Nuprl Lemma : hereditarily-mkterm
∀[opr:Type]. ∀[P:term(opr) ⟶ ℙ].
∀f:opr. ∀bts:bound-term(opr) List.
(hereditarily(opr;s.P[s];mkterm(f;bts))
⇐⇒ P[mkterm(f;bts)] ∧ (∀bt:bound-term(opr). ((bt ∈ bts) ⇒ hereditarily(opr;s.P[s];snd(bt)))))
Proof
Definitions occuring in Statement :
hereditarily: hereditarily(opr;s.P[s];t),
bound-term: bound-term(opr),
mkterm: mkterm(opr;bts),
term: term(opr),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
pi2: snd(t),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bound-term: bound-term(opr),
rev_implies: P ⇐ Q,
pi2: snd(t),
hereditarily: hereditarily(opr;s.P[s];t),
l_member: (x ∈ l),
exists: ∃x:A. B[x],
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
or: P ∨ Q,
cand: A c∧ B,
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
guard: {T}
Lemmas referenced :
l_member_wf,
bound-term_wf,
hereditarily_wf,
term_wf,
mkterm_wf,
list_wf,
istype-universe,
hereditarily_functionality_wrt_subterm,
subterm-mkterm,
istype-le,
istype-less_than,
length_wf,
subterm_wf,
select_wf,
nat_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,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
select_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
applyEquality,
productElimination,
productIsType,
functionIsType,
because_Cache,
universeEquality,
instantiate,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation_alt,
setElimination,
rename,
dependent_set_memberEquality_alt,
natural_numberEquality,
inlFormation_alt,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
inhabitedIsType,
applyLambdaEquality,
independent_isectElimination,
unionElimination,
approximateComputation,
int_eqEquality,
Error :memTop,
voidElimination,
unionIsType,
imageElimination,
hyp_replacement
Latex:
\mforall{}[opr:Type]. \mforall{}[P:term(opr) {}\mrightarrow{} \mBbbP{}].
\mforall{}f:opr. \mforall{}bts:bound-term(opr) List.
(hereditarily(opr;s.P[s];mkterm(f;bts))
\mLeftarrow{}{}\mRightarrow{} P[mkterm(f;bts)] \mwedge{} (\mforall{}bt:bound-term(opr). ((bt \mmember{} bts) {}\mRightarrow{} hereditarily(opr;s.P[s];snd(bt)))))
Date html generated:
2020_05_19-PM-09_54_37
Last ObjectModification:
2020_03_10-PM-03_49_56
Theory : terms
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