Nuprl Lemma : CCC-finite
∀[T:Type]. (finite(T) ⇒ CCC(T))
Proof
Definitions occuring in Statement :
contra-cc: CCC(T),
finite: finite(T),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
subtract: n - m,
primtailrec: primtailrec(n;i;b;f),
primrec: primrec(n;b;c),
exp: i^n,
true: True,
less_than': less_than'(a;b),
so_apply: x[s],
so_lambda: λ2x.t[x],
surject: Surj(A;B;f),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
squash: ↓T,
less_than: a < b,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
biject: Bij(A;B;f),
equipollent: A ~ B,
subtype_rel: A ⊆r B,
contra-cc: CCC(T),
guard: {T},
sq_type: SQType(T),
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
false: False,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
int_upper: {i...},
uimplies: b supposing a,
nat: ℕ,
all: ∀x:A. B[x],
prop: ℙ,
member: t ∈ T,
exists: ∃x:A. B[x],
finite: finite(T),
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
equipollent_inversion,
istype-false,
CCC-product,
int_formula_prop_eq_lemma,
intformeq_wf,
subtract-add-cancel,
iff_weakening_equal,
exp_add,
true_wf,
squash_wf,
equal_wf,
exp_wf4,
equipollent-multiply,
exp0_lemma,
primrec-wf2,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
contra-cc_wf,
lelt_wf,
le_wf,
set_subtype_base,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
CCC-bool,
surject_wf,
equipollent-two,
bool_wf,
CCC-surjection,
istype-less_than,
decidable__lt,
int_seg_cases,
int_seg_subtype_special,
subtype_rel_self,
int_seg_wf,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformless_wf,
intformand_wf,
int_seg_properties,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
istype-le,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
istype-void,
int_formula_prop_not_lemma,
istype-int,
itermConstant_wf,
intformle_wf,
intformnot_wf,
full-omega-unsat,
decidable__le,
nat_properties,
exp-greater,
exp_wf2,
le_weakening2,
istype-universe,
finite_wf
Rules used in proof :
imageMemberEquality,
productEquality,
Error :setIsType,
sqequalBase,
baseClosed,
closedConclusion,
baseApply,
promote_hyp,
Error :equalityIstype,
imageElimination,
equalityElimination,
hypothesis_subsumption,
applyEquality,
Error :productIsType,
Error :inhabitedIsType,
Error :functionIsType,
independent_pairFormation,
int_eqEquality,
equalitySymmetry,
equalityTransitivity,
intEquality,
cumulativity,
because_Cache,
sqequalRule,
voidElimination,
Error :isect_memberEquality_alt,
Error :lambdaEquality_alt,
independent_functionElimination,
approximateComputation,
unionElimination,
Error :dependent_set_memberEquality_alt,
independent_isectElimination,
natural_numberEquality,
rename,
setElimination,
dependent_functionElimination,
Error :dependent_pairFormation_alt,
universeEquality,
instantiate,
hypothesis,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
Error :universeIsType,
cut,
thin,
productElimination,
sqequalHypSubstitution,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. (finite(T) {}\mRightarrow{} CCC(T))
Date html generated:
2019_06_20-PM-03_01_10
Last ObjectModification:
2019_06_12-PM-09_48_12
Theory : continuity
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