Nuprl Lemma : case-real2_wf
∀[P:ℙ]. ∀[a,b:ℝ]. ∀[f:a ≠ b ⟶ ((↓P) ∨ (↓¬P))]. (case-real2(a;b;f) ∈ {z:ℝ| (P ⇒ (z = a)) ∧ ((¬P) ⇒ (z = b))} )
Proof
Definitions occuring in Statement :
case-real2: case-real2(a;b;f),
rneq: x ≠ y,
req: x = y,
real: ℝ,
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]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
case-real2: case-real2(a;b;f),
all: ∀x:A. B[x],
real: ℝ,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
gt: i > j,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
rneq: x ≠ y,
rless: x < y,
sq_exists: ∃x:A [B[x]],
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nat_plus: ℕ+,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
subtype_rel: A ⊆r B
Lemmas referenced :
case-real_wf,
absval_lbound,
subtract_wf,
istype-void,
istype-le,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-less_than,
rless_wf,
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_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
itermAdd_wf,
intformor_wf,
itermSubtract_wf,
int_term_value_add_lemma,
int_formula_prop_or_lemma,
int_term_value_subtract_lemma,
nat_plus_wf,
absval_wf,
rneq_wf,
squash_wf,
not_wf,
real_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality_alt,
setElimination,
rename,
dependent_functionElimination,
applyEquality,
because_Cache,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
sqequalRule,
lambdaFormation_alt,
voidElimination,
productElimination,
independent_functionElimination,
addEquality,
inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
inlEquality_alt,
universeIsType,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
inrEquality_alt,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
minusEquality,
setIsType,
functionIsType,
unionIsType,
universeEquality
Latex:
\mforall{}[P:\mBbbP{}]. \mforall{}[a,b:\mBbbR{}]. \mforall{}[f:a \mneq{} b {}\mrightarrow{} ((\mdownarrow{}P) \mvee{} (\mdownarrow{}\mneg{}P))].
(case-real2(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_59
Last ObjectModification:
2019_05_23-PM-05_48_44
Theory : reals
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