mathlib documentation


Strongly Connected Components #

This file defines tactics to construct proofs of equivalences between a set of mutually equivalent propositions. The tactics use implications transitively to find sets of equivalent propositions.

Implementation notes #

The tactics use a strongly connected components algorithm on a graph where propositions are vertices and edges are proofs that the source implies the target. The strongly connected components are therefore sets of propositions that are pairwise equivalent to each other.

The resulting strongly connected components are encoded in a disjoint set data structure to facilitate the construction of equivalence proofs between two arbitrary members of an equivalence class.

Possible generalizations #

Instead of reasoning about implications and equivalence, we could generalize the machinery to reason about arbitrary partial orders.

References #

Tags #

graphs, tactic, strongly connected components, disjoint sets

meta def tactic.closure  :

closure implements a disjoint set data structure using path compression optimization. For the sake of the scc algorithm, it also stores the preorder numbering of the equivalence graph of the local assumptions.

The expr_map encodes a directed forest by storing for every non-root node, a reference to its parent and a proof of equivalence between that node's expression and its parent's expression. Given that data structure, checking that two nodes belong to the same tree is easy and fast by repeatedly following the parent references until a root is reached. If both nodes have the same root, they belong to the same tree, i.e. their expressions are equivalent. The proof of equivalence can be formed by composing the proofs along the edges of the paths to the root.

More concretely, if we ignore preorder numbering, the set { {e₀,e₁,e₂,e₃}, {e₄,e₅} } is represented as:

e₀        -- no parent, i.e. e₀ is a root
e₁  e₀, p₁ -- with p₁ : e₁ ↔ e₀
e₂  e₁, p₂ -- with p₂ : e₂ ↔ e₁
e₃  e₀, p₃ -- with p₃ : e₃ ↔ e₀
e₄        -- no parent, i.e. e₄ is a root
e₅  e₄, p₅ -- with p₅ : e₅ ↔ e₄

We can check that e₂ and e₃ are equivalent by seeking the root of the tree of each. The parent of e₂ is e₁, the parent of e₁ is e₀ and e₀ does not have a parent, and thus, this is the root of its tree. The parent of e₃ is e₀ and it's also the root, the same as for e₂ and they are therefore equivalent. We can build a proof of that equivalence by using transitivity on p₂, p₁ and p₃.symm in that order.

Similarly, we can discover that e₂ and e₅ aren't equivalent.

A description of the path compression optimization can be found at:

Instances for tactic.closure
meta def tactic.closure.with_new_closure {α : Type u_1} :
(tactic.closuretactic α)tactic α

with_new_closure f creates an empty closure c, executes f on c, and then deletes c, returning the output of f.

to_tactic_format cl pretty-prints the closure cl as a list. Assuming cl was built by dfs_at, each element corresponds to a node pᵢ : expr and is one of the folllowing:

  • if pᵢ is a root: "pᵢ ⇐ i", where i is the preorder number of pᵢ,
  • otherwise: "(pᵢ, pⱼ) : P", where P is pᵢ ↔ pⱼ. Useful for debugging.

(n,r,p) ← root cl e returns r the root of the tree that e is a part of (which might be itself) along with p a proof of e ↔ r and n, the preorder numbering of the root.

meta def tactic.closure.merge_intl (cl : tactic.closure) (p e₀ p₀ e₁ p₁ : expr) :

(Implementation of merge.)

merge cl p, with p a proof of e₀ ↔ e₁ for some e₀ and e₁, merges the trees of e₀ and e₁ and keeps the root with the smallest preorder number as the root. This ensures that, in the depth-first traversal of the graph, when encountering an edge going into a vertex whose equivalence class includes a vertex that originated the current search, that vertex will be the root of the corresponding tree.

Sequentially assign numbers to the nodes of the graph as they are being visited.

meta def tactic.closure.prove_eqv (cl : tactic.closure) (e₀ e₁ : expr) :

prove_eqv cl e₀ e₁ constructs a proof of equivalence of e₀ and e₁ if they are equivalent.

meta def tactic.closure.prove_impl (cl : tactic.closure) (e₀ e₁ : expr) :

prove_impl cl e₀ e₁ constructs a proof of e₀ -> e₁ if they are equivalent.

meta def tactic.closure.is_eqv (cl : tactic.closure) (e₀ e₁ : expr) :

is_eqv cl e₀ e₁ checks whether e₀ and e₁ are equivalent without building a proof.

meta def tactic.impl_graph  :

mutable graphs between local propositions that imply each other with the proof of implication

meta def tactic.with_impl_graph {α : Type u_1} :

with_impl_graph f creates an empty impl_graph g, executes f on g, and then deletes g, returning the output of f.

add_edge g p, with p a proof of v₀ → v₁ or v₀ ↔ v₁, adds an edge to the implication graph g.

merge_path path e, where path and e forms a cycle with proofs of implication between consecutive vertices. The proofs are compiled into proofs of equivalences and added to the closure structure. e and the first vertex of path do not have to be the same but they have to be in the same equivalence class.

(implementation of collapse)

collapse path v, where v is a vertex that originated the current search (or a vertex in the same equivalence class as the one that originated the current search). It or its equivalent should be found in path. Since the vertices following v in the path form a cycle with v, they can all be added to an equivalence class.

Strongly connected component algorithm inspired by Tarjan's and Dijkstra's scc algorithm. Whereas they return strongly connected components by enumerating them, this algorithm returns a disjoint set data structure using path compression. This is a compact representation that allows us, after the fact, to construct a proof of equivalence between any two members of an equivalence class.

  • Tarjan, R. E. (1972), "Depth-first search and linear graph algorithms", SIAM Journal on Computing, 1 (2): 146–160, doi:10.1137/0201010
  • Dijkstra, Edsger (1976), A Discipline of Programming, NJ: Prentice Hall, Ch. 25.

Use the local assumptions to create a set of equivalence classes.

scc uses the available equivalences and implications to prove a goal of the form p ↔ q.

example (p q r : Prop) (hpq : p  q) (hqr : q  r) (hrp : r  p) : p  r :=
by scc

Collect all the available equivalences and implications and add assumptions for every equivalence that can be proven using the strongly connected components technique. Mostly useful for testing.

scc uses the available equivalences and implications to prove a goal of the form p ↔ q.

example (p q r : Prop) (hpq : p  q) (hqr : q  r) (hrp : r  p) : p  r :=
by scc

The variant scc' populates the local context with all equivalences that scc is able to prove. This is mostly useful for testing purposes.