Nuprl Lemma : assert-rat-term-eq
∀r1,r2:rat_term().
((↑rat-term-eq(r1;r2))
⇒ (∀f:ℤ ⟶ ℝ. let p,x = rat_term_to_real(f;r1) in let q,y = rat_term_to_real(f;r2) in p ⇒ q ⇒ (x = y)))
Proof
Definitions occuring in Statement :
rat-term-eq: rat-term-eq(r1;r2),
rat_term_to_real: rat_term_to_real(f;t),
rat_term: rat_term(),
req: x = y,
real: ℝ,
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
spread: spread def,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
rat-term-eq: rat-term-eq(r1;r2),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
iPolynomial: iPolynomial(),
so_lambda: λ2x.t[x],
uimplies: b supposing a,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
so_apply: x[s],
req_rat_term: r ≡ p/q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
real_term_value: real_term_value(f;t),
itermMinus: "-"num,
int_term_ind: int_term_ind,
itermAdd: left (+) right,
itermMultiply: left (*) right,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
ipolynomial-term: ipolynomial-term(p),
cons: [a / b],
bfalse: ff
Lemmas referenced :
rat_term_polynomial,
rat_term_to_ipolys_wf,
istype-int,
real_wf,
istype-assert,
null_wf3,
add-ipoly_wf,
mul-ipoly_wf,
minus-poly_wf,
subtype_rel_set,
all_wf,
imonomial-less_wf,
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
decidable__lt,
int_seg_wf,
length_wf,
iMonomial_wf,
subtype_rel_list,
top_wf,
req_rat_term_wf,
ipolynomial-term_wf,
rat_term_wf,
rat_term_to_real_wf,
rneq_wf,
real_term_value_wf,
int-to-real_wf,
req_wf,
uimplies_subtype,
rdiv_wf,
req_functionality,
rmul_preserves_req,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
req_transitivity,
rmul_functionality,
req_weakening,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
mul-ipoly-req,
add-ipoly-req,
minus-poly-req,
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
add-ipoly_wf1,
rminus_wf,
radd_wf,
radd_functionality,
radd-preserves-req,
itermAdd_wf,
itermMinus_wf,
real_term_value_add_lemma,
real_term_value_minus_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
inhabitedIsType,
productElimination,
sqequalRule,
functionIsType,
universeIsType,
applyEquality,
because_Cache,
lambdaEquality_alt,
independent_isectElimination,
setElimination,
rename,
imageElimination,
unionElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
productIsType,
promote_hyp,
hypothesis_subsumption
Latex:
\mforall{}r1,r2:rat\_term().
((\muparrow{}rat-term-eq(r1;r2))
{}\mRightarrow{} (\mforall{}f:\mBbbZ{} {}\mrightarrow{} \mBbbR{}
let p,x = rat\_term\_to\_real(f;r1)
in let q,y = rat\_term\_to\_real(f;r2)
in p {}\mRightarrow{} q {}\mRightarrow{} (x = y)))
Date html generated:
2019_10_29-AM-09_54_08
Last ObjectModification:
2019_04_01-PM-07_02_02
Theory : reals
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