Nuprl Lemma : dma-lift-compose-assoc
∀[I,J,K,H:Type]. ∀[eqi:EqDecider(I)]. ∀[eqj:EqDecider(J)]. ∀[eqk:EqDecider(K)].
∀[f:J ⟶ Point(free-DeMorgan-algebra(I;eqi))]. ∀[g:K ⟶ Point(free-DeMorgan-algebra(J;eqj))].
∀[h:H ⟶ Point(free-DeMorgan-algebra(K;eqk))].
(dma-lift-compose(I;K;eqi;eqk;dma-lift-compose(I;J;eqi;eqj;f;g);h)
= dma-lift-compose(I;J;eqi;eqj;f;dma-lift-compose(J;K;eqj;eqk;g;h))
∈ (H ⟶ 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],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
compose: f o g,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
top: Top,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
implies: P ⇒ Q,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
lattice-point_wf,
free-DeMorgan-algebra_wf,
deq_wf,
free-dma-point,
free-dml-deq_wf,
dminc_wf,
free-dma-lift-unique2,
compose_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,
free-dma-lift_wf,
dma-hom_wf,
all_wf,
set_wf,
compose-dma-hom,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionExtensionality,
sqequalRule,
hypothesisEquality,
hypothesis,
functionEquality,
cumulativity,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
because_Cache,
isect_memberEquality,
axiomEquality,
lambdaFormation,
voidElimination,
voidEquality,
universeEquality,
dependent_functionElimination,
instantiate,
lambdaEquality,
productEquality,
independent_isectElimination,
setElimination,
rename,
setEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}[I,J,K,H:Type]. \mforall{}[eqi:EqDecider(I)]. \mforall{}[eqj:EqDecider(J)]. \mforall{}[eqk:EqDecider(K)].
\mforall{}[f:J {}\mrightarrow{} Point(free-DeMorgan-algebra(I;eqi))]. \mforall{}[g:K {}\mrightarrow{} Point(free-DeMorgan-algebra(J;eqj))].
\mforall{}[h:H {}\mrightarrow{} Point(free-DeMorgan-algebra(K;eqk))].
(dma-lift-compose(I;K;eqi;eqk;dma-lift-compose(I;J;eqi;eqj;f;g);h)
= dma-lift-compose(I;J;eqi;eqj;f;dma-lift-compose(J;K;eqj;eqk;g;h)))
Date html generated:
2017_10_05-AM-00_43_13
Last ObjectModification:
2017_07_28-AM-09_17_52
Theory : lattices
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