Nuprl Lemma : simple-cbva-seq-list-case2
∀[F,L1,L2,G,a:Top]. ∀[m1:ℕ+]. ∀[m2:ℕ].
(simple-cbva-seq(λn.if (n) < (m1)
then L1 a n
else mk_lambdas(λout.if n - m1=m2
then mk_lambdas(λo1.G[out;o1];m2 - 1)
else (L2 a (n - m1));m1 - 1);F;(m1 + m2) + 1)
~ simple-cbva-seq(λn.if (n) < (m1)
then L1 a n
else if (n) < (m1 + m2)
then mk_lambdas(L2 a (n - m1);m1)
else mk_lambdas(λout1.mk_lambdas(λout2.G[out1;out2];m2 - 1);m1 - 1);F;(m1 + m2) + 1))
Proof
Definitions occuring in Statement :
simple-cbva-seq: simple-cbva-seq(L;F;m)
,
mk_lambdas: mk_lambdas(F;m)
,
nat_plus: ℕ+
,
nat: ℕ
,
uall: ∀[x:A]. B[x]
,
top: Top
,
so_apply: x[s1;s2]
,
less: if (a) < (b) then c else d
,
int_eq: if a=b then c else d
,
apply: f a
,
lambda: λx.A[x]
,
subtract: n - m
,
add: n + m
,
natural_number: $n
,
sqequal: s ~ t
Definitions unfolded in proof :
simple-cbva-seq: simple-cbva-seq(L;F;m)
,
cbva-seq: cbva-seq(L;F;m)
,
member: t ∈ T
,
uall: ∀[x:A]. B[x]
,
nat_plus: ℕ+
,
nat: ℕ
,
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
uiff: uiff(P;Q)
,
and: P ∧ Q
,
uimplies: b supposing a
,
ifthenelse: if b then t else f fi
,
ge: i ≥ j
,
not: ¬A
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
exists: ∃x:A. B[x]
,
false: False
,
top: Top
,
prop: ℙ
,
bfalse: ff
,
subtype_rel: A ⊆r B
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
or: P ∨ Q
,
sq_type: SQType(T)
,
guard: {T}
,
bnot: ¬bb
,
assert: ↑b
,
le: A ≤ B
,
less_than': less_than'(a;b)
,
decidable: Dec(P)
,
mk_applies: mk_applies(F;G;m)
,
int_seg: {i..j-}
,
lelt: i ≤ j < k
,
rev_implies: P
⇐ Q
,
iff: P
⇐⇒ Q
,
lt_int: i <z j
,
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w])
,
so_apply: x[s1;s2;s3;s4]
,
has-value: (a)↓
,
less_than: a < b
,
true: True
,
squash: ↓T
,
so_apply: x[s1;s2]
Lemmas referenced :
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
nat_properties,
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
eqff_to_assert,
set_subtype_base,
le_wf,
int_subtype_base,
less_than_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
nat_wf,
nat_plus_wf,
istype-top,
add-subtract-cancel,
istype-false,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
primrec0_lemma,
decidable__lt,
decidable__equal_int,
itermSubtract_wf,
int_term_value_subtract_lemma,
lt_int_wf,
assert_of_lt_int,
iff_weakening_uiff,
assert_wf,
callbyvalueall_seq-seq,
callbyvalueall_seq-decomp-last,
less_as_ite,
callbyvalueall_seq-fun2,
lifting-strict-less,
strict4-decide,
has-value_wf_base,
is-exception_wf,
callbyvalueall_seq-fun4,
mk_lambdas_unroll2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
addEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
equalityIsType2,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
promote_hyp,
instantiate,
cumulativity,
because_Cache,
equalityIsType1,
isect_memberFormation_alt,
axiomSqEquality,
dependent_set_memberEquality_alt,
productIsType,
int_eqReduceTrueSq,
sqequalSqle,
divergentSqle,
callbyvalueLess,
lessCases,
imageMemberEquality,
imageElimination,
sqleReflexivity,
lessExceptionCases,
axiomSqleEquality,
exceptionSqequal,
exceptionLess
Latex:
\mforall{}[F,L1,L2,G,a:Top]. \mforall{}[m1:\mBbbN{}\msupplus{}]. \mforall{}[m2:\mBbbN{}].
(simple-cbva-seq(\mlambda{}n.if (n) < (m1)
then L1 a n
else mk\_lambdas(\mlambda{}out.if n - m1=m2
then mk\_lambdas(\mlambda{}o1.G[out;o1];m2 - 1)
else (L2 a (n - m1));m1 - 1);F;(m1 + m2) + 1)
\msim{} simple-cbva-seq(\mlambda{}n.if (n) < (m1)
then L1 a n
else if (n) < (m1 + m2)
then mk\_lambdas(L2 a (n - m1);m1)
else mk\_lambdas(\mlambda{}out1.mk\_lambdas(\mlambda{}out2.G[out1;out2];m2 - 1);m1
- 1);F;(m1 + m2) + 1))
Date html generated:
2019_10_15-AM-10_59_15
Last ObjectModification:
2018_10_11-PM-09_47_36
Theory : untyped!computation
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