statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
coe_inv (U : unitary R) : ↑(U⁻¹) = (U⁻¹ : R) | eq_inv_of_mul_eq_one_right $ coe_mul_star_self _ | lemma | unitary.coe_inv | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"eq_inv_of_mul_eq_one_right",
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div (U₁ U₂ : unitary R) : ↑(U₁ / U₂) = (U₁ / U₂ : R) | by simp only [div_eq_mul_inv, coe_inv, submonoid.coe_mul] | lemma | unitary.coe_div | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"div_eq_mul_inv",
"submonoid.coe_mul",
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zpow (U : unitary R) (z : ℤ) : ↑(U ^ z) = (U ^ z : R) | begin
induction z,
{ simp [submonoid_class.coe_pow], },
{ simp [coe_inv] },
end | lemma | unitary.coe_zpow | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"submonoid_class.coe_pow",
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (U : unitary R) : ↑(-U) = (-U : R) | rfl | lemma | unitary.coe_neg | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tropical : Type u | R | def | tropical | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [] | The tropicalization of a type `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trop : R → tropical R | id | def | tropical.trop | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | Reinterpret `x : R` as an element of `tropical R`.
See `tropical.trop_equiv` for the equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
untrop : tropical R → R | id | def | tropical.untrop | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | Reinterpret `x : tropical R` as an element of `R`.
See `tropical.trop_equiv` for the equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trop_injective : function.injective (trop : R → tropical R) | λ _ _, id | lemma | tropical.trop_injective | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_injective : function.injective (untrop : tropical R → R) | λ _ _, id | lemma | tropical.untrop_injective | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_inj_iff (x y : R) : trop x = trop y ↔ x = y | iff.rfl | lemma | tropical.trop_inj_iff | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_inj_iff (x y : tropical R) : untrop x = untrop y ↔ x = y | iff.rfl | lemma | tropical.untrop_inj_iff | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_untrop (x : tropical R) : trop (untrop x) = x | rfl | lemma | tropical.trop_untrop | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_trop (x : R) : untrop (trop x) = x | rfl | lemma | tropical.untrop_trop | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inverse_trop : function.left_inverse (trop : R → tropical R) untrop | trop_untrop | lemma | tropical.left_inverse_trop | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_inverse_trop : function.right_inverse (trop : R → tropical R) untrop | trop_untrop | lemma | tropical.right_inverse_trop | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_equiv : R ≃ tropical R | { to_fun := trop,
inv_fun := untrop,
left_inv := untrop_trop,
right_inv := trop_untrop } | def | tropical.trop_equiv | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"inv_fun",
"tropical"
] | Reinterpret `x : R` as an element of `tropical R`.
See `tropical.trop_order_iso` for the order-preserving equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trop_equiv_coe_fn : (trop_equiv : R → tropical R) = trop | rfl | lemma | tropical.trop_equiv_coe_fn | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_equiv_symm_coe_fn : (trop_equiv.symm : tropical R → R) = untrop | rfl | lemma | tropical.trop_equiv_symm_coe_fn | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_eq_iff_eq_untrop {x : R} {y} : trop x = y ↔ x = untrop y | trop_equiv.apply_eq_iff_eq_symm_apply | lemma | tropical.trop_eq_iff_eq_untrop | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_eq_iff_eq_trop {x} {y : R} : untrop x = y ↔ x = trop y | trop_equiv.symm.apply_eq_iff_eq_symm_apply | lemma | tropical.untrop_eq_iff_eq_trop | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_trop : function.injective (trop : R → tropical R) | trop_equiv.injective | lemma | tropical.injective_trop | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_untrop : function.injective (untrop : tropical