Nuprl Lemma : extend-injection_wf
∀[a:ℕ]. ∀[f:ℕa →⟶ ℕa]. ∀[b:{a...}]. (extend-injection(a;f) ∈ ℕb →⟶ ℕb)
Proof
Definitions occuring in Statement :
extend-injection: extend-injection(a;f),
injection: A →⟶ B,
int_upper: {i...},
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
nat: ℕ,
extend-injection: extend-injection(a;f),
member: t ∈ T,
uall: ∀[x:A]. B[x],
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
inject: Inj(A;B;f),
bfalse: ff,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
guard: {T},
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
int_upper: {i...},
subtype_rel: A ⊆r B,
prop: ℙ,
lelt: i ≤ j < k,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
all: ∀x:A. B[x],
int_seg: {i..j-},
injection: A →⟶ B
Lemmas referenced :
nat_wf,
int_seg_wf,
injection_wf,
int_upper_wf,
inject_wf,
less_than_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
equal_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_upper_properties,
int_seg_properties,
false_wf,
int_seg_subtype,
lelt_wf,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_term_value_constant_lemma,
itermConstant_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int
Rules used in proof :
natural_numberEquality,
because_Cache,
isect_memberEquality,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
extract_by_obid,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
hypothesis,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cumulativity,
instantiate,
promote_hyp,
voidEquality,
voidElimination,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
dependent_functionElimination,
applyLambdaEquality,
independent_pairFormation,
functionExtensionality,
applyEquality,
independent_isectElimination,
productElimination,
equalityElimination,
unionElimination,
lambdaFormation,
lambdaEquality,
dependent_set_memberEquality
Latex:
\mforall{}[a:\mBbbN{}]. \mforall{}[f:\mBbbN{}a \mrightarrow{}{}\mrightarrow{} \mBbbN{}a]. \mforall{}[b:\{a...\}]. (extend-injection(a;f) \mmember{} \mBbbN{}b \mrightarrow{}{}\mrightarrow{} \mBbbN{}b)
Date html generated:
2018_05_21-PM-08_16_57
Last ObjectModification:
2017_12_15-AM-11_23_34
Theory : general
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