Nuprl Lemma : formal-sum-subtype
∀[K:RngSig]. ∀[S,T:Type]. formal-sum(K;S) ⊆r formal-sum(K;T) supposing S ⊆r T
Proof
Definitions occuring in Statement :
formal-sum: formal-sum(K;S),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
universe: Type,
rng_sig: RngSig
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
formal-sum: formal-sum(K;S),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
guard: {T},
implies: P ⇒ Q,
prop: ℙ,
bfs-reduce: bfs-reduce(K;S;as;bs),
or: P ∨ Q,
exists: ∃x:A. B[x],
basic-formal-sum: basic-formal-sum(K;S),
so_lambda: λ2x.t[x],
so_apply: x[s],
infix_ap: x f y,
cand: A c∧ B,
bfs-equiv: bfs-equiv(K;S;fs1;fs2)
Lemmas referenced :
formal-sum_wf,
quotient-member-eq,
basic-formal-sum_wf,
bfs-equiv_wf,
bfs-equiv-rel,
basic-formal-sum-subtype,
subtype_rel_wf,
bfs-equiv-implies,
bfs-reduce_wf,
implies-bfs-equiv,
subtype_rel_bag,
respects-equality-bag,
rng_car_wf,
respects-equality-product,
respects-equality-trivial,
subtype-respects-equality,
istype-base,
change-equality-type,
bag-append_wf,
subtype_rel_product,
zero-bfs_wf,
subtype_rel_self,
bag_wf,
formal-sum-mul_wf1,
rng_plus_wf,
least-equiv-is-equiv-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaEquality_alt,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
pertypeElimination,
promote_hyp,
productElimination,
inhabitedIsType,
universeIsType,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
independent_functionElimination,
lambdaFormation_alt,
equalityIstype,
productIsType,
sqequalBase,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
unionElimination,
inlFormation_alt,
dependent_pairFormation_alt,
productEquality,
inrFormation_alt,
independent_pairFormation
Latex:
\mforall{}[K:RngSig]. \mforall{}[S,T:Type]. formal-sum(K;S) \msubseteq{}r formal-sum(K;T) supposing S \msubseteq{}r T
Date html generated:
2019_10_31-AM-06_28_46
Last ObjectModification:
2019_08_22-AM-10_55_01
Theory : linear!algebra
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