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
value |
|---|---|---|---|---|---|---|---|---|---|---|
to_Ioc_div_neg' (a b : α) : to_Ioc_div hp (-a) b = -(to_Ico_div hp a (-b) + 1) | by simpa only [neg_neg] using to_Ioc_div_neg hp (-a) (-b) | lemma | to_Ioc_div_neg' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ioc_div",
"to_Ioc_div_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_add_zsmul (a b : α) (m : ℤ) :
to_Ico_mod hp a (b + m • p) = to_Ico_mod hp a b | by { rw [to_Ico_mod, to_Ico_div_add_zsmul, to_Ico_mod, add_smul], abel } | lemma | to_Ico_mod_add_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"add_smul",
"to_Ico_div_add_zsmul",
"to_Ico_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_add_zsmul' (a b : α) (m : ℤ) :
to_Ico_mod hp (a + m • p) b = to_Ico_mod hp a b + m • p | by simp only [to_Ico_mod, to_Ico_div_add_zsmul', sub_smul, sub_add] | lemma | to_Ico_mod_add_zsmul' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"sub_smul",
"to_Ico_div_add_zsmul'",
"to_Ico_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_add_zsmul (a b : α) (m : ℤ) :
to_Ioc_mod hp a (b + m • p) = to_Ioc_mod hp a b | by { rw [to_Ioc_mod, to_Ioc_div_add_zsmul, to_Ioc_mod, add_smul], abel } | lemma | to_Ioc_mod_add_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"add_smul",
"to_Ioc_div_add_zsmul",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_add_zsmul' (a b : α) (m : ℤ) :
to_Ioc_mod hp (a + m • p) b = to_Ioc_mod hp a b + m • p | by simp only [to_Ioc_mod, to_Ioc_div_add_zsmul', sub_smul, sub_add] | lemma | to_Ioc_mod_add_zsmul' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"sub_smul",
"to_Ioc_div_add_zsmul'",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_zsmul_add (a b : α) (m : ℤ) :
to_Ico_mod hp a (m • p + b) = to_Ico_mod hp a b | by rw [add_comm, to_Ico_mod_add_zsmul] | lemma | to_Ico_mod_zsmul_add | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_zsmul_add' (a b : α) (m : ℤ) :
to_Ico_mod hp (m • p + a) b = m • p + to_Ico_mod hp a b | by rw [add_comm, to_Ico_mod_add_zsmul', add_comm] | lemma | to_Ico_mod_zsmul_add' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_add_zsmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_zsmul_add (a b : α) (m : ℤ) :
to_Ioc_mod hp a (m • p + b) = to_Ioc_mod hp a b | by rw [add_comm, to_Ioc_mod_add_zsmul] | lemma | to_Ioc_mod_zsmul_add | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_mod",
"to_Ioc_mod_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_zsmul_add' (a b : α) (m : ℤ) :
to_Ioc_mod hp (m • p + a) b = m • p + to_Ioc_mod hp a b | by rw [add_comm, to_Ioc_mod_add_zsmul', add_comm] | lemma | to_Ioc_mod_zsmul_add' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_mod",
"to_Ioc_mod_add_zsmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_sub_zsmul (a b : α) (m : ℤ) :
to_Ico_mod hp a (b - m • p) = to_Ico_mod hp a b | by rw [sub_eq_add_neg, ←neg_smul, to_Ico_mod_add_zsmul] | lemma | to_Ico_mod_sub_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_sub_zsmul' (a b : α) (m : ℤ) :
to_Ico_mod hp (a - m • p) b = to_Ico_mod hp a b - m • p | by simp_rw [sub_eq_add_neg, ←neg_smul, to_Ico_mod_add_zsmul'] | lemma | to_Ico_mod_sub_zsmul' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_add_zsmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_sub_zsmul (a b : α) (m : ℤ) :
