Nuprl Lemma : term-accum_wf_wfterm_0
∀[opr,P:Type]. ∀[sort:term(opr) ⟶ ℕ]. ∀[arity:opr ⟶ ((ℕ × ℕ) List)]. ∀[R:P ⟶ wfterm(opr;sort;arity) ⟶ ℙ].
∀[Q:P ⟶ opr ⟶ (varname() List) ⟶ ((t:term(opr) × p:P × R[p;t] supposing ↑wf-term(arity;sort;t)) List) ⟶ P].
∀[varcase:p:P ⟶ v:{v:varname()| ¬(v = nullvar() ∈ varname())} ⟶ R[p;varterm(v)]].
∀[mktermcase: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)]]. ∀[t:wfterm(opr;sort;arity)]. ∀[p:P].
(term-accum(t with p)
p,f,vs,tr.Q[p;f;vs;tr]
varterm(x) with p ⇒ varcase[p;x]
mkterm(f,bts) with p ⇒ trs.mktermcase[p;f;bts;trs] ∈ 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),
wf-term: wf-term(arity;sort;t),
term-accum: term-accum,
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,
assert: ↑b,
uimplies: b supposing a,
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,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
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,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
wfterm: wfterm(opr;sort;arity),
so_apply: x[s],
so_apply: x[s1;s2;s3;s4],
all: ∀x:A. B[x],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
subtype_rel: A ⊆r B,
wf-term: wf-term(arity;sort;t),
varterm: varterm(v),
true: True,
not: ¬A,
implies: P ⇒ Q,
false: False,
bound-term: bound-term(opr),
and: P ∧ Q,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
pi1: fst(t),
pi2: snd(t),
sq_stable: SqStable(P),
squash: ↓T,
wfbts: wfbts(t),
term-bts: term-bts(t),
outr: outr(x),
mkterm: mkterm(opr;bts),
term-opr: term-opr(t),
mkwfterm: mkwfterm(f;bts),
wf-bound-terms: wf-bound-terms(opr;sort;arity;f),
isvarterm: isvarterm(t),
isl: isl(x),
bfalse: ff,
cand: A c∧ B,
l_member: (x ∈ l),
ge: i ≥ j ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
respects-equality: respects-equality(S;T)
Lemmas referenced :
term-accum_wf,
isect_wf,
assert_wf,
wf-term_wf,
istype-assert,
term_wf,
wf_term_var_lemma,
subtype_rel_dep_function,
varname_wf,
not_wf,
equal-wf-T-base,
varterm_wf,
nullvar_wf,
istype-void,
true_wf,
mkterm_wf,
list_wf,
equal-wf-base,
length_wf_nat,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
bound-term_wf,
all_wf,
int_seg_wf,
equal_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,
firstn_wf,
sq_stable__subtype_rel,
uimplies_subtype,
wfterm_wf,
subtype_rel_list,
subtype_rel_self,
mkwfterm_wf,
nat_wf,
istype-nat,
istype-universe,
wfbts_wf,
isvarterm_wf,
wf-bound-terms_wf,
subtype_rel_set,
length_wf,
subtype_rel_product,
list-subtype,
l_member_wf,
squash_wf,
nat_properties,
istype-le,
istype-less_than,
iff_weakening_equal,
pi1_wf,
assert_functionality_wrt_uiff,
assert-wf-mkterm,
respects-equality-set-trivial2
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality_alt,
sqequalRule,
applyEquality,
dependent_set_memberEquality_alt,
universeIsType,
dependent_functionElimination,
Error :memTop,
functionExtensionality,
setEquality,
baseClosed,
lambdaFormation_alt,
setElimination,
rename,
because_Cache,
natural_numberEquality,
independent_isectElimination,
setIsType,
functionIsType,
equalityIstype,
isectEquality,
isect_memberEquality_alt,
independent_functionElimination,
voidElimination,
inhabitedIsType,
productEquality,
intEquality,
closedConclusion,
equalityTransitivity,
equalitySymmetry,
productElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
imageMemberEquality,
imageElimination,
productIsType,
instantiate,
functionEquality,
cumulativity,
universeEquality,
dependent_pairEquality_alt,
isectIsType,
sqequalBase
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{} opr
{}\mrightarrow{} (varname() List)
{}\mrightarrow{} ((t:term(opr) \mtimes{} p:P \mtimes{} R[p;t] supposing \muparrow{}wf-term(arity;sort;t)) List)
{}\mrightarrow{} P]. \mforall{}[varcase:p:P {}\mrightarrow{} v:\{v:varname()| \mneg{}(v = nullvar())\} {}\mrightarrow{} R[p;varterm(v)]].
\mforall{}[mktermcase:p:P
{}\mrightarrow{} f:opr
{}\mrightarrow{} bts:wf-bound-terms(opr;sort;arity;f)
{}\mrightarrow{} 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)]))\}
{}\mrightarrow{} R[p;mkwfterm(f;bts)]]. \mforall{}[t:wfterm(opr;sort;arity)]. \mforall{}[p:P].
(term-accum(t with p)
p,f,vs,tr.Q[p;f;vs;tr]
varterm(x) with p {}\mRightarrow{} varcase[p;x]
mkterm(f,bts) with p {}\mRightarrow{} trs.mktermcase[p;f;bts;trs] \mmember{} R[p;t])
Date html generated:
2020_05_19-PM-09_59_02
Last ObjectModification:
2020_03_09-PM-04_10_30
Theory : terms
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