Nuprl Lemma : injection-is-surjection
∀n:ℕ. ∀f:ℕn ⟶ ℕn. Surj(ℕn;ℕn;f) supposing Inj(ℕn;ℕn;f)
Proof
Definitions occuring in Statement :
surject: Surj(A;B;f),
inject: Inj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
inject: Inj(A;B;f),
implies: P ⇒ Q,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
surject: Surj(A;B;f),
int_seg: {i..j-},
ge: i ≥ j ,
exists: ∃x:A. B[x],
lelt: i ≤ j < k,
and: P ∧ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
le: A ≤ B,
less_than': less_than'(a;b),
nequal: a ≠ b ∈ T ,
int_upper: {i...},
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
less_than: a < b,
squash: ↓T,
true: True
Lemmas referenced :
decidable__exists_int_seg,
all_wf,
int_seg_wf,
not_wf,
equal-wf-T-base,
decidable__all_int_seg,
decidable__not,
decidable__equal_int_seg,
inject_wf,
nat_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
injection_le,
subtract_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
le_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
upper_subtype_nat,
istype-false,
nequal-le-implies,
zero-add,
zero-le-nat,
int_seg_subtype_nat,
int_upper_properties,
decidable__lt,
less_than_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-base,
set_subtype_base,
lelt_wf,
bnot_wf,
bool_cases,
iff_transitivity,
assert_of_bnot,
iff_imp_equal_bool,
btrue_wf,
iff_functionality_wrt_iff,
true_wf,
iff_weakening_equal,
subtract-is-int-iff,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
Error :lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
rename,
instantiate,
extract_by_obid,
natural_numberEquality,
setElimination,
because_Cache,
isectElimination,
Error :universeIsType,
independent_functionElimination,
applyEquality,
unionElimination,
Error :functionIsType,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :dependent_set_memberEquality_alt,
equalityElimination,
hypothesis_subsumption,
Error :productIsType,
Error :equalityIsType1,
promote_hyp,
Error :equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
applyLambdaEquality,
imageElimination,
pointwiseFunctionality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n. Surj(\mBbbN{}n;\mBbbN{}n;f) supposing Inj(\mBbbN{}n;\mBbbN{}n;f)
Date html generated:
2019_06_20-PM-01_15_40
Last ObjectModification:
2018_10_07-PM-00_32_35
Theory : int_2
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