Nuprl Lemma : mk_applies_fun2
∀[F,K,v:Top]. ∀[p,n:ℕ]. ∀[m:ℕn + 1].
(mk_applies(F;λk.if (p + k =z p + n) then v else K (p + k) fi ;m) ~ mk_applies(F;λi.(K (p + i));m))
Proof
Definitions occuring in Statement :
mk_applies: mk_applies(F;G;m),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
lambda: λx.A[x],
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,
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
mk_applies: mk_applies(F;G;m),
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
int_seg_properties,
le_wf,
int_seg_wf,
nat_wf,
top_wf,
nat_properties,
full-omega-unsat,
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,
primrec0_lemma,
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,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
eq_int_wf,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
neg_assert_of_eq_int,
primrec-unroll
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,
hypothesis_subsumption,
lambdaEquality,
dependent_set_memberEquality,
isect_memberFormation,
sqequalAxiom,
sqequalRule,
isect_memberEquality,
intWeakElimination,
lambdaFormation,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[F,K,v:Top]. \mforall{}[p,n:\mBbbN{}]. \mforall{}[m:\mBbbN{}n + 1].
(mk\_applies(F;\mlambda{}k.if (p + k =\msubz{} p + n) then v else K (p + k) fi ;m)
\msim{} mk\_applies(F;\mlambda{}i.(K (p + i));m))
Date html generated:
2018_05_21-PM-06_21_44
Last ObjectModification:
2018_05_19-PM-05_27_53
Theory : untyped!computation
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