Nuprl Lemma : wf-term-accum-induction
∀[opr,P:Type]. ∀[sort:term(opr) ⟶ ℕ]. ∀[arity:opr ⟶ ((ℕ × ℕ) List)]. ∀[R:P ⟶ wfterm(opr;sort;arity) ⟶ ℙ].
∀Q:P
⟶ f:opr
⟶ vs:(varname() List)
⟶ {L:(t:wfterm(opr;sort;arity) × p:P × R[p;t]) List|
||L|| < ||arity f||
∧ (||vs|| = (fst(arity f[||L||])) ∈ ℤ)
∧ (∀i:ℕ||L||. ((sort (fst(L[i]))) = (snd(arity f[i])) ∈ ℤ))}
⟶ P
((∀p:P. ∀v:{v:varname()| ¬(v = nullvar() ∈ varname())} . R[p;varterm(v)])
⇒ (∀p:P. ∀f:opr. ∀bts:wf-bound-terms(opr;sort;arity;f). ∀L:{L:(t:wfterm(opr;sort;arity) × p:P × R[p;t]) List|
(||L|| = ||bts|| ∈ ℤ)
∧ (∀i:ℕ||L||
((fst(L[i])) = (snd(bts[i])) ∈ term(opr)))
∧ (∀i:ℕ||L||
((fst(snd(L[i])))
= Q[p;f;fst(bts[i]);firstn(i;L)]
∈ P))} .
R[p;mkwfterm(f;bts)])
⇒ (∀p:P. ∀t:wfterm(opr;sort;arity). R[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(),
firstn: firstn(n;as),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3;s4],
so_apply: x[s1;s2],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_apply: x[s1;s2],
so_apply: x[s1;s2;s3;s4],
and: P ∧ Q,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
wf-bound-terms: wf-bound-terms(opr;sort;arity;f),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
pi1: fst(t),
pi2: snd(t),
wfterm: wfterm(opr;sort;arity),
respects-equality: respects-equality(S;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,
less_than: a < b,
squash: ↓T,
int_iseg: {i...j},
cand: A c∧ B,
less_than': less_than'(a;b),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
term-accum_wf_wfterm,
wfterm_wf,
wf-bound-terms_wf,
list_wf,
istype-int,
length_wf_nat,
set_subtype_base,
le_wf,
int_subtype_base,
varname_wf,
int_seg_wf,
length_wf,
term_wf,
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
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,
intformeq_wf,
int_formula_prop_eq_lemma,
respects-equality-set-trivial2,
assert_wf,
wf-term_wf,
pi1_wf,
mkwfterm_wf,
nullvar_wf,
istype-void,
varterm_wf,
istype-assert,
subtype_rel_self,
istype-less_than,
nat_wf,
istype-nat,
istype-universe,
firstn_wf,
subtype_rel_sets_simple,
lelt_wf,
istype-le,
select-firstn,
subtype_rel_list,
top_wf,
int_seg_subtype_nat,
istype-false,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
length_firstn
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
rename,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
universeIsType,
functionIsType,
setIsType,
productEquality,
applyEquality,
because_Cache,
productIsType,
equalityIstype,
intEquality,
lambdaEquality_alt,
natural_numberEquality,
independent_isectElimination,
setElimination,
sqequalBase,
productElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
inhabitedIsType,
dependent_set_memberEquality_alt,
instantiate,
universeEquality,
imageElimination,
closedConclusion,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[opr,P:Type]. \mforall{}[sort:term(opr) {}\mrightarrow{} \mBbbN{}]. \mforall{}[arity:opr {}\mrightarrow{} ((\mBbbN{} \mtimes{} \mBbbN{}) List)]. \mforall{}[R:P
{}\mrightarrow{} wfterm(opr;sort;arity)
{}\mrightarrow{} \mBbbP{}].
\mforall{}Q:P
{}\mrightarrow{} f:opr
{}\mrightarrow{} vs:(varname() List)
{}\mrightarrow{} \{L:(t:wfterm(opr;sort;arity) \mtimes{} p:P \mtimes{} R[p;t]) List|
||L|| < ||arity f||
\mwedge{} (||vs|| = (fst(arity f[||L||])))
\mwedge{} (\mforall{}i:\mBbbN{}||L||. ((sort (fst(L[i]))) = (snd(arity f[i]))))\}
{}\mrightarrow{} P
((\mforall{}p:P. \mforall{}v:\{v:varname()| \mneg{}(v = nullvar())\} . R[p;varterm(v)])
{}\mRightarrow{} (\mforall{}p:P. \mforall{}f:opr. \mforall{}bts:wf-bound-terms(opr;sort;arity;f). \mforall{}L:\{L:(t:wfterm(opr;sort;arity)
\mtimes{} p:P
\mtimes{} R[p;t]) List|
(||L|| = ||bts||)
\mwedge{} (\mforall{}i:\mBbbN{}||L||
((fst(L[i])) = (snd(bts[i]))))
\mwedge{} (\mforall{}i:\mBbbN{}||L||
((fst(snd(L[i])))
= Q[p;f;fst(bts[i]);
firstn(i;L)]))\} .
R[p;mkwfterm(f;bts)])
{}\mRightarrow{} (\mforall{}p:P. \mforall{}t:wfterm(opr;sort;arity). R[p;t]))
Date html generated:
2020_05_19-PM-10_00_12
Last ObjectModification:
2020_03_09-PM-04_10_33
Theory : terms
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