Nuprl Lemma : rational-IVT-1
∀a,b:ℤ × ℕ+. ∀f:(ℤ × ℕ+) ⟶ (ℤ × ℕ+).
∀[g:{x:ℝ| x ∈ [ratreal(a), ratreal(b)]} ⟶ ℝ]
∃c:{c:ℝ| c ∈ [ratreal(a), ratreal(b)]} [(g[c] = r0)]
supposing (ratreal(a) ≤ ratreal(b))
∧ (ratreal(f[a]) ≤ r0)
∧ (r0 ≤ ratreal(f[b]))
∧ (∀x,y:{x:ℝ| x ∈ [ratreal(a), ratreal(b)]} . ((x = y) ⇒ (g[x] = g[y])))
∧ (∀r:ℤ × ℕ+. ((ratreal(r) ∈ [ratreal(a), ratreal(b)]) ⇒ (g[ratreal(r)] = ratreal(f[r]))))
Proof
Definitions occuring in Statement :
ratreal: ratreal(r),
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
pi1: fst(t),
pi2: snd(t),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
so_lambda: λ2x.t[x],
sq_stable: SqStable(P),
squash: ↓T,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T},
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
less_than': less_than'(a;b),
ravg: ravg(x;y),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
i-member: r ∈ I,
rccint: [l, u],
label: ...$L... t,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
nat: ℕ,
ge: i ≥ j ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
rbetween: x≤y≤z,
int_upper: {i...},
rless: x < y,
sq_exists: ∃x:A [B[x]],
int-to-real: r(n),
rinv: rinv(x),
mu-ge: mu-ge(f;n),
lt_int: i <z j,
absval: |i|,
eq_int: (i =z j),
accelerate: accelerate(k;f),
reg-seq-inv: reg-seq-inv(x),
reg-seq-adjust: reg-seq-adjust(n;x),
real: ℝ,
req: x = y,
rfun: I ⟶ℝ,
r-ap: f(x)
Lemmas referenced :
rleq_wf,
ratreal_wf,
int-to-real_wf,
req_wf,
i-member_wf,
rccint_wf,
real_wf,
istype-int,
nat_plus_wf,
member_rccint_lemma,
istype-void,
rat-nat-div_wf,
ratadd_wf,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
set-value-type,
equal_wf,
product-value-type,
ravg-dist-when-rleq,
sq_stable__rleq,
ravg-weak-between,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
nequal_wf,
radd_wf,
ravg_wf,
rdiv_wf,
rless-int,
rless_wf,
req_weakening,
rsub_wf,
rmul_wf,
req_functionality,
req_transitivity,
ratreal-rat-nat-div,
int-rdiv_functionality,
ratreal-ratadd,
int-rdiv-req,
iff_weakening_uiff,
rleq_functionality,
req_inversion,
squash_wf,
true_wf,
rsub_functionality,
rleq_transitivity,
sq_stable__req,
subtype_rel_self,
product_subtype_base,
set_subtype_base,
less_than_wf,
interval_wf,
iff_weakening_equal,
rmul_preserves_req,
rinv_wf2,
rat-zero-cases,
rleq_weakening,
rleq_weakening_equal,
pi1_wf_top,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
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,
rmul_functionality,
pi2_wf,
primrec_wf,
int_seg_wf,
istype-nat,
subtype_rel_product,
nat_wf,
nat_properties,
decidable__le,
intformand_wf,
intformle_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
istype-le,
rnexp_wf,
primrec-unroll,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
assert_wf,
istype-universe,
add-subtract-cancel,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
subtract_wf,
ge_wf,
req_witness,
rnexp_zero_lemma,
primrec0_lemma,
subtract-1-ge-0,
int_term_value_subtract_lemma,
rmul_comm,
rmul_assoc,
rnexp_step,
set_wf,
istype-top,
top_wf,
common-limit-squeeze,
le_witness_for_triv,
rinv-exp-converges-ext,
exp_wf2,
mul_bounds_1b,
exp_wf_nat_plus,
rnexp-positive,
rdiv_functionality,
req-int,
nat_plus_properties,
int_term_value_mul_lemma,
converges-to_functionality,
rnexp_functionality,
rinv-as-rdiv,
rnexp-rdiv,
rnexp-one,
rnexp-int,
const-rmul-limit-with-bound,
ratbound_wf,
ratsub_wf,
rabs_wf,
rleq-ratbound,
ratreal-ratsub,
rabs_functionality,
rmul-zero,
rmul-nonneg-case1,
rnexp-nonneg,
rleq_weakening_rless,
rleq-implies-rleq,
rleq-limit-constant,
constant-rleq-limit,
function-limit,
rfun_wf,
r-ap_wf,
req-iff-not-rneq,
rless_transitivity1,
rless_irreflexivity,
rneq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
sqequalRule,
productIsType,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
natural_numberEquality,
functionIsType,
because_Cache,
setElimination,
rename,
dependent_set_memberEquality_alt,
setIsType,
inhabitedIsType,
productElimination,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
productEquality,
intEquality,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
cutEval,
equalityIstype,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
cumulativity,
sqequalBase,
closedConclusion,
inrFormation_alt,
independent_pairFormation,
promote_hyp,
setEquality,
applyLambdaEquality,
universeEquality,
dependent_pairEquality_alt,
independent_pairEquality,
int_eqEquality,
functionExtensionality,
addEquality,
equalityElimination,
intWeakElimination,
functionIsTypeImplies,
multiplyEquality,
dependent_set_memberFormation_alt
Latex:
\mforall{}a,b:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}. \mforall{}f:(\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}) {}\mrightarrow{} (\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}).
\mforall{}[g:\{x:\mBbbR{}| x \mmember{} [ratreal(a), ratreal(b)]\} {}\mrightarrow{} \mBbbR{}]
\mexists{}c:\{c:\mBbbR{}| c \mmember{} [ratreal(a), ratreal(b)]\} [(g[c] = r0)]
supposing (ratreal(a) \mleq{} ratreal(b))
\mwedge{} (ratreal(f[a]) \mleq{} r0)
\mwedge{} (r0 \mleq{} ratreal(f[b]))
\mwedge{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [ratreal(a), ratreal(b)]\} . ((x = y) {}\mRightarrow{} (g[x] = g[y])))
\mwedge{} (\mforall{}r:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}. ((ratreal(r) \mmember{} [ratreal(a), ratreal(b)]) {}\mRightarrow{} (g[ratreal(r)] = ratreal(f[r]))))
Date html generated:
2019_10_30-AM-10_00_13
Last ObjectModification:
2019_01_11-PM-03_39_13
Theory : reals
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