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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
order: Order(T;x,y.R[x; y]),
and: P ∧ Q,
anti_sym: AntiSym(T;x,y.R[x; y]),
implies: P ⇒ Q,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
exists: ∃x:A. B[x],
cand: A c∧ B,
squash: ↓T,
face-lattice: face-lattice(T;eq),
face-lattice0: (x=0),
fset-constrained-ac-glb: glb(P;ac1;ac2),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uiff: uiff(P;Q),
fset-constrained-image: f"(s) s.t. P,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
fset-member: a ∈ s,
deq-member: x ∈b L,
reduce: reduce(f;k;as),
list_ind: list_ind,
empty-fset: {},
nil: [],
f-proper-subset: xs ⊆≠ ys,
f-subset: xs ⊆ ys,
face-lattice1: (x=1),
fl-all: (∀i.phi),
fl-filter: fl-filter(s;x.Q[x]),
cal-filter: cal-filter(s;x.P[x])
Lemmas referenced :
lattice-le-order,
face-lattice_wf,
bdd-distributive-lattice-subtype-lattice,
lattice-join_wf,
fl-all_wf,
lattice-meet_wf,
face-lattice0_wf,
face-lattice1_wf,
lattice-point_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,
deq_wf,
fl-point-sq,
fset_wf,
assert_wf,
fset-antichain_wf,
union-deq_wf,
fset-all_wf,
fset-contains-none_wf,
face-lattice-constraints_wf,
implies-le-face-lattice-join3,
fset-member_wf,
deq-fset_wf,
decidable__fset-member,
squash_wf,
exists_wf,
f-subset_wf,
f-subset_weakening,
free-dlwc-meet,
member-fset-minimals,
f-proper-subset-dec_wf,
f-union_wf,
fset-constrained-image_wf,
fset-union_wf,
fset-singleton_wf,
ifthenelse_wf,
empty-fset_wf,
member-f-union,
member-fset-singleton,
fset-extensionality,
member-fset-union,
or_wf,
and_wf,
fset-member_witness,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
mem_empty_lemma,
fset-all-iff,
iff_weakening_uiff,
bnot_wf,
isect_wf,
iff_transitivity,
not_wf,
f-proper-subset_wf,
assert_of_bnot,
assert-f-proper-subset-dec,
assert_witness,
bool_cases,
f-subset_transitivity,
f-subset-union,
assert-fset-antichain,
member-fset-filter,
band_wf,
deq-fset-member_wf,
assert_of_band,
assert-deq-fset-member,
lattice-join-le,
lattice-meet-le,
face-lattice-subset-le,
fl-filter-subset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
productElimination,
because_Cache,
independent_functionElimination,
isect_memberEquality,
axiomEquality,
instantiate,
lambdaEquality,
productEquality,
universeEquality,
independent_isectElimination,
voidElimination,
voidEquality,
setElimination,
rename,
setEquality,
unionEquality,
lambdaFormation,
inlEquality,
unionElimination,
inrFormation,
inlFormation,
dependent_pairFormation,
independent_pairFormation,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
addLevel,
orFunctionality,
promote_hyp,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality,
levelHypothesis,
independent_pairEquality,
equalityElimination,
impliesFunctionality,
imageElimination,
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:
2017_10_05-AM-00_40_59
Last ObjectModification:
2017_07_28-AM-09_16_18
Theory : lattices
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