Nuprl Lemma : req-vec-extend
∀[k:ℕ+]. ∀[a,b:ℝ^k - 1]. ∀[z,u:ℝ]. uiff(req-vec(k;a++z;b++u);(z = u) ∧ req-vec(k - 1;a;b))
Proof
Definitions occuring in Statement :
real-vec-extend: a++z,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
req: x = y,
real: ℝ,
nat_plus: ℕ+,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
lelt: i ≤ j < k,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
nat: ℕ,
not: ¬A,
false: False,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
int_seg: {i..j-},
real-vec-extend: a++z,
subtype_rel: A ⊆r B,
prop: ℙ,
nat_plus: ℕ+,
real-vec: ℝ^n,
all: ∀x:A. B[x],
req-vec: req-vec(n;x;y),
implies: P ⇒ Q,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
int_seg_properties,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
bool_cases,
not_wf,
bnot_wf,
assert_wf,
decidable__lt,
lelt_wf,
nat_plus_wf,
real-vec_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_plus_properties,
req_wf,
real_wf,
subtype_rel_self,
less_than_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
real-vec-extend_wf,
nat_plus_subtype_nat,
req-vec_wf,
subtract_wf,
int_seg_wf,
req_witness
Rules used in proof :
impliesFunctionality,
voidEquality,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
dependent_set_memberEquality,
productEquality,
functionEquality,
voidElimination,
cumulativity,
instantiate,
promote_hyp,
dependent_pairFormation,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
because_Cache,
lambdaFormation,
rename,
setElimination,
natural_numberEquality,
applyEquality,
dependent_functionElimination,
lambdaEquality,
hypothesis,
independent_functionElimination,
hypothesisEquality,
isectElimination,
extract_by_obid,
independent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
sqequalRule,
independent_pairFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[a,b:\mBbbR{}\^{}k - 1]. \mforall{}[z,u:\mBbbR{}]. uiff(req-vec(k;a++z;b++u);(z = u) \mwedge{} req-vec(k - 1;a;b))
Date html generated:
2018_07_29-AM-09_44_17
Last ObjectModification:
2018_07_02-PM-00_43_31
Theory : reals
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