Nuprl Lemma : provisional-equiv
∀[T:𝕌']. EquivRel(ok:ℙ × T supposing ↓ok;x,y.(↓fst(x) ⇐⇒ ↓fst(y)) ∧ ((↓fst(x)) ⇒ ((snd(x)) = (snd(y)) ∈ T)))
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
pi1: fst(t),
pi2: snd(t),
iff: P ⇐⇒ Q,
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
sym: Sym(T;x,y.E[x; y]),
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
respects-equality: respects-equality(S;T),
trans: Trans(T;x,y.E[x; y])
Lemmas referenced :
squash_wf,
uimplies_subtype,
pi1_wf,
pi2_wf,
subtype-respects-equality,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaFormation_alt,
productElimination,
thin,
sqequalRule,
hypothesis,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
because_Cache,
applyEquality,
instantiate,
cumulativity,
independent_isectElimination,
lambdaEquality_alt,
productIsType,
universeEquality,
isectIsType,
imageElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination,
equalitySymmetry,
functionIsType,
isectEquality,
equalityIstype,
isect_memberEquality_alt,
inhabitedIsType,
dependent_functionElimination,
equalityTransitivity,
independent_pairEquality,
functionIsTypeImplies,
axiomEquality
Latex:
\mforall{}[T:\mBbbU{}']
EquivRel(ok:\mBbbP{} \mtimes{} T supposing \mdownarrow{}ok;x,y.(\mdownarrow{}fst(x) \mLeftarrow{}{}\mRightarrow{} \mdownarrow{}fst(y)) \mwedge{} ((\mdownarrow{}fst(x)) {}\mRightarrow{} ((snd(x)) = (snd(y)))))
Date html generated:
2020_05_20-AM-08_00_35
Last ObjectModification:
2020_05_17-PM-06_47_39
Theory : monads
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