Nuprl Lemma : int_seg_ind
∀i:ℤ. ∀j:{i + 1...}. ∀[E:{i..j-} ⟶ ℙ{u}]. (E[i] ⇒ (∀k:{i + 1..j-}. (E[k - 1] ⇒ E[k])) ⇒ {∀k:{i..j-}. E[k]})
Proof
Definitions occuring in Statement :
int_upper: {i...},
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
member: t ∈ T,
int_upper: {i...},
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
label: ...$L... t
Lemmas referenced :
int_seg_wf,
int_seg_properties,
int_upper_properties,
decidable__equal_int,
subtract_wf,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__le,
intformand_wf,
intformle_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
le_wf,
less_than_wf,
int_upper_wf,
int_seg_well_founded_up,
upper_subtype_upper,
istype-false,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
sqequalRule,
Error :functionIsType,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
addEquality,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
hypothesis,
applyEquality,
because_Cache,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
Error :dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
Error :productIsType,
universeEquality,
instantiate,
minusEquality,
multiplyEquality,
Error :inhabitedIsType
Latex:
\mforall{}i:\mBbbZ{}. \mforall{}j:\{i + 1...\}.
\mforall{}[E:\{i..j\msupminus{}\} {}\mrightarrow{} \mBbbP{}\{u\}]. (E[i] {}\mRightarrow{} (\mforall{}k:\{i + 1..j\msupminus{}\}. (E[k - 1] {}\mRightarrow{} E[k])) {}\mRightarrow{} \{\mforall{}k:\{i..j\msupminus{}\}. E[k]\})
Date html generated:
2019_06_20-PM-01_15_26
Last ObjectModification:
2018_10_06-AM-11_22_04
Theory : int_2
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