Nuprl Lemma : select_fun_ap_wf
∀[n:ℕ]. ∀[m:ℕn]. ∀[A:ℕn ⟶ Type]. ∀[g:∀[T:Type]. (funtype(n;A;T) ⟶ T)]. (select_fun_ap(g;n;m) ∈ A m)
Proof
Definitions occuring in Statement :
select_fun_ap: select_fun_ap(g;n;m),
funtype: funtype(n;A;T),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
select_fun_ap: select_fun_ap(g;n;m),
subtype_rel: A ⊆r B,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than: a < b,
guard: {T},
uiff: uiff(P;Q),
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
subtract: n - m
Lemmas referenced :
int_seg_wf,
funtype_wf,
funtype-split,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
lelt_wf,
mk_lambdas_wf,
subtract_wf,
int_seg_properties,
decidable__le,
intformle_wf,
itermSubtract_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
le_wf,
add-member-int_seg1,
int_seg_subtype_nat,
false_wf,
subtype_rel_dep_function,
int_seg_subtype,
subtype_rel_self,
funtype-unroll,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_subtype_base,
zero-add,
subtype_rel_weakening,
ext-eq_weakening,
uall_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
applyEquality,
hypothesisEquality,
lambdaEquality,
isectElimination,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesis,
equalityTransitivity,
equalitySymmetry,
isectEquality,
universeEquality,
cumulativity,
functionEquality,
because_Cache,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
dependent_functionElimination,
addEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
lambdaFormation,
instantiate,
equalityElimination,
promote_hyp,
independent_functionElimination,
axiomEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[m:\mBbbN{}n]. \mforall{}[A:\mBbbN{}n {}\mrightarrow{} Type]. \mforall{}[g:\mforall{}[T:Type]. (funtype(n;A;T) {}\mrightarrow{} T)].
(select\_fun\_ap(g;n;m) \mmember{} A m)
Date html generated:
2017_10_01-AM-08_40_11
Last ObjectModification:
2017_07_26-PM-04_27_56
Theory : untyped!computation
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