Nuprl Lemma : mk_lambdas-sqequal-n2
∀[F1,F2:Base]. ∀n,m:ℕ. (((m ≤ n) ⇒ (F1 ~n - m F2)) ⇒ (mk_lambdas(F1;m) ~n mk_lambdas(F2;m)))
Proof
Definitions occuring in Statement :
mk_lambdas: mk_lambdas(F;m),
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtract: n - m,
base: Base,
sqequal_n: s ~n t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
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,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
mk_lambdas: mk_lambdas(F;m),
and: P ∧ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
primrec0_lemma,
decidable__le,
intformand_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
le_wf,
sqequal_n_wf,
subtract_wf,
all_wf,
nat_wf,
set_wf,
less_than_wf,
primrec-wf2,
set_subtype_base,
base_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformless_wf,
int_formula_prop_less_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
decidable__lt,
primrec-unroll
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
hypothesisEquality,
setElimination,
rename,
dependent_functionElimination,
because_Cache,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
independent_pairFormation,
functionEquality,
dependent_set_memberEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
equalityElimination,
productElimination,
promote_hyp,
sqequal_n rule,
sqequalZero
Latex:
\mforall{}[F1,F2:Base]. \mforall{}n,m:\mBbbN{}. (((m \mleq{} n) {}\mRightarrow{} (F1 \msim{}n - m F2)) {}\mRightarrow{} (mk\_lambdas(F1;m) \msim{}n mk\_lambdas(F2;m)))
Date html generated:
2017_10_01-AM-08_43_10
Last ObjectModification:
2017_07_26-PM-04_29_36
Theory : untyped!computation
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