Nuprl Lemma : firstn-mklist
∀[m,n:ℕ]. ∀[f:ℕm ⟶ Top]. (firstn(n;mklist(m;f)) ~ mklist(imin(n;m);f))
Proof
Definitions occuring in Statement :
mklist: mklist(n;f),
imin: imin(a;b),
firstn: firstn(n;as),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
function: x:A ⟶ B[x],
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
sq_type: SQType(T),
mklist: mklist(n;f),
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
firstn: firstn(n;as),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
int_seg_wf,
top_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
int_seg_properties,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
first0,
mklist_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
primrec0_lemma,
decidable__lt,
lelt_wf,
firstn_decomp2,
int_seg_subtype_nat,
false_wf,
mklist_length,
primrec-unroll,
eq_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
list_ind_nil_lemma,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
firstn_append,
cons_wf,
nil_wf,
subtract-add-cancel,
length_wf,
select-mklist,
imin_unfold,
set_subtype_base,
iff_weakening_equal,
le_int_wf,
assert_of_le_int,
firstn_all
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
functionEquality,
unionElimination,
because_Cache,
productElimination,
instantiate,
cumulativity,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
applyEquality,
equalityElimination,
baseClosed,
impliesFunctionality,
promote_hyp,
addEquality,
isect_memberFormation,
sqequalIntensionalEquality
Latex:
\mforall{}[m,n:\mBbbN{}]. \mforall{}[f:\mBbbN{}m {}\mrightarrow{} Top]. (firstn(n;mklist(m;f)) \msim{} mklist(imin(n;m);f))
Date html generated:
2017_04_17-AM-08_01_29
Last ObjectModification:
2017_02_27-PM-04_32_48
Theory : list_1
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