feat(library): weak duality of max-flow min-cut theorem #18996
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LaTeX is used.
Implemented in LaTeX
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eric-wieser
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leanpkg.toml
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| @@ -5,3 +5,4 @@ lean_version = "leanprover-community/lean:3.50.3" | |||
| path = "src" | |||
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| [dependencies] | |||
| mathlib = {git = "https://github.com/leanprover-community/mathlib", rev = "290a7ba01fbcab1b64757bdaa270d28f4dcede35"} | |||
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This doesn't make any sense in mathlib itself!
eric-wieser
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| -- /-! | ||
| -- Here is our second big lemma, if the active flow is maximum, | ||
| -- then no augmenting path exists in the residual network. | ||
| -- -/ | ||
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| -- lemma no_augm_path {V : Type*} [inst' : fintype V] | ||
| -- (rsn : residual_network V) : (is_max_flow rsn.afn) -> no_augmenting_path rsn := | ||
| -- begin |
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If this is intended to be part of a later PR, you should just delete it from this one.
eric-wieser
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Max-Flow_Min-Cut_Proof.pdf
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This should either be:
- Hosted on your own website / arxiv, and linked from
references.bib - Put in the module docstring, or in various smaller docstrings throughout the file. Note that the doc website supports LaTeX in
$$s
eric-wieser
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May 12, 2023
| (is_edge : V → V → Prop) | ||
| (nonsymmetric : ∀ u v : V, ¬ ((is_edge u v) ∧ (is_edge v u))) | ||
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| structure capacity (V : Type*) [inst : fintype V] |
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To me this looks like it would work better as
Suggested change
| structure capacity (V : Type*) [inst : fintype V] | |
| structure capacity {V : Type*} [inst : fintype V] (g : digraph V) |
Which says "a capacity on a particular graph". I assume you don't ever need to compare capacities between different graphs?
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Formalise and validate the weak duality of the max-flow min-cut theorem, i.e. the value of every flow is less than or equal to the capacity of every cut.
The rest of the max-flow min-cut theorem proof is commented out because it contains sorry.