Nuprl Lemma : sq_stable_double_ex_rneq
∀m,n:ℕ. ∀a,b:ℕm ⟶ ℕn ⟶ ℝ. SqStable(∃i:ℕm. ∃j:ℕn. a[i;j] ≠ b[i;j])
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
sq_stable: SqStable(P),
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
uall: ∀[x:A]. B[x],
nat: ℕ,
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than: a < b,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
rev_implies: P ⇐ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+
Lemmas referenced :
exists-rneq-iff,
int_seg_wf,
rless_wf,
int-to-real_wf,
rsum_wf,
subtract_wf,
rabs_wf,
rsub_wf,
subtract-add-cancel,
int_seg_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
lelt_wf,
rsum-of-nonneg-positive-iff,
rsum_nonneg,
zero-rleq-rabs,
intformle_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
le_wf,
exists_wf,
rneq_wf,
squash_wf,
real_wf,
nat_wf,
sq_stable__rless,
nat_plus_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
addLevel,
sqequalHypSubstitution,
imageElimination,
introduction,
existsFunctionality,
extract_by_obid,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
isectElimination,
natural_numberEquality,
setElimination,
rename,
because_Cache,
productElimination,
independent_functionElimination,
dependent_set_memberEquality,
independent_pairFormation,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
addEquality,
imageMemberEquality,
baseClosed,
levelHypothesis,
promote_hyp,
functionEquality
Latex:
\mforall{}m,n:\mBbbN{}. \mforall{}a,b:\mBbbN{}m {}\mrightarrow{} \mBbbN{}n {}\mrightarrow{} \mBbbR{}. SqStable(\mexists{}i:\mBbbN{}m. \mexists{}j:\mBbbN{}n. a[i;j] \mneq{} b[i;j])
Date html generated:
2017_10_03-AM-09_00_51
Last ObjectModification:
2017_06_16-PM-00_07_06
Theory : reals
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