Nuprl Lemma : A-loop_wf
∀[Val:Type]. ∀[n:ℕ]. ∀[AType:array{i:l}(Val;n)].
∀lo:ℕn. ∀k:ℕ. (k < n - lo ⇒ (∀[body:{lo..lo + k-} ⟶ (A-map Unit)]. (A-loop(AType;lo;lo + k;body) ∈ A-map Unit)))
Proof
Definitions occuring in Statement :
A-loop: A-loop(AType;lo;hi;body),
A-map: A-map,
array-model: array-model(AType),
array: array{i:l}(Val;n),
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
unit: Unit,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
A-loop: A-loop(AType;lo;hi;body),
guard: {T},
lelt: i ≤ j < k,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
label: ...$L... t,
subtract: n - m,
sq_type: SQType(T),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
int_seg_wf,
A-map_wf,
unit_wf2,
less_than_wf,
subtract_wf,
nat_wf,
array_wf,
nat_properties,
satisfiable-full-omega-tt,
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,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
le_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
le_wf,
A-null_wf,
lt_int_wf,
bnot_wf,
int_seg_properties,
itermAdd_wf,
int_term_value_add_lemma,
uiff_transitivity,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
equal_wf,
decidable__lt,
subtype_rel_dep_function,
int_seg_subtype,
subtype_rel_self,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
member_wf,
squash_wf,
true_wf,
add-swap,
add-commutes,
and_wf,
subtype_base_sq,
int_subtype_base,
iff_weakening_equal,
lelt_wf,
A-bind_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
extract_by_obid,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
addEquality,
because_Cache,
applyEquality,
cumulativity,
natural_numberEquality,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
universeEquality,
intWeakElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
unionElimination,
baseClosed,
productElimination,
equalityElimination,
imageElimination,
minusEquality,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality,
instantiate,
imageMemberEquality,
functionExtensionality
Latex:
\mforall{}[Val:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[AType:array\{i:l\}(Val;n)].
\mforall{}lo:\mBbbN{}n. \mforall{}k:\mBbbN{}.
(k < n - lo
{}\mRightarrow{} (\mforall{}[body:\{lo..lo + k\msupminus{}\} {}\mrightarrow{} (A-map Unit)]. (A-loop(AType;lo;lo + k;body) \mmember{} A-map Unit)))
Date html generated:
2017_10_01-AM-08_44_13
Last ObjectModification:
2017_07_26-PM-04_30_10
Theory : monads
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