Nuprl Lemma : bfs-reduce-subtype1
∀[K:RngSig]. ∀[S,T:Type].
∀[as,bs:basic-formal-sum(K;S)]. (bfs-reduce(K;S;as;bs) ⇒ bfs-reduce(K;T;as;bs)) supposing S ⊆r T
Proof
Definitions occuring in Statement :
bfs-reduce: bfs-reduce(K;S;as;bs),
basic-formal-sum: basic-formal-sum(K;S),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
universe: Type,
rng_sig: RngSig
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
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],
all: ∀x:A. B[x],
and: P ∧ Q,
infix_ap: x f y,
cand: A c∧ B,
prop: ℙ
Lemmas referenced :
subtype_rel_bag,
basic-formal-sum-subtype,
respects-equality-bag,
rng_car_wf,
respects-equality-product,
respects-equality-trivial,
subtype-respects-equality,
istype-base,
change-equality-type,
basic-formal-sum_wf,
bag-append_wf,
subtype_rel_product,
zero-bfs_wf,
subtype_rel_self,
bag_wf,
formal-sum-mul_wf1,
rng_plus_wf,
bfs-reduce_wf,
subtype_rel_wf,
istype-universe,
rng_sig_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
axiomEquality,
hypothesis,
thin,
rename,
lambdaFormation_alt,
sqequalHypSubstitution,
unionElimination,
inlFormation_alt,
productElimination,
dependent_pairFormation_alt,
hypothesisEquality,
applyEquality,
extract_by_obid,
isectElimination,
independent_isectElimination,
productEquality,
because_Cache,
lambdaEquality_alt,
universeIsType,
independent_functionElimination,
inhabitedIsType,
equalityIstype,
sqequalBase,
equalitySymmetry,
dependent_functionElimination,
equalityTransitivity,
productIsType,
inrFormation_alt,
independent_pairFormation,
promote_hyp,
instantiate,
universeEquality
Latex:
\mforall{}[K:RngSig]. \mforall{}[S,T:Type].
\mforall{}[as,bs:basic-formal-sum(K;S)]. (bfs-reduce(K;S;as;bs) {}\mRightarrow{} bfs-reduce(K;T;as;bs))
supposing S \msubseteq{}r T
Date html generated:
2019_10_31-AM-06_28_30
Last ObjectModification:
2019_08_15-PM-02_20_27
Theory : linear!algebra
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