Nuprl Lemma : int_upper_ind
∀i:ℤ. ∀[E:{i...} ⟶ ℙ{u}]. (E[i] ⇒ (∀k:{i + 1...}. (E[k - 1] ⇒ E[k])) ⇒ {∀k:{i...}. E[k]})
Proof
Definitions occuring in Statement :
int_upper: {i...},
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,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
wellfounded: WellFnd{i}(A;x,y.R[x; y])
Lemmas referenced :
all_wf,
int_upper_wf,
subtract_wf,
int_upper_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
le_wf,
int_upper_subtype_int_upper,
set_wf,
int_upper_well_founded,
less_than_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
addEquality,
hypothesisEquality,
natural_numberEquality,
hypothesis,
applyEquality,
lambdaEquality,
cumulativity,
universeEquality,
sqequalRule,
functionEquality,
functionExtensionality,
because_Cache,
dependent_set_memberEquality,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality
Latex:
\mforall{}i:\mBbbZ{}. \mforall{}[E:\{i...\} {}\mrightarrow{} \mBbbP{}\{u\}]. (E[i] {}\mRightarrow{} (\mforall{}k:\{i + 1...\}. (E[k - 1] {}\mRightarrow{} E[k])) {}\mRightarrow{} \{\mforall{}k:\{i...\}. E[k]\})
Date html generated:
2016_10_21-AM-09_59_13
Last ObjectModification:
2016_07_12-AM-05_13_17
Theory : int_2
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