to_Ioc_mod hp a (b - m • p) = to_Ioc_mod hp a b | by rw [sub_eq_add_neg, ←neg_smul, to_Ioc_mod_add_zsmul] | lemma | to_Ioc_mod_sub_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_mod",
"to_Ioc_mod_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_sub_zsmul' (a b : α) (m : ℤ) :
to_Ioc_mod hp (a - m • p) b = to_Ioc_mod hp a b - m • p | by simp_rw [sub_eq_add_neg, ←neg_smul, to_Ioc_mod_add_zsmul'] | lemma | to_Ioc_mod_sub_zsmul' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_mod",
"to_Ioc_mod_add_zsmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_add_right (a b : α) : to_Ico_mod hp a (b + p) = to_Ico_mod hp a b | by simpa only [one_zsmul] using to_Ico_mod_add_zsmul hp a b 1 | lemma | to_Ico_mod_add_right | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_add_right' (a b : α) : to_Ico_mod hp (a + p) b = to_Ico_mod hp a b + p | by simpa only [one_zsmul] using to_Ico_mod_add_zsmul' hp a b 1 | lemma | to_Ico_mod_add_right' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_add_zsmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_add_right (a b : α) : to_Ioc_mod hp a (b + p) = to_Ioc_mod hp a b | by simpa only [one_zsmul] using to_Ioc_mod_add_zsmul hp a b 1 | lemma | to_Ioc_mod_add_right | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_mod",
"to_Ioc_mod_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_add_right' (a b : α) : to_Ioc_mod hp (a + p) b = to_Ioc_mod hp a b + p | by simpa only [one_zsmul] using to_Ioc_mod_add_zsmul' hp a b 1 | lemma | to_Ioc_mod_add_right' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_mod",
"to_Ioc_mod_add_zsmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_add_left (a b : α) : to_Ico_mod hp a (p + b) = to_Ico_mod hp a b | by rw [add_comm, to_Ico_mod_add_right] | lemma | to_Ico_mod_add_left | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_add_left' (a b : α) : to_Ico_mod hp (p + a) b = p + to_Ico_mod hp a b | by rw [add_comm, to_Ico_mod_add_right', add_comm] | lemma | to_Ico_mod_add_left' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_add_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_add_left (a b : α) : to_Ioc_mod hp a (p + b) = to_Ioc_mod hp a b | by rw [add_comm, to_Ioc_mod_add_right] | lemma | to_Ioc_mod_add_left | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_mod",
"to_Ioc_mod_add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_add_left' (a b : α) : to_Ioc_mod hp (p + a) b = p + to_Ioc_mod hp a b | by rw [add_comm, to_Ioc_mod_add_right', add_comm] | lemma | to_Ioc_mod_add_left' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_mod",
"to_Ioc_mod_add_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_sub (a b : α) : to_Ico_mod hp a (b - p) = to_Ico_mod hp a b | by simpa only [one_zsmul] using to_Ico_mod_sub_zsmul hp a b 1 | lemma | to_Ico_mod_sub | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_sub_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_sub' (a b : α) : to_Ico_mod hp (a - p) b = to_Ico_mod hp a b - p | by simpa only [one_zsmul] using to_Ico_mod_sub_zsmul' hp a b 1 | lemma | to_Ico_mod_sub' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_sub_zsmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_sub (a b : α) : to_Ioc_mod hp a (b - p) = to_Ioc_mod hp a b | by simpa only [one_zsmul] using to_Ioc_mod_sub_zsmul hp a b 1 | lemma | to_Ioc_mod_sub | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_mod",
