Nuprl Lemma : equal-partial
∀[T:Type]. ∀[x,y:partial(T)]. uiff(x = y ∈ partial(T);uiff((x)↓;(y)↓) ∧ ((x)↓ ⇒ (x = y ∈ T))) supposing value-type(T)
Proof
Definitions occuring in Statement :
partial: partial(T),
value-type: value-type(T),
has-value: (a)↓,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
member: t ∈ T,
has-value: (a)↓,
implies: P ⇒ Q,
prop: ℙ,
respects-equality: respects-equality(S;T),
all: ∀x:A. B[x],
cand: A c∧ B,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
label: ...$L... t,
true: True,
partial: partial(T),
quotient: x,y:A//B[x; y],
so_lambda: λ2x y.t[x; y],
base-partial: base-partial(T),
so_apply: x[s1;s2],
equiv_rel: EquivRel(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
per-partial: per-partial(T;x;y)
Lemmas referenced :
has-value_wf-partial,
respects-equality-partial,
partial_wf,
value-type_wf,
istype-universe,
termination,
squash_wf,
subtype_rel_self,
iff_weakening_equal,
equal_wf,
true_wf,
termination-equality,
quotient-member-eq,
per-partial_wf,
per-partial-equiv_rel,
base-partial_wf,
has-value_wf_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
independent_pairFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
Error :isect_memberEquality_alt,
isectElimination,
hypothesisEquality,
axiomSqleEquality,
hypothesis,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
Error :functionIsTypeImplies,
Error :equalityIstype,
because_Cache,
Error :productIsType,
Error :isectIsType,
Error :universeIsType,
extract_by_obid,
independent_isectElimination,
Error :functionIsType,
independent_functionElimination,
instantiate,
universeEquality,
Error :lambdaFormation_alt,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
promote_hyp,
natural_numberEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
setElimination,
rename,
sqequalBase
Latex:
\mforall{}[T:Type]
\mforall{}[x,y:partial(T)]. uiff(x = y;uiff((x)\mdownarrow{};(y)\mdownarrow{}) \mwedge{} ((x)\mdownarrow{} {}\mRightarrow{} (x = y))) supposing value-type(T)
Date html generated:
2019_06_20-PM-00_34_01
Last ObjectModification:
2018_11_23-PM-01_15_52
Theory : partial_1
Home
Index