Nuprl Lemma : provisional-type-equality
∀[T:𝕌']. ∀[x,y:Provisional(T)].
(x = y ∈ Provisional(T)) supposing ((allowed(x) ⇒ (allow(x) = allow(y) ∈ T)) and (allowed(x) ⇐⇒ allowed(y)))
Proof
Definitions occuring in Statement :
allow: allow(x),
allowed: allowed(x),
provisional-type: Provisional(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
provisional-type: Provisional(T),
quotient: x,y:A//B[x; y],
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
cand: A c∧ B,
squash: ↓T,
allow: allow(x),
pi1: fst(t),
pi2: snd(t),
respects-equality: respects-equality(S;T),
guard: {T},
allowed: allowed(x),
true: True
Lemmas referenced :
quotient-member-eq,
squash_wf,
iff_wf,
pi1_wf,
equal_wf,
pi2_wf,
uimplies_subtype,
provisional-equiv,
subtype-respects-equality,
allowed_wf,
allow_wf,
provisional-type_wf,
istype-universe,
squash-implies-usquash,
usquash-implies-squash,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
pointwiseFunctionalityForEquality,
because_Cache,
hypothesis,
sqequalRule,
pertypeElimination,
promote_hyp,
instantiate,
extract_by_obid,
isectElimination,
productEquality,
universeEquality,
isectEquality,
cumulativity,
hypothesisEquality,
lambdaEquality_alt,
universeIsType,
functionEquality,
applyEquality,
independent_functionElimination,
independent_isectElimination,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_pairFormation,
lambdaFormation_alt,
imageElimination,
imageMemberEquality,
baseClosed,
equalityIstype,
productIsType,
sqequalBase,
functionIsType,
isect_memberEquality_alt,
hyp_replacement,
dependent_set_memberEquality_alt,
applyLambdaEquality,
setElimination,
rename,
isectIsType,
axiomEquality,
isectIsTypeImplies,
natural_numberEquality
Latex:
\mforall{}[T:\mBbbU{}']. \mforall{}[x,y:Provisional(T)].
(x = y) supposing ((allowed(x) {}\mRightarrow{} (allow(x) = allow(y))) and (allowed(x) \mLeftarrow{}{}\mRightarrow{} allowed(y)))
Date html generated:
2020_05_20-AM-08_01_01
Last ObjectModification:
2020_05_17-PM-07_17_16
Theory : monads
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