Nuprl Lemma : mk_array_wf
∀[Val:Type]. ∀[n:ℕ]. ∀[Arr:Type]. ∀[idx:ℕn ⟶ Arr ⟶ Val]. ∀[upd:ℕn ⟶ Val ⟶ Arr ⟶ Arr]. ∀[newarray:Val ⟶ Arr].
mk_array(Arr;idx;upd;newarray) ∈ array{i:l}(Val;n)
supposing (∀i:ℕn. ∀v:Val. ((idx i (newarray v)) = v ∈ Val))
∧ (∀i,j:ℕn. ∀x:Arr. ∀v:Val. ((idx i (upd j v x)) = if (i =z j) then v else idx i x fi ∈ Val))
Proof
Definitions occuring in Statement :
mk_array: mk_array(Arr;idx;upd;newarray),
array: array{i:l}(Val;n),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
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,
uimplies: b supposing a,
and: P ∧ Q,
mk_array: mk_array(Arr;idx;upd;newarray),
array: array{i:l}(Val;n),
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
eq_int_wf,
bool_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
all_wf,
int_seg_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
dependent_pairEquality,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
because_Cache,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
lambdaEquality,
imageElimination,
extract_by_obid,
isectElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
independent_functionElimination,
setElimination,
rename,
lambdaFormation,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
voidElimination,
productEquality,
isectEquality,
functionEquality
Latex:
\mforall{}[Val:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[Arr:Type]. \mforall{}[idx:\mBbbN{}n {}\mrightarrow{} Arr {}\mrightarrow{} Val]. \mforall{}[upd:\mBbbN{}n {}\mrightarrow{} Val {}\mrightarrow{} Arr {}\mrightarrow{} Arr].
\mforall{}[newarray:Val {}\mrightarrow{} Arr].
mk\_array(Arr;idx;upd;newarray) \mmember{} array\{i:l\}(Val;n)
supposing (\mforall{}i:\mBbbN{}n. \mforall{}v:Val. ((idx i (newarray v)) = v))
\mwedge{} (\mforall{}i,j:\mBbbN{}n. \mforall{}x:Arr. \mforall{}v:Val. ((idx i (upd j v x)) = if (i =\msubz{} j) then v else idx i x fi ))
Date html generated:
2017_10_01-AM-08_43_52
Last ObjectModification:
2017_07_26-PM-04_29_59
Theory : monads
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