Nuprl Lemma : circle-param-onto
∀p:{p:ℝ^2| r2-unit-circle(p)} . ((r(-1) < (p 0)) ⇒ (∃t:ℝ. req-vec(2;circle-param(t);p)))
Proof
Definitions occuring in Statement :
circle-param: circle-param(t),
r2-unit-circle: r2-unit-circle(p),
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
r2-unit-circle: r2-unit-circle(p),
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
sq_stable: SqStable(P),
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
exists: ∃x:A. B[x],
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
req-vec: req-vec(n;x;y),
decidable: Dec(P),
sq_type: SQType(T),
circle-param: circle-param(t),
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
rev_uimplies: rev_uimplies(P;Q),
bfalse: ff,
satisfiable_int_formula: satisfiable_int_formula(fmla),
rge: x ≥ y,
rdiv: (x/y)
Lemmas referenced :
sq_stable__req,
radd_wf,
rnexp_wf,
false_wf,
le_wf,
lelt_wf,
int-to-real_wf,
rless_wf,
set_wf,
real-vec_wf,
r2-unit-circle_wf,
rless-implies-rless,
rdiv_wf,
req-vec_wf,
circle-param_wf,
rsub_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
itermAdd_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_wf,
equal_wf,
rmul-is-positive,
square-nonneg,
req-vec_inversion,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
req-rdiv,
rmul_wf,
rmul_preserves_req,
int_seg_subtype,
int_seg_cases,
full-omega-unsat,
intformand_wf,
intformless_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
trivial-rless-radd,
rless-int,
rminus_wf,
rinv_wf2,
itermMultiply_wf,
itermMinus_wf,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
req_functionality,
req_transitivity,
radd_functionality,
rmul_functionality,
req_weakening,
rmul-rinv3,
rmul-rinv,
rminus_functionality,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
req_wf,
squash_wf,
true_wf,
iff_weakening_equal,
radd-preserves-req,
radd-rminus-both,
rnexp2,
rmul-identity1,
req-implies-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
sqequalRule,
setElimination,
thin,
rename,
introduction,
extract_by_obid,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
hypothesis,
hypothesisEquality,
applyEquality,
because_Cache,
imageMemberEquality,
baseClosed,
independent_functionElimination,
imageElimination,
minusEquality,
lambdaEquality,
dependent_functionElimination,
independent_isectElimination,
dependent_pairFormation,
inrFormation,
setEquality,
productElimination,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
inlFormation,
productEquality,
unionElimination,
instantiate,
cumulativity,
hypothesis_subsumption,
addEquality,
universeEquality
Latex:
\mforall{}p:\{p:\mBbbR{}\^{}2| r2-unit-circle(p)\} . ((r(-1) < (p 0)) {}\mRightarrow{} (\mexists{}t:\mBbbR{}. req-vec(2;circle-param(t);p)))
Date html generated:
2017_10_03-AM-10_52_24
Last ObjectModification:
2017_06_18-PM-01_24_56
Theory : reals
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