Nuprl Lemma : free-DeMorgan-algebra-hom-unique
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[dm:DeMorganAlgebra]. ∀[eq2:EqDecider(Point(dm))].
∀f:T ⟶ Point(dm)
∀[g,h:dma-hom(free-DeMorgan-algebra(T;eq);dm)].
g = h ∈ dma-hom(free-DeMorgan-algebra(T;eq);dm) supposing ∀i:T. ((g <i>) = (h <i>) ∈ Point(dm))
Proof
Definitions occuring in Statement :
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),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
btrue: tt,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
free-dist-lattice: free-dist-lattice(T; eq),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
bfalse: ff,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
record-update: r[x := v],
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
record-select: r.x,
lattice-point: Point(l),
lattice-hom: Hom(l1;l2),
bounded-lattice-hom: Hom(l1;l2),
dma-hom: dma-hom(dma1;dma2),
so_apply: x[s],
guard: {T},
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
DeMorgan-algebra: DeMorganAlgebra,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
dmopp: <1-i>,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
squash: ↓T,
dminc: <i>,
top: Top,
cons: [a / b],
fset-singleton: {x},
lattice-1: 1,
it: ⋅,
nil: [],
empty-fset: {},
lattice-0: 0,
l-union: as ⋃ bs,
fset-union: x ⋃ y,
fset-ac-lub: fset-ac-lub(eq;ac1;ac2),
lattice-join: a ∨ b,
list_accum: list_accum,
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_ind: list_ind,
reduce: reduce(f;k;as),
filter: filter(P;l),
fset-filter: {x ∈ s | P[x]},
fset-minimals: fset-minimals(x,y.less[x; y]; s),
fset-ac-glb: fset-ac-glb(eq;ac1;ac2),
lattice-meet: a ∧ b
Lemmas referenced :
istype-universe,
DeMorgan-algebra_wf,
deq_wf,
free-DeMorgan-algebra_wf,
dma-hom_wf,
dminc_wf,
DeMorgan-algebra-axioms_wf,
lattice-join_wf,
lattice-meet_wf,
equal_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure_wf,
subtype_rel_transitivity,
DeMorgan-algebra-structure-subtype,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
DeMorgan-algebra-structure_wf,
subtype_rel_set,
lattice-point_wf,
DeMorgan-algebra-subtype,
union-deq_wf,
free-dist-lattice-hom-unique2,
free-dma-hom-is-lattice-hom,
iff_weakening_equal,
subtype_rel_self,
trivial-equal,
true_wf,
squash_wf,
dma-neg_wf,
free-dma-point-subtype,
dm-neg-inc,
free-dma-neg,
istype-void
Rules used in proof :
universeEquality,
functionIsTypeImplies,
dependent_functionElimination,
inhabitedIsType,
isectIsTypeImplies,
axiomEquality,
isect_memberEquality_alt,
rename,
setElimination,
because_Cache,
isectEquality,
independent_isectElimination,
cumulativity,
productEquality,
lambdaEquality_alt,
instantiate,
equalityIstype,
universeIsType,
functionIsType,
sqequalRule,
applyEquality,
hypothesis,
hypothesisEquality,
unionEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaFormation_alt,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
hyp_replacement,
equalityTransitivity,
equalitySymmetry,
unionIsType,
independent_functionElimination,
productElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
unionElimination,
voidElimination,
dependent_set_memberEquality_alt,
isectIsType
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[dm:DeMorganAlgebra]. \mforall{}[eq2:EqDecider(Point(dm))].
\mforall{}f:T {}\mrightarrow{} Point(dm)
\mforall{}[g,h:dma-hom(free-DeMorgan-algebra(T;eq);dm)]. g = h supposing \mforall{}i:T. ((g <i>) = (h <i>))
Date html generated:
2020_05_20-AM-08_57_08
Last ObjectModification:
2020_02_05-AM-08_06_34
Theory : lattices
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