Nuprl Lemma : strongwellfounded_rel_exp
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[swf:SWellFounded(x R y)]. ∀[n:ℕ+]. ∀[x,y:T].
(((fst(swf)) x) + n) ≤ ((fst(swf)) y) supposing x R^n y
Proof
Definitions occuring in Statement :
strongwellfounded: SWellFounded(R[x; y]),
rel_exp: R^n,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
pi1: fst(t),
le: A ≤ B,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
le: A ≤ B,
and: P ∧ Q,
infix_ap: x f y,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
so_apply: x[s],
top: Top,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
nat_plus: ℕ+,
bfalse: ff,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
rel_exp: R^n,
all: ∀x:A. B[x],
pi1: fst(t),
exists: ∃x:A. B[x],
strongwellfounded: SWellFounded(R[x; y]),
squash: ↓T,
less_than: a < b,
btrue: tt,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
nequal: a ≠ b ∈ T
Lemmas referenced :
le_witness_for_triv,
rel_exp_wf,
nat_plus_subtype_nat,
istype-universe,
nat_plus_wf,
strongwellfounded_wf,
primrec-wf-nat-plus,
nat_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
infix_ap_wf,
all_wf,
nat_plus_properties,
equal_wf,
exists_wf,
int_term_value_add_lemma,
itermAdd_wf,
less_than_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
int_formula_prop_eq_lemma,
intformeq_wf,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
independent_isectElimination,
Error :universeIsType,
applyEquality,
hypothesisEquality,
sqequalRule,
because_Cache,
Error :inhabitedIsType,
Error :lambdaEquality_alt,
Error :functionIsType,
universeEquality,
independent_functionElimination,
addEquality,
functionExtensionality,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
natural_numberEquality,
dependent_functionElimination,
dependent_set_memberEquality,
instantiate,
functionEquality,
lambdaEquality,
cumulativity,
setElimination,
lambdaFormation,
rename,
productEquality,
imageElimination,
applyLambdaEquality,
hyp_replacement,
promote_hyp,
equalityElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[swf:SWellFounded(x R y)]. \mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[x,y:T].
(((fst(swf)) x) + n) \mleq{} ((fst(swf)) y) supposing x rel\_exp(T; R; n) y
Date html generated:
2019_06_20-PM-02_01_50
Last ObjectModification:
2018_10_06-AM-11_23_54
Theory : relations2
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