Nuprl Lemma : natset-transitive
∀n:ℕ. transitive-set(natset(n))
Proof
Definitions occuring in Statement :
natset: natset(n),
transitive-set: transitive-set(s),
nat: ℕ,
all: ∀x:A. B[x]
Definitions unfolded in proof :
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
so_apply: x[s],
so_lambda: λ2x.t[x],
and: P ∧ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
top: Top,
member: t ∈ T,
all: ∀x:A. B[x],
natset: natset(n)
Lemmas referenced :
plus-set-transitive,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
primrec-unroll,
nat_wf,
primrec-wf2,
less_than_wf,
set_wf,
int_seg_wf,
plus-set_wf,
emptyset_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
subtract_wf,
decidable__le,
Set_wf,
primrec_wf,
transitive-set_wf,
emptyset-transitive,
primrec0_lemma
Rules used in proof :
cumulativity,
promote_hyp,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
independent_pairFormation,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
hypothesisEquality,
natural_numberEquality,
because_Cache,
dependent_set_memberEquality,
instantiate,
isectElimination,
setElimination,
rename,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}n:\mBbbN{}. transitive-set(natset(n))
Date html generated:
2018_05_29-PM-01_49_35
Last ObjectModification:
2018_05_24-PM-11_31_52
Theory : constructive!set!theory
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