R → R) | trop_equiv.symm.injective | lemma | tropical.injective_untrop | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_trop : function.surjective (trop : R → tropical R) | trop_equiv.surjective | lemma | tropical.surjective_trop | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_untrop : function.surjective (untrop : tropical R → R) | trop_equiv.symm.surjective | lemma | tropical.surjective_untrop | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_rec {F : Π (X : tropical R), Sort v} (h : Π X, F (trop X)) : Π X, F X | λ X, h (untrop X) | def | tropical.trop_rec | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | Recursing on a `x' : tropical R` is the same as recursing on an `x : R` reinterpreted
as a term of `tropical R` via `trop x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
untrop_le_iff [has_le R] {x y : tropical R} :
untrop x ≤ untrop y ↔ x ≤ y | iff.rfl | lemma | tropical.untrop_le_iff | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decidable_le [has_le R] [decidable_rel ((≤) : R → R → Prop)] :
decidable_rel ((≤) : tropical R → tropical R → Prop) | λ x y, ‹decidable_rel (≤)› (untrop x) (untrop y) | instance | tropical.decidable_le | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_lt_iff [has_lt R] {x y : tropical R} :
untrop x < untrop y ↔ x < y | iff.rfl | lemma | tropical.untrop_lt_iff | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decidable_lt [has_lt R] [decidable_rel ((<) : R → R → Prop)] :
decidable_rel ((<) : tropical R → tropical R → Prop) | λ x y, ‹decidable_rel (<)› (untrop x) (untrop y) | instance | tropical.decidable_lt | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_order_iso [preorder R] : R ≃o tropical R | { map_rel_iff' := λ _ _, untrop_le_iff,
..trop_equiv } | def | tropical.trop_order_iso | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | Reinterpret `x : R` as an element of `tropical R`, preserving the order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trop_order_iso_coe_fn [preorder R] : (trop_order_iso : R → tropical R) = trop | rfl | lemma | tropical.trop_order_iso_coe_fn | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_order_iso_symm_coe_fn [preorder R] : (trop_order_iso.symm : tropical R → R) = untrop | rfl | lemma | tropical.trop_order_iso_symm_coe_fn | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_monotone [preorder R] : monotone (trop : R → tropical R) | λ _ _, id | lemma | tropical.trop_monotone | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"monotone",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_monotone [preorder R] : monotone (untrop : tropical R → R) | λ _ _, id | lemma | tropical.untrop_monotone | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"monotone",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_zero [has_top R] : untrop (0 : tropical R) = ⊤ | rfl | lemma | tropical.untrop_zero | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"has_top",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_top [has_top R] : trop (⊤ : R) = 0 | rfl | lemma | tropical.trop_top | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"has_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_coe_ne_zero (x : R) : trop (x : with_top R) ≠ 0 | lemma | tropical.trop_coe_ne_zero | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | ||
zero_ne_trop_coe (x : R) : (0 : tropical (with_top R)) ≠ trop x | lemma | tropical.zero_ne_trop_coe | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | ||
le_zero [has_le R] [order_top R] (x : tropical R) : x ≤ 0 | le_top | lemma | tropical.le_zero | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"le_top",
