Nuprl Lemma : nth_tl-mklist
∀[n:ℕ]. ∀[f:Top]. ∀[k:ℕ]. (nth_tl(k;mklist(n;f)) ~ mklist(n - k;λi.(f (i + k))))
Proof
Definitions occuring in Statement :
mklist: mklist(n;f),
nth_tl: nth_tl(n;as),
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
lambda: λx.A[x],
subtract: n - m,
add: n + m,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
mklist: mklist(n;f),
lt_int: i <z j,
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
decidable: Dec(P),
nth_tl: nth_tl(n;as),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
has-value: (a)↓,
subtype_rel: A ⊆r B,
le_int: i ≤z j
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
subtract-1-ge-0,
istype-nat,
istype-top,
subtract_wf,
nth_tl_nil,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
primrec-unroll,
decidable__le,
mklist-prepend1,
istype-le,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
le_int_wf,
assert_of_le_int,
list_ind_cons_lemma,
list_ind_nil_lemma,
le_wf,
reduce_tl_cons_lemma,
add-associates,
nat_wf,
set_subtype_base,
value-type-has-value,
int-value-type,
exception-not-value,
has-value_wf_base,
is-exception_wf,
primrec0_lemma,
general_arith_equation2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
axiomSqEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
because_Cache,
unionElimination,
equalityElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
Error :equalityIstype,
promote_hyp,
instantiate,
cumulativity,
Error :dependent_set_memberEquality_alt,
intEquality,
addEquality,
sqequalSqle,
divergentSqle,
callbyvalueAdd,
baseClosed,
baseApply,
closedConclusion,
applyEquality,
sqleReflexivity,
addExceptionCases,
axiomSqleEquality,
exceptionSqequal
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:Top]. \mforall{}[k:\mBbbN{}]. (nth\_tl(k;mklist(n;f)) \msim{} mklist(n - k;\mlambda{}i.(f (i + k))))
Date html generated:
2019_06_20-PM-01_31_51
Last ObjectModification:
2019_01_21-PM-09_27_30
Theory : list_1
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