Nuprl Lemma : rotate-as-flips
∀n:ℕ. ∃flips:(ℕn × ℕn) List. (rot(n) = compose-flips(flips) ∈ (ℕn ⟶ ℕn))
Proof
Definitions occuring in Statement :
compose-flips: compose-flips(flips),
rotate: rot(n),
list: T List,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
and: P ∧ Q,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
less_than': less_than'(a;b),
le: A ≤ B,
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
subtract: n - m,
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
sq_type: SQType(T),
compose: f o g,
rotate: rot(n),
flip: (i, j),
true: True,
squash: ↓T,
less_than: a < b,
eq_int: (i =z j),
compose-flips: compose-flips(flips),
nil: [],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
colength: colength(L),
cons: [a / b],
ge: i ≥ j ,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs
Lemmas referenced :
nat_wf,
rotate_wf,
equal_wf,
primrec-wf2,
less_than_wf,
set_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
compose-flips_wf,
equal-wf-base-T,
subtract_wf,
int_seg_wf,
list_wf,
exists_wf,
false_wf,
lelt_wf,
decidable__equal_int,
int_seg_properties,
nil_wf,
decidable__lt,
add-member-int_seg2,
int_term_value_add_lemma,
itermAdd_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
int_subtype_base,
subtype_base_sq,
int_seg_cases,
int_seg_subtype,
reduce_nil_lemma,
map_nil_lemma,
cons_wf,
subtype_rel_product,
subtype_rel_list,
append_wf,
set_subtype_base,
spread_cons_lemma,
product_subtype_list,
list-cases,
less_than_irreflexivity,
less_than_transitivity1,
colength_wf_list,
equal-wf-T-base,
ge_wf,
nat_properties,
reduce_cons_lemma,
map_cons_lemma,
list_ind_nil_lemma,
list_ind_cons_lemma,
ifthenelse_wf,
nequal_wf
Rules used in proof :
functionEquality,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_set_memberEquality,
dependent_functionElimination,
lambdaEquality,
sqequalRule,
because_Cache,
hypothesis,
hypothesisEquality,
natural_numberEquality,
productEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
setElimination,
rename,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
baseClosed,
applyEquality,
productElimination,
functionExtensionality,
addEquality,
promote_hyp,
equalityElimination,
equalitySymmetry,
equalityTransitivity,
cumulativity,
instantiate,
hypothesis_subsumption,
imageMemberEquality,
independent_pairEquality,
imageElimination,
applyLambdaEquality,
axiomEquality,
intWeakElimination
Latex:
\mforall{}n:\mBbbN{}. \mexists{}flips:(\mBbbN{}n \mtimes{} \mBbbN{}n) List. (rot(n) = compose-flips(flips))
Date html generated:
2018_05_21-PM-00_42_23
Last ObjectModification:
2017_12_10-PM-03_29_29
Theory : list_1
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