Nuprl Lemma : pcw-path-coPath_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. ∀[p:Path].
∀[n:ℕ]
((pcw-path-coPath(n;p) ∈ copath(a.B[a];w))
∧ ((copath-length(pcw-path-coPath(n;p)) = n ∈ ℤ)
⇒ (copath-at(w;pcw-path-coPath(n;p)) = (fst(snd((p n)))) ∈ coW(A;a.B[a]))))
supposing StepAgree(p 0;⋅;w)
Proof
Definitions occuring in Statement :
pcw-path-coPath: pcw-path-coPath(n;p),
copath-length: copath-length(p),
copath-at: copath-at(w;p),
copath: copath(a.B[a];w),
coW: coW(A;a.B[a]),
pcw-path: Path,
pcw-step-agree: StepAgree(s;p1;w),
nat: ℕ,
it: ⋅,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
implies: P ⇒ Q,
and: P ∧ Q,
unit: Unit,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
coW: coW(A;a.B[a]),
less_than': less_than'(a;b),
le: A ≤ B,
pcw-path: Path,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
and: P ∧ Q,
top: Top,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
coPath-at: coPath-at(n;w;p),
pi1: fst(t),
pi2: snd(t),
copath-nil: (),
spreadn: spread3,
pcw-step-agree: StepAgree(s;p1;w),
copath-at: copath-at(w;p),
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
pcw-path-coPath: pcw-path-coPath(n;p),
guard: {T},
or: P ∨ Q,
decidable: Dec(P),
let: let,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
pcw-steprel: StepRel(s1;s2),
cand: A c∧ B,
true: True,
coW-dom: coW-dom(a.B[a];w),
squash: ↓T,
ext-eq: A ≡ B,
sq_type: SQType(T),
coW-item: coW-item(w;b)
Lemmas referenced :
istype-universe,
coW_wf,
pcw-path_wf,
istype-le,
it_wf,
unit_wf2,
pcw-step-agree_wf,
istype-nat,
subtract-1-ge-0,
istype-less_than,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
equal-wf-base,
and_wf,
top_wf,
param-co-W_wf,
subtype_rel-equal,
equal_wf,
equal-wf-T-base,
pcw-step_wf,
le_wf,
false_wf,
copath-nil_wf,
copath_length_nil_lemma,
int_subtype_base,
set_subtype_base,
copath-length_wf,
subtract-add-cancel,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
decidable__le,
subtract_wf,
eq_int_wf,
bool_wf,
assert_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
bnot_wf,
not_wf,
istype-assert,
pcw-steprel_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
istype-top,
copath-extend_wf,
coW-ext,
subtype_rel_weakening,
pi1_wf,
coW-dom_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
copath-at-extend,
subtype_base_sq,
unit_subtype_base,
coW-item_wf
Rules used in proof :
universeEquality,
Error :functionIsType,
Error :dependent_set_memberEquality_alt,
applyEquality,
cumulativity,
instantiate,
equalitySymmetry,
equalityTransitivity,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
axiomEquality,
Error :universeIsType,
independent_pairFormation,
sqequalRule,
voidElimination,
Error :isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
Error :lambdaEquality_alt,
Error :dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
Error :lambdaFormation_alt,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
intEquality,
applyLambdaEquality,
voidEquality,
isect_memberEquality,
baseClosed,
productEquality,
productElimination,
lambdaFormation,
dependent_set_memberEquality,
because_Cache,
functionExtensionality,
lambdaEquality,
independent_pairEquality,
sqequalBase,
closedConclusion,
baseApply,
Error :equalityIstype,
unionElimination,
equalityElimination,
hyp_replacement,
promote_hyp,
hypothesis_subsumption,
functionEquality,
imageElimination,
imageMemberEquality,
Error :productIsType
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])]. \mforall{}[p:Path].
\mforall{}[n:\mBbbN{}]
((pcw-path-coPath(n;p) \mmember{} copath(a.B[a];w))
\mwedge{} ((copath-length(pcw-path-coPath(n;p)) = n)
{}\mRightarrow{} (copath-at(w;pcw-path-coPath(n;p)) = (fst(snd((p n)))))))
supposing StepAgree(p 0;\mcdot{};w)
Date html generated:
2019_06_20-PM-00_57_04
Last ObjectModification:
2019_04_11-AM-09_56_17
Theory : co-recursion-2
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