"order_top",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_add (x y : tropical R) : untrop (x + y) = min (untrop x) (untrop y) | rfl | lemma | tropical.untrop_add | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_min (x y : R) : trop (min x y) = trop x + trop y | rfl | lemma | tropical.trop_min | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_inf (x y : R) : trop (x ⊓ y) = trop x + trop y | rfl | lemma | tropical.trop_inf | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_add_def (x y : tropical R) : x + y = trop (min (untrop x) (untrop y)) | rfl | lemma | tropical.trop_add_def | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_sup (x y : tropical R) : untrop (x ⊔ y) = untrop x ⊔ untrop y | rfl | lemma | tropical.untrop_sup | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_max (x y : tropical R) : untrop (max x y) = max (untrop x) (untrop y) | rfl | lemma | tropical.untrop_max | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_eq_add : (min : tropical R → tropical R → tropical R) = (+) | rfl | lemma | tropical.min_eq_add | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_eq_add : ((⊓) : tropical R → tropical R → tropical R) = (+) | rfl | lemma | tropical.inf_eq_add | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_max_def (x y : tropical R) : max x y = trop (max (untrop x) (untrop y)) | rfl | lemma | tropical.trop_max_def | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_sup_def (x y : tropical R) : x ⊔ y = trop (untrop x ⊔ untrop y) | rfl | lemma | tropical.trop_sup_def | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_left ⦃x y : tropical R⦄ (h : x ≤ y) :
x + y = x | untrop_injective (by simpa using h) | lemma | tropical.add_eq_left | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_right ⦃x y : tropical R⦄ (h : y ≤ x) :
x + y = y | untrop_injective (by simpa using h) | lemma | tropical.add_eq_right | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_left_iff {x y : tropical R} : x + y = x ↔ x ≤ y | by rw [trop_add_def, trop_eq_iff_eq_untrop, ←untrop_le_iff, min_eq_left_iff] | lemma | tropical.add_eq_left_iff | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"min_eq_left_iff",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_right_iff {x y : tropical R} : x + y = y ↔ y ≤ x | by rw [trop_add_def, trop_eq_iff_eq_untrop, ←untrop_le_iff, min_eq_right_iff] | lemma | tropical.add_eq_right_iff | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"min_eq_right_iff",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_self (x : tropical R) : x + x = x | untrop_injective (min_eq_right le_rfl) | lemma | tropical.add_self | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"add_self",
"le_rfl",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0 (x : tropical R) : bit0 x = x | add_self x | lemma | tropical.bit0 | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"add_self",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_iff {x y z : tropical R} :
x + y = z ↔ x = z ∧ x ≤ y ∨ y = z ∧ y ≤ x | by { rw [trop_add_def, trop_eq_iff_eq_untrop], simp [min_eq_iff] } | lemma | tropical.add_eq_iff | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"min_eq_iff",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_zero_iff {a b : tropical (with_top R)} :
a + b = 0 ↔ a = 0 ∧ b = 0 | begin
rw add_eq_iff,
split,
{ rintro (⟨rfl, h⟩|⟨rfl, h⟩),
{ exact ⟨rfl, le_antisymm (le_zero _) h⟩ },
{ exact ⟨le_antisymm (le_zero _) h, rfl⟩ } },
{ rintro ⟨rfl, rfl⟩,
simp }
end | lemma | tropical.add_eq_zero_iff | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_add [has_add R] (x y : R) :
trop (x + y) = trop x * trop y | rfl | lemma | tropical.trop_add | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_mul [has_add R] (x y : tropical R) :
untrop (x * y) = untrop x + untrop y | rfl | lemma | tropical.untrop_mul | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_mul_def [has_add R] (x y : tropical R) :
x * y = trop (untrop x + untrop y) | rfl | lemma | tropical.trop_mul_def | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_zero [has_zero R] : trop (0 : R) = 1 | rfl | lemma | tropical.trop_zero | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_one [has_zero R] : untrop (1 : tropical R) = 0 | rfl | lemma | tropical.untrop_one | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_inv [has_neg R] (x : tropical R) : untrop x⁻¹ = - untrop x | rfl | lemma | tropical.untrop_inv | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_div [has_sub R] (x y : tropical R) :
untrop (x / y) = untrop x - untrop y | rfl | lemma | tropical.untrop_div | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_pow {α : Type*} [has_smul α R] (x : tropical R) (n : α) :
untrop (x ^ n) = n • untrop x | rfl | lemma | tropical.untrop_pow | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"has_smul",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_smul {α : Type*} [has_smul α R] (x : R) (n : α) :
