Nuprl Lemma : bound-terms-accum_wf
∀[opr:Type]. ∀[P:term(opr) ⟶ ℙ]. ∀[Param:Type]. ∀[C:Param ⟶ (varname() List × hered-term(opr;t.P[t])) ⟶ Type].
∀[bts:(varname() List × hered-term(opr;t.P[t])) List].
∀[f:very-dep-fun(Param;varname() List × hered-term(opr;t.P[t]);a,bt.C[a;bt])]. ∀[g:a:Param
⟶ bt:{bt:varname() List
× hered-term(opr;t.P[t])|
(bt ∈ bts)}
⟶ C[a;bt]].
(bound-terms-accum(L,bt.f[L;bt];a,bt.g[a;bt];bts)
∈ {L:(a:Param × b:varname() List × hered-term(opr;t.P[t]) × C[a;b]) List|
vdf-eq(Param;f;L) ∧ (map(λx.(fst(snd(x)));L) = bts ∈ ((varname() List × hered-term(opr;t.P[t])) List))} )
Proof
Definitions occuring in Statement :
bound-terms-accum: bound-terms-accum(L,bt.f[L; bt];param,bndtrm.g[param; bndtrm];bts),
hered-term: hered-term(opr;t.P[t]),
term: term(opr),
varname: varname(),
very-dep-fun: very-dep-fun(A;B;a,b.C[a; b]),
vdf-eq: vdf-eq(A;f;L),
l_member: (x ∈ l),
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bound-terms-accum: bound-terms-accum(L,bt.f[L; bt];param,bndtrm.g[param; bndtrm];bts),
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
and: P ∧ Q,
istype: istype(T),
pi1: fst(t),
pi2: snd(t),
implies: P ⇒ Q,
cand: A c∧ B,
respects-equality: respects-equality(S;T)
Lemmas referenced :
dep-accum_wf,
list_wf,
varname_wf,
hered-term_wf,
term_wf,
l_member_wf,
subtype_rel_dep_function,
very-dep-fun-subtype-domain,
list-subtype,
subtype_rel_sets,
vdf-eq_wf,
subtype_rel_list,
subtype_rel_product,
equal_wf,
map_wf,
pi1_wf,
pi2_wf,
subtype_rel_set,
equal_functionality_wrt_subtype_rel2,
respects-equality-list,
respects-equality-set-trivial,
very-dep-fun_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setEquality,
productEquality,
hypothesis,
lambdaEquality_alt,
applyEquality,
universeIsType,
because_Cache,
functionExtensionality,
instantiate,
cumulativity,
universeEquality,
productIsType,
setIsType,
independent_isectElimination,
setElimination,
rename,
lambdaFormation_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
productElimination,
independent_pairFormation,
independent_functionElimination,
equalityIstype,
axiomEquality,
functionIsType,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[opr:Type]. \mforall{}[P:term(opr) {}\mrightarrow{} \mBbbP{}]. \mforall{}[Param:Type]. \mforall{}[C:Param
{}\mrightarrow{} (varname() List \mtimes{} hered-term(opr;t.P[t]))
{}\mrightarrow{} Type].
\mforall{}[bts:(varname() List \mtimes{} hered-term(opr;t.P[t])) List].
\mforall{}[f:very-dep-fun(Param;varname() List \mtimes{} hered-term(opr;t.P[t]);a,bt.C[a;bt])].
\mforall{}[g:a:Param {}\mrightarrow{} bt:\{bt:varname() List \mtimes{} hered-term(opr;t.P[t])| (bt \mmember{} bts)\} {}\mrightarrow{} C[a;bt]].
(bound-terms-accum(L,bt.f[L;bt];a,bt.g[a;bt];bts) \mmember{} \{L:(a:Param
\mtimes{} b:varname() List \mtimes{} hered-term(opr;t.P[t])
\mtimes{} C[a;b]) List|
vdf-eq(Param;f;L)
\mwedge{} (map(\mlambda{}x.(fst(snd(x)));L) = bts)\} )
Date html generated:
2020_05_19-PM-09_54_43
Last ObjectModification:
2020_03_10-PM-05_34_52
Theory : terms
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