Nuprl Lemma : cardinality-le_functionality
∀[T:Type]. ∀n:ℕ+. ∀[m:ℕ]. {|T| ≤ n ⇒ |T| ≤ m} supposing n ≤ m
Proof
Definitions occuring in Statement :
cardinality-le: |T| ≤ n,
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
nat_plus: ℕ+,
prop: ℙ,
guard: {T},
cardinality-le: |T| ≤ n,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
lelt: i ≤ j < k,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
less_than': less_than'(a;b),
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
surject: Surj(A;B;f)
Lemmas referenced :
less_than'_wf,
cardinality-le_wf,
nat_plus_subtype_nat,
le_wf,
nat_wf,
nat_plus_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
int_seg_wf,
lelt_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
false_wf,
int_seg_properties,
nat_properties,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
surject_wf,
intformle_wf,
int_formula_prop_le_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
universeEquality,
dependent_pairFormation,
because_Cache,
unionElimination,
equalityElimination,
independent_isectElimination,
functionExtensionality,
natural_numberEquality,
dependent_set_memberEquality,
independent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality
Latex:
\mforall{}[T:Type]. \mforall{}n:\mBbbN{}\msupplus{}. \mforall{}[m:\mBbbN{}]. \{|T| \mleq{} n {}\mRightarrow{} |T| \mleq{} m\} supposing n \mleq{} m
Date html generated:
2018_05_21-PM-00_39_32
Last ObjectModification:
2018_05_19-AM-06_45_12
Theory : list_1
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