y)<f> = 1)))
Date html generated:
2019_11_04-PM-05_35_48
Last ObjectModification:
2018_12_13-PM-00_50_46
Theory : cubical!type!theory
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