"to_Ioc_mod_sub_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_sub' (a b : α) : to_Ioc_mod hp (a - p) b = to_Ioc_mod hp a b - p | by simpa only [one_zsmul] using to_Ioc_mod_sub_zsmul' hp a b 1 | lemma | to_Ioc_mod_sub' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_mod",
"to_Ioc_mod_sub_zsmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_sub_eq_sub (a b c : α) : to_Ico_mod hp a (b - c) = to_Ico_mod hp (a + c) b - c | by simp_rw [to_Ico_mod, to_Ico_div_sub_eq_to_Ico_div_add, sub_right_comm] | lemma | to_Ico_mod_sub_eq_sub | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div_sub_eq_to_Ico_div_add",
"to_Ico_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_sub_eq_sub (a b c : α) : to_Ioc_mod hp a (b - c) = to_Ioc_mod hp (a + c) b - c | by simp_rw [to_Ioc_mod, to_Ioc_div_sub_eq_to_Ioc_div_add, sub_right_comm] | lemma | to_Ioc_mod_sub_eq_sub | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div_sub_eq_to_Ioc_div_add",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_add_right_eq_add (a b c : α) :
to_Ico_mod hp a (b + c) = to_Ico_mod hp (a - c) b + c | by simp_rw [to_Ico_mod, to_Ico_div_sub_eq_to_Ico_div_add', sub_add_eq_add_sub] | lemma | to_Ico_mod_add_right_eq_add | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div_sub_eq_to_Ico_div_add'",
"to_Ico_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_add_right_eq_add (a b c : α) :
to_Ioc_mod hp a (b + c) = to_Ioc_mod hp (a - c) b + c | by simp_rw [to_Ioc_mod, to_Ioc_div_sub_eq_to_Ioc_div_add', sub_add_eq_add_sub] | lemma | to_Ioc_mod_add_right_eq_add | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div_sub_eq_to_Ioc_div_add'",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_neg (a b : α) : to_Ico_mod hp a (-b) = p - to_Ioc_mod hp (-a) b | by { simp_rw [to_Ico_mod, to_Ioc_mod, to_Ico_div_neg, neg_smul, add_smul], abel } | lemma | to_Ico_mod_neg | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"add_smul",
"neg_smul",
"to_Ico_div_neg",
"to_Ico_mod",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_neg' (a b : α) : to_Ico_mod hp (-a) b = p - to_Ioc_mod hp a (-b) | by simpa only [neg_neg] using to_Ico_mod_neg hp (-a) (-b) | lemma | to_Ico_mod_neg' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_neg",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_neg (a b : α) : to_Ioc_mod hp a (-b) = p - to_Ico_mod hp (-a) b | by { simp_rw [to_Ioc_mod, to_Ico_mod, to_Ioc_div_neg, neg_smul, add_smul], abel } | lemma | to_Ioc_mod_neg | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"add_smul",
"neg_smul",
"to_Ico_mod",
"to_Ioc_div_neg",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_neg' (a b : α) : to_Ioc_mod hp (-a) b = p - to_Ico_mod hp a (-b) | by simpa only [neg_neg] using to_Ioc_mod_neg hp (-a) (-b) | lemma | to_Ioc_mod_neg' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ioc_mod",
"to_Ioc_mod_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_eq_to_Ico_mod : to_Ico_mod hp a b = to_Ico_mod hp a c ↔ ∃ n : ℤ, c - b = n • p | begin
refine ⟨λ h, ⟨to_Ico_div hp a c - to_Ico_div hp a b, _⟩, λ h, _⟩,
{ conv_lhs { rw [←to_Ico_mod_add_to_Ico_div_zsmul hp a b,
←to_Ico_mod_add_to_Ico_div_zsmul hp a c] },
rw [h, sub_smul],
abel },
{ rcases h with ⟨z, hz⟩,
rw sub_eq_iff_eq_add at hz,
