Nuprl Lemma : case-real_wf
∀[P:ℙ]. ∀[a,b:ℝ]. ∀[f:{n:ℕ+| 4 < |(a n) - b n|} ⟶ ((↓P) ∨ (↓¬P))].
(case-real(a;b;f) ∈ {z:ℝ| (P ⇒ (z = a)) ∧ ((¬P) ⇒ (z = b))} )
Proof
Definitions occuring in Statement :
case-real: case-real(a;b;f),
req: x = y,
real: ℝ,
absval: |i|,
nat_plus: ℕ+,
less_than: a < b,
uall: ∀[x:A]. B[x],
prop: ℙ,
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
real: ℝ,
case-real: case-real(a;b;f),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
nat: ℕ,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
or: P ∨ Q,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
regular-int-seq: k-regular-seq(f),
squash: ↓T,
nat_plus: ℕ+,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
stable: Stable{P},
rneq: x ≠ y,
int_upper: {i...},
le: A ≤ B,
less_than': less_than'(a;b),
gt: i > j,
less_than: a < b,
req: x = y,
bdd-diff: bdd-diff(f;g),
absval: |i|,
subtract: n - m
Lemmas referenced :
lt_int_wf,
absval_wf,
subtract_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-less_than,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
nat_plus_wf,
squash_wf,
not_wf,
real_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermMultiply_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_functionality,
le_weakening,
int-triangle-inequality,
int_subtype_base,
decidable__equal_int,
decidable__lt,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
mul_preserves_le,
le_wf,
true_wf,
absval_mul,
subtype_rel_self,
iff_weakening_equal,
nat_wf,
set_subtype_base,
absval-non-neg,
equal_wf,
istype-universe,
left_mul_subtract_distrib,
istype-le,
mul-commutes,
absval_pos,
nat_plus_subtype_nat,
add_functionality_wrt_le,
absval-diff-symmetry,
add-commutes,
accelerate_wf,
regular-int-seq_wf,
real-regular,
req_wf,
stable_req,
false_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
req-iff-bdd-diff,
eventually-equal-implies-bdd-diff,
bdd-diff_functionality,
accelerate-bdd-diff,
bdd-diff_weakening,
rneq-iff,
rless-iff4,
absval_lbound,
int_upper_properties,
gt_wf,
istype-int_upper,
subtype_rel_sets_simple,
less_than_transitivity1,
req_functionality,
req_weakening,
istype-false,
minus-one-mul,
add-mul-special,
zero-mul,
le_weakening2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
lambdaEquality_alt,
extract_by_obid,
isectElimination,
natural_numberEquality,
applyEquality,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalRule,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_set_memberEquality_alt,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation_alt,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
universeIsType,
axiomEquality,
functionIsType,
setIsType,
unionIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
universeEquality,
imageElimination,
multiplyEquality,
addEquality,
approximateComputation,
int_eqEquality,
independent_pairFormation,
intEquality,
imageMemberEquality,
baseClosed,
productIsType,
unionEquality,
productEquality,
functionEquality,
inlFormation_alt,
minusEquality,
pointwiseFunctionality,
inrFormation_alt,
sqequalBase
Latex:
\mforall{}[P:\mBbbP{}]. \mforall{}[a,b:\mBbbR{}]. \mforall{}[f:\{n:\mBbbN{}\msupplus{}| 4 < |(a n) - b n|\} {}\mrightarrow{} ((\mdownarrow{}P) \mvee{} (\mdownarrow{}\mneg{}P))].
(case-real(a;b;f) \mmember{} \{z:\mBbbR{}| (P {}\mRightarrow{} (z = a)) \mwedge{} ((\mneg{}P) {}\mRightarrow{} (z = b))\} )
Date html generated:
2019_10_29-AM-09_36_43
Last ObjectModification:
2019_05_23-PM-05_35_58
Theory : reals
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