Nuprl Lemma : rational-IVT
∀a,b:ℝ. ∀f:(ℤ × ℕ+) ⟶ (ℤ × ℕ+).
∀[g:{x:ℝ| x ∈ [a, b]} ⟶ ℝ]
∃c:{c:ℝ| c ∈ (a, b)} [(g[c] = r0)]
supposing (a < b)
∧ ((g[a] * g[b]) < r0)
∧ (∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ (g[x] = g[y])))
∧ (∀r:ℤ × ℕ+. ((ratreal(r) ∈ [a, b]) ⇒ (g[ratreal(r)] = ratreal(f[r]))))
Proof
Definitions occuring in Statement :
ratreal: ratreal(r),
rooint: (l, u),
rccint: [l, u],
i-member: r ∈ I,
rless: x < y,
req: x = y,
rmul: a * b,
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],
top: Top,
cand: A c∧ B,
guard: {T},
implies: P ⇒ Q,
real: ℝ,
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,
rless: x < y,
sq_exists: ∃x:A [B[x]],
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rational-upper-approx: above x within 1/n,
has-value: (a)↓,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rational-lower-approx: (below x within 1/n),
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
req_int_terms: t1 ≡ t2,
int_upper: {i...},
sq_stable: SqStable(P),
rge: x ≥ y,
rdiv: (x/y),
rgt: x > y,
le: A ≤ B,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
sq_type: SQType(T),
pi1: fst(t),
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
i-member: r ∈ I,
rccint: [l, u]
Lemmas referenced :
rless_wf,
rmul_wf,
member_rccint_lemma,
istype-void,
rleq_weakening_equal,
rleq_weakening_rless,
rleq_wf,
int-to-real_wf,
i-member_wf,
rccint_wf,
req_wf,
istype-int,
nat_plus_wf,
ratreal_wf,
real_wf,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_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_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
rdiv_wf,
mul_nat_plus,
rless-int,
rational-upper-approx_wf,
value-type-has-value,
int-value-type,
int-rdiv_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
nequal_wf,
req_weakening,
req-int,
decidable__equal_int,
subtract_wf,
rational-lower-approx_wf,
req_functionality,
ratreal-req,
int-rdiv-req,
rdiv_functionality,
rless-implies-rless,
rminus_wf,
rmul-is-positive,
rsub_wf,
itermSubtract_wf,
itermMinus_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
function-values-near-same-sign,
rccint-icompact,
rabs-positive-iff,
small-reciprocal-real,
imax_wf,
imax_nat_plus,
multiply_nat_plus,
imax_ub,
sq_stable__less_than,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
rational-upper-approx-property,
int_upper_properties,
rational-lower-approx-property,
rleq-int-fractions,
rless_transitivity2,
radd_wf,
radd-preserves-rleq,
rinv_wf2,
itermAdd_wf,
rleq_functionality_wrt_implies,
rleq_functionality,
rleq-implies-rleq,
real_term_value_add_lemma,
rleq_transitivity,
rabs-difference-bound-rleq,
sq_stable__rless,
trivial-rsub-rleq,
rsub_functionality_wrt_rleq,
req_transitivity,
rinv-as-rdiv,
rminus_functionality,
rmul_reverses_rleq_iff,
trivial-rleq-radd,
rmul-negative-iff,
subtype_rel_sets_simple,
less_than_wf,
le_wf,
istype-assert,
le_int_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
lt_int_wf,
ratmul_wf,
bfalse_wf,
assert_wf,
assert_of_le_int,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert_of_lt_int,
mul_cancel_in_le,
multiply-is-int-iff,
subtract-is-int-iff,
add-is-int-iff,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
false_wf,
req_inversion,
ratreal-negative,
rless_functionality,
ratreal-ratmul,
i-member_functionality,
rmul_functionality,
mu-ge_wf,
istype-int_upper,
istype-false,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
sq_stable__and,
sq_stable__le,
le_witness_for_triv,
mu-ge-property,
set_subtype_base,
pi1_wf_top,
subtype_rel_product,
top_wf,
rless-int-fractions3,
set-value-type,
equal_wf,
mul_preserves_le,
nat_plus_subtype_nat,
rmul-is-negative,
rat-zero-cases,
sq_stable__rleq,
rless_transitivity1,
rless_irreflexivity,
rmul_reverses_rleq,
rational-fun-zero_wf,
subtype_rel_dep_function,
member_rooint_lemma,
sq_stable__req,
rleq_weakening,
rabs_wf,
rabs-of-nonpos,
rleq-iff-not-rless,
trivial-rless-radd,
rless_functionality_wrt_implies,
rabs-of-nonneg,
radd-preserves-rless,
int-rat-mul_wf,
int-rmul_wf,
ratreal-int-rat-mul,
int-rmul-req,
subtype_rel_self,
rminus-as-rmul,
radd-preserves-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalRule,
productIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
because_Cache,
independent_isectElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
natural_numberEquality,
functionIsType,
setIsType,
inhabitedIsType,
setElimination,
rename,
independent_pairEquality,
addEquality,
multiplyEquality,
productElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
closedConclusion,
inrFormation_alt,
callbyvalueReduce,
intEquality,
equalityIstype,
baseApply,
baseClosed,
sqequalBase,
equalitySymmetry,
imageElimination,
imageMemberEquality,
inlFormation_alt,
equalityTransitivity,
applyLambdaEquality,
minusEquality,
promote_hyp,
instantiate,
cumulativity,
productEquality,
pointwiseFunctionality,
functionIsTypeImplies,
cutEval,
setEquality,
dependent_set_memberFormation_alt
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}f:(\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}) {}\mrightarrow{} (\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}).
\mforall{}[g:\{x:\mBbbR{}| x \mmember{} [a, b]\} {}\mrightarrow{} \mBbbR{}]
\mexists{}c:\{c:\mBbbR{}| c \mmember{} (a, b)\} [(g[c] = r0)]
supposing (a < b)
\mwedge{} ((g[a] * g[b]) < r0)
\mwedge{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [a, b]\} . ((x = y) {}\mRightarrow{} (g[x] = g[y])))
\mwedge{} (\mforall{}r:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}. ((ratreal(r) \mmember{} [a, b]) {}\mRightarrow{} (g[ratreal(r)] = ratreal(f[r]))))
Date html generated:
2019_10_30-AM-10_04_01
Last ObjectModification:
2019_01_14-PM-00_29_17
Theory : reals
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