Nuprl Lemma : bfs-reduce-strong-subtype
∀[K:RngSig]. ∀[S,T:Type].
∀[as,bs:basic-formal-sum(K;S)]. (bfs-reduce(K;T;as;bs) ⇒ bfs-reduce(K;S;as;bs)) supposing strong-subtype(S;T)
Proof
Definitions occuring in Statement :
bfs-reduce: bfs-reduce(K;S;as;bs),
basic-formal-sum: basic-formal-sum(K;S),
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
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,
implies: P ⇒ Q,
guard: {T},
bfs-reduce: bfs-reduce(K;S;as;bs),
or: P ∨ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
basic-formal-sum: basic-formal-sum(K;S),
infix_ap: x f y,
subtype_rel: A ⊆r B,
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_or: a ↓∨ b,
squash: ↓T,
bag-member: x ↓∈ bs,
zero-bfs: 0 * ss,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
pi2: snd(t),
top: Top,
pi1: fst(t),
formal-sum-add: x + y,
label: ...$L... t,
formal-sum-mul: k * x
Lemmas referenced :
strong-subtype_witness,
basic-formal-sum-strong-subtype,
strong-subtype-implies,
basic-formal-sum_wf,
rng_car_wf,
bag-append_wf,
formal-sum-mul_wf1,
rng_plus_wf,
subtype_rel_self,
bag_wf,
zero-bfs_wf,
bfs-reduce_wf,
basic-formal-sum-subtype,
strong-subtype_wf,
istype-universe,
rng_sig_wf,
bag-member-append,
bag-member_wf,
bag-in-subtype2,
strong-subtype-product,
strong-subtype-self,
bag-in-subtype,
bag-member-map,
rng_zero_wf,
bag-member-strong-subtype,
pi2_wf,
pi1_wf_top,
istype-void,
formal-sum-add_wf1,
infix_ap_wf,
rng_times_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
lambdaFormation_alt,
because_Cache,
independent_isectElimination,
unionElimination,
inlFormation_alt,
productElimination,
sqequalRule,
productIsType,
universeIsType,
equalityIstype,
productEquality,
applyEquality,
inrFormation_alt,
inhabitedIsType,
instantiate,
universeEquality,
promote_hyp,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
dependent_pairFormation_alt,
independent_pairEquality,
lambdaEquality_alt,
independent_pairFormation,
equalityTransitivity,
isect_memberEquality_alt,
voidElimination,
spreadEquality
Latex:
\mforall{}[K:RngSig]. \mforall{}[S,T:Type].
\mforall{}[as,bs:basic-formal-sum(K;S)]. (bfs-reduce(K;T;as;bs) {}\mRightarrow{} bfs-reduce(K;S;as;bs))
supposing strong-subtype(S;T)
Date html generated:
2019_10_31-AM-06_29_21
Last ObjectModification:
2019_08_15-PM-04_10_49
Theory : linear!algebra
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