Nuprl Lemma : swap_id
∀n:ℕ. ∀i,j:ℕn. ((i = j ∈ ℤ) ⇒ (swap(i;j) = Id ∈ (ℕn ⟶ ℕn)))
Proof
Definitions occuring in Statement :
swap: swap(i;j),
identity: Id,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
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,
identity: Id,
swap: swap(i;j),
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
int_seg_wf,
nat_wf,
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,
ifthenelse_wf,
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
lambdaEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
because_Cache,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
equalityTransitivity,
baseClosed,
equalityElimination,
independent_functionElimination,
impliesFunctionality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}i,j:\mBbbN{}n. ((i = j) {}\mRightarrow{} (swap(i;j) = Id))
Date html generated:
2017_10_01-AM-09_52_29
Last ObjectModification:
2017_03_03-PM-00_47_18
Theory : perms_1
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