[Merged by Bors] - feat(data/dfinsupp): Port over the finsupp.lift_add_hom API
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| top_unique $ λ x hx, (begin | ||
| apply dfinsupp.induction x, | ||
| exact add_submonoid.zero_mem _, | ||
| exact λ a b f ha hb hf, add_submonoid.add_mem _ | ||
| (add_submonoid.subset_closure $ set.mem_Union.2 ⟨a, set.mem_range_self _⟩) hf | ||
| end) |
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This worked in term mode for finsupp, but the elaborator couldn't fill out the underscores unless I did it in tactic mode.
src/data/dfinsupp.lean
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| /-- | ||
| When summing over an `add_monoid_hom`, the decidability assumption is not needed, and the result is | ||
| also an `add_monoid_hom` -/ | ||
| def sum_add_hom [Π i, add_monoid (β i)] [add_comm_monoid γ] (φ : Π i, β i →+ γ) : ((Π₀ i, β i) →+ γ) := | ||
| { to_fun := (λ f, | ||
| quotient.lift_on f (λ x, ∑ i in x.2.to_finset, φ i (x.1 i)) $ λ x y H, | ||
| begin | ||
| have H1 : x.2.to_finset ∩ y.2.to_finset ⊆ x.2.to_finset, from finset.inter_subset_left _ _, | ||
| have H2 : x.2.to_finset ∩ y.2.to_finset ⊆ y.2.to_finset, from finset.inter_subset_right _ _, | ||
| refine (finset.sum_subset H1 _).symm.trans ((finset.sum_congr rfl _).trans (finset.sum_subset H2 _)), | ||
| { intros i H1 H2, rw finset.mem_inter at H2, rw H i, | ||
| simp only [multiset.mem_to_finset] at H1 H2, | ||
| rw [(y.3 i).resolve_left (mt (and.intro H1) H2), add_monoid_hom.map_zero] }, | ||
| { intros i H1, rw H i }, | ||
| { intros i H1 H2, rw finset.mem_inter at H2, rw ← H i, | ||
| simp only [multiset.mem_to_finset] at H1 H2, | ||
| rw [(x.3 i).resolve_left (mt (λ H3, and.intro H3 H1) H2), add_monoid_hom.map_zero] } | ||
| end), | ||
| map_add' := assume f g, | ||
| begin | ||
| refine quotient.induction_on f (λ x, _), | ||
| refine quotient.induction_on g (λ y, _), | ||
| change ∑ i in _, _ = (∑ i in _, _) + (∑ i in _, _), | ||
| simp only, conv { to_lhs, congr, skip, funext, rw add_monoid_hom.map_add }, | ||
| simp only [finset.sum_add_distrib], | ||
| congr' 1, | ||
| { refine (finset.sum_subset _ _).symm, | ||
| { intro i, simp only [multiset.mem_to_finset, multiset.mem_add], exact or.inl }, | ||
| { intros i H1 H2, simp only [multiset.mem_to_finset, multiset.mem_add] at H2, | ||
| rw [(x.3 i).resolve_left H2, add_monoid_hom.map_zero] } }, | ||
| { refine (finset.sum_subset _ _).symm, | ||
| { intro i, simp only [multiset.mem_to_finset, multiset.mem_add], exact or.inr }, | ||
| { intros i H1 H2, simp only [multiset.mem_to_finset, multiset.mem_add] at H2, | ||
| rw [(y.3 i).resolve_left H2, add_monoid_hom.map_zero] } } | ||
| end, | ||
| map_zero' := rfl } | ||
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| @[simp] lemma sum_add_hom_single [Π i, add_monoid (β i)] [add_comm_monoid γ] | ||
| (φ : Π i, β i →+ γ) (i) (x : β i) : sum_add_hom φ (single i x) = φ i x := | ||
| (add_zero _).trans $ congr_arg (φ i) $ show (if H : i ∈ ({i} : finset _) then x else 0) = x, | ||
| from dif_pos $ finset.mem_singleton_self i |
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These proofs are taken verbatim from direct_sum
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| left_inv := λ x, by { ext, simp }, | ||
| right_inv := λ ψ, by { ext, simp }, | ||
| map_add' := λ F G, by { ext, simp } } |
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Thanks to @urkud's awesome ext lemma, the longer proofs of these from direct_sum can be discarded.
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These lemmas are mostly copied with minimal translation from `finsupp`. A few proofs are taken from `direct_sum`. The API of `direct_sum` is deliberately unchanged in this PR.
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…over-community#4821) These lemmas are mostly copied with minimal translation from `finsupp`. A few proofs are taken from `direct_sum`. The API of `direct_sum` is deliberately unchanged in this PR.
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These lemmas are mostly copied with minimal translation from
finsupp.A few proofs are taken from
direct_sum.The API of
direct_sumis deliberately unchanged in this PR.cc @urkud and @kckennylau, who wrote parts of this moved code.