Nuprl Lemma : flip-conjugation
∀[n:ℕ]. ∀[k:ℕn]. ∀[j:ℕk]. ∀[l:ℕn - k].  ((j, k + l) = ((k, k + l) o ((j, k) o (k, k + l))) ∈ (ℕn ⟶ ℕn))
Proof
Definitions occuring in Statement : 
flip: (i, j)
, 
compose: f o g
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
uall: ∀[x:A]. B[x]
, 
function: x:A ⟶ B[x]
, 
subtract: n - m
, 
add: n + m
, 
natural_number: $n
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
nat: ℕ
, 
int_seg: {i..j-}
, 
flip: (i, j)
, 
compose: f o g
, 
lelt: i ≤ j < k
, 
and: P ∧ Q
, 
uimplies: b supposing a
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
false: False
, 
all: ∀x:A. B[x]
, 
top: Top
, 
prop: ℙ
, 
nequal: a ≠ b ∈ T 
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
bfalse: ff
, 
or: P ∨ Q
, 
sq_type: SQType(T)
, 
guard: {T}
, 
bnot: ¬bb
, 
ifthenelse: if b then t else f fi 
, 
assert: ↑b
, 
ge: i ≥ j 
, 
decidable: Dec(P)
Lemmas referenced : 
int_seg_wf, 
subtract_wf, 
nat_wf, 
eq_int_wf, 
bool_wf, 
full-omega-unsat, 
intformand_wf, 
intformeq_wf, 
itermVar_wf, 
intformless_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_eq_lemma, 
int_term_value_var_lemma, 
int_formula_prop_less_lemma, 
int_formula_prop_wf, 
itermAdd_wf, 
intformle_wf, 
itermConstant_wf, 
int_term_value_add_lemma, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
ifthenelse_wf, 
eqtt_to_assert, 
assert_of_eq_int, 
eqff_to_assert, 
equal_wf, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_subtype_base, 
assert-bnot, 
neg_assert_of_eq_int, 
intformnot_wf, 
int_formula_prop_not_lemma, 
int_seg_properties, 
nat_properties, 
decidable__le, 
decidable__lt, 
itermSubtract_wf, 
int_term_value_subtract_lemma, 
lelt_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
functionExtensionality, 
because_Cache, 
hypothesis, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
natural_numberEquality, 
setElimination, 
rename, 
hypothesisEquality, 
sqequalRule, 
isect_memberEquality, 
axiomEquality, 
dependent_set_memberEquality, 
productElimination, 
independent_isectElimination, 
approximateComputation, 
independent_functionElimination, 
dependent_pairFormation, 
lambdaEquality, 
int_eqEquality, 
intEquality, 
dependent_functionElimination, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
addEquality, 
lambdaFormation, 
unionElimination, 
equalityElimination, 
equalityTransitivity, 
equalitySymmetry, 
promote_hyp, 
instantiate, 
cumulativity
Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[k:\mBbbN{}n].  \mforall{}[j:\mBbbN{}k].  \mforall{}[l:\mBbbN{}n  -  k].    ((j,  k  +  l)  =  ((k,  k  +  l)  o  ((j,  k)  o  (k,  k  +  l))))
Date html generated:
2018_05_21-PM-00_41_12
Last ObjectModification:
2018_05_19-AM-06_48_36
Theory : list_1
Home
Index