Nuprl Lemma : wf-term-induction
∀[opr:Type]
  ∀sort:term(opr) ⟶ ℕ. ∀arity:opr ⟶ ((ℕ × ℕ) List).
    ∀[P:wfterm(opr;sort;arity) ⟶ ℙ]
      ((∀v:{v:varname()| ¬(v = nullvar() ∈ varname())} . P[varterm(v)])
      
⇒ (∀f:opr. ∀bts:wf-bound-terms(opr;sort;arity;f).  ((∀i:ℕ||bts||. P[snd(bts[i])]) 
⇒ P[mkwfterm(f;bts)]))
      
⇒ {∀t:wfterm(opr;sort;arity). P[t]})
Proof
Definitions occuring in Statement : 
mkwfterm: mkwfterm(f;bts)
, 
wf-bound-terms: wf-bound-terms(opr;sort;arity;f)
, 
wfterm: wfterm(opr;sort;arity)
, 
varterm: varterm(v)
, 
term: term(opr)
, 
nullvar: nullvar()
, 
varname: varname()
, 
select: L[n]
, 
length: ||as||
, 
list: T List
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
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]
, 
product: x:A × B[x]
, 
natural_number: $n
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
all: ∀x:A. B[x]
, 
guard: {T}
, 
implies: P 
⇒ Q
, 
so_apply: x[s]
, 
wf-bound-terms: wf-bound-terms(opr;sort;arity;f)
, 
prop: ℙ
, 
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
, 
pi2: snd(t)
, 
assert: ↑b
, 
ifthenelse: if b then t else f fi 
, 
wf-term: wf-term(arity;sort;t)
, 
varterm: varterm(v)
, 
btrue: tt
, 
true: True
, 
wfterm: wfterm(opr;sort;arity)
Lemmas referenced : 
term-ind_wf_wfterm, 
wfterm_wf, 
wf-bound-terms_wf, 
int_seg_wf, 
length_wf, 
list_wf, 
varname_wf, 
select_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, 
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, 
intformless_wf, 
int_formula_prop_less_lemma, 
mkwfterm_wf, 
nullvar_wf, 
istype-void, 
varterm_wf, 
istype-assert, 
wf-term_wf, 
nat_wf, 
term_wf, 
istype-nat, 
istype-universe
Rules used in proof : 
cut, 
introduction, 
extract_by_obid, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
hypothesis, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
lambdaFormation_alt, 
sqequalRule, 
equalityTransitivity, 
equalitySymmetry, 
universeIsType, 
functionIsType, 
natural_numberEquality, 
productEquality, 
setElimination, 
rename, 
applyEquality, 
functionExtensionality, 
because_Cache, 
independent_isectElimination, 
productElimination, 
imageElimination, 
dependent_functionElimination, 
unionElimination, 
approximateComputation, 
independent_functionElimination, 
dependent_pairFormation_alt, 
lambdaEquality_alt, 
int_eqEquality, 
Error :memTop, 
independent_pairFormation, 
voidElimination, 
inhabitedIsType, 
equalityIstype, 
setIsType, 
dependent_set_memberEquality_alt, 
universeEquality, 
instantiate
Latex:
\mforall{}[opr:Type]
    \mforall{}sort:term(opr)  {}\mrightarrow{}  \mBbbN{}.  \mforall{}arity:opr  {}\mrightarrow{}  ((\mBbbN{}  \mtimes{}  \mBbbN{})  List).
        \mforall{}[P:wfterm(opr;sort;arity)  {}\mrightarrow{}  \mBbbP{}]
            ((\mforall{}v:\{v:varname()|  \mneg{}(v  =  nullvar())\}  .  P[varterm(v)])
            {}\mRightarrow{}  (\mforall{}f:opr.  \mforall{}bts:wf-bound-terms(opr;sort;arity;f).
                        ((\mforall{}i:\mBbbN{}||bts||.  P[snd(bts[i])])  {}\mRightarrow{}  P[mkwfterm(f;bts)]))
            {}\mRightarrow{}  \{\mforall{}t:wfterm(opr;sort;arity).  P[t]\})
Date html generated:
2020_05_19-PM-09_58_56
Last ObjectModification:
2020_03_09-PM-04_10_28
Theory : terms
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