[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"xHdajZ8XYd":3},"# logic_induction\n\nA Lean 4 formalization of the exercises in \"Induction: Well-founded relations and generalized induction\", lecture notes by Prof. Mohamed Amer for a logic course at Cairo University.\n\nThe project has no dependencies, not even Mathlib. It defines its own `Set` and the few operations it needs, so that the statements stay close to the notes.\n\n## Contents\n\n- `LogicInduction/Set.lean`: sets as predicates, membership, subset, intersection, difference, `univ`, `range`, and `predR`, which is the notes' `ρ[x]`.\n- `LogicInduction/IndN.lean`: Section 1. Peano-like structures and the equivalence of the two induction principles.\n- `LogicInduction/Relations.lean`: relative product, transitivity, powers `ρⁿ`, the closures `ρ⁺` and `ρ*`, and their properties.\n- `LogicInduction/WellFounded.lean`: well-founded relations, the principle of ρ,A-induction, and the equivalence theorem.\n- `LogicInduction/SuccWF.lean`: the successor relation is well founded.\n- `LogicInduction/WfClosure.lean`: ρ is well founded iff ρ⁺ is.\n- `LogicInduction/NatOrder.lean`: `≤ = S*`, `\u003C = S⁺`, and course of values induction on `ℕ`.\n- `LogicInduction/WellOrder.lean`: well orderings and well-foundedness.\n- `docs/Logic_Induction.pdf`: my written proofs, which the Lean code follows (not updated).\n\nEach theorem carries a comment with its number in the notes where it has one.\n\n## Progress\n\n- [x] Peano-like structures and Theorem 1.5 (the two forms of induction are equivalent)\n- [x] Well-founded relations and the principle of ρ,A-induction\n- [x] Equivalence of well-foundedness, ρ,A-induction, and no infinite descending sequence\n- [x] ρ is well founded iff it is well founded on Rρ\n- [x] ρ,A-induction holds for every A iff it holds for Rρ\n- [x] The successor relation is well founded\n- [x] If σ ⊆ ρ and ρ is well founded, then σ is well founded\n- [x] Relative product: transitivity, functions, associativity, failure of commutativity\n- [x] Powers of a relation, including `ρ^(m+n) = ρ^m | ρ^n`\n- [x] `TC(ρ) = ρ⁺` and `RTC(ρ) = ρ*`\n- [x] `ρ*|ρ = ρ⁺ = ρ|ρ*`\n- [x] ρ is well founded iff ρ⁺ is\n- [x] `≤ = S*`, `\u003C = S⁺`, `\u003C` is well founded, course of values induction\n- [x] Well orderings: a strict linear order is a well ordering iff it is well founded, with a counterexample when linearity is dropped\n- [x] Remarks 1 to 3 on domains, ranges, and powers\n- [x] Uniqueness in the recursion theorem\n- [x] Other proof of Claim (⋆)\n\n## Building\n\nInstall Lean through [elan](https://github.com/leanprover/elan). The toolchain version is pinned in `lean-toolchain`.\n\n```\nlake build\n```\n\nNo network access is needed after elan has fetched the toolchain.\n\n## Proof Atlas\n\n[`atlas/`](./atlas) contains an interactive companion site that presents every declaration three ways: its informal pen-and-paper proof (transcribed from `docs/Logic_Induction.pdf` [which is not upated; I know]), the exact machine-checked Lean source, and its position in the proof dependency tree. It is a static Next.js site generated from the Lean sources; see [`atlas/README.md`](./atlas/README.md) to run it locally or just access it from [https://MinaBasilious.github.io/logic_induction/](https://MinaBasilious.github.io/logic_induction/).\n\n## Authorship\n\n- The **Lean formalization** (all definitions, theorems and proofs) is entirely human work, designed and written by me (Mina Basilious) and only *checked* by the machine (`lake build`, plus an axiom audit with `#print axioms`: every declaration depends solely on the standard axioms).\n- The **Proof Atlas** website was built with AI assistance; parsing, informal proof transcriptions, and UI. It is a presentation layer over the human-written mathematics; no mathematical statement or proof originates from AI.\n\n## Source\n\nThe statements follow the notes by Prof. Mohamed Amer, which are not included in this repository. The proofs and any mistakes in them are mine.\n\n## License\n\nSee [MIT License](./LICENSE).\n",1794686186823]