Nuprl Lemma : fs-in-subtype-basic
∀[K:RngSig]. ∀[S,T:Type].
∀[f:formal-sum(K;S)]. ↓∃b:basic-formal-sum(K;T). (f = b ∈ formal-sum(K;S)) supposing 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),
basic-formal-sum: basic-formal-sum(K;S),
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
universe: Type,
equal: s = t ∈ T,
rng_sig: RngSig
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
fs-in-subtype: fs-in-subtype(K;S;T;f),
fs-predicate: fs-predicate(K;S;p.P[p];f),
squash: ↓T,
exists: ∃x:A. B[x],
and: P ∧ Q,
bfs-predicate: bfs-predicate(K;S;p.P[p];b),
basic-formal-sum: basic-formal-sum(K;S),
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
formal-sum: formal-sum(K;S),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
respects-equality: respects-equality(S;T),
guard: {T},
pi2: snd(t),
true: True,
subtype_rel: A ⊆r B
Lemmas referenced :
bag-in-subtype,
rng_car_wf,
strong-subtype-product,
strong-subtype-self,
bag-member_wf,
respects-equality-quotient1,
basic-formal-sum_wf,
bfs-equiv_wf,
bfs-equiv-rel,
respects-equality-bag,
respects-equality-product,
respects-equality-trivial,
subtype-respects-equality,
istype-base,
fs-in-subtype_wf,
formal-sum_wf,
strong-subtype_wf,
istype-universe,
rng_sig_wf,
trivial-equal,
member_wf,
squash_wf,
true_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
imageElimination,
productElimination,
thin,
extract_by_obid,
isectElimination,
productEquality,
hypothesisEquality,
hypothesis,
independent_isectElimination,
because_Cache,
lambdaFormation_alt,
independent_pairEquality,
universeIsType,
productIsType,
dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
inhabitedIsType,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
independent_functionElimination,
sqequalBase,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
hyp_replacement,
applyEquality,
natural_numberEquality
Latex:
\mforall{}[K:RngSig]. \mforall{}[S,T:Type].
\mforall{}[f:formal-sum(K;S)]. \mdownarrow{}\mexists{}b:basic-formal-sum(K;T). (f = b) supposing fs-in-subtype(K;S;T;f)
supposing strong-subtype(T;S)
Date html generated:
2019_10_31-AM-06_29_14
Last ObjectModification:
2019_08_20-PM-05_07_53
Theory : linear!algebra
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