Nuprl Lemma : app_permf_id
∀m,n:ℕ. (app_permf(m;n;Id{ℕm};Id{ℕn}) = Id{ℕm + n} ∈ (ℕm + n ⟶ ℕm + n))
Proof
Definitions occuring in Statement :
app_permf: app_permf(m;n;p;q),
tidentity: Id{T},
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
tidentity: Id{T},
app_permf: app_permf(m;n;p;q),
identity: Id,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
int_seg: {i..j-},
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
int_seg_wf,
nat_wf,
lt_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
less_than_wf,
le_int_wf,
le_wf,
bnot_wf,
subtract-add-cancel,
int_seg_properties,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
intformand_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
lelt_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
lambdaEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
baseClosed,
because_Cache,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality,
independent_pairFormation,
equalityElimination,
independent_functionElimination
Latex:
\mforall{}m,n:\mBbbN{}. (app\_permf(m;n;Id\{\mBbbN{}m\};Id\{\mBbbN{}n\}) = Id\{\mBbbN{}m + n\})
Date html generated:
2017_10_01-AM-09_52_48
Last ObjectModification:
2017_03_03-PM-00_47_32
Theory : perms_1
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