rw [hz, to_Ico_mod_zsmul_a... | lemma | to_Ico_mod_eq_to_Ico_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"sub_smul",
"to_Ico_div",
"to_Ico_mod",
"to_Ico_mod_zsmul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_eq_to_Ioc_mod : to_Ioc_mod hp a b = to_Ioc_mod hp a c ↔ ∃ n : ℤ, c - b = n • p | begin
refine ⟨λ h, ⟨to_Ioc_div hp a c - to_Ioc_div hp a b, _⟩, λ h, _⟩,
{ conv_lhs { rw [←to_Ioc_mod_add_to_Ioc_div_zsmul hp a b,
←to_Ioc_mod_add_to_Ioc_div_zsmul hp a c] },
rw [h, sub_smul],
abel },
{ rcases h with ⟨z, hz⟩,
rw sub_eq_iff_eq_add at hz,
rw [hz, to_Ioc_mod_zsmul_a... | lemma | to_Ioc_mod_eq_to_Ioc_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"sub_smul",
"to_Ioc_div",
"to_Ioc_mod",
"to_Ioc_mod_zsmul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_iff_to_Ico_mod_eq_left : a ≡ b [PMOD p] ↔ to_Ico_mod hp a b = a | modeq_iff_eq_add_zsmul.trans ⟨by { rintro ⟨n, rfl⟩,
rw [to_Ico_mod_add_zsmul, to_Ico_mod_apply_left] }, λ h, ⟨to_Ico_div hp a b, eq_add_of_sub_eq h⟩⟩ | lemma | add_comm_group.modeq_iff_to_Ico_mod_eq_left | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_add_zsmul",
"to_Ico_mod_apply_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_iff_to_Ioc_mod_eq_right : a ≡ b [PMOD p] ↔ to_Ioc_mod hp a b = a + p | begin
refine modeq_iff_eq_add_zsmul.trans ⟨_, λ h, ⟨to_Ioc_div hp a b + 1, _⟩⟩,
{ rintro ⟨z, rfl⟩,
rw [to_Ioc_mod_add_zsmul, to_Ioc_mod_apply_left] },
{ rwa [add_one_zsmul, add_left_comm, ←sub_eq_iff_eq_add'] }
end | lemma | add_comm_group.modeq_iff_to_Ioc_mod_eq_right | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_mod",
"to_Ioc_mod_add_zsmul",
"to_Ioc_mod_apply_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tfae_modeq :
tfae [
a ≡ b [PMOD p],
∀ z : ℤ, b - z • p ∉ set.Ioo a (a + p),
to_Ico_mod hp a b ≠ to_Ioc_mod hp a b,
to_Ico_mod hp a b + p = to_Ioc_mod hp a b] | begin
rw modeq_iff_to_Ico_mod_eq_left hp,
tfae_have : 3 → 2,
{ rw [←not_exists, not_imp_not],
exact λ ⟨i, hi⟩,
((to_Ico_mod_eq_iff hp).2 ⟨set.Ioo_subset_Ico_self hi, i, (sub_add_cancel b _).symm⟩).trans
((to_Ioc_mod_eq_iff hp).2 ⟨set.Ioo_subset_Ioc_self hi, i, (sub_add_cancel b _).symm⟩).symm },
... | lemma | add_comm_group.tfae_modeq | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"not_imp_comm",
"not_imp_not",
"set.Ioo",
"set.right_mem_Ioc",
"to_Ico_div",
"to_Ico_mod",
"to_Ico_mod_add_to_Ico_div_zsmul",
"to_Ico_mod_eq_iff",
"to_Ico_mod_mem_Ico",
"to_Ioc_mod",
"to_Ioc_mod_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_iff_not_forall_mem_Ioo_mod :
a ≡ b [PMOD p] ↔ ∀ z : ℤ, b - z • p ∉ set.Ioo a (a + p) | (tfae_modeq hp a b).out 0 1 | lemma | add_comm_group.modeq_iff_not_forall_mem_Ioo_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"set.Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_iff_to_Ico_mod_ne_to_Ioc_mod :
a ≡ b [PMOD p] ↔ to_Ico_mod hp a b ≠ to_Ioc_mod hp a b | (tfae_modeq hp a b).out 0 2 | lemma | add_comm_group.modeq_iff_to_Ico_mod_ne_to_Ioc_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_iff_to_Ico_mod_add_period_eq_to_Ioc_mod :
a ≡ b [PMOD p] ↔ to_Ico_mod hp a b + p = to_Ioc_mod hp a b | (tfae_modeq hp a b).out 0 3 | lemma | add_comm_group.modeq_iff_to_Ico_mod_add_period_eq_to_Ioc_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_modeq_iff_to_Ico_mod_eq_to_Ioc_mod :