trop (n • x) = trop x ^ n | rfl | lemma | tropical.trop_smul | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_nsmul [add_monoid R] (x : R) (n : ℕ) :
trop (n • x) = trop x ^ n | rfl | lemma | tropical.trop_nsmul | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"add_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_zpow [add_group R] (x : tropical R) (n : ℤ) :
untrop (x ^ n) = n • untrop x | rfl | lemma | tropical.untrop_zpow | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"add_group",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_zsmul [add_group R] (x : R) (n : ℤ) :
trop (n • x) = trop x ^ n | rfl | lemma | tropical.trop_zsmul | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_mul [has_le R] [has_add R] [covariant_class R R (+) (≤)] :
covariant_class (tropical R) (tropical R) (*) (≤) | ⟨λ x y z h, add_le_add_left h _⟩ | instance | tropical.covariant_mul | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"covariant_class",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_swap_mul [has_le R] [has_add R] [covariant_class R R (function.swap (+)) (≤)] :
covariant_class (tropical R) (tropical R) (function.swap (*)) (≤) | ⟨λ x y z h, add_le_add_right h _⟩ | instance | tropical.covariant_swap_mul | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"covariant_class",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_add [linear_order R] : covariant_class (tropical R) (tropical R) (+) (≤) | ⟨λ x y z h, begin
cases le_total x y with hx hy,
{ rw [add_eq_left hx, add_eq_left (hx.trans h)] },
{ rw [add_eq_right hy],
cases le_total x z with hx hx,
{ rwa [add_eq_left hx] },
{ rwa [add_eq_right hx] } }
end⟩ | instance | tropical.covariant_add | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"covariant_class",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_mul_lt [has_lt R] [has_add R] [covariant_class R R (+) (<)] :
covariant_class (tropical R) (tropical R) (*) (<) | ⟨λ x y z h, add_lt_add_left h _⟩ | instance | tropical.covariant_mul_lt | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"covariant_class",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_swap_mul_lt [preorder R] [has_add R]
[covariant_class R R (function.swap (+)) (<)] :
covariant_class (tropical R) (tropical R) (function.swap (*)) (<) | ⟨λ x y z h, add_lt_add_right h _⟩ | instance | tropical.covariant_swap_mul_lt | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"covariant_class",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_pow [linear_order R] [add_monoid R]
[covariant_class R R (+) (≤)] [covariant_class R R (function.swap (+)) (≤)]
(x y : tropical R) (n : ℕ) :
(x + y) ^ n = x ^ n + y ^ n | begin
cases le_total x y with h h,
{ rw [add_eq_left h, add_eq_left (pow_le_pow_of_le_left' h _)] },
{ rw [add_eq_right h, add_eq_right (pow_le_pow_of_le_left' h _)] }
end | lemma | tropical.add_pow | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"add_monoid",
"add_pow",
"covariant_class",
"pow_le_pow_of_le_left'",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_nsmul {R} [linear_order R] [order_top R] (x : tropical R) (n : ℕ) :
(n + 1) • x = x | begin
induction n with n IH,
{ simp },
{ rw [add_nsmul, IH, one_nsmul, add_self] }
end | lemma | tropical.succ_nsmul | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"add_self",
"order_top",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_zero_iff {R : Type*} [linear_ordered_add_comm_monoid R]
{a b : tropical (with_top R)} :
a * b = 0 ↔ a = 0 ∨ b = 0 | by simp [←untrop_inj_iff, with_top.add_eq_top] | lemma | tropical.mul_eq_zero_iff | algebra.tropical | src/algebra/tropical/basic.lean | [
"algebra.group_power.order",
"algebra.order.monoid.with_top",
"algebra.smul_with_zero",
"algebra.order.monoid.min_max"
] | [
"linear_ordered_add_comm_monoid",
"tropical",
"with_top",
"with_top.add_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list.trop_sum [add_monoid R] (l : list R) : trop l.sum = list.prod (l.map trop) | begin
induction l with hd tl IH,
{ simp },
{ simp [←IH] }
end | lemma | list.trop_sum | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"add_monoid",
