[Merged by Bors] - chore: improve defeq for Sup and sSup of LieSubmodules
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and some tiny additional golfing
Unfortunately because `C` is dependent, its type cannot be passed to `Submodule.iSup_induction'` so I have to provide a proof here. I have just copied the proof of `Submodule.iSup_induction'`.
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The point is that the following four lemmas are now all true by definition: - `LieSubmodule.inf_coe_toSubmodule` - `LieSubmodule.sInf_coe_toSubmodule` - `LieSubmodule.sup_coe_toSubmodule` [previously existed but not true by definition] - `LieSubmodule.sSup_coe_toSubmodule` [previously did not exist]
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| @[elab_as_elim] | ||
| theorem iSup_induction' {ι} (N : ι → LieSubmodule R L M) {C : (x : M) → (x ∈ ⨆ i, N i) → Prop} | ||
| (hN : ∀ (i) (x) (hx : x ∈ N i), C x (mem_iSup_of_mem i hx)) (h0 : C 0 (zero_mem _)) | ||
| (hadd : ∀ x y hx hy, C x hx → C y hy → C (x + y) (add_mem ‹_› ‹_›)) {x : M} | ||
| (hx : x ∈ ⨆ i, N i) : C x hx := by | ||
| refine' Exists.elim _ fun (hx : x ∈ ⨆ i, N i) (hc : C x hx) => hc | ||
| refine' iSup_induction N (C := fun x : M ↦ ∃ (hx : x ∈ ⨆ i, N i), C x hx) hx | ||
| (fun i x hx => _) _ fun x y => _ | ||
| · exact ⟨_, hN _ _ hx⟩ | ||
| · exact ⟨_, h0⟩ | ||
| · rintro ⟨_, Cx⟩ ⟨_, Cy⟩ | ||
| refine' ⟨_, hadd _ _ _ _ Cx Cy⟩ |
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I tried to golf this by proving it in terms of the submodule version, but ran afoul of leanprover/lean4#1926
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Thank you for trying all the same!
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Sup and sSup of LieSubmodulesSup and sSup of LieSubmodules
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The point is that the following four lemmas are now all true by definition: - `LieSubmodule.inf_coe_toSubmodule` - `LieSubmodule.sInf_coe_toSubmodule` - `LieSubmodule.sup_coe_toSubmodule` [previously existed but not true by definition] - `LieSubmodule.sSup_coe_toSubmodule` [previously did not exist]
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The point is that the following four lemmas are now all true by definition:
LieSubmodule.inf_coe_toSubmoduleLieSubmodule.sInf_coe_toSubmoduleLieSubmodule.sup_coe_toSubmodule[previously existed but not true by definition]LieSubmodule.sSup_coe_toSubmodule[previously did not exist]