Nuprl Lemma : last_mklist
∀T:Type. ∀f:ℕ ⟶ T. ∀n:ℕ+. (last(mklist(n;f)) = (f (n - 1)) ∈ T)
Proof
Definitions occuring in Statement :
mklist: mklist(n;f),
last: last(L),
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
apply: f a,
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],
member: t ∈ T,
last: last(L),
uall: ∀[x:A]. B[x],
top: Top,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat_plus: ℕ+,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
nat: ℕ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_plus_wf,
nat_wf,
mklist_length,
nat_plus_subtype_nat,
equal_wf,
squash_wf,
true_wf,
mklist_select,
subtype_rel_dep_function,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
subtract_wf,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
decidable__lt,
lelt_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
functionEquality,
cumulativity,
hypothesisEquality,
universeEquality,
sqequalRule,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
functionExtensionality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}T:Type. \mforall{}f:\mBbbN{} {}\mrightarrow{} T. \mforall{}n:\mBbbN{}\msupplus{}. (last(mklist(n;f)) = (f (n - 1)))
Date html generated:
2017_04_17-AM-07_59_51
Last ObjectModification:
2017_02_27-PM-04_30_22
Theory : list_1
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