Nuprl Lemma : mk_applies_fun
∀[F,K,v:Top]. ∀[n:ℕ]. ∀[m:ℕn + 1]. (mk_applies(F;λk.if (k =z n) then v else K k fi ;m) ~ mk_applies(F;K;m))
Proof
Definitions occuring in Statement :
mk_applies: mk_applies(F;G;m),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
lambda: λx.A[x],
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),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
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,
istype-le,
subtract-1-ge-0,
int_seg_wf,
istype-nat,
istype-top,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
eq_int_wf,
subtract_wf,
assert_of_eq_int,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
neg_assert_of_eq_int,
primrec-unroll
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,
because_Cache,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
isectIsTypeImplies,
equalityElimination,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[F,K,v:Top]. \mforall{}[n:\mBbbN{}]. \mforall{}[m:\mBbbN{}n + 1].
(mk\_applies(F;\mlambda{}k.if (k =\msubz{} n) then v else K k fi ;m) \msim{} mk\_applies(F;K;m))
Date html generated:
2020_05_20-AM-07_49_12
Last ObjectModification:
2019_11_27-PM-04_17_37
Theory : untyped!computation
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