Nuprl Lemma : flattice-equiv-equiv
∀[X:Type]. EquivRel(Point(free-dl(X + X));x,y.flattice-equiv(X;x;y))
Proof
Definitions occuring in Statement :
flattice-equiv: flattice-equiv(X;x;y),
free-dl: free-dl(X),
lattice-point: Point(l),
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
lattice-point: Point(l),
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,
free-dl-type: free-dl-type(X),
quotient: x,y:A//B[x; y],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
cand: A c∧ B,
sym: Sym(T;x,y.E[x; y]),
implies: P ⇒ Q,
flattice-equiv: flattice-equiv(X;x;y),
squash: ↓T,
trans: Trans(T;x,y.E[x; y]),
guard: {T},
exists: ∃x:A. B[x],
flattice-order: flattice-order(X;as;bs),
l_all: (∀x∈L.P[x]),
or: P ∨ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
l_exists: (∃x∈L. P[x]),
dlattice-eq: dlattice-eq(X;as;bs),
dlattice-order: as ⇒ bs
Lemmas referenced :
subtype_quotient,
list_wf,
dlattice-eq_wf,
dlattice-eq-equiv,
lattice-point_wf,
free-dl_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,
flattice-equiv_wf,
equal-wf-base,
member_wf,
flattice-order_wf,
exists_wf,
l_exists_wf,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
l_member_wf,
flip-union_wf,
int_seg_wf,
l_contains_weakening,
l_contains_wf,
flattice-order_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
unionEquality,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
because_Cache,
applyEquality,
instantiate,
productEquality,
universeEquality,
imageElimination,
imageMemberEquality,
baseClosed,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
inrFormation,
setElimination,
rename,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
setEquality,
independent_functionElimination
Latex:
\mforall{}[X:Type]. EquivRel(Point(free-dl(X + X));x,y.flattice-equiv(X;x;y))
Date html generated:
2020_05_20-AM-08_59_49
Last ObjectModification:
2017_07_28-AM-09_18_26
Theory : lattices
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