Nuprl Lemma : provisional-subtype
∀[T,S:𝕌']. Provisional(T) ⊆r Provisional(S) supposing T ⊆r S
Proof
Definitions occuring in Statement :
provisional-type: Provisional(T),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
provisional-type: Provisional(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
guard: {T},
cand: A c∧ B,
squash: ↓T,
pi1: fst(t),
pi2: snd(t),
respects-equality: respects-equality(S;T)
Lemmas referenced :
provisional-type_wf,
quotient-member-eq,
squash_wf,
iff_wf,
pi1_wf,
equal_wf,
pi2_wf,
uimplies_subtype,
provisional-equiv,
subtype_rel_transitivity,
equal_functionality_wrt_subtype_rel2,
subtype-respects-equality,
subtype_rel_wf
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,
instantiate,
productEquality,
universeEquality,
isectEquality,
cumulativity,
universeIsType,
functionEquality,
applyEquality,
independent_functionElimination,
because_Cache,
independent_isectElimination,
inhabitedIsType,
productIsType,
isectIsType,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairEquality_alt,
isect_memberEquality_alt,
independent_pairFormation,
lambdaFormation_alt,
imageElimination,
imageMemberEquality,
baseClosed,
equalityIstype,
sqequalBase,
functionIsType,
hyp_replacement,
dependent_set_memberEquality_alt,
applyLambdaEquality,
setElimination,
rename,
axiomEquality,
isectIsTypeImplies
Latex:
\mforall{}[T,S:\mBbbU{}']. Provisional(T) \msubseteq{}r Provisional(S) supposing T \msubseteq{}r S
Date html generated:
2020_05_20-AM-08_01_16
Last ObjectModification:
2020_05_17-PM-08_08_25
Theory : monads
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