Nuprl Lemma : provision-equality
∀[T:𝕌']. ∀[ok1,ok2:ℙ]. ∀[v1:⋂:↓ok1. T]. ∀[v2:⋂:↓ok2. T].
(provision(ok1; v1) = provision(ok2; v2) ∈ Provisional(T)) supposing (((↓ok1) ⇒ (v1 = v2 ∈ T)) and (↓ok1 ⇐⇒ ↓ok2))
Proof
Definitions occuring in Statement :
provision: provision(ok; v),
provisional-type: Provisional(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
iff: P ⇐⇒ Q,
squash: ↓T,
implies: P ⇒ Q,
isect: ⋂x:A. B[x],
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),
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],
provision: provision(ok; v),
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B,
squash: ↓T,
exists: ∃x:A. B[x]
Lemmas referenced :
quotient-member-eq,
squash_wf,
iff_wf,
pi1_wf,
equal_wf,
pi2_wf,
uimplies_subtype,
provisional-equiv,
isect_subtype_rel_trivial
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
instantiate,
extract_by_obid,
isectElimination,
productEquality,
universeEquality,
sqequalRule,
isectEquality,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaEquality_alt,
inhabitedIsType,
functionEquality,
applyEquality,
independent_functionElimination,
because_Cache,
independent_isectElimination,
universeIsType,
productIsType,
isectIsType,
dependent_functionElimination,
dependent_pairEquality_alt,
isect_memberEquality_alt,
rename,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
lambdaFormation_alt,
imageElimination,
imageMemberEquality,
baseClosed,
functionIsType,
equalityIstype,
axiomEquality,
isectIsTypeImplies
Latex:
\mforall{}[T:\mBbbU{}']. \mforall{}[ok1,ok2:\mBbbP{}]. \mforall{}[v1:\mcap{}:\mdownarrow{}ok1. T]. \mforall{}[v2:\mcap{}:\mdownarrow{}ok2. T].
(provision(ok1; v1) = provision(ok2; v2)) supposing (((\mdownarrow{}ok1) {}\mRightarrow{} (v1 = v2)) and (\mdownarrow{}ok1 \mLeftarrow{}{}\mRightarrow{} \mdownarrow{}ok2))
Date html generated:
2020_05_20-AM-08_00_47
Last ObjectModification:
2020_05_17-PM-06_52_38
Theory : monads
Home
Index