Nuprl Lemma : mk_applies_lambdas_fun
∀[F,G:Top]. ∀[m:ℕ]. ∀[n:ℕm + 1].
(mk_applies(mk_lambdas_fun(F;m);G;n) ~ mk_lambdas_fun(λg.(F (λx.(g mk_applies(x;G;n))));m - n))
Proof
Definitions occuring in Statement :
mk_applies: mk_applies(F;G;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,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
mk_applies: mk_applies(F;G;m),
sq_type: SQType(T),
guard: {T},
nat_plus: ℕ+
Lemmas referenced :
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
itermAdd_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
int_seg_subtype_nat,
istype-false,
ge_wf,
istype-less_than,
primrec0_lemma,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
mk_lambdas_fun-eta,
istype-le,
subtract-1-ge-0,
int_seg_wf,
istype-nat,
istype-top,
primrec1_lemma,
mk_lambdas_fun-unroll-first,
decidable__lt,
subtract_wf,
mk_applies_unroll,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setElimination,
rename,
productElimination,
hypothesis,
imageElimination,
hypothesisEquality,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
applyEquality,
addEquality,
lambdaFormation_alt,
inhabitedIsType,
intWeakElimination,
axiomSqEquality,
functionIsTypeImplies,
instantiate,
cumulativity,
intEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
isectIsTypeImplies,
dependent_set_memberEquality_alt
Latex:
\mforall{}[F,G:Top]. \mforall{}[m:\mBbbN{}]. \mforall{}[n:\mBbbN{}m + 1].
(mk\_applies(mk\_lambdas\_fun(F;m);G;n) \msim{} mk\_lambdas\_fun(\mlambda{}g.(F (\mlambda{}x.(g mk\_applies(x;G;n))));m - n))
Date html generated:
2020_05_20-AM-07_49_17
Last ObjectModification:
2019_11_27-PM-04_19_28
Theory : untyped!computation
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