Nuprl Lemma : sq_stable__fs-in-subtype
∀[K:RngSig]. ∀[S,T:Type]. ∀[f:formal-sum(K;S)]. SqStable(fs-in-subtype(K;S;T;f)) supposing strong-subtype(T;S)
Proof
Definitions occuring in Statement :
fs-in-subtype: fs-in-subtype(K;S;T;f),
formal-sum: formal-sum(K;S),
strong-subtype: strong-subtype(A;B),
sq_stable: SqStable(P),
uimplies: b supposing a,
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,
uiff: uiff(P;Q),
and: P ∧ Q,
sq_stable: SqStable(P),
implies: P ⇒ Q,
fs-in-subtype: fs-in-subtype(K;S;T;f),
fs-predicate: fs-predicate(K;S;p.P[p];f),
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
formal-sum: formal-sum(K;S),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
so_apply: x[s]
Lemmas referenced :
strong-subtype-iff-respects-equality,
formal-sum_wf,
strong-subtype_wf,
istype-universe,
rng_sig_wf,
sq_stable__squash,
exists_wf,
basic-formal-sum_wf,
equal_wf,
subtype_quotient,
bfs-equiv_wf,
bfs-equiv-rel,
bfs-predicate_wf,
pi2_wf,
rng_car_wf,
equal-wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
functionIsTypeImplies,
inhabitedIsType,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate,
universeEquality,
productEquality,
because_Cache,
applyEquality,
productIsType,
independent_functionElimination
Latex:
\mforall{}[K:RngSig]. \mforall{}[S,T:Type].
\mforall{}[f:formal-sum(K;S)]. SqStable(fs-in-subtype(K;S;T;f)) supposing strong-subtype(T;S)
Date html generated:
2019_10_31-AM-06_29_10
Last ObjectModification:
2019_08_19-PM-01_18_26
Theory : linear!algebra
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