Nuprl Lemma : fdl-hom_wf
∀[X:Type]. ∀[L:BoundedDistributiveLattice]. ∀[f:X ⟶ Point(L)]. (fdl-hom(L;f) ∈ Hom(free-dl(X);L))
Proof
Definitions occuring in Statement :
fdl-hom: fdl-hom(L;f),
free-dl: free-dl(X),
bdd-distributive-lattice: BoundedDistributiveLattice,
bounded-lattice-hom: Hom(l1;l2),
lattice-point: Point(l),
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
fdl-hom: fdl-hom(L;f),
list_accum: list_accum,
lattice-0: 0,
record-select: r.x,
free-dl: free-dl(X),
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,
nil: [],
it: ⋅,
lattice-1: 1,
cons: [a / b],
lattice-join: a ∨ b,
all: ∀x:A. B[x],
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
free-dl-join: free-dl-join(as;bs),
append: as @ bs,
list_ind: list_ind,
implies: P ⇒ Q,
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
guard: {T},
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
free-dl-type: free-dl-type(X),
cand: A c∧ B,
dlattice-eq: dlattice-eq(X;as;bs),
quotient: x,y:A//B[x; y],
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
dlattice-order: as ⇒ bs,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
select: L[n],
lattice-le: a ≤ b,
l_contains: A ⊆ B,
nat_plus: ℕ+,
uiff: uiff(P;Q),
istype: istype(T),
order: Order(T;x,y.R[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y]),
lattice-point: Point(l),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
sym: Sym(T;x,y.E[x; y]),
lattice-meet: a ∧ b,
free-dl-meet: free-dl-meet(as;bs),
map: map(f;as),
listp: A List+,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
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,
lattice-meet_wf,
lattice-join_wf,
bdd-distributive-lattice_wf,
istype-universe,
lattice-0_wf,
lattice-join-0,
bdd-distributive-lattice-subtype-bdd-lattice,
lattice-1_wf,
list_accum_nil_lemma,
istype-void,
list_wf,
list_accum_append,
subtype_rel_list,
top_wf,
fdl-hom_wf1,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
list_accum_cons_lemma,
istype-nat,
unit_wf2,
unit_subtype_base,
it_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
cons_wf,
list_accum_wf,
lattice_properties,
bdd-distributive-lattice-subtype-lattice,
dlattice-eq-equiv,
dlattice-eq_wf,
member_wf,
free-dl-type_wf,
last_induction,
all_wf,
dlattice-order_wf,
lattice-le_wf,
nil_wf,
append_wf,
lattice-0-le,
lattice-join-le,
l_all_append,
l_exists_wf,
l_member_wf,
l_contains_wf,
length_of_cons_lemma,
length_of_nil_lemma,
l_exists_iff,
lattice-le_transitivity,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
cons_member,
lattice-le_weakening,
le-lattice-1,
l_all_cons,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
false_wf,
length_wf,
lattice-meet-le,
lattice-le-meet,
lattice-le-order,
free-dl_wf,
quotient-member-eq,
subtype_quotient,
free-dl-meet_wf,
lattice-meet-0,
cons_wf_listp,
less_than_wf,
distributive-lattice-distrib,
bdd-distributive-lattice-subtype-distributive-lattice,
free-dl-meet_wf_list,
list_ind_nil_lemma,
map_nil_lemma,
lattice-0-meet,
map_cons_lemma,
map_wf,
lattice-meet-1,
lattice-1-meet
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsType,
universeIsType,
hypothesisEquality,
extract_by_obid,
isectElimination,
thin,
applyEquality,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
inhabitedIsType,
because_Cache,
independent_isectElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
universeEquality,
independent_pairFormation,
setElimination,
rename,
productElimination,
dependent_functionElimination,
voidElimination,
lambdaFormation_alt,
equalityIsType1,
independent_functionElimination,
intWeakElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
functionIsTypeImplies,
unionElimination,
promote_hyp,
hypothesis_subsumption,
dependent_set_memberEquality_alt,
applyLambdaEquality,
imageElimination,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
intEquality,
imageMemberEquality,
functionExtensionality,
pointwiseFunctionality,
pertypeElimination,
productIsType,
functionEquality,
setIsType,
hyp_replacement,
addEquality,
independent_pairEquality,
isectIsType
Latex:
\mforall{}[X:Type]. \mforall{}[L:BoundedDistributiveLattice]. \mforall{}[f:X {}\mrightarrow{} Point(L)]. (fdl-hom(L;f) \mmember{} Hom(free-dl(X);L))
Date html generated:
2019_10_31-AM-07_20_49
Last ObjectModification:
2018_11_13-AM-10_25_36
Theory : lattices
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