¬a ≡ b [PMOD p] ↔ to_Ico_mod hp a b = to_Ioc_mod hp a b | (modeq_iff_to_Ico_mod_ne_to_Ioc_mod _).not_left | lemma | add_comm_group.not_modeq_iff_to_Ico_mod_eq_to_Ioc_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_modeq_iff_to_Ico_div_eq_to_Ioc_div :
¬a ≡ b [PMOD p] ↔ to_Ico_div hp a b = to_Ioc_div hp a b | by rw [not_modeq_iff_to_Ico_mod_eq_to_Ioc_mod hp,
to_Ico_mod, to_Ioc_mod, sub_right_inj, (zsmul_strict_mono_left hp).injective.eq_iff] | lemma | add_comm_group.not_modeq_iff_to_Ico_div_eq_to_Ioc_div | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_mod",
"to_Ioc_div",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_iff_to_Ico_div_eq_to_Ioc_div_add_one :
a ≡ b [PMOD p] ↔ to_Ico_div hp a b = to_Ioc_div hp a b + 1 | by rw [modeq_iff_to_Ico_mod_add_period_eq_to_Ioc_mod hp, to_Ico_mod, to_Ioc_mod,
← eq_sub_iff_add_eq, sub_sub, sub_right_inj, ← add_one_zsmul,
(zsmul_strict_mono_left hp).injective.eq_iff] | lemma | add_comm_group.modeq_iff_to_Ico_div_eq_to_Ioc_div_add_one | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_mod",
"to_Ioc_div",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_inj {c : α} : to_Ico_mod hp c a = to_Ico_mod hp c b ↔ a ≡ b [PMOD p] | by simp_rw [to_Ico_mod_eq_to_Ico_mod, modeq_iff_eq_add_zsmul, sub_eq_iff_eq_add'] | lemma | to_Ico_mod_inj | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_eq_to_Ico_mod"
] | If `a` and `b` fall within the same cycle WRT `c`, then they are congruent modulo `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Ico_eq_locus_Ioc_eq_Union_Ioo :
{b | to_Ico_mod hp a b = to_Ioc_mod hp a b} = ⋃ z : ℤ, set.Ioo (a + z • p) (a + p + z • p) | begin
ext1, simp_rw [set.mem_set_of, set.mem_Union, ← set.sub_mem_Ioo_iff_left,
←not_modeq_iff_to_Ico_mod_eq_to_Ioc_mod, modeq_iff_not_forall_mem_Ioo_mod hp, not_forall,
not_not],
end | lemma | Ico_eq_locus_Ioc_eq_Union_Ioo | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"not_forall",
"not_not",
"set.Ioo",
"set.mem_Union",
"set.mem_set_of",
"set.sub_mem_Ioo_iff_left",
"to_Ico_mod",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_wcovby_to_Ico_div (a b : α) : to_Ioc_div hp a b ⩿ to_Ico_div hp a b | begin
suffices : to_Ioc_div hp a b = to_Ico_div hp a b ∨ to_Ioc_div hp a b + 1 = to_Ico_div hp a b,
{ rwa [wcovby_iff_eq_or_covby, ←order.succ_eq_iff_covby] },
rw [eq_comm, ←not_modeq_iff_to_Ico_div_eq_to_Ioc_div,
eq_comm, ←modeq_iff_to_Ico_div_eq_to_Ioc_div_add_one],
exact em' _,
end | lemma | to_Ioc_div_wcovby_to_Ico_div | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"em'",
"to_Ico_div",
"to_Ioc_div",
"wcovby_iff_eq_or_covby"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_le_to_Ioc_mod (a b : α) : to_Ico_mod hp a b ≤ to_Ioc_mod hp a b | begin
rw [to_Ico_mod, to_Ioc_mod, sub_le_sub_iff_left],
exact zsmul_mono_left hp.le (to_Ioc_div_wcovby_to_Ico_div _ _ _).le
end | lemma | to_Ico_mod_le_to_Ioc_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ioc_div_wcovby_to_Ico_div",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_le_to_Ico_mod_add (a b : α) : to_Ioc_mod hp a b ≤ to_Ico_mod hp a b + p | begin
rw [to_Ico_mod, to_Ioc_mod, sub_add, sub_le_sub_iff_left, sub_le_iff_le_add, ←add_one_zsmul,
(zsmul_strict_mono_left hp).le_iff_le],
apply (to_Ioc_div_wcovby_to_Ico_div _ _ _).le_succ,
