Nuprl Lemma : free-dma-lift_wf
∀T:Type. ∀eq:EqDecider(T). ∀dm:DeMorganAlgebra. ∀eq2:EqDecider(Point(dm)). ∀f:T ⟶ Point(dm).
(free-dma-lift(T;eq;dm;eq2;f) ∈ {g:dma-hom(free-DeMorgan-algebra(T;eq);dm)| ∀i:T. ((g <i>) = (f i) ∈ Point(dm))} )
Proof
Definitions occuring in Statement :
free-dma-lift: free-dma-lift(T;eq;dm;eq2;f),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
dma-hom: dma-hom(dma1;dma2),
DeMorgan-algebra: DeMorganAlgebra,
dminc: <i>,
lattice-point: Point(l),
deq: EqDecider(T),
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
free-dma-lift: free-dma-lift(T;eq;dm;eq2;f),
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
free-DeMorgan-algebra-property,
free-dist-lattice-property,
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
lattice-point: Point(l),
record-select: r.x,
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
btrue: tt,
sq_exists: ∃x:A [B[x]],
implies: P ⇒ Q
Lemmas referenced :
lattice-point_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
deq_wf,
DeMorgan-algebra_wf,
istype-universe,
free-DeMorgan-algebra-property,
subtype_rel_self,
all_wf,
sq_exists_wf,
dma-hom_wf,
free-DeMorgan-algebra_wf,
dminc_wf,
subtype_rel_function,
free-dist-lattice-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
hypothesis,
functionIsType,
universeIsType,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
thin,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality_alt,
productEquality,
independent_isectElimination,
cumulativity,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
because_Cache,
universeEquality,
functionEquality,
dependent_functionElimination,
setElimination,
rename,
equalityIsType1,
independent_functionElimination,
functionExtensionality
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}dm:DeMorganAlgebra. \mforall{}eq2:EqDecider(Point(dm)). \mforall{}f:T {}\mrightarrow{} Point(dm).
(free-dma-lift(T;eq;dm;eq2;f) \mmember{} \{g:dma-hom(free-DeMorgan-algebra(T;eq);dm)|
\mforall{}i:T. ((g <i>) = (f i))\} )
Date html generated:
2020_05_20-AM-08_56_53
Last ObjectModification:
2018_11_08-PM-06_00_25
Theory : lattices
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