Nuprl Lemma : lattice-extend-wc_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[Cs:T ⟶ fset(fset(T))]. ∀[L:BoundedDistributiveLattice]. ∀[eqL:EqDecider(Point(L))].
∀[f:T ⟶ Point(L)]. ∀[ac:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))].
(lattice-extend-wc(L;eq;eqL;f;ac) ∈ Point(L))
Proof
Definitions occuring in Statement :
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-point: Point(l),
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
bdd-distributive-lattice: BoundedDistributiveLattice,
prop: ℙ,
and: P ∧ Q,
uimplies: b supposing a
Lemmas referenced :
lattice-extend_wf,
free-dlwc-point-subtype,
lattice-point_wf,
free-dist-lattice-with-constraints_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,
deq_wf,
bdd-distributive-lattice_wf,
fset_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
lambdaEquality,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity,
because_Cache,
instantiate,
productEquality,
universeEquality,
independent_isectElimination,
isect_memberEquality,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[Cs:T {}\mrightarrow{} fset(fset(T))]. \mforall{}[L:BoundedDistributiveLattice].
\mforall{}[eqL:EqDecider(Point(L))]. \mforall{}[f:T {}\mrightarrow{} Point(L)].
\mforall{}[ac:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))].
(lattice-extend-wc(L;eq;eqL;f;ac) \mmember{} Point(L))
Date html generated:
2020_05_20-AM-08_48_56
Last ObjectModification:
2015_12_28-PM-01_58_51
Theory : lattices
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