Nuprl Lemma : lattice-extend-wc-meet
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[Cs:T ⟶ fset(fset(T))]. ∀[L:BoundedDistributiveLattice]. ∀[eqL:EqDecider(Point(L))].
∀[f:T ⟶ Point(L)].
∀[a,b:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))].
lattice-extend-wc(L;eq;eqL;f;a) ∧ lattice-extend-wc(L;eq;eqL;f;b) ≤ lattice-extend-wc(L;eq;eqL;f;a ∧ b)
supposing ∀x:T. ∀c:fset(T). (c ∈ Cs[x] ⇒ (/\(f"(c)) = 0 ∈ 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]),
lattice-fset-meet: /\(s),
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-0: 0,
lattice-le: a ≤ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
fset-image: f"(s),
deq-fset: deq-fset(eq),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
prop: ℙ,
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-le: a ≤ b,
subtype_rel: A ⊆r B,
lattice-extend': lattice-extend'(L;eq;eqL;f;ac),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
so_apply: x[s],
so_lambda: λ2x.t[x],
top: Top,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
fset-constrained-ac-glb: glb(P;ac1;ac2),
guard: {T},
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
squash: ↓T,
list_accum: list_accum,
f-union: f-union(domeq;rngeq;s;x.g[x]),
fset-image: f"(s),
list_ind: list_ind,
reduce: reduce(f;k;as),
lattice-fset-meet: /\(s),
cand: A c∧ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
bdd-lattice: BoundedLattice,
true: True,
compose: f o g,
f-subset: xs ⊆ ys,
decidable: Dec(P),
or: P ∨ Q,
fset-union: x ⋃ y,
l-union: as ⋃ bs,
not: ¬A,
false: False,
bounded-lattice-axioms: bounded-lattice-axioms(l)
Lemmas referenced :
bdd-distributive-lattice_wf,
deq_wf,
lattice-0_wf,
fset-image_wf,
decidable-equal-deq,
bdd-distributive-lattice-subtype-bdd-lattice,
lattice-fset-meet_wf,
fset-member_wf,
all_wf,
lattice-join_wf,
equal_wf,
uall_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
free-dist-lattice-with-constraints_wf,
lattice-point_wf,
fset-constrained-ac-glb_wf,
fset-contains-none_wf,
fset-union_wf,
fset-constrained-image_wf,
deq-fset_wf,
fset_wf,
f-union_wf,
lattice-extend'_wf,
lattice-meet_wf,
bdd-distributive-lattice-subtype-lattice,
lattice-le_transitivity,
free-dlwc-point,
free-dlwc-meet,
f-proper-subset-dec_wf,
fset-minimals_wf,
fset-minimals-ac-le,
fset-ac-le_wf,
fset-ac-le-implies2,
member-fset-image-iff,
lattice-fset-join_wf,
lattice-fset-join-is-lub,
lattice-le_wf,
sq_stable_from_decidable,
fset-image_functionality_wrt_subset,
lattice-fset-meet_functionality_wrt_subset,
iff_weakening_equal,
bdd-lattice_wf,
decidable_wf,
true_wf,
squash_wf,
fset-image-compose,
lattice-meet-join-images-distrib,
lattice-fset-join_functionality_wrt_subset2,
fset-member_witness,
fset-singleton_wf,
sq_stable__fset-member,
member-f-union,
istype-universe,
fset-image-union,
subtype_rel_self,
member-fset-union,
decidable__assert,
member-fset-constrained-image-iff,
member-fset-singleton,
iff_weakening_uiff,
assert_wf,
rev_implies_wf,
not_wf,
f-subset_wf,
assert-fset-contains-none,
istype-void,
deq-implies,
lattice-le-iff
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
lambdaFormation,
independent_functionElimination,
functionEquality,
universeEquality,
productEquality,
instantiate,
cumulativity,
axiomEquality,
independent_isectElimination,
lambdaEquality,
rename,
setElimination,
functionExtensionality,
applyEquality,
hypothesisEquality,
because_Cache,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
baseClosed,
imageMemberEquality,
applyLambdaEquality,
hyp_replacement,
imageElimination,
independent_pairFormation,
dependent_pairFormation,
natural_numberEquality,
lambdaEquality_alt,
universeIsType,
inhabitedIsType,
isectEquality,
lambdaFormation_alt,
isect_memberFormation_alt,
unionElimination,
inrFormation_alt,
dependent_pairFormation_alt,
productIsType,
equalityIstype,
inlFormation_alt,
functionIsType
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{}[a,b:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))].
lattice-extend-wc(L;eq;eqL;f;a) \mwedge{} lattice-extend-wc(L;eq;eqL;f;b)
\mleq{} lattice-extend-wc(L;eq;eqL;f;a \mwedge{} b)
supposing \mforall{}x:T. \mforall{}c:fset(T). (c \mmember{} Cs[x] {}\mRightarrow{} (/\mbackslash{}(f"(c)) = 0))
Date html generated:
2020_05_20-AM-08_50_20
Last ObjectModification:
2020_02_03-PM-03_27_53
Theory : lattices
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