Nuprl Lemma : radd-list_functionality
∀[L1,L2:ℝ List]. radd-list(L1) = radd-list(L2) supposing (||L1|| = ||L2|| ∈ ℤ) ∧ (∀i:ℕ||L1||. (L1[i] = L2[i]))
Proof
Definitions occuring in Statement :
req: x = y,
radd-list: radd-list(L),
real: ℝ,
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
subtract: n - m,
cons: [a / b],
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
select: L[n],
nil: [],
list_ind: list_ind,
map: map(f;as),
lelt: i ≤ j < k,
int_seg: {i..j-},
true: True,
real: ℝ,
squash: ↓T,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
decidable: Dec(P),
nat_plus: ℕ+,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
false: False,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
prop: ℙ,
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
nat: ℕ,
has-valueall: has-valueall(a),
has-value: (a)↓,
callbyvalueall: callbyvalueall,
radd-list: radd-list(L),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
add-subtract-cancel,
select-cons-tl,
real-regular,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
add-member-int_seg2,
decidable__equal_int,
lelt_wf,
nat_plus_properties,
add_nat_plus,
bdd-diff-add,
bdd-diff_weakening,
equal-wf-T-base,
false_wf,
add-is-int-iff,
cons_wf,
int_term_value_add_lemma,
itermAdd_wf,
l_sum_cons_lemma,
map_cons_lemma,
length_of_cons_lemma,
equal-wf-base,
l_sum_nil_lemma,
map_nil_lemma,
base_wf,
stuck-spread,
length_of_nil_lemma,
nil_wf,
equal-wf-base-T,
map_wf,
l_sum_wf,
list_induction,
decidable__le,
int_seg_properties,
select_wf,
req_wf,
int_seg_wf,
all_wf,
list-subtype-bag,
radd-list_wf-bag,
req_witness,
iff_weakening_equal,
subtype_rel_self,
subtype_rel_list,
reg-seq-list-add-as-l_sum,
true_wf,
squash_wf,
bdd-diff_wf,
accelerate-bdd-diff,
regular-int-seq_wf,
nat_plus_wf,
bdd-diff_functionality,
reg-seq-list-add_wf,
less_than_wf,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
intformle_wf,
intformless_wf,
decidable__lt,
accelerate_wf,
req-iff-bdd-diff,
int-to-real_wf,
req_weakening,
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformnot_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
non_neg_length,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
length_wf,
eq_int_wf,
length_wf_nat,
int-value-type,
le_wf,
set-value-type,
nat_wf,
value-type-has-value,
valueall-type-real-list,
evalall-reduce,
real-valueall-type,
list-valueall-type,
real_wf,
list_wf,
valueall-type-has-valueall
Rules used in proof :
functionExtensionality,
hyp_replacement,
applyLambdaEquality,
closedConclusion,
baseApply,
pointwiseFunctionality,
addEquality,
productEquality,
universeEquality,
baseClosed,
imageMemberEquality,
imageElimination,
functionEquality,
setEquality,
rename,
setElimination,
applyEquality,
dependent_set_memberEquality,
independent_pairFormation,
voidEquality,
isect_memberEquality,
int_eqEquality,
approximateComputation,
voidElimination,
independent_functionElimination,
cumulativity,
instantiate,
dependent_functionElimination,
promote_hyp,
dependent_pairFormation,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
lambdaFormation,
natural_numberEquality,
lambdaEquality,
intEquality,
because_Cache,
callbyvalueReduce,
hypothesisEquality,
independent_isectElimination,
hypothesis,
isectElimination,
extract_by_obid,
sqequalRule,
thin,
productElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[L1,L2:\mBbbR{} List].
radd-list(L1) = radd-list(L2) supposing (||L1|| = ||L2||) \mwedge{} (\mforall{}i:\mBbbN{}||L1||. (L1[i] = L2[i]))
Date html generated:
2018_05_22-PM-01_20_45
Last ObjectModification:
2018_05_21-AM-00_04_41
Theory : reals
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