"list.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset.trop_sum [add_comm_monoid R] (s : multiset R) :
trop s.sum = multiset.prod (s.map trop) | quotient.induction_on s (by simpa using list.trop_sum) | lemma | multiset.trop_sum | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"add_comm_monoid",
"list.trop_sum",
"multiset",
"multiset.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_sum [add_comm_monoid R] (s : finset S) (f : S → R) :
trop (∑ i in s, f i) = ∏ i in s, trop (f i) | begin
cases s,
convert multiset.trop_sum _,
simp
end | lemma | trop_sum | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"add_comm_monoid",
"finset",
"multiset.trop_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list.untrop_prod [add_monoid R] (l : list (tropical R)) :
untrop l.prod = list.sum (l.map untrop) | begin
induction l with hd tl IH,
{ simp },
{ simp [←IH] }
end | lemma | list.untrop_prod | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"add_monoid",
"list.sum",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset.untrop_prod [add_comm_monoid R] (s : multiset (tropical R)) :
untrop s.prod = multiset.sum (s.map untrop) | quotient.induction_on s (by simpa using list.untrop_prod) | lemma | multiset.untrop_prod | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"add_comm_monoid",
"list.untrop_prod",
"multiset",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_prod [add_comm_monoid R] (s : finset S) (f : S → tropical R) :
untrop (∏ i in s, f i) = ∑ i in s, untrop (f i) | begin
cases s,
convert multiset.untrop_prod _,
simp
end | lemma | untrop_prod | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"add_comm_monoid",
"finset",
"multiset.untrop_prod",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list.trop_minimum [linear_order R] (l : list R) :
trop l.minimum = list.sum (l.map (trop ∘ coe)) | begin
induction l with hd tl IH,
{ simp },
{ simp [list.minimum_cons, ←IH] }
end | lemma | list.trop_minimum | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"list.minimum_cons",
"list.sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset.trop_inf [linear_order R] [order_top R] (s : multiset R) :
trop s.inf = multiset.sum (s.map trop) | begin
induction s using multiset.induction with s x IH,
{ simp },
{ simp [←IH] }
end | lemma | multiset.trop_inf | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"multiset",
"multiset.induction",
"order_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.trop_inf [linear_order R] [order_top R] (s : finset S) (f : S → R) :
trop (s.inf f) = ∑ i in s, trop (f i) | begin
cases s,
convert multiset.trop_inf _,
simp
end | lemma | finset.trop_inf | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"finset",
"multiset.trop_inf",
"order_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_Inf_image [conditionally_complete_linear_order R] (s : finset S)
(f : S → with_top R) : trop (Inf (f '' s)) = ∑ i in s, trop (f i) | begin
rcases s.eq_empty_or_nonempty with rfl|h,
{ simp only [set.image_empty, coe_empty, sum_empty, with_top.cInf_empty, trop_top] },
rw [←inf'_eq_cInf_image _ h, inf'_eq_inf, s.trop_inf],
end | lemma | trop_Inf_image | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"conditionally_complete_linear_order",
"finset",
"set.image_empty",
"with_top",
"with_top.cInf_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trop_infi [conditionally_complete_linear_order R] [fintype S] (f : S → with_top R) :
trop (⨅ (i : S), f i) = ∑ (i : S), trop (f i) | by rw [infi, ←set.image_univ, ←coe_univ, trop_Inf_image] | lemma | trop_infi | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"conditionally_complete_linear_order",
"fintype",
"infi",
"trop_Inf_image",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset.untrop_sum [linear_order R] [order_top R] (s : multiset (tropical R)) :
untrop s.sum = multiset.inf (s.map untrop) | begin
induction s using multiset.induction with s x IH,
{ simp },
{ simpa [←IH] }
end | lemma | multiset.untrop_sum | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"multiset",
"multiset.induction",
"multiset.inf",
"order_top",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.untrop_sum' [linear_order R] [order_top R] (s : finset S)
(f : S → tropical R) : untrop (∑ i in s, f i) = s.inf (untrop ∘ f) | begin
