Nuprl Lemma : callbyvalueall_seq-combine3
∀[F,L1,L2,K:Top]. ∀[m1,m2:ℕ]. ∀[n:ℕm1 + 1].
(callbyvalueall_seq(L1;λf.mk_applies(f;K;n);λg.callbyvalueall_seq(L2[g];λx.x;F[g];0;m2);n;m1)
~ callbyvalueall_seq(λi.if i <z m1 then L1 i else mk_lambdas_fun(λg.(L2[g] (i - m1));m1) fi ;λf.mk_applies(f;K;n)
;λg.(F[partial_ap(g;m1 + m2;m1)] partial_ap_gen(g;m1 + m2;m1;m2));n;m1 + m2))
Proof
Definitions occuring in Statement :
mk_applies: mk_applies(F;G;m),
partial_ap: partial_ap(g;n;m),
partial_ap_gen: partial_ap_gen(g;n;s;m),
mk_lambdas_fun: mk_lambdas_fun(F;m),
callbyvalueall_seq: callbyvalueall_seq(L;G;F;n;m),
int_seg: {i..j-},
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: 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,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
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 ,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
bfalse: ff,
partial_ap_gen: partial_ap_gen(g;n;s;m),
partial_ap: partial_ap(g;n;m),
mk_lambdas: mk_lambdas(F;m),
bnot: ¬bb
Lemmas referenced :
int_seg_properties,
subtract_wf,
nat_properties,
decidable__le,
full-omega-unsat,
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,
nat_wf,
top_wf,
add-zero,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
zero-add,
callbyvalueall_seq-shift,
false_wf,
callbyvalueall_seq-shift-init0,
mk_applies_ite,
callbyvalueall_seq-fun1,
lt_int_wf,
iff_imp_equal_bool,
bfalse_wf,
assert_of_lt_int,
assert_wf,
iff_wf,
mk_applies_lambdas_fun0,
callbyvalueall_seq-eta,
add-subtract-cancel,
mk_applies_lambdas,
decidable__lt,
lelt_wf,
primrec0_lemma,
callbyvalueall_seq-lambdas-all0,
callbyvalueall_seq-partial-ap-all0,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
mk_applies_roll
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
natural_numberEquality,
addEquality,
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,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
intWeakElimination,
lambdaFormation,
sqequalAxiom,
isect_memberFormation,
equalityElimination,
addLevel,
impliesFunctionality,
promote_hyp
Latex:
\mforall{}[F,L1,L2,K:Top]. \mforall{}[m1,m2:\mBbbN{}]. \mforall{}[n:\mBbbN{}m1 + 1].
(callbyvalueall\_seq(L1;\mlambda{}f.mk\_applies(f;K;n);\mlambda{}g.callbyvalueall\_seq(L2[g];\mlambda{}x.x;F[g];0;m2);n;m1)
\msim{} callbyvalueall\_seq(\mlambda{}i.if i <z m1 then L1 i else mk\_lambdas\_fun(\mlambda{}g.(L2[g] (i - m1));m1) fi
;\mlambda{}f.mk\_applies(f;K;n);\mlambda{}g.(F[partial\_ap(g;m1 + m2;m1)]
partial\_ap\_gen(g;m1 + m2;m1;m2));n;m1 + m2))
Date html generated:
2018_05_21-PM-06_23_34
Last ObjectModification:
2018_05_19-PM-05_31_53
Theory : untyped!computation
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