Nuprl Lemma : seq-adjoin-is-seq-add
∀[n:ℕ]. ∀[s:ℕn ⟶ ℕ]. ∀[m:ℕ]. (s.m@n = s++m ∈ (ℕn + 1 ⟶ ℕ))
Proof
Definitions occuring in Statement :
seq-adjoin: s++t,
seq-add: s.x@n,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
seq-adjoin: s++t,
seq-add: s.x@n,
seq-append: seq-append(n;m;s1;s2),
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
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,
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
lt_int_wf,
assert_of_lt_int,
top_wf,
less_than_wf,
int_seg_properties,
nat_properties,
decidable__equal_int,
int_seg_wf,
lelt_wf,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
le_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
intformnot_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
neg_assert_of_eq_int,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
int_eqReduceTrueSq,
lessCases,
axiomSqEquality,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
addEquality,
dependent_functionElimination,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
promote_hyp,
instantiate,
cumulativity,
int_eqReduceFalseSq,
axiomEquality,
functionEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}]. \mforall{}[m:\mBbbN{}]. (s.m@n = s++m)
Date html generated:
2019_06_20-PM-02_54_48
Last ObjectModification:
2018_08_20-PM-09_34_23
Theory : continuity
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