Nuprl Lemma : proof-tree-ext
∀[Sequent,Rule:Type]. ∀[effect:(Sequent × Rule) ⟶ (Sequent List?)].
proof-tree(Sequent;Rule;effect) ≡ sr:Sequent × Rule × (case effect sr
of inl(subgoals) =>
ℕ||subgoals||
| inr(x) =>
Void ⟶ proof-tree(Sequent;Rule;effect))
Proof
Definitions occuring in Statement :
proof-tree: proof-tree(Sequent;Rule;effect),
length: ||as||,
list: T List,
int_seg: {i..j-},
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
union: left + right,
natural_number: $n,
void: Void,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
proof-tree: proof-tree(Sequent;Rule;effect),
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B
Lemmas referenced :
W-ext,
list_wf,
unit_wf2,
int_seg_wf,
length_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
unionEquality,
hypothesis,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
unionElimination,
natural_numberEquality,
voidEquality,
dependent_functionElimination,
independent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
functionEquality,
isect_memberEquality,
because_Cache,
universeEquality
Latex:
\mforall{}[Sequent,Rule:Type]. \mforall{}[effect:(Sequent \mtimes{} Rule) {}\mrightarrow{} (Sequent List?)].
proof-tree(Sequent;Rule;effect) \mequiv{} sr:Sequent \mtimes{} Rule \mtimes{} (case effect sr
of inl(subgoals) =>
\mBbbN{}||subgoals||
| inr(x) =>
Void {}\mrightarrow{} proof-tree(Sequent;Rule;effect))
Date html generated:
2019_10_15-AM-11_06_10
Last ObjectModification:
2018_08_21-PM-01_57_53
Theory : general
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