Nuprl Lemma : circle-param_wf
∀[t:ℝ]. (circle-param(t) ∈ {p:ℝ^2| r2-unit-circle(p)} )
Proof
Definitions occuring in Statement :
circle-param: circle-param(t),
r2-unit-circle: r2-unit-circle(p),
real-vec: ℝ^n,
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rge: x ≥ y,
guard: {T},
circle-param: circle-param(t),
real-vec: ℝ^n,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
rneq: x ≠ y,
or: P ∨ Q,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
r2-unit-circle: r2-unit-circle(p),
eq_int: (i =z j),
nat: ℕ,
le: A ≤ B,
not: ¬A,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
square-nonneg,
trivial-rleq-radd,
int-to-real_wf,
rmul_wf,
r2-unit-circle_wf,
real_wf,
radd_wf,
rless-int,
rless_functionality_wrt_implies,
rleq_weakening_equal,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
rdiv_wf,
rsub_wf,
rless_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_seg_wf,
rnexp_wf,
false_wf,
le_wf,
req_functionality,
radd_functionality,
rnexp2,
req_weakening,
rmul_preserves_req,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
req-iff-rsub-is-0,
rmul-identity1,
req_transitivity,
rmul-rinv3,
rmul_functionality,
rinv-mul-as-rdiv,
rinv-as-rdiv,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rmul-rinv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
productElimination,
independent_isectElimination,
dependent_set_memberEquality,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
lambdaEquality,
setElimination,
rename,
lambdaFormation,
unionElimination,
equalityElimination,
inrFormation,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality
Latex:
\mforall{}[t:\mBbbR{}]. (circle-param(t) \mmember{} \{p:\mBbbR{}\^{}2| r2-unit-circle(p)\} )
Date html generated:
2017_10_03-AM-10_51_29
Last ObjectModification:
2017_06_18-PM-01_24_17
Theory : reals
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