Nuprl Lemma : rv-T'-implies-rv-T
∀n:ℕ+. ∀a,b,c:ℝ^n. (rv-T'(n;a;b;c) ⇒ rv-T(n;a;b;c))
Proof
Definitions occuring in Statement :
rv-T': rv-T'(n;a;b;c),
rv-T: rv-T(n;a;b;c),
real-vec: ℝ^n,
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rv-T: rv-T(n;a;b;c),
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
not: ¬A,
false: False,
rv-T': rv-T'(n;a;b;c),
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
real-vec-sep: a ≠ b,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
rv-between: a-b-c,
real-vec-between: a-b-c,
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
rgt: x > y,
rge: x ≥ y,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
real-vec-mul: a*X,
real-vec-add: X + Y,
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y,
real-vec-dist: d(x;y),
real-vec-sub: X - Y,
cand: A c∧ B,
real-vec-be: real-vec-be(n;a;b;c),
so_apply: x[s],
so_lambda: λ2x.t[x],
sq_stable: SqStable(P),
real: ℝ,
le: A ≤ B
Lemmas referenced :
real-vec-sep_wf,
not_wf,
rv-T'_wf,
nat_plus_subtype_nat,
real-vec_wf,
nat_plus_wf,
real-vec-add_wf,
real-vec-mul_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
rv-between-small-expand,
rv-between-symmetry,
real-vec-sep-symmetry,
rsub_wf,
radd_wf,
rmul_wf,
i-member_wf,
rooint_wf,
req-vec_wf,
member_rooint_lemma,
rmul_preserves_rless,
rinv_wf2,
rminus_wf,
rless_functionality,
req_transitivity,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
rminus_functionality,
rmul-rinv,
itermAdd_wf,
real_term_value_add_lemma,
radd_functionality,
rmul_functionality,
req_weakening,
rmul-rinv3,
req_inversion,
radd-int,
radd_functionality_wrt_rless2,
radd_functionality_wrt_rless1,
rleq_weakening_rless,
rleq_weakening_equal,
rless_functionality_wrt_implies,
rmul_comm,
rmul-int,
radd-assoc,
rmul-distrib2,
rmul-identity1,
rminus-as-rmul,
rmul-zero-both,
radd-zero-both,
iff_weakening_equal,
rminus-int,
true_wf,
squash_wf,
equal_wf,
radd-preserves-rless,
real_wf,
zero-mul,
rless-implies-rless,
int_seg_wf,
int_seg_properties,
req_functionality,
rmul-distrib1,
rmul-assoc,
rmul_preserves_req,
req_wf,
req-implies-req,
req-vec_weakening,
req-vec_functionality,
real-vec-sub_wf,
real-vec-norm-positive-iff,
radd_comm,
radd-rminus-assoc,
radd-rminus-both,
rminus-rminus,
radd-ac,
rmul-one-both,
rmul_over_rminus,
rmul-distrib,
uiff_transitivity,
radd-preserves-req,
rneq_wf,
rccint_wf,
less_than_wf,
i-member_functionality,
set_wf,
small-reciprocal-real,
rleq-iff-all-rless,
member_rccint_lemma,
rleq_functionality_wrt_implies,
rmul-rdiv-cancel2,
rleq_functionality,
int_term_value_mul_lemma,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
real-vec-dist_wf,
sq_stable__less_than,
rleq-int,
rleq_wf,
rmul_preserves_rleq,
not-real-vec-sep-iff-eq,
rv-between_wf,
rv-between_functionality,
req-iff-not-rneq,
rneq-iff-rabs,
rabs_wf,
rv-between-simple,
int_formula_prop_eq_lemma,
intformeq_wf,
rneq-int,
rless-int-fractions2,
lelt_wf,
false_wf,
rinv-mul-as-rdiv,
rinv-as-rdiv,
trivial-rless-radd,
rabs-difference-bound-iff,
rleq-int-fractions2,
rabs-rmul,
rabs-of-nonneg,
rless_transitivity2,
rless_irreflexivity,
rabs_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
addEquality,
setElimination,
rename,
natural_numberEquality,
independent_isectElimination,
inrFormation,
productElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
minusEquality,
productEquality,
equalitySymmetry,
equalityTransitivity,
promote_hyp,
multiplyEquality,
levelHypothesis,
baseClosed,
imageMemberEquality,
addLevel,
universeEquality,
imageElimination,
inlFormation,
dependent_set_memberEquality,
setEquality,
allFunctionality,
impliesFunctionality,
allLevelFunctionality,
impliesLevelFunctionality,
functionEquality
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}a,b,c:\mBbbR{}\^{}n. (rv-T'(n;a;b;c) {}\mRightarrow{} rv-T(n;a;b;c))
Date html generated:
2017_10_03-AM-11_25_09
Last ObjectModification:
2017_07_28-AM-08_26_50
Theory : reals
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