Nuprl Lemma : proof-tree-induction-ext
∀[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 :
member: t ∈ T,
proof_tree_ind: proof_tree_ind(effect;abort;progress;pf),
proof-tree-induction,
W-induction,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
top: Top,
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
guard: {T},
or: P ∨ Q,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
proof-tree-induction,
lifting-strict-spread,
has-value_wf_base,
base_wf,
is-exception_wf,
top_wf,
equal_wf,
lifting-strict-decide,
W-induction
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
callbyvalueApply,
baseApply,
closedConclusion,
hypothesisEquality,
applyExceptionCases,
inrFormation,
imageMemberEquality,
imageElimination,
inlFormation,
because_Cache,
sqequalSqle,
divergentSqle,
callbyvalueSpread,
productEquality,
productElimination,
sqleReflexivity,
dependent_functionElimination,
independent_functionElimination,
spreadExceptionCases,
axiomSqleEquality,
exceptionSqequal
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:
2018_05_21-PM-06_28_50
Last ObjectModification:
2018_05_19-PM-04_40_00
Theory : general
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