Nuprl Lemma : flip-adjacent
∀n:ℕ. ∀i,j:ℕn. ∃L:ℕn - 1 List. ((i, j) = reduce(λi,g. ((i, i + 1) o g);λx.x;L) ∈ (ℕn ⟶ ℕn))
Proof
Definitions occuring in Statement :
flip: (i, j),
reduce: reduce(f;k;as),
list: T List,
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
subtract: n - m,
uiff: uiff(P;Q),
or: P ∨ Q,
decidable: Dec(P),
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
and: P ∧ Q,
lelt: i ≤ j < k,
ge: i ≥ j ,
nat: ℕ,
int_seg: {i..j-},
guard: {T},
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda3,
append: as @ bs,
compose: f o g,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
le: A ≤ B,
less_than: a < b,
flip: (i, j),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
istype-nat,
equal_wf,
exists_wf,
le_wf,
all_wf,
primrec-wf2,
add-member-int_seg2,
reduce_wf,
istype-less_than,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__lt,
flip_wf,
compose_wf,
list_wf,
decidable__le,
int_seg_wf,
subtract_wf,
istype-le,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_formula_prop_le_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermSubtract_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
int_seg_properties,
subtract-add-cancel,
nil_wf,
cons_wf,
append_wf,
reduce_nil_lemma,
reduce-append,
reduce_cons_lemma,
list_ind_nil_lemma,
list_ind_cons_lemma,
flip-conjugation1,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
decidable__equal_int,
int_subtype_base,
lelt_wf,
set_subtype_base,
subtype_base_sq,
list_subtype_base,
equal-wf-base-T,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
ifthenelse_wf,
flip_identity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
hypothesis,
inhabitedIsType,
setIsType,
functionEquality,
dependent_set_memberEquality_alt,
closedConclusion,
equalityIstype,
productIsType,
functionIsType,
unionElimination,
because_Cache,
universeIsType,
independent_pairFormation,
sqequalRule,
voidElimination,
isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
productElimination,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
isectElimination,
sqequalHypSubstitution,
thin,
addEquality,
equalitySymmetry,
equalityTransitivity,
applyLambdaEquality,
hyp_replacement,
applyEquality,
functionExtensionality_alt,
equalityIsType1,
imageElimination,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
intEquality,
cumulativity,
sqequalBase,
baseApply,
Error :memTop,
functionExtensionality,
equalityElimination,
promote_hyp
Latex:
\mforall{}n:\mBbbN{}. \mforall{}i,j:\mBbbN{}n. \mexists{}L:\mBbbN{}n - 1 List. ((i, j) = reduce(\mlambda{}i,g. ((i, i + 1) o g);\mlambda{}x.x;L))
Date html generated:
2020_05_19-PM-09_44_10
Last ObjectModification:
2020_01_04-PM-08_18_51
Theory : list_1
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