Nuprl Lemma : proof_tree_ind_wf
∀[Sequent,Rule:Type]. ∀[effect:(Sequent × Rule) ⟶ (Sequent List?)]. ∀[Q:proof-tree(Sequent;Rule;effect) ⟶ ℙ].
∀[abort:∀s:Sequent. ∀r:Rule. Q[proof-abort(s;r)] supposing ↑isr(effect <s, r>)].
∀[progress:∀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)].
(proof_tree_ind(effect;abort;progress;pf) ∈ Q[pf])
Proof
Definitions occuring in Statement :
proof_tree_ind: proof_tree_ind(effect;abort;progress;pf),
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,
member: t ∈ T,
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 :
false: False,
not: ¬A,
and: P ∧ Q,
isl: isl(x),
outl: outl(x),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
proof-tree-induction-ext
Lemmas referenced :
all_wf,
make-proof-tree_wf,
l_member_wf,
l_all_wf,
btrue_neq_bfalse,
and_wf,
bfalse_wf,
assert_elim,
length_wf,
equal_wf,
isl_wf,
proof-abort_wf,
isr_wf,
assert_wf,
proof-tree_wf,
unit_wf2,
list_wf,
isect_wf,
proof-tree-induction-ext
Rules used in proof :
isect_memberEquality,
axiomEquality,
levelHypothesis,
addLevel,
setEquality,
dependent_functionElimination,
voidElimination,
independent_functionElimination,
productElimination,
rename,
setElimination,
applyLambdaEquality,
independent_pairFormation,
dependent_set_memberEquality,
unionElimination,
lambdaFormation,
intEquality,
independent_isectElimination,
independent_pairEquality,
functionExtensionality,
isectEquality,
because_Cache,
unionEquality,
cumulativity,
productEquality,
functionEquality,
universeEquality,
sqequalHypSubstitution,
equalitySymmetry,
equalityTransitivity,
hypothesisEquality,
isectElimination,
lambdaEquality,
sqequalRule,
applyEquality,
hypothesis,
extract_by_obid,
instantiate,
thin,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[Sequent,Rule:Type]. \mforall{}[effect:(Sequent \mtimes{} Rule) {}\mrightarrow{} (Sequent List?)].
\mforall{}[Q:proof-tree(Sequent;Rule;effect) {}\mrightarrow{} \mBbbP{}]. \mforall{}[abort:\mforall{}s:Sequent. \mforall{}r:Rule.
Q[proof-abort(s;r)]
supposing \muparrow{}isr(effect <s, r>)].
\mforall{}[progress:\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>)]. \mforall{}[pf:proof-tree(Sequent;Rule;effect)].
(proof\_tree\_ind(effect;abort;progress;pf) \mmember{} Q[pf])
Date html generated:
2020_05_20-AM-08_04_56
Last ObjectModification:
2020_02_04-PM-02_15_40
Theory : general
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