Nuprl Lemma : mk_applies_ite
∀[G,b,x,y:Top]. ∀[m:ℕ].
(mk_applies(if b then x else y fi ;G;m) ~ if b then mk_applies(x;G;m) else mk_applies(y;G;m) fi )
Proof
Definitions occuring in Statement :
mk_applies: mk_applies(F;G;m),
nat: ℕ,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
top: Top,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
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,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
mk_applies: mk_applies(F;G;m),
decidable: Dec(P),
or: P ∨ Q,
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,
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
strict4: strict4(F),
has-value: (a)↓,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
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,
primrec0_lemma,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
top_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lifting-strict-decide,
has-value_wf_base,
base_wf,
is-exception_wf,
primrec-unroll
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
unionElimination,
because_Cache,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
promote_hyp,
instantiate,
cumulativity,
baseClosed,
callbyvalueApply,
baseApply,
closedConclusion,
applyExceptionCases,
inrFormation,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation
Latex:
\mforall{}[G,b,x,y:Top]. \mforall{}[m:\mBbbN{}].
(mk\_applies(if b then x else y fi ;G;m) \msim{} if b then mk\_applies(x;G;m) else mk\_applies(y;G;m) fi )
Date html generated:
2017_10_01-AM-08_41_02
Last ObjectModification:
2017_07_26-PM-04_28_21
Theory : untyped!computation
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