Nuprl Lemma : decidable__squash-list-match
∀[A,B:Type]. ∀[R:A ⟶ B ⟶ ℙ].
((∀a:A. ∀b:B. Dec(R[a;b])) ⇒ (∀as:A List. ∀bs:B List. Dec(↓list-match(as;bs;a,b.R[a;b]))))
Proof
Definitions occuring in Statement :
list-match: list-match(L1;L2;a,b.R[a; b]),
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
guard: {T},
not: ¬A,
false: False,
list-match-aux: list-match-aux(L1;L2;used;a,b.R[a; b]),
sq_exists: ∃x:A [B[x]],
list-match: list-match(L1;L2;a,b.R[a; b]),
and: P ∧ Q,
cand: A c∧ B,
int_seg: {i..j-},
uimplies: b supposing a,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
less_than: a < b,
subtype_rel: A ⊆r B,
ge: i ≥ j ,
nat: ℕ
Lemmas referenced :
decidable__squash-list-match-aux-ext,
all_wf,
decidable_wf,
nil_wf,
not_wf,
squash_wf,
list-match_wf,
list_wf,
int_seg_wf,
length_wf,
inject_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,
non_neg_length,
lelt_wf,
length_wf_nat,
nat_properties,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
l_member_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
intEquality,
unionElimination,
inlFormation,
imageElimination,
imageMemberEquality,
baseClosed,
inrFormation,
voidElimination,
because_Cache,
setElimination,
rename,
dependent_set_memberFormation,
productElimination,
independent_pairFormation,
natural_numberEquality,
productEquality,
functionExtensionality,
independent_isectElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[R:A {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}].
((\mforall{}a:A. \mforall{}b:B. Dec(R[a;b])) {}\mRightarrow{} (\mforall{}as:A List. \mforall{}bs:B List. Dec(\mdownarrow{}list-match(as;bs;a,b.R[a;b]))))
Date html generated:
2018_05_21-PM-00_48_47
Last ObjectModification:
2018_05_19-AM-06_51_46
Theory : list_1
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