Nuprl Lemma : funtype-split
∀[T:Type]. ∀[n:ℕ]. ∀[m:ℕn + 1]. ∀[A:ℕn ⟶ Type]. (funtype(n;A;T) = funtype(m;A;funtype(n - m;λk.(A (k + m));T)) ∈ Type)
Proof
Definitions occuring in Statement :
funtype: funtype(n;A;T),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
nat: ℕ,
int_seg: {i..j-},
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
sq_type: SQType(T),
funtype: funtype(n;A;T),
squash: ↓T,
le: A ≤ B,
uiff: uiff(P;Q),
less_than: a < b,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than': less_than'(a;b),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
subtract_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
le_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
subtype_base_sq,
int_subtype_base,
ge_wf,
less_than_wf,
int_seg_wf,
add-zero,
primrec0_lemma,
funtype_wf,
squash_wf,
true_wf,
funtype-unroll-last-eq,
add-member-int_seg2,
decidable__lt,
lelt_wf,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
subtype_rel_self,
iff_weakening_equal,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
add-commutes
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_pairFormation,
dependent_set_memberEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
natural_numberEquality,
addEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
lambdaFormation,
intWeakElimination,
axiomEquality,
functionEquality,
universeEquality,
applyEquality,
imageElimination,
functionExtensionality,
imageMemberEquality,
baseClosed,
equalityElimination,
promote_hyp
Latex:
\mforall{}[T:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[m:\mBbbN{}n + 1]. \mforall{}[A:\mBbbN{}n {}\mrightarrow{} Type].
(funtype(n;A;T) = funtype(m;A;funtype(n - m;\mlambda{}k.(A (k + m));T)))
Date html generated:
2017_10_01-AM-08_39_46
Last ObjectModification:
2017_07_26-PM-04_27_42
Theory : untyped!computation
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