Nuprl Lemma : extend_permf_over_comp
∀n:ℕ. ∀f,g:ℕn ⟶ ℕn. (extend_permf(g o f;n) = (extend_permf(g;n) o extend_permf(f;n)) ∈ (ℕn + 1 ⟶ ℕn + 1))
Proof
Definitions occuring in Statement :
extend_permf: extend_permf(pf;n),
compose: f o g,
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 :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
extend_permf: extend_permf(pf;n),
compose: f o g,
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 ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
nequal: a ≠ b ∈ T ,
not: ¬A,
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
prop: ℙ,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
subtype_rel: A ⊆r B
Lemmas referenced :
int_seg_wf,
istype-nat,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformnot_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
decidable__lt,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
istype-le,
istype-less_than
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesis,
inhabitedIsType,
hypothesisEquality,
functionIsType,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
lambdaEquality_alt,
addEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
approximateComputation,
int_eqEquality,
Error :memTop,
independent_pairFormation,
imageElimination,
applyEquality,
dependent_set_memberEquality_alt,
productIsType,
applyLambdaEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}f,g:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n. (extend\_permf(g o f;n) = (extend\_permf(g;n) o extend\_permf(f;n)))
Date html generated:
2020_05_20-AM-09_35_26
Last ObjectModification:
2020_01_08-PM-06_17_03
Theory : perms_1
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