Nuprl Lemma : mk_lambdas-fun_wf
∀[T,U:Type]. ∀[m:ℕ]. ∀[n:ℕm + 1]. ∀[A:ℕm ⟶ Type]. ∀[F:(funtype(m;A;T) ⟶ T) ⟶ U].
∀[G:∀[T:Type]. (funtype(n;A;T) ⟶ T)].
(mk_lambdas-fun(F;G;n;m) ∈ funtype(m - n;λi.(A (i + n));U))
Proof
Definitions occuring in Statement :
mk_lambdas-fun: mk_lambdas-fun(F;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,
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
exists: ∃x:A. B[x],
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),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
le: A ≤ B,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
less_than': less_than'(a;b),
funtype: funtype(n;A;T),
mk_lambdas-fun: mk_lambdas-fun(F;G;n;m),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
subtract: n - m,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b
Lemmas referenced :
int_seg_properties,
subtract_wf,
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,
funtype_wf,
int_seg_wf,
uall_wf,
less_than_transitivity1,
less_than_irreflexivity,
add-zero,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
subtype_rel_self,
int_seg_subtype_nat,
nat_wf,
primrec0_lemma,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
funtype-unroll,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
zero-add,
add-commutes,
add-associates,
add-swap,
decidable__lt,
lelt_wf,
member_wf,
squash_wf,
true_wf,
funtype-unroll-last-eq,
iff_weakening_equal,
add-subtract-cancel,
subtype_rel-equal,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
add-member-int_seg2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
addEquality,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
productElimination,
dependent_pairFormation,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
intWeakElimination,
lambdaFormation,
axiomEquality,
functionEquality,
functionExtensionality,
applyEquality,
universeEquality,
equalityElimination,
isectEquality,
promote_hyp,
imageElimination,
imageMemberEquality,
baseClosed,
baseApply,
closedConclusion,
impliesFunctionality
Latex:
\mforall{}[T,U:Type]. \mforall{}[m:\mBbbN{}]. \mforall{}[n:\mBbbN{}m + 1]. \mforall{}[A:\mBbbN{}m {}\mrightarrow{} Type]. \mforall{}[F:(funtype(m;A;T) {}\mrightarrow{} T) {}\mrightarrow{} U].
\mforall{}[G:\mforall{}[T:Type]. (funtype(n;A;T) {}\mrightarrow{} T)].
(mk\_lambdas-fun(F;G;n;m) \mmember{} funtype(m - n;\mlambda{}i.(A (i + n));U))
Date html generated:
2017_10_01-AM-08_40_02
Last ObjectModification:
2017_07_26-PM-04_27_50
Theory : untyped!computation
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