Nuprl Lemma : nat-prop-dep-all-wf
∀[n:ℕ]. ((nat-prop{i:l}(n) ∈ 𝕌') ∧ (∀P:nat-prop{i:l}(n). ∀j:ℕn + 1. (dep-all(j;i.P[i]) ∈ ℙ)))
Proof
Definitions occuring in Statement :
dep-all: dep-all(n;i.P[i]),
nat-prop: nat-prop{i:l}(n),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
prop: ℙ,
cand: A c∧ B,
nat-prop: nat-prop{i:l}(n),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
dep-all: dep-all(n;i.P[i]),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
int_seg_wf,
subtract-1-ge-0,
istype-nat,
top_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
true_wf,
int_seg_subtype_special,
int_seg_cases,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
subtype_rel_universe1,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-top,
dep-isect_wf,
subtract-add-cancel,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
subtract_wf,
dep-isect-wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
itermAdd_wf,
int_term_value_add_lemma,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
addEquality,
because_Cache,
cumulativity,
unionElimination,
instantiate,
intEquality,
hypothesis_subsumption,
closedConclusion,
equalityElimination,
lessCases,
axiomSqEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
equalityIstype,
promote_hyp,
functionEquality,
isectEquality,
universeEquality,
dependent_set_memberEquality_alt,
productIsType,
dependentIntersectionElimination
Latex:
\mforall{}[n:\mBbbN{}]. ((nat-prop\{i:l\}(n) \mmember{} \mBbbU{}') \mwedge{} (\mforall{}P:nat-prop\{i:l\}(n). \mforall{}j:\mBbbN{}n + 1. (dep-all(j;i.P[i]) \mmember{} \mBbbP{})))
Date html generated:
2020_05_19-PM-09_39_49
Last ObjectModification:
2020_03_05-PM-02_52_37
Theory : co-recursion-2
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