Nuprl Lemma : cWO-induction-extract-sqequal
λf.fix((λF,t. (f t F))) ~ TERMOF{cWO-induction_1-ext:o, \\v:l, i:l}
Proof
Definitions occuring in Statement :
apply: f a,
fix: fix(F),
lambda: λx.A[x],
sqequal: s ~ t
Definitions unfolded in proof :
cWO-induction_1-ext,
bar_recursion: bar_recursion,
exists: ∃x:A. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
top: Top,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
compose: f o g,
nat_plus: ℕ+,
exposed-it: exposed-it,
less_than: a < b,
squash: ↓T,
nequal: a ≠ b ∈ T ,
so_lambda: λ2x.t[x],
so_apply: x[s],
has-value: (a)↓
Lemmas referenced :
base_wf,
nat_plus_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
fun_exp_unroll,
false_wf,
le_wf,
decidable__le,
subtract_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_weakening2,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
le_weakening,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lt_int_wf,
assert_of_lt_int,
strictness-apply,
bottom-sqle,
has-value_wf_base,
is-exception_wf,
decidable__lt,
not-lt-2,
not-equal-2,
set_subtype_base,
int_subtype_base,
add-subtract-cancel,
exception-not-value,
top_wf,
bottom_diverge,
exception-not-bottom,
cWO-induction_1-ext
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
dependent_pairFormation,
thin,
baseClosed,
sqequalIntensionalEquality,
hypothesisEquality,
sqequalHypSubstitution,
productElimination,
hypothesis,
rename,
lambdaFormation,
introduction,
extract_by_obid,
sqequalSqle,
fixpointLeast,
isectElimination,
setElimination,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
lambdaEquality,
dependent_functionElimination,
axiomSqleEquality,
dependent_set_memberEquality,
independent_pairFormation,
isect_memberEquality,
voidEquality,
unionElimination,
addEquality,
applyEquality,
intEquality,
minusEquality,
because_Cache,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
int_eqReduceTrueSq,
lessCases,
isect_memberFormation,
sqequalAxiom,
imageMemberEquality,
imageElimination,
sqleRule,
sqleReflexivity,
divergentSqle,
baseApply,
closedConclusion
Latex:
\mlambda{}f.fix((\mlambda{}F,t. (f t F))) \msim{} TERMOF\{cWO-induction\_1-ext:o, \mbackslash{}\mbackslash{}v:l, i:l\}
Date html generated:
2017_04_14-AM-07_34_33
Last ObjectModification:
2017_02_27-PM-03_08_37
Theory : fun_1
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