Nuprl Lemma : sq-all-large-and
∀[P,Q:ℕ ⟶ ℙ]. (∀large(n).{P[n]} ⇒ ∀large(n).{Q[n]} ⇒ ∀large(n).{P[n] ∧ Q[n]})
Proof
Definitions occuring in Statement :
sq-all-large: ∀large(n).{P[n]},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
sq-all-large: ∀large(n).{P[n]},
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
sq_exists: ∃x:A [B[x]],
member: t ∈ T,
nat: ℕ,
all: ∀x:A. B[x],
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
cand: A c∧ B,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B
Lemmas referenced :
imax_wf,
imax_nat,
nat_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_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_eq_lemma,
int_formula_prop_wf,
equal_wf,
le_wf,
imax_lb,
all_wf,
sq_exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberFormation,
dependent_set_memberEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
because_Cache,
productElimination,
functionEquality,
productEquality,
applyEquality,
functionExtensionality,
universeEquality,
cumulativity
Latex:
\mforall{}[P,Q:\mBbbN{} {}\mrightarrow{} \mBbbP{}]. (\mforall{}large(n).\{P[n]\} {}\mRightarrow{} \mforall{}large(n).\{Q[n]\} {}\mRightarrow{} \mforall{}large(n).\{P[n] \mwedge{} Q[n]\})
Date html generated:
2018_05_21-PM-07_59_51
Last ObjectModification:
2017_07_26-PM-05_36_43
Theory : general
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