Nuprl Lemma : extensional-discrete-real-fun-is-constant
∀a,b:ℝ. ∀f:{x:ℝ| x ∈ [a, b]} ⟶ ℤ.
∀x,y:{x:ℝ| x ∈ [a, b]} . ((f x) = (f y) ∈ ℤ) supposing ∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ ((f x) = (f y) ∈ ℤ))
Proof
Definitions occuring in Statement :
rccint: [l, u],
i-member: r ∈ I,
req: x = y,
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
top: Top,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
guard: {T},
rfun: I ⟶ℝ,
prop: ℙ,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
real-fun: real-fun(f;a;b),
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
real-cont: real-cont(f;a;b),
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
false: False,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
rneq: x ≠ y,
rev_implies: P ⇐ Q,
le: A ≤ B,
rleq: x ≤ y,
rnonneg: rnonneg(x),
subtype_rel: A ⊆r B,
nat: ℕ,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
sq_type: SQType(T),
real: ℝ,
sq_stable: SqStable(P),
rbetween: x≤y≤z,
rev_uimplies: rev_uimplies(P;Q),
i-member: r ∈ I,
rccint: [l, u],
label: ...$L... t,
full-partition: full-partition(I;p),
partition: partition(I)
Lemmas referenced :
real-fun-iff-continuous,
member_rccint_lemma,
rleq_transitivity,
int-to-real_wf,
real_wf,
rleq_wf,
i-member_wf,
rccint_wf,
req-int,
req_wf,
set_wf,
all_wf,
equal_wf,
less_than_wf,
rabs_wf,
rsub_wf,
decidable__equal_int,
subtract-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
false_wf,
subtract_wf,
rdiv_wf,
rless-int,
rless_wf,
rleq_functionality,
rabs_functionality,
rsub-int,
req_weakening,
rmul_preserves_rleq2,
rleq-int,
less_than'_wf,
rmul_wf,
nat_plus_wf,
absval_wf,
nat_wf,
itermMultiply_wf,
rinv_wf2,
req_transitivity,
squash_wf,
true_wf,
rabs-int,
rmul-int,
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,
absval_unfold,
decidable__lt,
top_wf,
intformless_wf,
intformle_wf,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
itermMinus_wf,
int_term_value_minus_lemma,
partition-exists,
rccint-icompact,
list_set_type,
full-partition_wf,
full-partition-point-member,
mesh-property,
int_seg_properties,
length_wf,
int_seg_wf,
le_wf,
nat_properties,
ge_wf,
decidable__le,
select_wf,
nat_plus_properties,
subtype_base_sq,
int_subtype_base,
sq_stable__less_than,
adjacent-full-partition-points,
lelt_wf,
subtract-add-cancel,
rbetween_wf,
partition-mesh_wf,
rabs-of-nonneg,
sq_stable__rleq,
iff_weakening_equal,
length_of_cons_lemma,
right_endpoint_rccint_lemma,
add_nat_plus,
length_wf_nat,
append_wf,
cons_wf,
nil_wf,
length-append,
length_of_nil_lemma,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isect_memberFormation,
setElimination,
rename,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_isectElimination,
productElimination,
isectElimination,
lambdaEquality,
applyEquality,
functionExtensionality,
setEquality,
productEquality,
independent_functionElimination,
dependent_set_memberEquality,
axiomEquality,
because_Cache,
functionEquality,
intEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
unionElimination,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
inrFormation,
independent_pairEquality,
minusEquality,
multiplyEquality,
imageElimination,
lessCases,
sqequalAxiom,
intWeakElimination,
instantiate,
cumulativity,
addEquality,
universeEquality,
applyLambdaEquality
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}f:\{x:\mBbbR{}| x \mmember{} [a, b]\} {}\mrightarrow{} \mBbbZ{}.
\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [a, b]\} . ((f x) = (f y))
supposing \mforall{}x,y:\{x:\mBbbR{}| x \mmember{} [a, b]\} . ((x = y) {}\mRightarrow{} ((f x) = (f y)))
Date html generated:
2017_10_03-AM-09_59_57
Last ObjectModification:
2017_06_01-PM-04_32_03
Theory : reals
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