Nuprl Lemma : not-inject
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ (∀n:ℕ. ∀f:ℕn ⟶ T. ∃i:ℕn. ∃j:ℕi. ((f i) = (f j) ∈ T) supposing ¬Inj(ℕn;T;f)))
Proof
Definitions occuring in Statement :
inject: Inj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
false: False,
nat: ℕ,
prop: ℙ,
inject: Inj(A;B;f),
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
itermConstant_wf,
intformle_wf,
decidable__le,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
exists_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
int_seg_properties,
lelt_wf,
decidable__lt,
decidable_wf,
nat_wf,
not_wf,
decidable__equal_int_seg,
decidable__implies,
decidable__all_int_seg,
equal_wf,
all_wf,
not-all-int_seg,
int_seg_wf,
inject_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
lemma_by_obid,
isectElimination,
natural_numberEquality,
setElimination,
rename,
hypothesis,
functionEquality,
applyEquality,
independent_functionElimination,
instantiate,
because_Cache,
isect_memberEquality,
productElimination,
universeEquality,
unionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
independent_pairFormation,
equalitySymmetry,
cumulativity,
independent_isectElimination,
int_eqEquality,
intEquality,
voidEquality,
computeAll
Latex:
\mforall{}[T:Type]
((\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}f:\mBbbN{}n {}\mrightarrow{} T. \mexists{}i:\mBbbN{}n. \mexists{}j:\mBbbN{}i. ((f i) = (f j)) supposing \mneg{}Inj(\mBbbN{}n;T;f)))
Date html generated:
2016_05_14-AM-07_27_06
Last ObjectModification:
2016_01_14-PM-09_59_50
Theory : int_2
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