Nuprl Lemma : term-ind_wf_wfterm
∀[opr:Type]. ∀[sort:term(opr) ⟶ ℕ]. ∀[arity:opr ⟶ ((ℕ × ℕ) List)]. ∀[P:wfterm(opr;sort;arity) ⟶ ℙ].
∀[varcase:∀v:{v:varname()| ¬(v = nullvar() ∈ varname())} . P[varterm(v)]].
∀[mktermcase:∀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)].
(term-ind(x.varcase[x];f,bts,r.mktermcase[f;bts;r];t) ∈ 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),
term-ind: term-ind,
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,
so_apply: x[s1;s2;s3],
so_apply: x[s],
pi2: snd(t),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
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,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s],
wfterm: wfterm(opr;sort;arity),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
so_lambda: so_lambda3,
bound-term: bound-term(opr),
wfbts: wfbts(t),
term-bts: term-bts(t),
pi2: snd(t),
outr: outr(x),
mkterm: mkterm(opr;bts),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
so_apply: x[s1;s2;s3],
subtype_rel: A ⊆r B,
term-opr: term-opr(t),
pi1: fst(t),
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
mkwfterm: mkwfterm(f;bts),
wf-bound-terms: wf-bound-terms(opr;sort;arity;f),
nat: ℕ,
sq_type: SQType(T),
guard: {T},
true: True,
wf-term: wf-term(arity;sort;t),
varterm: varterm(v),
isvarterm: isvarterm(t),
isl: isl(x),
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
term-ind_wf,
isect_wf,
assert_wf,
wf-term_wf,
istype-assert,
not_wf,
equal-wf-T-base,
varname_wf,
equal_wf,
nullvar_wf,
istype-void,
varterm_wf,
mkterm_wf,
wf_term_var_lemma,
wfbts_wf,
isvarterm_wf,
subtype_rel_self,
wf-bound-terms_wf,
subtype_rel_function,
int_seg_wf,
length_wf,
list_wf,
wfterm_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,
uimplies_subtype,
subtype_rel_set,
equal-wf-base,
set_subtype_base,
le_wf,
int_subtype_base,
subtype_rel_list,
term_wf,
subtype_rel_product,
bound-term_wf,
assert_elim,
subtype_base_sq,
bool_wf,
bool_subtype_base,
nat_wf,
istype-nat,
istype-universe,
iff_imp_equal_bool,
btrue_wf,
istype-true,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
rename,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
hypothesis,
applyEquality,
dependent_set_memberEquality_alt,
because_Cache,
lambdaFormation_alt,
baseClosed,
universeIsType,
setElimination,
isect_memberEquality_alt,
functionExtensionality,
closedConclusion,
setEquality,
functionIsType,
equalityIstype,
inhabitedIsType,
independent_isectElimination,
dependent_functionElimination,
Error :memTop,
functionEquality,
natural_numberEquality,
productEquality,
productElimination,
imageElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
voidElimination,
equalityTransitivity,
equalitySymmetry,
isectIsType,
productIsType,
applyLambdaEquality,
instantiate,
cumulativity,
setIsType,
universeEquality
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{}[varcase:\mforall{}v:\{v:varname()| \mneg{}(v = nullvar())\} . P[varterm(v)]].
\mforall{}[mktermcase:\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)])]. \mforall{}[t:wfterm(opr;sort;arity)].
(term-ind(x.varcase[x];f,bts,r.mktermcase[f;bts;r];t) \mmember{} P[t])
Date html generated:
2020_05_19-PM-09_58_52
Last ObjectModification:
2020_03_09-PM-04_10_27
Theory : terms
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