Nuprl Lemma : flip-generators
∀n:ℕ
∀i,j:ℕn. ∃L:𝔹 List. ((i, j) = reduce(λi,g. (if i then rot(n) else (0, 1) fi o g);λx.x;L) ∈ (ℕn ⟶ ℕn)) supposing 1 <\000C n
Proof
Definitions occuring in Statement :
flip: (i, j),
rotate: rot(n),
reduce: reduce(f;k;as),
list: T List,
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
nat: ℕ,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
compose: f o g,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
pi1: fst(t),
concat: concat(ll),
cons: [a / b],
colength: colength(L),
nil: [],
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
flip-adjacent,
member-less_than,
int_seg_wf,
istype-less_than,
istype-nat,
flip-conjugate-rotate,
subtract_wf,
append_wf,
bool_wf,
primrec_wf,
list_wf,
int_seg_subtype_nat,
istype-false,
nil_wf,
cons_wf,
btrue_wf,
bfalse_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-le,
set_subtype_base,
lelt_wf,
int_subtype_base,
reduce_wf,
eqtt_to_assert,
compose_wf,
rotate_wf,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
assert-bnot,
flip_wf,
decidable__lt,
equal-wf-T-base,
le_wf,
reduce_cons_lemma,
reduce_nil_lemma,
reduce-append,
ge_wf,
primrec0_lemma,
fun_exp0_lemma,
subtract-1-ge-0,
primrec-unroll,
lt_int_wf,
equal-wf-base,
assert_wf,
less_than_wf,
le_int_wf,
bnot_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
fun_exp_wf,
subtype_rel_self,
iff_weakening_equal,
fun_exp_add_apply1,
add-associates,
add-swap,
add-commutes,
zero-add,
uiff_transitivity,
assert_of_lt_int,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
concat_wf,
map_wf,
list-cases,
map_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
intformeq_wf,
int_formula_prop_eq_lemma,
spread_cons_lemma,
decidable__equal_int,
itermAdd_wf,
int_term_value_add_lemma,
map_cons_lemma,
equal-wf-base-T
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
Error :isect_memberFormation_alt,
isectElimination,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
promote_hyp,
productElimination,
Error :inhabitedIsType,
Error :universeIsType,
Error :dependent_pairFormation_alt,
because_Cache,
applyEquality,
sqequalRule,
independent_pairFormation,
Error :lambdaEquality_alt,
Error :dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
independent_functionElimination,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
baseApply,
closedConclusion,
baseClosed,
intEquality,
functionEquality,
equalityElimination,
Error :equalityIsType4,
instantiate,
cumulativity,
Error :productIsType,
Error :equalityIsType1,
sqequalBase,
hyp_replacement,
applyLambdaEquality,
intWeakElimination,
axiomEquality,
Error :functionIsTypeImplies,
Error :functionExtensionality_alt,
imageElimination,
universeEquality,
imageMemberEquality,
Error :functionIsType,
functionExtensionality,
hypothesis_subsumption
Latex:
\mforall{}n:\mBbbN{}
\mforall{}i,j:\mBbbN{}n. \mexists{}L:\mBbbB{} List. ((i, j) = reduce(\mlambda{}i,g. (if i then rot(n) else (0, 1) fi o g);\mlambda{}x.x;L))
supposing 1 < n
Date html generated:
2019_06_20-PM-01_36_43
Last ObjectModification:
2018_11_22-PM-10_09_18
Theory : list_1
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