Nuprl Lemma : dma-lift-compose_wf
∀[I,J,K:Type]. ∀[eqi:EqDecider(I)]. ∀[eqj:EqDecider(J)]. ∀[f:J ⟶ Point(free-DeMorgan-algebra(I;eqi))].
∀[g:K ⟶ Point(free-DeMorgan-algebra(J;eqj))].
(dma-lift-compose(I;J;eqi;eqj;f;g) ∈ K ⟶ Point(free-DeMorgan-algebra(I;eqi)))
Proof
Definitions occuring in Statement :
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
lattice-point: Point(l),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
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],
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
rev_implies: P ⇐ Q,
assert: ↑b,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
bool: 𝔹,
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),
deq: EqDecider(T),
all: ∀x:A. B[x],
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2)
Lemmas referenced :
lattice-point_wf,
free-DeMorgan-algebra_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,
istype-universe,
free-dml-deq_wf,
free-dma-lift_wf,
compose_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsType,
universeIsType,
hypothesisEquality,
extract_by_obid,
isectElimination,
thin,
applyEquality,
instantiate,
lambdaEquality_alt,
productEquality,
independent_isectElimination,
cumulativity,
inhabitedIsType,
because_Cache,
isect_memberEquality_alt,
isectIsTypeImplies,
universeEquality,
functionExtensionality,
dependent_functionElimination,
setElimination,
rename,
lambdaEquality
Latex:
\mforall{}[I,J,K:Type]. \mforall{}[eqi:EqDecider(I)]. \mforall{}[eqj:EqDecider(J)].
\mforall{}[f:J {}\mrightarrow{} Point(free-DeMorgan-algebra(I;eqi))]. \mforall{}[g:K {}\mrightarrow{} Point(free-DeMorgan-algebra(J;eqj))].
(dma-lift-compose(I;J;eqi;eqj;f;g) \mmember{} K {}\mrightarrow{} Point(free-DeMorgan-algebra(I;eqi)))
Date html generated:
2019_10_31-AM-07_23_06
Last ObjectModification:
2018_11_08-PM-05_59_51
Theory : lattices
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