Nuprl Lemma : radd_rcos_wf
∀[x:ℝ]. (radd_rcos(x) ∈ {y:ℝ| y = (x + rcos(x))} )
Proof
Definitions occuring in Statement :
radd_rcos: radd_rcos(x),
rcos: rcos(x),
req: x = y,
radd: a + b,
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
radd_rcos: radd_rcos(x),
req: x = y,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
real: ℝ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
prop: ℙ,
implies: P ⇒ Q,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
sq_stable: SqStable(P),
regular-int-seq: k-regular-seq(f),
uimplies: b supposing a,
le: A ≤ B,
guard: {T},
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
sq_type: SQType(T),
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
bdd-diff: bdd-diff(f;g)
Lemmas referenced :
addrcos_wf2,
real_wf,
set_wf,
nat_plus_wf,
all_wf,
le_wf,
absval_wf,
subtract_wf,
radd_wf,
rcos_wf,
nat_wf,
sq_stable__regular-int-seq,
less_than_wf,
equal_wf,
mul_preserves_le,
squash_wf,
true_wf,
absval_mul,
iff_weakening_equal,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
multiply-is-int-iff,
add-is-int-iff,
minus-one-mul,
mul-distributes,
mul-swap,
minus-one-mul-top,
mul-commutes,
absval_pos,
nat_plus_subtype_nat,
absval-non-neg,
absval-diff-symmetry,
left_mul_subtract_distrib,
nat_plus_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
le_functionality,
le_transitivity,
int-triangle-inequality,
add_functionality_wrt_le,
le_weakening,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
accelerate_wf,
regular-int-seq_wf,
req_wf,
req-iff-bdd-diff,
accelerate-bdd-diff,
bdd-diff_functionality,
bdd-diff_weakening,
false_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
functionEquality,
intEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
setElimination,
rename,
natural_numberEquality,
lambdaFormation,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
independent_functionElimination,
imageElimination,
dependent_functionElimination,
because_Cache,
independent_isectElimination,
productElimination,
universeEquality,
instantiate,
cumulativity,
baseApply,
closedConclusion,
multiplyEquality,
isect_memberEquality,
voidElimination,
voidEquality,
minusEquality,
addEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
setEquality,
applyLambdaEquality
Latex:
\mforall{}[x:\mBbbR{}]. (radd\_rcos(x) \mmember{} \{y:\mBbbR{}| y = (x + rcos(x))\} )
Date html generated:
2017_10_04-PM-10_22_25
Last ObjectModification:
2017_07_28-AM-08_48_29
Theory : reals_2
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