cases s,
convert multiset.untrop_sum _,
simpa
end | lemma | finset.untrop_sum' | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"finset",
"multiset.untrop_sum",
"order_top",
"tropical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_sum_eq_Inf_image [conditionally_complete_linear_order R] (s : finset S)
(f : S → tropical (with_top R)) :
untrop (∑ i in s, f i) = Inf (untrop ∘ f '' s) | begin
rcases s.eq_empty_or_nonempty with rfl|h,
{ simp only [set.image_empty, coe_empty, sum_empty, with_top.cInf_empty, untrop_zero] },
rw [←inf'_eq_cInf_image _ h, inf'_eq_inf, finset.untrop_sum'],
end | lemma | untrop_sum_eq_Inf_image | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"conditionally_complete_linear_order",
"finset",
"finset.untrop_sum'",
"set.image_empty",
"tropical",
"with_top",
"with_top.cInf_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untrop_sum [conditionally_complete_linear_order R] [fintype S]
(f : S → tropical (with_top R)) :
untrop (∑ i : S, f i) = ⨅ i : S, untrop (f i) | by rw [infi, ←set.image_univ, ←coe_univ, untrop_sum_eq_Inf_image] | lemma | untrop_sum | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"conditionally_complete_linear_order",
"fintype",
"infi",
"tropical",
"untrop_sum_eq_Inf_image",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.untrop_sum [conditionally_complete_linear_order R] (s : finset S)
(f : S → tropical (with_top R)) : untrop (∑ i in s, f i) = ⨅ i : s, untrop (f i) | by simpa [←untrop_sum] using sum_attach.symm | lemma | finset.untrop_sum | algebra.tropical | src/algebra/tropical/big_operators.lean | [
"algebra.big_operators.basic",
"data.list.min_max",
"algebra.tropical.basic",
"order.conditionally_complete_lattice.finset"
] | [
"conditionally_complete_linear_order",
"finset",
"tropical",
"with_top"
] | Note we cannot use `i ∈ s` instead of `i : s` here
as it is simply not true on conditionally complete lattices! | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
AffineScheme | Scheme.Spec.ess_image_subcategory | def | algebraic_geometry.AffineScheme | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | The category of affine schemes | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_affine (X : Scheme) : Prop | (affine : is_iso (Γ_Spec.adjunction.unit.app X)) | class | algebraic_geometry.is_affine | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | A Scheme is affine if the canonical map `X ⟶ Spec Γ(X)` is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme.iso_Spec (X : Scheme) [is_affine X] :
X ≅ Scheme.Spec.obj (op $ Scheme.Γ.obj $ op X) | as_iso (Γ_Spec.adjunction.unit.app X) | def | algebraic_geometry.Scheme.iso_Spec | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | The canonical isomorphism `X ≅ Spec Γ(X)` for an affine scheme. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
AffineScheme.mk (X : Scheme) (h : is_affine X) : AffineScheme | ⟨X, @@mem_ess_image_of_unit_is_iso _ _ _ _ h.1⟩ | def | algebraic_geometry.AffineScheme.mk | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | Construct an affine scheme from a scheme and the information that it is affine.
Also see `AffineScheme.of` for a typclass version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
AffineScheme.of (X : Scheme) [h : is_affine X] : AffineScheme | AffineScheme.mk X h | def | algebraic_geometry.AffineScheme.of | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | Construct an affine scheme from a scheme. Also see `AffineScheme.mk` for a non-typeclass
version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
AffineScheme.of_hom {X Y : Scheme} [is_affine X] [is_affine Y] (f : X ⟶ Y) :
AffineScheme.of X ⟶ AffineScheme.of Y | f | def | algebraic_geometry.AffineScheme.of_hom | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | Type check a morphism of schemes as a morphism in `AffineScheme`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_Spec_ess_image (X : Scheme) : X ∈ Scheme.Spec.ess_image ↔ is_affine X | ⟨λ h, ⟨functor.ess_image.unit_is_iso h⟩, λ h, @@mem_ess_image_of_unit_is_iso _ _ _ X h.1⟩ | lemma | algebraic_geometry.mem_Spec_ess_image | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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