Nuprl Lemma : rng_lsum-partial-permutations
∀[r:CRng]. ∀[n:{2...}]. ∀[i:ℕn]. ∀[F:(ℕn ⟶ ℕn) ⟶ |r|].
(Σ{r} p ∈ partial-permutations-list(n;i). F[p]
= Σ{r} f ∈ permutations-list(n - 1). F[(i, n - 1) o extend-injection(n - 1;f)]
∈ |r|)
Proof
Definitions occuring in Statement :
rng_lsum: Σ{r} x ∈ as. f[x],
partial-permutations-list: partial-permutations-list(n;i),
permutations-list: permutations-list(n),
extend-injection: extend-injection(a;f),
flip: (i, j),
compose: f o g,
int_upper: {i...},
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
equal: s = t ∈ T,
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
injection: A →⟶ B,
int_seg: {i..j-},
int_upper: {i...},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
nat: ℕ,
subtype_rel: A ⊆r B,
crng: CRng,
so_lambda: λ2x.t[x],
rng: Rng,
so_apply: x[s],
istype: istype(T),
nat_plus: ℕ+,
le: A ≤ B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
partial-permutations-list: partial-permutations-list(n;i),
less_than': less_than'(a;b),
uiff: uiff(P;Q),
sq_type: SQType(T),
inject: Inj(A;B;f),
cand: A c∧ B,
compose: f o g,
extend-injection: extend-injection(a;f),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
flip: (i, j),
nequal: a ≠ b ∈ T ,
squash: ↓T,
less_than: a < b
Lemmas referenced :
compose-injections,
flip-injection,
subtract_wf,
int_seg_properties,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
decidable__lt,
istype-le,
istype-less_than,
flip_wf,
inject_wf,
int_seg_wf,
extend-injection_wf,
subtype_rel_self,
injection_wf,
rng_lsum_map,
subtype_rel_dep_function,
rng_car_wf,
permutations-list_wf,
rng_lsum_functionality_wrt_permutation,
map_wf,
partial-permutations-list_wf,
less_than_transitivity2,
istype-int_upper,
crng_wf,
permutation-when-no_repeats,
l_member_wf,
no_repeats-partial-permutations-list,
member_filter,
eq_int_wf,
upper_subtype_nat,
istype-false,
assert_of_eq_int,
member_map,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
lelt_wf,
member-permutations-list,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
neg_assert_of_eq_int,
flip_twice,
member-map,
equal-wf-base,
member-partial-permutations-list,
set_wf,
equal_wf,
all_wf,
no_repeats_wf,
list_wf,
no_repeats-permutations-list,
le_wf,
no_repeats_map,
false_wf,
int_seg_subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
hypothesis,
natural_numberEquality,
independent_pairFormation,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
productIsType,
applyEquality,
functionEquality,
functionIsType,
inhabitedIsType,
closedConclusion,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
isectIsTypeImplies,
promote_hyp,
instantiate,
cumulativity,
intEquality,
functionExtensionality,
applyLambdaEquality,
equalityIstype,
sqequalBase,
equalityElimination,
imageMemberEquality,
baseClosed,
imageElimination,
hyp_replacement,
productEquality,
setEquality,
lambdaFormation,
voidEquality,
isect_memberEquality,
lambdaEquality,
dependent_pairFormation,
dependent_set_memberEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\{2...\}]. \mforall{}[i:\mBbbN{}n]. \mforall{}[F:(\mBbbN{}n {}\mrightarrow{} \mBbbN{}n) {}\mrightarrow{} |r|].
(\mSigma{}\{r\} p \mmember{} partial-permutations-list(n;i). F[p]
= \mSigma{}\{r\} f \mmember{} permutations-list(n - 1). F[(i, n - 1) o extend-injection(n - 1;f)])
Date html generated:
2019_10_16-AM-11_29_55
Last ObjectModification:
2018_11_30-PM-01_22_54
Theory : matrices
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