Nuprl Lemma : lattice-extend-wc-order-preserving
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[Cs:T ⟶ fset(fset(T))]. ∀[L:BoundedDistributiveLattice]. ∀[eqL:EqDecider(Point(L))].
∀[f:T ⟶ Point(L)]. ∀[x,y:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))].
lattice-extend-wc(L;eq;eqL;f;x) ≤ lattice-extend-wc(L;eq;eqL;f;y) supposing x ≤ y
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-le: a ≤ b,
lattice-point: Point(l),
fset: fset(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
lattice-le: a ≤ b,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
prop: ℙ,
and: P ∧ Q,
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
lattice-le_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,
lattice-point_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
deq_wf,
bdd-distributive-lattice_wf,
fset_wf,
lattice-extend-order-preserving,
free-dlwc-point-subtype,
free-dlwc-le,
free-dl-le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
axiomEquality,
hypothesis,
lemma_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
lambdaEquality,
applyEquality,
because_Cache,
instantiate,
productEquality,
universeEquality,
independent_isectElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination
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{}[x,y:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))].
lattice-extend-wc(L;eq;eqL;f;x) \mleq{} lattice-extend-wc(L;eq;eqL;f;y) supposing x \mleq{} y
Date html generated:
2020_05_20-AM-08_50_02
Last ObjectModification:
2015_12_28-PM-01_58_34
Theory : lattices
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