Nuprl Lemma : mk-general-bounded-lattice_wf
∀[T:Type]. ∀[m,j:T ⟶ T ⟶ T]. ∀[z,o:T]. ∀[E:T ⟶ T ⟶ ℙ].
mk-general-bounded-lattice(T;m;j;z;o;E) ∈ GeneralBoundedLattice
supposing EquivRel(T;x,y.E x y)
∧ (∀[a,b:T]. (E m[a;b] m[b;a]))
∧ (∀[a,b:T]. (E j[a;b] j[b;a]))
∧ (∀[a,b,c:T]. (E m[a;m[b;c]] m[m[a;b];c]))
∧ (∀[a,b,c:T]. (E j[a;j[b;c]] j[j[a;b];c]))
∧ (∀[a,b:T]. (E j[a;m[a;b]] a))
∧ (∀[a,b:T]. (E m[a;j[a;b]] a))
∧ (∀[a:T]. (E m[a;o] a))
∧ (∀[a:T]. (E j[a;z] a))
Proof
Definitions occuring in Statement :
mk-general-bounded-lattice: mk-general-bounded-lattice(T;m;j;z;o;E),
general-bounded-lattice: GeneralBoundedLattice,
equiv_rel: EquivRel(T;x,y.E[x; y]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
and: P ∧ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
mk-general-bounded-lattice: mk-general-bounded-lattice(T;m;j;z;o;E),
general-bounded-lattice: GeneralBoundedLattice,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
general-bounded-lattice-structure: GeneralBoundedLatticeStructure,
record+: record+,
record-update: r[x := v],
record: record(x.T[x]),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
guard: {T},
record-select: r.x,
top: Top,
eq_atom: x =a y,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
general-lattice-axioms: general-lattice-axioms(l),
lattice-meet: a ∧ b,
lattice-point: Point(l),
lattice-join: a ∨ b,
lattice-1: 1,
lattice-equiv: a ≡ b,
lattice-0: 0,
cand: A c∧ B
Lemmas referenced :
general-lattice-axioms_wf,
equiv_rel_wf,
uall_wf,
eq_atom_wf,
uiff_transitivity,
equal-wf-base,
bool_wf,
assert_wf,
atom_subtype_base,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
rec_select_update_lemma,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
dependent_set_memberEquality,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
cumulativity,
lambdaEquality,
applyEquality,
functionExtensionality,
because_Cache,
isect_memberEquality,
functionEquality,
universeEquality,
dependentIntersection_memberEquality,
tokenEquality,
lambdaFormation,
unionElimination,
equalityElimination,
baseApply,
closedConclusion,
baseClosed,
atomEquality,
independent_functionElimination,
independent_isectElimination,
instantiate,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[m,j:T {}\mrightarrow{} T {}\mrightarrow{} T]. \mforall{}[z,o:T]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
mk-general-bounded-lattice(T;m;j;z;o;E) \mmember{} GeneralBoundedLattice
supposing EquivRel(T;x,y.E x y)
\mwedge{} (\mforall{}[a,b:T]. (E m[a;b] m[b;a]))
\mwedge{} (\mforall{}[a,b:T]. (E j[a;b] j[b;a]))
\mwedge{} (\mforall{}[a,b,c:T]. (E m[a;m[b;c]] m[m[a;b];c]))
\mwedge{} (\mforall{}[a,b,c:T]. (E j[a;j[b;c]] j[j[a;b];c]))
\mwedge{} (\mforall{}[a,b:T]. (E j[a;m[a;b]] a))
\mwedge{} (\mforall{}[a,b:T]. (E m[a;j[a;b]] a))
\mwedge{} (\mforall{}[a:T]. (E m[a;o] a))
\mwedge{} (\mforall{}[a:T]. (E j[a;z] a))
Date html generated:
2020_05_20-AM-08_58_23
Last ObjectModification:
2017_07_28-AM-09_17_59
Theory : lattices
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