Nuprl Lemma : mrec-induction2
∀L:MutualRectypeSpec. ∀[P:mobj(L) ⟶ TYPE]. (mrecind(L;x.P[x]) ⇒ (∀x:mobj(L). P[x]))
Proof
Definitions occuring in Statement :
mrecind: mrecind(L;x.P[x]),
mobj: mobj(L),
mrec_spec: MutualRectypeSpec,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
mrecind: mrecind(L;x.P[x]),
mkinds: mKinds,
int_seg: {i..j-},
uimplies: b supposing a,
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
true: True,
ext-eq: A ≡ B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
mrec: mrec(L;i),
outl: outl(x),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
sq_type: SQType(T),
guard: {T},
prec-arg-types: prec-arg-types(lbl,p.a[lbl; p];i;lbl),
less_than': less_than'(a;b),
cand: A c∧ B,
mrec-lt: x < y,
prec_sub+: prec_sub+(P;lbl,p.a[lbl; p]),
prec_sub: prec_sub(P;lbl,p.a[lbl; p]),
prec-sub: prec-sub(P;lbl,p.a[lbl; p];j;x;i;y),
dest-prec: dest-prec(x),
mk-prec: mk-prec(lbl;x),
let: let,
bfalse: ff,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
outr: outr(x),
bnot: ¬bb,
l_all: (∀x∈L.P[x])
Lemmas referenced :
istype-mrecind,
mobj_wf,
mrec_spec_wf,
mrec-induction,
select_wf,
mrec-spec_wf,
int_seg_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,
decidable__lt,
length_wf,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
mrec-lt_wf,
mobj-ext,
mk-prec_wf,
subtype_rel_self,
mrec_wf,
tuple-type_wf,
prec-arg-types_wf,
istype-atom,
istype-less_than,
mkinds_wf,
subtype_base_sq,
union_subtype_base,
atom_subtype_base,
outl_wf,
equal_wf,
assert_wf,
btrue_wf,
bfalse_wf,
btrue_neq_bfalse,
select-map,
subtype_rel_list,
top_wf,
select-tuple_wf,
int_seg_subtype_nat,
istype-false,
length-map,
implies-rel_plus,
prec_wf,
prec_sub_wf,
int_subtype_base,
istype-true,
squash_wf,
true_wf,
istype-universe,
inl-one-one,
not-0-eq-1,
inr-one-one,
iff_weakening_equal,
outr_wf,
bnot_wf,
list_wf,
l_member_wf,
select_member,
mobj-sq,
mobj-kind_wf,
mobj-label_wf,
mobj-tuple_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
Error :universeIsType,
hypothesis,
Error :functionIsType,
Error :TYPEIsType,
independent_functionElimination,
dependent_functionElimination,
setElimination,
rename,
instantiate,
closedConclusion,
unionEquality,
cumulativity,
atomEquality,
universeEquality,
because_Cache,
independent_isectElimination,
productElimination,
imageElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :inhabitedIsType,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
Error :setIsType,
Error :dependent_pairEquality_alt,
Error :TYPEMemberIsType,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
applyLambdaEquality,
promote_hyp,
hyp_replacement,
Error :inlEquality_alt,
baseApply,
baseClosed,
sqequalBase,
productEquality,
Error :inlFormation_alt,
Error :unionIsType,
intEquality,
imageMemberEquality,
Error :inrEquality_alt,
Error :inrFormation_alt
Latex:
\mforall{}L:MutualRectypeSpec. \mforall{}[P:mobj(L) {}\mrightarrow{} TYPE]. (mrecind(L;x.P[x]) {}\mRightarrow{} (\mforall{}x:mobj(L). P[x]))
Date html generated:
2019_06_20-PM-02_16_16
Last ObjectModification:
2019_03_12-PM-11_30_30
Theory : tuples
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