Nuprl Lemma : fl-all-decomp
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[phi:Point(face-lattice(T;eq))]. ∀[i:T].
(phi = (∀i.phi) ∨ phi ∧ (i=0) ∨ phi ∧ (i=1) ∈ Point(face-lattice(T;eq)))
Proof
Definitions occuring in Statement :
fl-all: (∀i.phi),
face-lattice1: (x=1),
face-lattice0: (x=0),
face-lattice: face-lattice(T;eq),
lattice-join: a ∨ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uimplies: b supposing a,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
bdd-distributive-lattice: BoundedDistributiveLattice,
implies: P ⇒ Q,
anti_sym: AntiSym(T;x,y.R[x; y]),
and: P ∧ Q,
order: Order(T;x,y.R[x; y]),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
top: Top,
or: P ∨ Q,
decidable: Dec(P),
guard: {T},
squash: ↓T,
cand: A c∧ B,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
fset-constrained-ac-glb: glb(P;ac1;ac2),
face-lattice0: (x=0),
face-lattice: face-lattice(T;eq),
fset-constrained-image: f"(s) s.t. P,
false: False,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
not: ¬A,
nil: [],
empty-fset: {},
list_ind: list_ind,
reduce: reduce(f;k;as),
deq-member: x ∈b L,
fset-member: a ∈ s,
f-proper-subset: xs ⊆≠ ys,
f-subset: xs ⊆ ys,
face-lattice1: (x=1),
cal-filter: cal-filter(s;x.P[x]),
fl-filter: fl-filter(s;x.Q[x]),
fl-all: (∀i.phi)
Lemmas referenced :
deq_wf,
equal_wf,
uall_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
lattice-point_wf,
face-lattice1_wf,
face-lattice0_wf,
lattice-meet_wf,
fl-all_wf,
lattice-join_wf,
bdd-distributive-lattice-subtype-lattice,
face-lattice_wf,
lattice-le-order,
deq-fset_wf,
fset-member_wf,
implies-le-face-lattice-join3,
face-lattice-constraints_wf,
fset-contains-none_wf,
fset-all_wf,
union-deq_wf,
fset-antichain_wf,
assert_wf,
fset_wf,
fl-point-sq,
decidable__fset-member,
f-subset_wf,
exists_wf,
squash_wf,
f-subset_weakening,
fset-singleton_wf,
fset-union_wf,
fset-constrained-image_wf,
f-union_wf,
f-proper-subset-dec_wf,
member-fset-minimals,
free-dlwc-meet,
member-f-union,
empty-fset_wf,
ifthenelse_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
eqtt_to_assert,
bool_wf,
fset-member_witness,
and_wf,
or_wf,
member-fset-union,
fset-extensionality,
member-fset-singleton,
mem_empty_lemma,
fset-all-iff,
assert_witness,
assert-f-proper-subset-dec,
assert_of_bnot,
f-proper-subset_wf,
not_wf,
iff_transitivity,
isect_wf,
bnot_wf,
iff_weakening_uiff,
bool_cases,
f-subset-union,
f-subset_transitivity,
assert-fset-antichain,
assert-deq-fset-member,
assert_of_band,
deq-fset-member_wf,
band_wf,
member-fset-filter,
lattice-meet-le,
lattice-join-le,
face-lattice-subset-le,
fl-filter-subset
Rules used in proof :
independent_isectElimination,
universeEquality,
productEquality,
lambdaEquality,
instantiate,
axiomEquality,
isect_memberEquality,
independent_functionElimination,
because_Cache,
productElimination,
sqequalRule,
applyEquality,
hypothesis,
hypothesisEquality,
cumulativity,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lambdaFormation,
unionEquality,
setEquality,
rename,
setElimination,
voidEquality,
voidElimination,
unionElimination,
inlEquality,
inlFormation,
inrFormation,
baseClosed,
imageMemberEquality,
independent_pairFormation,
dependent_pairFormation,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
independent_pairEquality,
levelHypothesis,
applyLambdaEquality,
dependent_set_memberEquality,
hyp_replacement,
promote_hyp,
orFunctionality,
addLevel,
imageElimination,
impliesFunctionality,
Error :memTop,
inrEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[phi:Point(face-lattice(T;eq))]. \mforall{}[i:T].
(phi = (\mforall{}i.phi) \mvee{} phi \mwedge{} (i=0) \mvee{} phi \mwedge{} (i=1))
Date html generated:
2020_05_20-AM-08_53_05
Last ObjectModification:
2020_02_04-PM-01_48_08
Theory : lattices
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