Nuprl Lemma : def-cont-induction-lemma
∀[P:ℕ ⟶ ℙ]
((∀n:ℕ. (P[n] ⇒ P[n + 1])) ⇒ (∀x:ℤ List. ∀[n,m:ℕ]. P[n] ⇒ P[m] supposing (x = [n, m) ∈ (ℤ List)) ∧ (n ≤ m)))
Proof
Definitions occuring in Statement :
from-upto: [n, m),
list: T List,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
and: P ∧ Q,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
le: A ≤ B,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
sq_stable: SqStable(P),
squash: ↓T,
from-upto: [n, m),
cand: A c∧ B,
has-value: (a)↓
Lemmas referenced :
list_induction,
uall_wf,
isect_wf,
equal-wf-base-T,
length_wf,
list_wf,
all_wf,
nat_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
less_than'_wf,
from-upto_wf,
subtype_rel_list,
less_than_wf,
length_of_nil_lemma,
length-from-upto,
lt_int_wf,
decidable__equal_int,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
intformless_wf,
int_formula_prop_less_lemma,
assert_wf,
bnot_wf,
not_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
list_subtype_base,
int_subtype_base,
length_of_cons_lemma,
squash_wf,
sq_stable__and,
sq_stable__equal,
value-type-has-value,
int-value-type,
cons_one_one,
null_nil_lemma,
btrue_wf,
and_wf,
equal_wf,
null_wf3,
top_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
le_weakening2,
non_neg_length,
decidable__lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
intEquality,
sqequalRule,
lambdaEquality,
because_Cache,
productEquality,
hypothesis,
applyEquality,
hypothesisEquality,
functionEquality,
functionExtensionality,
independent_functionElimination,
rename,
dependent_functionElimination,
universeEquality,
dependent_set_memberEquality,
addEquality,
setElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
cumulativity,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
setEquality,
hyp_replacement,
Error :applyLambdaEquality,
instantiate,
impliesFunctionality,
baseApply,
closedConclusion,
imageMemberEquality,
imageElimination,
callbyvalueReduce
Latex:
\mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbP{}]
((\mforall{}n:\mBbbN{}. (P[n] {}\mRightarrow{} P[n + 1]))
{}\mRightarrow{} (\mforall{}x:\mBbbZ{} List. \mforall{}[n,m:\mBbbN{}]. P[n] {}\mRightarrow{} P[m] supposing (x = [n, m)) \mwedge{} (n \mleq{} m)))
Date html generated:
2016_10_25-AM-10_52_14
Last ObjectModification:
2016_07_12-AM-07_00_51
Theory : general
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