Nuprl Lemma : face-lattice-basis
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:Point(face-lattice(T;eq))].
(x = \/(λs./\(λu.{{u}}"(s))"(x)) ∈ Point(face-lattice(T;eq)))
Proof
Definitions occuring in Statement :
face-lattice: face-lattice(T;eq),
lattice-fset-join: \/(s),
lattice-fset-meet: /\(s),
lattice-point: Point(l),
fset-image: f"(s),
deq-fset: deq-fset(eq),
fset-singleton: {x},
union-deq: union-deq(A;B;a;b),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
face-lattice: face-lattice(T;eq),
prop: ℙ,
squash: ↓T,
top: Top,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
and: P ∧ Q,
uimplies: b supposing a,
free-dlwc-inc: free-dlwc-inc(eq;a.Cs[a];x),
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
true: True,
bdd-distributive-lattice: BoundedDistributiveLattice,
face-lattice0: (x=0),
face-lattice1: (x=1),
not: ¬A,
face-lattice-constraints: face-lattice-constraints(x),
fset-singleton: {x},
fset-filter: {x ∈ s | P[x]},
fset-null: fset-null(s),
isl: isl(x),
iff_weakening_uiff,
iff: P ⇐⇒ Q,
deq-f-subset: deq-f-subset(eq),
decidable__f-subset,
decidable__all_fset,
decidable_functionality,
iff_preserves_decidability,
fset-all-iff,
decidable__assert,
null: null(as),
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-pair: {a,b},
cons: [a / b],
decidable__fset-member,
assert-deq-fset-member,
deq-fset-member: a ∈b s,
deq-member: x ∈b L,
bor: p ∨bq,
union-deq: union-deq(A;B;a;b),
sumdeq: sumdeq(a;b),
nil: [],
f-subset: xs ⊆ ys,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
free-dlwc-basis,
union-deq_wf,
face-lattice-constraints_wf,
equal_wf,
squash_wf,
true_wf,
fl-point-sq,
istype-void,
lattice-fset-join_wf,
face-lattice_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
fset-image_wf,
fset_wf,
assert_wf,
fset-antichain_wf,
fset-all_wf,
fset-contains-none_wf,
deq-fset_wf,
strong-subtype-deq-subtype,
strong-subtype-set2,
lattice-fset-meet_wf,
fset-null_wf,
fset-filter_wf,
deq-f-subset_wf,
fset-singleton_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
decidable__equal_set,
decidable__equal_fset,
decidable__equal_union,
decidable-equal-deq,
istype-assert,
lattice-point_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
deq_wf,
istype-universe,
face-lattice0_wf,
face-lattice1_wf,
filter_cons_lemma,
filter_nil_lemma,
fset-pair_wf,
bfalse_wf,
btrue_wf,
btrue_neq_bfalse,
iff_weakening_uiff,
f-subset_wf,
assert-deq-f-subset,
equal-wf-T-base,
null_nil_lemma,
int_subtype_base,
member-fset-singleton,
member-fset-pair,
decidable__f-subset,
decidable__all_fset,
decidable_functionality,
iff_preserves_decidability,
fset-all-iff,
decidable__assert,
decidable__fset-member,
assert-deq-fset-member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
unionEquality,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
unionIsType,
universeIsType,
hyp_replacement,
equalitySymmetry,
applyEquality,
imageElimination,
equalityTransitivity,
because_Cache,
isect_memberEquality_alt,
voidElimination,
setElimination,
rename,
independent_functionElimination,
productElimination,
setEquality,
productEquality,
independent_isectElimination,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
imageMemberEquality,
baseClosed,
natural_numberEquality,
setIsType,
productIsType,
isectEquality,
universeEquality,
inlEquality_alt,
inrEquality_alt,
dependent_set_memberEquality_alt,
independent_pairFormation,
applyLambdaEquality,
functionIsType,
intEquality,
inrFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:Point(face-lattice(T;eq))]. (x = \mbackslash{}/(\mlambda{}s./\mbackslash{}(\mlambda{}u.\{\{u\}\}"(s))"(x)))
Date html generated:
2020_05_20-AM-08_51_43
Last ObjectModification:
2019_12_08-PM-07_01_19
Theory : lattices
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