Nuprl Lemma : injections-combinations
∀n:ℕ. ∀[T:Type]. ℕn →⟶ T ~ Combination(n;T)
Proof
Definitions occuring in Statement :
injection: A →⟶ B,
combination: Combination(n;T),
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
injection: A →⟶ B,
combination: Combination(n;T),
and: P ∧ Q,
cand: A c∧ B,
no_repeats: no_repeats(T;l),
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
inject: Inj(A;B;f),
le: A ≤ B,
less_than': less_than'(a;b),
equipollent: A ~ B,
biject: Bij(A;B;f),
respects-equality: respects-equality(S;T),
surject: Surj(A;B;f),
rev_implies: P ⇐ Q
Lemmas referenced :
istype-universe,
istype-nat,
mklist_wf,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
istype-void,
istype-less_than,
length_wf,
mklist_length,
no_repeats_wf,
length_wf_nat,
injection_wf,
int_seg_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
select_wf,
equal_wf,
squash_wf,
true_wf,
mklist_select,
subtype_rel_self,
iff_weakening_equal,
int_seg_subtype_nat,
istype-false,
biject_wf,
combination_wf,
respects-equality-set-trivial,
list_wf,
equal-wf-base,
inject_wf,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
less_than_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__equal_int_seg,
lelt_wf,
list_extensionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
universeEquality,
hypothesis,
lambdaEquality_alt,
setElimination,
rename,
dependent_set_memberEquality_alt,
hypothesisEquality,
sqequalRule,
dependent_functionElimination,
because_Cache,
functionIsTypeImplies,
inhabitedIsType,
functionIsType,
equalityIstype,
applyEquality,
intEquality,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
sqequalBase,
equalitySymmetry,
isect_memberEquality_alt,
isectIsTypeImplies,
voidElimination,
independent_pairFormation,
productIsType,
universeIsType,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
imageElimination,
equalityTransitivity,
imageMemberEquality,
baseClosed,
productElimination,
productEquality,
functionExtensionality_alt,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}n:\mBbbN{}. \mforall{}[T:Type]. \mBbbN{}n \mrightarrow{}{}\mrightarrow{} T \msim{} Combination(n;T)
Date html generated:
2019_10_15-AM-11_20_32
Last ObjectModification:
2018_11_27-AM-00_31_20
Theory : general
Home
Index