Nuprl Lemma : term-accum_wf_wfterm
∀[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]. ∀[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),
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,
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,
and: P ∧ Q,
member: t ∈ T,
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 :
varterm: varterm(v),
wf-term: wf-term(arity;sort;t),
less_than': less_than'(a;b),
guard: {T},
int_iseg: {i...j},
cand: A c∧ B,
wf-bound-terms: wf-bound-terms(opr;sort;arity;f),
bfalse: ff,
true: True,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
isl: isl(x),
squash: ↓T,
less_than: a < b,
pi2: snd(t),
lelt: i ≤ j < k,
int_seg: {i..j-},
pi1: fst(t),
nat: ℕ,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
le: A ≤ B,
false: False,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
all: ∀x:A. B[x],
so_apply: x[s1;s2;s3;s4],
so_apply: x[s],
wfterm: wfterm(opr;sort;arity),
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
prop: ℙ,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
l_member: (x ∈ l),
bound-term: bound-term(opr),
mkwfterm: mkwfterm(f;bts),
sq_type: SQType(T),
respects-equality: respects-equality(S;T),
rev_uimplies: rev_uimplies(P;Q),
mkterm: mkterm(opr;bts),
so_lambda: λ2x y.t[x; y],
so_lambda: so_lambda4,
coterm-fun: coterm-fun(opr;T),
isvarterm: isvarterm(t),
outr: outr(x),
term-bts: term-bts(t),
wfbts: wfbts(t),
sq_stable: SqStable(P),
let: let,
nil: [],
select: L[n],
term-opr: term-opr(t),
bnot: ¬bb,
cons: [a / b]
Lemmas referenced :
varterm_wf,
nullvar_wf,
mkwfterm_wf,
istype-universe,
true_wf,
squash_wf,
equal_wf,
istype-false,
int_seg_subtype_nat,
top_wf,
subtype_rel_list,
select-firstn,
length_firstn,
int_formula_prop_eq_lemma,
intformeq_wf,
iff_weakening_equal,
istype-le,
lelt_wf,
subtype_rel_sets_simple,
length_firstn_eq,
firstn_wf,
wfterm_wf,
istype-void,
istype-true,
decidable_wf,
subtype_rel_self,
list_wf,
decidable__assert,
decidable__all_int_seg,
decidable__equal_int,
varname_wf,
length_wf_nat,
istype-less_than,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_seg_properties,
int_seg_wf,
all_wf,
int_subtype_base,
le_wf,
set_subtype_base,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
non_neg_length,
select_wf,
istype-nat,
pi1_wf,
equal-wf-base,
nat_wf,
length_wf,
less_than_wf,
decidable__and2,
wf_term_var_lemma,
term_wf,
istype-assert,
wf-term_wf,
assert_wf,
isect_wf,
term-accum_wf,
assert_of_bnot,
eqff_to_assert,
uiff_transitivity,
eqtt_to_assert,
not_wf,
bnot_wf,
bool_wf,
equal-wf-T-base,
l_member_wf,
list-subtype,
bfalse_wf,
btrue_wf,
assert_functionality_wrt_uiff,
nat_properties,
uimplies_subtype,
subtype_rel_dep_function,
bound-term_wf,
mkterm_wf,
bool_subtype_base,
subtype_base_sq,
assert_elim,
assert-wf-mkterm,
respects-equality-set-trivial2,
iff_imp_equal_bool,
assert_of_tt,
int_term_value_add_lemma,
itermAdd_wf,
pi2_wf,
term-size_wf,
lsum_wf,
term_accum_mkterm_lemma,
term_size_mkterm_lemma,
term_accum_varterm_lemma,
term_size_var_lemma,
coterm-fun_wf,
subtype_rel_weakening,
term-ext,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
subtract-1-ge-0,
ge_wf,
subtype_rel_product,
subtype_rel_set,
isvarterm_wf,
wfbts_wf,
term-size-positive,
false_wf,
add-is-int-iff,
select_member,
sq_stable__le,
summand-le-lsum,
firstn_all,
list_accum_append,
first0,
list_accum_cons_lemma,
list_accum_nil_lemma,
firstn_length,
firstn_decomp,
istype-base,
stuck-spread,
length_of_nil_lemma,
nil_wf,
length-append,
length_of_cons_lemma,
cons_wf,
append_wf,
iff_weakening_uiff,
assert-bnot,
bool_cases_sqequal,
assert_of_lt_int,
lt_int_wf,
select-append
Rules used in proof :
universeEquality,
baseClosed,
imageMemberEquality,
setIsType,
productIsType,
functionEquality,
functionExtensionality,
instantiate,
functionIsType,
sqequalBase,
intEquality,
imageElimination,
rename,
setElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
inhabitedIsType,
voidElimination,
independent_pairFormation,
int_eqEquality,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
natural_numberEquality,
productElimination,
unionElimination,
independent_isectElimination,
isect_memberEquality_alt,
closedConclusion,
because_Cache,
isectEquality,
productEquality,
lambdaFormation_alt,
Error :memTop,
dependent_functionElimination,
universeIsType,
dependent_set_memberEquality_alt,
applyEquality,
sqequalRule,
lambdaEquality_alt,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
equalityElimination,
isectIsType,
dependent_pairEquality_alt,
setEquality,
independent_pairEquality,
cumulativity,
applyLambdaEquality,
hyp_replacement,
addEquality,
hypothesis_subsumption,
promote_hyp,
functionIsTypeImplies,
axiomEquality,
intWeakElimination,
axiomSqEquality,
baseApply,
pointwiseFunctionality,
minusEquality
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{}[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-10_00_09
Last ObjectModification:
2020_03_11-PM-09_42_33
Theory : terms
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