Nuprl Lemma : lattice-hom-fset-join
∀[l1,l2:BoundedLattice]. ∀[eq1:EqDecider(Point(l1))]. ∀[eq2:EqDecider(Point(l2))]. ∀[f:Hom(l1;l2)].
∀[s:fset(Point(l1))].
((f \/(s)) = \/(f"(s)) ∈ Point(l2))
Proof
Definitions occuring in Statement :
lattice-fset-join: \/(s),
bounded-lattice-hom: Hom(l1;l2),
bdd-lattice: BoundedLattice,
lattice-point: Point(l),
fset-image: f"(s),
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fset: fset(T),
subtype_rel: A ⊆r B,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
bdd-lattice: BoundedLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
or: P ∨ Q,
cons: [a / b],
decidable: Dec(P),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
empty-fset: {},
lattice-fset-join: \/(s),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
fset-add: fset-add(eq;x;s),
fset-union: x ⋃ y,
l-union: as ⋃ bs,
reduce: reduce(f;k;as),
list_ind: list_ind
Lemmas referenced :
lattice-point_wf,
list_wf,
set-equal_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
fset_wf,
bounded-lattice-hom_wf,
deq_wf,
bdd-lattice_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
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,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
le_wf,
istype-nat,
fset-image-empty,
reduce_nil_lemma,
reduce_cons_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
lattice-hom-join,
lattice-fset-join_wf,
decidable-equal-deq,
list_subtype_fset,
lattice-join_wf,
subtype_rel_self,
iff_weakening_equal,
fset-image_wf,
fset-add-as-cons,
fset-union_wf,
fset-singleton_wf,
decidable_wf,
fset-image-union,
lattice-fset-join-union,
fset-image-singleton,
lattice-fset-join-singleton,
quotient-member-eq,
set-equal-equiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
pertypeElimination,
promote_hyp,
productElimination,
productIsType,
equalityIstype,
universeIsType,
sqequalBase,
equalitySymmetry,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
independent_isectElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
setElimination,
rename,
independent_pairFormation,
equalityTransitivity,
lambdaFormation_alt,
dependent_functionElimination,
independent_functionElimination,
intWeakElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
functionIsTypeImplies,
unionElimination,
hypothesis_subsumption,
dependent_set_memberEquality_alt,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
intEquality,
voidEquality,
isect_memberEquality,
universeEquality,
imageMemberEquality,
hyp_replacement,
functionEquality,
functionIsType
Latex:
\mforall{}[l1,l2:BoundedLattice]. \mforall{}[eq1:EqDecider(Point(l1))]. \mforall{}[eq2:EqDecider(Point(l2))]. \mforall{}[f:Hom(l1;l2)].
\mforall{}[s:fset(Point(l1))].
((f \mbackslash{}/(s)) = \mbackslash{}/(f"(s)))
Date html generated:
2020_05_20-AM-08_44_42
Last ObjectModification:
2018_12_13-PM-02_29_47
Theory : lattices
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