Nuprl Lemma : subtype_rel_record
∀[T1,T2:Atom ⟶ Type]. uiff(record(x.T1[x]) ⊆r record(x.T2[x]);∀[x:Atom]. (T1[x] ⊆r T2[x]) supposing record(x.T1[x]))
Proof
Definitions occuring in Statement :
record: record(x.T[x]),
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
atom: Atom,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
false: False,
not: ¬A,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
top: Top,
all: ∀x:A. B[x],
record: record(x.T[x])
Lemmas referenced :
istype-atom,
record_wf,
subtype_rel_wf,
istype-universe,
istype-assert,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
btrue_neq_bfalse,
eq_atom-reflexive,
not_wf,
bnot_wf,
iff_transitivity,
assert_of_eq_atom,
eqtt_to_assert,
assert_wf,
atom_subtype_base,
bool_wf,
equal-wf-base,
uiff_transitivity,
eq_atom_wf,
istype-void,
rec_select_update_lemma,
record-select_wf,
record-update_wf,
subtype_rel_dep_function
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
sqequalRule,
axiomEquality,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isect_memberEquality_alt,
isectElimination,
thin,
hypothesisEquality,
isectIsTypeImplies,
inhabitedIsType,
universeIsType,
lambdaEquality_alt,
cumulativity,
applyEquality,
instantiate,
isectIsType,
productElimination,
independent_pairEquality,
functionIsType,
universeEquality,
rename,
sqequalBase,
equalityIstype,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
independent_functionElimination,
because_Cache,
atomEquality,
baseClosed,
closedConclusion,
baseApply,
equalityElimination,
unionElimination,
lambdaFormation_alt,
voidElimination,
dependent_functionElimination,
lambdaEquality,
functionExtensionality,
lambdaFormation
Latex:
\mforall{}[T1,T2:Atom {}\mrightarrow{} Type].
uiff(record(x.T1[x]) \msubseteq{}r record(x.T2[x]);\mforall{}[x:Atom]. (T1[x] \msubseteq{}r T2[x]) supposing record(x.T1[x]))
Date html generated:
2020_05_20-AM-08_17_29
Last ObjectModification:
2019_11_27-PM-02_35_17
Theory : general
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