Nuprl Lemma : mk_applies_wf
∀[T:Type]. ∀[m,n:ℕ]. ∀[A:ℕm + n ⟶ Type]. ∀[G:k:ℕm ⟶ (A k)]. ∀[F:funtype(m + n;A;T)].
(mk_applies(F;G;m) ∈ funtype(n;λk.(A (m + k));T))
Proof
Definitions occuring in Statement :
mk_applies: mk_applies(F;G;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],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
mk_applies: mk_applies(F;G;m),
subtype_rel: A ⊆r B,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
decidable: Dec(P),
or: P ∨ Q,
subtract: n - m,
squash: ↓T,
true: True,
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 ,
so_lambda: λ2x.t[x],
less_than: a < b,
so_apply: x[s],
less_than': less_than'(a;b),
funtype: funtype(n;A;T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
less_than_wf,
funtype_wf,
int_seg_wf,
lelt_wf,
zero-add,
primrec0_lemma,
subtype_rel-equal,
subtype_base_sq,
int_subtype_base,
add-member-int_seg1,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
le_wf,
primrec-unroll,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
nat_wf,
add-associates,
add-swap,
add-commutes,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
subtype_rel_self,
intformeq_wf,
int_formula_prop_eq_lemma,
assert_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
add-subtract-cancel,
minus-add,
minus-minus,
minus-one-mul,
add-mul-special,
zero-mul,
add-zero,
primrec_wf,
int_seg_properties,
decidable__equal_int,
itermMultiply_wf,
int_term_value_mul_lemma,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity,
functionExtensionality,
applyEquality,
productElimination,
addEquality,
functionEquality,
because_Cache,
dependent_set_memberEquality,
universeEquality,
instantiate,
unionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
equalityElimination,
promote_hyp,
multiplyEquality,
minusEquality,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[m,n:\mBbbN{}]. \mforall{}[A:\mBbbN{}m + n {}\mrightarrow{} Type]. \mforall{}[G:k:\mBbbN{}m {}\mrightarrow{} (A k)]. \mforall{}[F:funtype(m + n;A;T)].
(mk\_applies(F;G;m) \mmember{} funtype(n;\mlambda{}k.(A (m + k));T))
Date html generated:
2017_10_01-AM-08_40_14
Last ObjectModification:
2017_07_26-PM-04_27_58
Theory : untyped!computation
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