Nuprl Lemma : firstn_upto
∀[n,m:ℕ]. (firstn(n;upto(m)) ~ if n ≤z m then upto(n) else upto(m) fi )
Proof
Definitions occuring in Statement :
upto: upto(n),
firstn: firstn(n;as),
le_int: i ≤z j,
nat: ℕ,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
sqequal: s ~ t
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
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,
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T},
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
le: A ≤ B,
subtype_rel: A ⊆r B
Lemmas referenced :
le_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
le_wf,
lt_int_wf,
less_than_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
equal_wf,
nat_wf,
upto_decomp,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
lelt_wf,
firstn_append,
upto_wf,
subtype_rel_list,
int_seg_wf,
top_wf,
map_wf,
subtract_wf,
length_upto,
length_wf,
firstn_all,
decidable__le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
baseClosed,
because_Cache,
isect_memberFormation,
lambdaFormation,
unionElimination,
equalityElimination,
independent_functionElimination,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
sqequalAxiom,
isect_memberEquality,
dependent_set_memberEquality,
independent_pairFormation,
addEquality,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
applyEquality
Latex:
\mforall{}[n,m:\mBbbN{}]. (firstn(n;upto(m)) \msim{} if n \mleq{}z m then upto(n) else upto(m) fi )
Date html generated:
2017_04_17-AM-08_00_55
Last ObjectModification:
2017_02_27-PM-04_32_07
Theory : list_1
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