Nuprl Lemma : mk_applies_roll
∀[F,G,x:Top]. ∀[m:ℕ]. (mk_applies(F;G;m) x ~ mk_applies(F;λi.if (i =z m) then x else G i fi ;m + 1))
Proof
Definitions occuring in Statement :
mk_applies: mk_applies(F;G;m),
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,
top: Top,
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
ge: i ≥ j
Lemmas referenced :
mk_applies_unroll,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
eq_int_wf,
subtract_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
add-subtract-cancel,
mk_applies_fun,
satisfiable-full-omega-tt,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
lelt_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
nat_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
nat_wf,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
hypothesis,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
lambdaFormation,
independent_functionElimination,
independent_isectElimination,
applyEquality,
because_Cache,
minusEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll,
promote_hyp,
instantiate,
cumulativity,
sqequalAxiom
Latex:
\mforall{}[F,G,x:Top]. \mforall{}[m:\mBbbN{}].
(mk\_applies(F;G;m) x \msim{} mk\_applies(F;\mlambda{}i.if (i =\msubz{} m) then x else G i fi ;m + 1))
Date html generated:
2017_10_01-AM-08_40_28
Last ObjectModification:
2017_07_26-PM-04_28_07
Theory : untyped!computation
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