Nuprl Lemma : mk_lambdas_fun_compose1
∀[f:Top]. ∀[k,m:ℕ]. ∀[n:ℕm + 1].
(mk_lambdas_fun(λh.mk_lambdas(h mk_lambdas_fun(f;n);k);m)
~ mk_lambdas_fun(λg.mk_lambdas_fun(λh.mk_lambdas(h (f g);k);m - n);n))
Proof
Definitions occuring in Statement :
mk_lambdas: mk_lambdas(F;m),
mk_lambdas_fun: mk_lambdas_fun(F;m),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
lambda: λx.A[x],
subtract: n - m,
add: n + m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
mk_lambdas_fun: mk_lambdas_fun(F;m),
mk_lambdas-fun: mk_lambdas-fun(F;G;n;m),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m
Lemmas referenced :
int_seg_properties,
nat_wf,
le_wf,
int_seg_wf,
top_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
base_wf,
btrue_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
subtract_wf,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
itermAdd_wf,
int_term_value_add_lemma,
eqff_to_assert,
le_int_wf,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
bfalse_wf,
mk_lambdas_fun-unroll-first,
decidable__lt,
false_wf,
not-lt-2,
le_antisymmetry_iff,
less-iff-le,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
zero-add,
add_functionality_wrt_le,
add-commutes,
le-add-cancel2,
add-zero,
subtract-add-cancel
Rules used in proof :
cut,
sqequalRule,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
natural_numberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
hypothesis_subsumption,
lambdaEquality,
dependent_set_memberEquality,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
isect_memberFormation,
sqequalAxiom,
isect_memberEquality,
intWeakElimination,
lambdaFormation,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
promote_hyp,
cumulativity,
applyEquality,
minusEquality,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[f:Top]. \mforall{}[k,m:\mBbbN{}]. \mforall{}[n:\mBbbN{}m + 1].
(mk\_lambdas\_fun(\mlambda{}h.mk\_lambdas(h mk\_lambdas\_fun(f;n);k);m)
\msim{} mk\_lambdas\_fun(\mlambda{}g.mk\_lambdas\_fun(\mlambda{}h.mk\_lambdas(h (f g);k);m - n);n))
Date html generated:
2017_10_01-AM-08_40_56
Last ObjectModification:
2017_07_26-PM-04_28_18
Theory : untyped!computation
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