Nuprl Lemma : swap_order_2
∀n:ℕ. ∀i,j:ℕn. ((swap(i;j) o swap(i;j)) = Id ∈ (ℕn ⟶ ℕn))
Proof
Definitions occuring in Statement :
swap: swap(i;j),
compose: f o g,
identity: Id,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[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,
uall: ∀[x:A]. B[x],
nat: ℕ,
identity: Id,
compose: f o g,
swap: swap(i;j),
int_seg: {i..j-},
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
guard: {T},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
int_seg_wf,
nat_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
int_seg_properties,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
ifthenelse_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
functionExtensionality,
sqequalRule,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}i,j:\mBbbN{}n. ((swap(i;j) o swap(i;j)) = Id)
Date html generated:
2017_10_01-AM-09_52_24
Last ObjectModification:
2017_03_03-PM-00_47_20
Theory : perms_1
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