Nuprl Lemma : proof-tree-induction
∀[Sequent,Rule:Type].
∀effect:(Sequent × Rule) ⟶ (Sequent List?)
∀[Q:proof-tree(Sequent;Rule;effect) ⟶ ℙ]
((∀s:Sequent. ∀r:Rule. Q[proof-abort(s;r)] supposing ↑isr(effect <s, r>))
⇒ (∀s:Sequent. ∀r:Rule.
∀L:proof-tree(Sequent;Rule;effect) List
(∀pf∈L.Q[pf]) ⇒ Q[make-proof-tree(s;r;L)] supposing ||L|| = ||outl(effect <s, r>)|| ∈ ℤ
supposing ↑isl(effect <s, r>))
⇒ (∀pf:proof-tree(Sequent;Rule;effect). Q[pf]))
Proof
Definitions occuring in Statement :
proof-abort: proof-abort(s;r),
make-proof-tree: make-proof-tree(s;r;L),
proof-tree: proof-tree(Sequent;Rule;effect),
l_all: (∀x∈L.P[x]),
length: ||as||,
list: T List,
outl: outl(x),
assert: ↑b,
isr: isr(x),
isl: isl(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
union: left + right,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
proof-tree: proof-tree(Sequent;Rule;effect),
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
and: P ∧ Q,
isl: isl(x),
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
subtype_rel: A ⊆r B,
isr: isr(x),
prop: ℙ,
nat: ℕ,
outl: outl(x),
not: ¬A,
false: False,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
squash: ↓T,
l_all: (∀x∈L.P[x]),
top: Top,
Wsup: Wsup(a;b),
make-proof-tree: make-proof-tree(s;r;L),
lelt: i ≤ j < k,
int_seg: {i..j-},
proof-abort: proof-abort(s;r)
Lemmas referenced :
W-induction,
int_seg_wf,
length_wf,
btrue_wf,
bfalse_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert_wf,
isr_wf,
list_wf,
unit_wf2,
istype-void,
proof-tree_wf,
istype-assert,
istype-int,
length_wf_nat,
set_subtype_base,
le_wf,
int_subtype_base,
outl_wf,
l_all_wf,
l_member_wf,
make-proof-tree_wf,
subtype_rel_self,
proof-abort_wf,
istype-universe,
nat_wf,
non_neg_length,
btrue_neq_bfalse,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
squash_wf,
isl_wf,
istype-le,
mklist_wf,
mklist_length,
trivial-equal,
select-mklist,
W_wf,
true_wf,
Wsup_wf,
less_than_wf,
equal_wf,
istype-less_than,
lelt_wf,
subtype_rel_sets,
unit_subtype_base,
subtype_rel-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
productEquality,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
inhabitedIsType,
hypothesis,
unionElimination,
natural_numberEquality,
voidEquality,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
productIsType,
universeIsType,
equalityElimination,
productElimination,
rename,
independent_isectElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
applyLambdaEquality,
setElimination,
instantiate,
cumulativity,
functionIsType,
because_Cache,
hyp_replacement,
voidElimination,
isectIsType,
independent_pairEquality,
intEquality,
sqequalBase,
setIsType,
universeEquality,
unionIsType,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
imageElimination,
imageMemberEquality,
baseClosed,
isect_memberEquality_alt,
closedConclusion,
functionExtensionality,
equalityIsType1,
baseApply,
equalityIsType4,
unionEquality,
promote_hyp,
equalityIsType3
Latex:
\mforall{}[Sequent,Rule:Type].
\mforall{}effect:(Sequent \mtimes{} Rule) {}\mrightarrow{} (Sequent List?)
\mforall{}[Q:proof-tree(Sequent;Rule;effect) {}\mrightarrow{} \mBbbP{}]
((\mforall{}s:Sequent. \mforall{}r:Rule. Q[proof-abort(s;r)] supposing \muparrow{}isr(effect <s, r>))
{}\mRightarrow{} (\mforall{}s:Sequent. \mforall{}r:Rule.
\mforall{}L:proof-tree(Sequent;Rule;effect) List
(\mforall{}pf\mmember{}L.Q[pf]) {}\mRightarrow{} Q[make-proof-tree(s;r;L)] supposing ||L|| = ||outl(effect <s, r>)||
supposing \muparrow{}isl(effect <s, r>))
{}\mRightarrow{} (\mforall{}pf:proof-tree(Sequent;Rule;effect). Q[pf]))
Date html generated:
2020_05_20-AM-08_04_51
Last ObjectModification:
2019_12_26-PM-04_07_32
Theory : general
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