Nuprl Lemma : sqs-rneq-or
∀x,y:ℝ. SqStable(x ≠ r0 ∨ y ≠ r0)
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
int-to-real: r(n),
real: ℝ,
sq_stable: SqStable(P),
all: ∀x:A. B[x],
or: P ∨ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
member: t ∈ T,
prop: ℙ,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
has-value: (a)↓,
real: ℝ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
nat: ℕ,
int_upper: {i...},
rneq: x ≠ y,
iff: P ⇐⇒ Q,
int-to-real: r(n),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rless: x < y,
sq_exists: ∃x:A [B[x]],
cand: A c∧ B,
le: A ≤ B,
pi1: fst(t),
pi2: snd(t),
gt: i > j
Lemmas referenced :
le_wf,
equal-wf-base-T,
product_subtype_base,
int_subtype_base,
value-type-has-value,
int-value-type,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-less_than,
equal-wf-base,
bool_wf,
find-ge-val_wf,
product-value-type,
bor_wf,
lt_int_wf,
absval_wf,
int_upper_properties,
istype-int_upper,
set-value-type,
equal_wf,
squash_wf,
rneq_wf,
int-to-real_wf,
real_wf,
rless-iff4,
subtype_rel_sets_simple,
less_than_wf,
decidable__le,
spread_wf,
nat_plus_properties,
eqtt_to_assert,
assert_wf,
istype-assert,
mul-commutes,
mul-swap,
mul-associates,
zero-mul,
add-commutes,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_lt_int,
zero-add,
absval_unfold,
itermAdd_wf,
int_term_value_add_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
istype-top,
add-is-int-iff,
itermMinus_wf,
int_term_value_minus_lemma,
false_wf,
true_wf,
absval_pos,
istype-le,
subtype_rel_self,
iff_weakening_equal,
rless_wf,
istype-false,
not-lt-2,
add_functionality_wrt_le,
le-add-cancel,
itermMultiply_wf,
int_term_value_mul_lemma,
absval_lbound,
intformor_wf,
int_formula_prop_or_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
imageElimination,
productEquality,
intEquality,
thin,
setEquality,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
lambdaEquality_alt,
inhabitedIsType,
independent_isectElimination,
callbyvalueReduce,
setElimination,
rename,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
because_Cache,
independent_pairEquality,
baseApply,
closedConclusion,
baseClosed,
productElimination,
equalityTransitivity,
equalitySymmetry,
productIsType,
cutEval,
equalityIstype,
unionEquality,
functionIsType,
sqequalBase,
unionIsType,
inlFormation_alt,
inrFormation_alt,
minusEquality,
equalityElimination,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
isectIsTypeImplies,
imageMemberEquality,
promote_hyp,
instantiate,
cumulativity,
pointwiseFunctionality,
universeEquality,
dependent_set_memberFormation_alt,
addEquality,
multiplyEquality,
applyLambdaEquality
Latex:
\mforall{}x,y:\mBbbR{}. SqStable(x \mneq{} r0 \mvee{} y \mneq{} r0)
Date html generated:
2019_10_29-AM-09_35_58
Last ObjectModification:
2019_01_09-PM-05_13_38
Theory : reals
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