Nuprl Lemma : callbyvalueall_seq-combine
∀[F,L1,L2,K:Top]. ∀[m1:ℕ+]. ∀[m2:ℕ]. ∀[n:ℕm1].
(callbyvalueall_seq(L1;λf.mk_applies(f;K;n);λg.(g
mk_lambdas(λout.callbyvalueall_seq(L2[out];λx.x;λg.(g F[m2]);0;m2);m1
- 1));n;m1)
~ callbyvalueall_seq(λi.if i <z m1 then L1 i else mk_lambdas(λout.(L2[out] (i - m1));m1 - 1) fi ;λf.mk_applies(f;K;n)
;λg.(g mk_lambdas(F[m2];m1));n;m1 + m2))
Proof
Definitions occuring in Statement :
mk_applies: mk_applies(F;G;m),
mk_lambdas: mk_lambdas(F;m),
callbyvalueall_seq: callbyvalueall_seq(L;G;F;n;m),
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
apply: f a,
lambda: λx.A[x],
subtract: n - m,
add: n + m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
exists: ∃x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
callbyvalueall_seq: callbyvalueall_seq(L;G;F;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,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
subtype_rel: A ⊆r B,
true: True,
nequal: a ≠ b ∈ T ,
mk_applies: mk_applies(F;G;m),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
int_seg_properties,
subtract_wf,
nat_properties,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_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_formula_prop_wf,
le_wf,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
equal_wf,
decidable__lt,
subtype_base_sq,
int_subtype_base,
ge_wf,
less_than_wf,
int_seg_wf,
nat_wf,
nat_plus_wf,
top_wf,
add-subtract-cancel,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
mk_applies_lambdas2,
lt_int_wf,
assert_of_lt_int,
zero-add,
callbyvalueall_seq-shift,
false_wf,
mk_applies_unroll,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
eq_int_wf,
assert_of_eq_int,
mk_applies_fun,
lelt_wf,
neg_assert_of_eq_int,
callbyvalueall_seq-shift-init0,
mk_applies_ite,
mk_applies_lambdas1,
primrec1_lemma,
callbyvalueall_seq-fun1,
iff_imp_equal_bool,
bfalse_wf,
assert_wf,
iff_wf,
callbyvalueall_seq-eta,
mk_applies_lambdas,
bnot_wf,
not_wf,
equal-wf-base,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
productElimination,
dependent_pairFormation,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
addEquality,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
intWeakElimination,
lambdaFormation,
sqequalAxiom,
imageElimination,
isect_memberFormation,
equalityElimination,
promote_hyp,
applyEquality,
minusEquality,
addLevel,
impliesFunctionality,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[F,L1,L2,K:Top]. \mforall{}[m1:\mBbbN{}\msupplus{}]. \mforall{}[m2:\mBbbN{}]. \mforall{}[n:\mBbbN{}m1].
(callbyvalueall\_seq(L1;\mlambda{}f.mk\_applies(f;K;n);\mlambda{}g.(g
mk\_lambdas(\mlambda{}out.callbyvalueall\_seq(L2[out];\mlambda{}x.x
;\mlambda{}g.(g F[m2]);0
;m2);m1 - 1));n
;m1) \msim{} callbyvalueall\_seq(\mlambda{}i.if i <z m1
then L1 i
else mk\_lambdas(\mlambda{}out.(L2[out] (i - m1));m1 - 1)
fi ;\mlambda{}f.mk\_applies(f;K;n);\mlambda{}g.(g
mk\_lambdas(F[m2];m1))
;n;m1 + m2))
Date html generated:
2018_05_21-PM-06_23_18
Last ObjectModification:
2018_05_19-PM-05_31_41
Theory : untyped!computation
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