Nuprl Lemma : dM-subobject
∀[I,J:fset(ℕ)]. λv.v ∈ dma-hom(dM(I);dM(J)) supposing I ⊆ J
Proof
Definitions occuring in Statement :
dM: dM(I),
dma-hom: dma-hom(dma1;dma2),
f-subset: xs ⊆ ys,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
lambda: λx.A[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
dma-hom: dma-hom(dma1;dma2),
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
prop: ℙ,
nat: ℕ,
and: P ∧ Q,
cand: A c∧ B,
dM: dM(I),
lattice-meet: a ∧ b,
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
all: ∀x:A. B[x],
top: Top,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
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,
union-deq: union-deq(A;B;a;b),
lattice-join: a ∨ b,
lattice-0: 0,
record-select: r.x,
record-update: r[x := v],
empty-fset: {},
nil: [],
it: ⋅,
lattice-1: 1,
fset-singleton: {x},
cons: [a / b],
dma-neg: ¬(x),
dm-neg: ¬(x),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-meet: /\(s),
lattice-fset-join: \/(s),
opposite-lattice: opposite-lattice(L),
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
lattice-point_wf,
dM_wf,
subtype_rel_set,
lattice-structure_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
uall_wf,
equal_wf,
dma-neg_wf,
DeMorgan-algebra_wf,
f-subset_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
fset_wf,
dM-point-subtype,
rec_select_update_lemma,
lattice-meet_wf,
lattice-join_wf,
lattice-0_wf,
DeMorgan-algebra-structure_wf,
bounded-lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-axioms_wf,
DeMorgan-algebra-axioms_wf,
lattice-1_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
sqequalRule,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
instantiate,
because_Cache,
independent_isectElimination,
lambdaEquality,
setElimination,
rename,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
natural_numberEquality,
isect_memberEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
productElimination,
independent_pairEquality,
productEquality,
functionExtensionality,
cumulativity,
universeEquality
Latex:
\mforall{}[I,J:fset(\mBbbN{})]. \mlambda{}v.v \mmember{} dma-hom(dM(I);dM(J)) supposing I \msubseteq{} J
Date html generated:
2017_10_05-AM-01_00_48
Last ObjectModification:
2017_07_28-AM-09_25_53
Theory : cubical!type!theory
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