Nuprl Lemma : sq_stable__ex_nat
∀[P:ℕ ⟶ ℙ]. ((∀m:ℕ. Dec(P[m])) ⇒ SqStable(∃m:ℕ. P[m]))
Proof
Definitions occuring in Statement :
nat: ℕ,
sq_stable: SqStable(P),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
nat: ℕ,
int_upper: {i...},
so_apply: x[s],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
sq_stable__ex_int_upper,
subtype_rel_function,
nat_wf,
int_upper_wf,
upper_subtype_nat,
istype-false,
subtype_rel_self,
istype-int_upper,
int_upper_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
squash_wf,
decidable_wf,
istype-nat
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
natural_numberEquality,
Error :isect_memberFormation_alt,
hypothesis,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
cumulativity,
universeEquality,
because_Cache,
independent_isectElimination,
sqequalRule,
independent_pairFormation,
Error :lambdaFormation_alt,
independent_functionElimination,
imageElimination,
productElimination,
Error :dependent_pairFormation_alt,
Error :universeIsType,
Error :dependent_set_memberEquality_alt,
setElimination,
rename,
unionElimination,
approximateComputation,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
imageMemberEquality,
baseClosed,
productEquality,
Error :functionIsType
Latex:
\mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}m:\mBbbN{}. Dec(P[m])) {}\mRightarrow{} SqStable(\mexists{}m:\mBbbN{}. P[m]))
Date html generated:
2019_06_20-PM-01_17_02
Last ObjectModification:
2019_01_09-PM-03_01_35
Theory : int_2
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