Nuprl Lemma : replace-seq-from-succ
∀T:Type. ∀f:ℕ ⟶ T. ∀m:ℕ. ∀k:T.
(0 < m ⇒ (replace-seq-from(f;m;k) = (λx.if x=m - 1 then f x else (replace-seq-from(f;m - 1;k) x)) ∈ (ℕ ⟶ T)))
Proof
Definitions occuring in Statement :
replace-seq-from: replace-seq-from(s;n;k),
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
int_eq: if a=b then c else d,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
replace-seq-from: replace-seq-from(s;n;k),
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
eq_int_wf,
subtract_wf,
assert_of_eq_int,
nat_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
functionExtensionality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
lessCases,
isect_memberFormation,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
int_eqReduceTrueSq,
applyEquality,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
int_eqReduceFalseSq,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll,
functionEquality,
universeEquality
Latex:
\mforall{}T:Type. \mforall{}f:\mBbbN{} {}\mrightarrow{} T. \mforall{}m:\mBbbN{}. \mforall{}k:T.
(0 < m
{}\mRightarrow{} (replace-seq-from(f;m;k) = (\mlambda{}x.if x=m - 1 then f x else (replace-seq-from(f;m - 1;k) x))))
Date html generated:
2017_04_20-AM-07_22_51
Last ObjectModification:
2017_02_27-PM-05_58_17
Theory : continuity
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