Nuprl Lemma : term-induction1
∀[opr:Type]. ∀[P:term(opr) ⟶ ℙ].
  ((∀v:{v:varname()| ¬(v = nullvar() ∈ varname())} . P[varterm(v)])
  
⇒ (∀f:opr. ∀bts:bound-term(opr) List.  ((∀i:ℕ||bts||. P[snd(bts[i])]) 
⇒ P[mkterm(f;bts)]))
  
⇒ {∀t:term(opr). P[t]})
Proof
Definitions occuring in Statement : 
bound-term: bound-term(opr)
, 
mkterm: mkterm(opr;bts)
, 
varterm: varterm(v)
, 
term: term(opr)
, 
nullvar: nullvar()
, 
varname: varname()
, 
select: L[n]
, 
length: ||as||
, 
list: T List
, 
int_seg: {i..j-}
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
guard: {T}
, 
so_apply: x[s]
, 
pi2: snd(t)
, 
all: ∀x:A. B[x]
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
set: {x:A| B[x]} 
, 
function: x:A ⟶ B[x]
, 
natural_number: $n
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
guard: {T}
, 
uall: ∀[x:A]. B[x]
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
so_lambda: λ2x.t[x]
, 
member: t ∈ T
, 
prop: ℙ
, 
subtype_rel: A ⊆r B
, 
nat: ℕ
, 
so_apply: x[s]
, 
sq_stable: SqStable(P)
, 
squash: ↓T
, 
uimplies: b supposing a
, 
coterm-fun: coterm-fun(opr;T)
, 
varterm: varterm(v)
, 
mkterm: mkterm(opr;bts)
, 
int_seg: {i..j-}
, 
ge: i ≥ j 
, 
lelt: i ≤ j < k
, 
and: P ∧ Q
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
false: False
, 
le: A ≤ B
, 
uiff: uiff(P;Q)
, 
not: ¬A
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
bound-term: bound-term(opr)
, 
less_than: a < b
, 
pi2: snd(t)
Lemmas referenced : 
uniform-comp-nat-induction, 
term_wf, 
le_wf, 
term-size_wf, 
istype-nat, 
term-ext, 
sq_stable__le, 
istype-le, 
subtype_rel_transitivity, 
coterm-fun_wf, 
subtype_rel_weakening, 
ext-eq_inversion, 
term_size_var_lemma, 
term_size_mkterm_lemma, 
term-size-positive, 
mkterm_wf, 
subtract_wf, 
nat_properties, 
decidable__le, 
add-is-int-iff, 
full-omega-unsat, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermSubtract_wf, 
itermVar_wf, 
itermAdd_wf, 
istype-int, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_term_value_subtract_lemma, 
int_term_value_var_lemma, 
int_term_value_add_lemma, 
int_formula_prop_wf, 
false_wf, 
decidable__lt, 
intformless_wf, 
int_formula_prop_less_lemma, 
istype-less_than, 
int_seg_wf, 
length_wf, 
list_wf, 
varname_wf, 
lsum_wf, 
pi2_wf, 
l_member_wf, 
bound-term_wf, 
select_wf, 
int_seg_properties, 
subtype_rel_self, 
nullvar_wf, 
istype-void, 
varterm_wf, 
istype-universe, 
summand-le-lsum, 
non_neg_length, 
select_member
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
isect_memberFormation_alt, 
lambdaFormation_alt, 
cut, 
thin, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
lambdaEquality_alt, 
functionEquality, 
setEquality, 
hypothesisEquality, 
hypothesis, 
applyEquality, 
because_Cache, 
setElimination, 
rename, 
independent_functionElimination, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
setIsType, 
universeIsType, 
independent_isectElimination, 
inhabitedIsType, 
unionElimination, 
dependent_functionElimination, 
Error :memTop, 
natural_numberEquality, 
productElimination, 
dependent_set_memberEquality_alt, 
independent_pairFormation, 
pointwiseFunctionality, 
equalityTransitivity, 
equalitySymmetry, 
promote_hyp, 
baseApply, 
closedConclusion, 
approximateComputation, 
dependent_pairFormation_alt, 
int_eqEquality, 
voidElimination, 
productIsType, 
productEquality, 
addEquality, 
equalityIstype, 
isectIsType, 
functionIsType, 
instantiate, 
universeEquality, 
applyLambdaEquality
Latex:
\mforall{}[opr:Type].  \mforall{}[P:term(opr)  {}\mrightarrow{}  \mBbbP{}].
    ((\mforall{}v:\{v:varname()|  \mneg{}(v  =  nullvar())\}  .  P[varterm(v)])
    {}\mRightarrow{}  (\mforall{}f:opr.  \mforall{}bts:bound-term(opr)  List.    ((\mforall{}i:\mBbbN{}||bts||.  P[snd(bts[i])])  {}\mRightarrow{}  P[mkterm(f;bts)]))
    {}\mRightarrow{}  \{\mforall{}t:term(opr).  P[t]\})
Date html generated:
2020_05_19-PM-09_54_24
Last ObjectModification:
2020_03_09-PM-04_08_33
Theory : terms
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