Nuprl Lemma : mk_lambdas-fun-eta
∀[F,K:Top]. ∀[m,n:ℕ].
(mk_lambdas-fun(F;λf.mk_applies(f;K;n);n;m) ~ mk_lambdas-fun(λg.(F (λx.(g x)));λf.mk_applies(f;K;n);n;m))
Proof
Definitions occuring in Statement :
mk_applies: mk_applies(F;G;m),
mk_lambdas-fun: mk_lambdas-fun(F;G;n;m),
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
lambda: λx.A[x],
sqequal: s ~ t
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
sq_type: SQType(T),
guard: {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,
nat_plus: ℕ+,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
decidable__le,
subtract_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_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_wf,
le_wf,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
equal_wf,
subtype_base_sq,
int_subtype_base,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
less_than_wf,
nat_wf,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
eq_int_wf,
add-subtract-cancel,
lelt_wf,
assert_wf,
bnot_wf,
not_wf,
top_wf,
mk_applies_unroll,
bool_cases,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
mk_applies_fun
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
unionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
isectElimination,
because_Cache,
natural_numberEquality,
independent_isectElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
addEquality,
productElimination,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
intWeakElimination,
lambdaFormation,
sqequalAxiom,
equalityElimination,
promote_hyp,
applyEquality,
minusEquality,
isect_memberFormation,
impliesFunctionality
Latex:
\mforall{}[F,K:Top]. \mforall{}[m,n:\mBbbN{}].
(mk\_lambdas-fun(F;\mlambda{}f.mk\_applies(f;K;n);n;m)
\msim{} mk\_lambdas-fun(\mlambda{}g.(F (\mlambda{}x.(g x)));\mlambda{}f.mk\_applies(f;K;n);n;m))
Date html generated:
2017_10_01-AM-08_40_31
Last ObjectModification:
2017_07_26-PM-04_28_10
Theory : untyped!computation
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