Nuprl Lemma : real-vec-extend_wf_interval
∀[I:Interval]. ∀[k:ℕ]. ∀[a:I^k]. ∀[z:{z:ℝ| z ∈ I} ]. (a++z ∈ I^k + 1)
Proof
Definitions occuring in Statement :
interval-vec: I^n,
real-vec-extend: a++z,
i-member: r ∈ I,
interval: Interval,
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
interval-vec: I^n,
all: ∀x:A. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
real-vec: ℝ^n,
so_apply: x[s],
prop: ℙ,
nat_plus: ℕ+,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
guard: {T},
sq_type: SQType(T),
squash: ↓T,
sq_stable: SqStable(P),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
label: ...$L... t,
lelt: i ≤ j < k,
assert: ↑b,
bnot: ¬bb,
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
Lemmas referenced :
int_seg_wf,
all_wf,
i-member_wf,
set_wf,
real_wf,
interval-vec_wf,
nat_wf,
interval_wf,
real-vec-extend_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
add-swap,
member_wf,
real-vec_wf,
add-subtract-cancel,
sq_stable__and,
equal_wf,
subtract_wf,
sq_stable__equal,
squash_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
iff_weakening_equal,
and_wf,
true_wf,
subtype_rel_self,
lelt_wf,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
lambdaFormation,
extract_by_obid,
isectElimination,
natural_numberEquality,
addEquality,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
voidEquality,
intEquality,
minusEquality,
instantiate,
cumulativity,
imageElimination,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
hyp_replacement,
universeEquality,
promote_hyp,
equalityElimination
Latex:
\mforall{}[I:Interval]. \mforall{}[k:\mBbbN{}]. \mforall{}[a:I\^{}k]. \mforall{}[z:\{z:\mBbbR{}| z \mmember{} I\} ]. (a++z \mmember{} I\^{}k + 1)
Date html generated:
2019_10_30-AM-08_23_16
Last ObjectModification:
2018_08_23-PM-01_45_25
Theory : reals
Home
Index