Nuprl Lemma : tuple-type-monotone
∀[P:Type]. ∀[F,G:P ⟶ Type]. ∀[v:P List]. (tuple-type(map(F;v)) ⊆r tuple-type(map(G;v))) supposing F ⊆ G
Proof
Definitions occuring in Statement :
tuple-type: tuple-type(L),
map: map(f;as),
list: T List,
sub-family: F ⊆ G,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
istype: istype(T),
cons: [a / b],
decidable: Dec(P),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
sub-family: F ⊆ G,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
list-cases,
map_nil_lemma,
tupletype_nil_lemma,
unit_wf2,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
le_wf,
map_cons_lemma,
tupletype_cons_lemma,
istype-nat,
list_wf,
sub-family_wf,
istype-universe,
null_wf,
eqtt_to_assert,
assert_of_null,
length_wf,
length_of_nil_lemma,
subtype_rel_ifthenelse,
btrue_wf,
tuple-type_wf,
map_wf,
subtype_rel_product,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
nil_wf,
null-map
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
Error :equalityIstype,
because_Cache,
Error :dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
sqequalBase,
Error :isectIsTypeImplies,
Error :functionIsType,
universeEquality,
equalityElimination,
productEquality,
cumulativity
Latex:
\mforall{}[P:Type]. \mforall{}[F,G:P {}\mrightarrow{} Type].
\mforall{}[v:P List]. (tuple-type(map(F;v)) \msubseteq{}r tuple-type(map(G;v))) supposing F \msubseteq{} G
Date html generated:
2019_06_20-PM-02_03_56
Last ObjectModification:
2019_02_20-PM-01_03_47
Theory : tuples
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