Nuprl Lemma : A-null-loop
∀[Val:Type]. ∀[n:ℕ]. ∀[AType:array{i:l}(Val;n)].
∀lo,hi:ℕn. ∀body:{lo..hi-} ⟶ (A-map Unit). ((hi ≤ lo) ⇒ (A-loop(AType;lo;hi;body) = A-null(AType) ∈ (A-map Unit)))
Proof
Definitions occuring in Statement :
A-loop: A-loop(AType;lo;hi;body),
A-null: A-null(AType),
A-map: A-map,
array-model: array-model(AType),
array: array{i:l}(Val;n),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
A-loop: A-loop(AType;lo;hi;body),
prop: ℙ,
int_seg: {i..j-},
nat: ℕ,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
le_wf,
int_seg_wf,
A-map_wf,
unit_wf2,
array_wf,
nat_wf,
le_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
A-null_wf,
lt_int_wf,
less_than_wf,
bnot_wf,
int_seg_properties,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
functionEquality,
applyEquality,
cumulativity,
natural_numberEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
because_Cache,
isect_memberEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
productElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
unionElimination,
equalityElimination,
independent_functionElimination
Latex:
\mforall{}[Val:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[AType:array\{i:l\}(Val;n)].
\mforall{}lo,hi:\mBbbN{}n. \mforall{}body:\{lo..hi\msupminus{}\} {}\mrightarrow{} (A-map Unit).
((hi \mleq{} lo) {}\mRightarrow{} (A-loop(AType;lo;hi;body) = A-null(AType)))
Date html generated:
2017_10_01-AM-08_44_18
Last ObjectModification:
2017_07_26-PM-04_30_13
Theory : monads
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