Nuprl Lemma : finite-function-equipollent
∀n:ℕ+. ∀[F:ℕn ⟶ Type]. i:ℕn ⟶ F[i] ~ i:ℕn - 1 ⟶ F[i] × F[n - 1]
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
subtract: n - m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
equipollent: A ~ B,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
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),
false: False,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
istype: istype(T),
biject: Bij(A;B;f),
inject: Inj(A;B;f),
guard: {T},
respects-equality: respects-equality(S;T),
surject: Surj(A;B;f),
pi1: fst(t),
pi2: snd(t),
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
squash: ↓T,
sq_stable: SqStable(P),
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
int_seg_wf,
istype-universe,
nat_plus_wf,
subtype_rel_dep_function,
subtract_wf,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
int_seg_subtype,
istype-false,
decidable__le,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
intformle_wf,
int_formula_prop_le_lemma,
biject_wf,
respects-equality-product,
respects-equality-function,
respects-equality-trivial,
istype-base,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
eq_int_wf,
subtype_rel-equal,
eqtt_to_assert,
assert_of_eq_int,
set_subtype_base,
less_than_wf,
sq_stable__and,
equal-wf-base,
sq_stable__equal,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
assert_wf,
equal_wf,
squash_wf,
true_wf,
eq_int_eq_true,
btrue_wf,
subtype_rel_self,
iff_weakening_equal,
btrue_neq_bfalse,
bnot_wf,
not_wf,
istype-assert,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
Error :functionIsType,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
instantiate,
universeEquality,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
independent_pairEquality,
applyEquality,
sqequalRule,
because_Cache,
Error :dependent_set_memberEquality_alt,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
Error :productIsType,
addEquality,
minusEquality,
multiplyEquality,
functionEquality,
productEquality,
closedConclusion,
Error :equalityIstype,
Error :inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalBase,
applyLambdaEquality,
Error :functionExtensionality_alt,
cumulativity,
intEquality,
equalityElimination,
baseApply,
baseClosed,
imageMemberEquality,
imageElimination,
axiomEquality,
Error :functionIsTypeImplies,
promote_hyp
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}[F:\mBbbN{}n {}\mrightarrow{} Type]. i:\mBbbN{}n {}\mrightarrow{} F[i] \msim{} i:\mBbbN{}n - 1 {}\mrightarrow{} F[i] \mtimes{} F[n - 1]
Date html generated:
2019_06_20-PM-02_19_19
Last ObjectModification:
2018_12_19-PM-05_13_29
Theory : equipollence!!cardinality!
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