Nuprl Lemma : subtype_rel_record+
∀[T1,T2:𝕌']. ∀[B1:T1 ⟶ 𝕌']. ∀[B2:T2 ⟶ 𝕌'].
(∀[z:Atom]. (T1z:B1[self] ⊆r T2z:B2[self])) supposing ((∀x:T1. (B1[x] ⊆r B2[x])) and (T1 ⊆r T2))
Proof
Definitions occuring in Statement :
record+: record+,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
atom: Atom,
universe: Type
Definitions unfolded in proof :
record+: record+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
so_apply: x[s],
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
subtype_rel: A ⊆r B
Lemmas referenced :
dep-isect-subtype2,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
top_wf,
subtype_rel_self,
all_wf,
subtype_rel_wf,
subtype_rel_dep_function
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
atomEquality,
hypothesis,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
because_Cache,
applyEquality,
functionExtensionality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
universeEquality,
isect_memberFormation,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[T1,T2:\mBbbU{}']. \mforall{}[B1:T1 {}\mrightarrow{} \mBbbU{}']. \mforall{}[B2:T2 {}\mrightarrow{} \mBbbU{}'].
(\mforall{}[z:Atom]. (T1z:B1[self] \msubseteq{}r T2z:B2[self])) supposing ((\mforall{}x:T1. (B1[x] \msubseteq{}r B2[x])) and (T1 \msubseteq{}r T2))
Date html generated:
2018_05_21-PM-08_38_32
Last ObjectModification:
2017_07_26-PM-06_02_50
Theory : general
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