Nuprl Lemma : derivative-id
∀[I:Interval]. λx.r1 = d(x)/dx on I
Proof
Definitions occuring in Statement :
derivative: λz.g[z] = d(f[x])/dx on I,
interval: Interval,
int-to-real: r(n),
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
rless: x < y,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
prop: ℙ,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
and: P ∧ Q,
member: t ∈ T,
sq_exists: ∃x:{A| B[x]},
all: ∀x:A. B[x],
derivative: λz.g[z] = d(f[x])/dx on I,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
nat: ℕ,
subtype_rel: A ⊆r B,
rsub: x - y,
absval: |i|,
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
real: ℝ
Lemmas referenced :
rleq_functionality,
rmul-nonneg-case1,
rleq-int-fractions2,
sq_stable__less_than,
sq_stable__icompact,
decidable__le,
intformle_wf,
itermMultiply_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
zero-rleq-rabs,
rabs-abs,
radd-zero-both,
rminus-zero,
rmul-zero-both,
radd-int,
rmul_functionality,
rminus-as-rmul,
rmul-distrib2,
rmul-identity1,
radd-ac,
radd-assoc,
req_inversion,
rminus-radd,
rmul-one-both,
req_weakening,
rmul_over_rminus,
rmul-distrib,
req_transitivity,
rminus_functionality,
radd_functionality,
rabs_functionality,
req_functionality,
uiff_transitivity,
req_wf,
radd_wf,
rminus_wf,
absval_wf,
nat_wf,
req-int,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
int-to-real_wf,
rless-int,
rleq_wf,
rabs_wf,
rsub_wf,
i-member_wf,
i-approx_wf,
real_wf,
rless_wf,
all_wf,
less_than_wf,
rmul_wf,
rdiv_wf,
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,
set_wf,
nat_plus_wf,
icompact_wf,
interval_wf
Rules used in proof :
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
inrFormation,
independent_isectElimination,
dependent_set_memberEquality,
functionEquality,
because_Cache,
lambdaEquality,
productEquality,
rename,
setElimination,
baseClosed,
hypothesisEquality,
imageMemberEquality,
introduction,
independent_pairFormation,
sqequalRule,
independent_functionElimination,
productElimination,
dependent_functionElimination,
hypothesis,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
cut,
dependent_set_memberFormation,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
applyEquality,
addEquality,
minusEquality,
multiplyEquality,
imageElimination
Latex:
\mforall{}[I:Interval]. \mlambda{}x.r1 = d(x)/dx on I
Date html generated:
2016_05_18-AM-10_06_01
Last ObjectModification:
2016_01_17-AM-00_39_15
Theory : reals
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