Nuprl Lemma : triple_swap
∀n:ℕ. ∀i,j,k:ℕn. ((¬(i = j ∈ ℤ)) ⇒ (¬(j = k ∈ ℤ)) ⇒ (swap(i;j) = (swap(i;k) o (swap(j;k) o swap(i;k))) ∈ (ℕn ⟶ ℕn)))
Proof
Definitions occuring in Statement :
swap: swap(i;j),
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
nat: ℕ,
swap: swap(i;j),
compose: f o g,
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},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
not_wf,
equal_wf,
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,
intformeq_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
ifthenelse_wf,
lelt_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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
sqequalRule,
lambdaEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
promote_hyp,
instantiate,
independent_functionElimination,
cumulativity
Latex:
\mforall{}n:\mBbbN{}. \mforall{}i,j,k:\mBbbN{}n. ((\mneg{}(i = j)) {}\mRightarrow{} (\mneg{}(j = k)) {}\mRightarrow{} (swap(i;j) = (swap(i;k) o (swap(j;k) o swap(i;k)))))
Date html generated:
2017_10_01-AM-09_52_37
Last ObjectModification:
2017_03_03-PM-00_49_30
Theory : perms_1
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