end | lemma | to_Ioc_mod_le_to_Ico_mod_add | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ioc_div_wcovby_to_Ico_div",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_eq_self : to_Ico_mod hp a b = b ↔ b ∈ set.Ico a (a + p) | by { rw [to_Ico_mod_eq_iff, and_iff_left], exact ⟨0, by simp⟩ } | lemma | to_Ico_mod_eq_self | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"set.Ico",
"to_Ico_mod",
"to_Ico_mod_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_eq_self : to_Ioc_mod hp a b = b ↔ b ∈ set.Ioc a (a + p) | by { rw [to_Ioc_mod_eq_iff, and_iff_left], exact ⟨0, by simp⟩ } | lemma | to_Ioc_mod_eq_self | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"set.Ioc",
"to_Ioc_mod",
"to_Ioc_mod_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_to_Ico_mod (a₁ a₂ b : α) :
to_Ico_mod hp a₁ (to_Ico_mod hp a₂ b) = to_Ico_mod hp a₁ b | (to_Ico_mod_eq_to_Ico_mod _).2 ⟨to_Ico_div hp a₂ b, self_sub_to_Ico_mod hp a₂ b⟩ | lemma | to_Ico_mod_to_Ico_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"self_sub_to_Ico_mod",
"to_Ico_mod",
"to_Ico_mod_eq_to_Ico_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_to_Ioc_mod (a₁ a₂ b : α) :
to_Ico_mod hp a₁ (to_Ioc_mod hp a₂ b) = to_Ico_mod hp a₁ b | (to_Ico_mod_eq_to_Ico_mod _).2 ⟨to_Ioc_div hp a₂ b, self_sub_to_Ioc_mod hp a₂ b⟩ | lemma | to_Ico_mod_to_Ioc_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"self_sub_to_Ioc_mod",
"to_Ico_mod",
"to_Ico_mod_eq_to_Ico_mod",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_to_Ioc_mod (a₁ a₂ b : α) :
to_Ioc_mod hp a₁ (to_Ioc_mod hp a₂ b) = to_Ioc_mod hp a₁ b | (to_Ioc_mod_eq_to_Ioc_mod _).2 ⟨to_Ioc_div hp a₂ b, self_sub_to_Ioc_mod hp a₂ b⟩ | lemma | to_Ioc_mod_to_Ioc_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"self_sub_to_Ioc_mod",
"to_Ioc_mod",
"to_Ioc_mod_eq_to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_to_Ico_mod (a₁ a₂ b : α) :
to_Ioc_mod hp a₁ (to_Ico_mod hp a₂ b) = to_Ioc_mod hp a₁ b | (to_Ioc_mod_eq_to_Ioc_mod _).2 ⟨to_Ico_div hp a₂ b, self_sub_to_Ico_mod hp a₂ b⟩ | lemma | to_Ioc_mod_to_Ico_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"self_sub_to_Ico_mod",
"to_Ico_mod",
"to_Ioc_mod",
"to_Ioc_mod_eq_to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_periodic (a : α) : function.periodic (to_Ico_mod hp a) p | to_Ico_mod_add_right hp a | lemma | to_Ico_mod_periodic | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"function.periodic",
"to_Ico_mod",
"to_Ico_mod_add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_periodic (a : α) : function.periodic (to_Ioc_mod hp a) p | to_Ioc_mod_add_right hp a | lemma | to_Ioc_mod_periodic | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"function.periodic",
"to_Ioc_mod",
"to_Ioc_mod_add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_zero_sub_comm (a b : α) : to_Ico_mod hp 0 (a - b) = p - to_Ioc_mod hp 0 (b - a) | by rw [←neg_sub, to_Ico_mod_neg, neg_zero] | lemma | to_Ico_mod_zero_sub_comm | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_neg",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_zero_sub_comm (a b : α) : to_Ioc_mod hp 0 (a - b) = p - to_Ico_mod hp 0 (b - a) | by rw [←neg_sub, to_Ioc_mod_neg, neg_zero] | lemma | to_Ioc_mod_zero_sub_comm | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ioc_mod",
"to_Ioc_mod_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_eq_sub (a b : α) : to_Ico_div hp a b = to_Ico_div hp 0 (b - a) | by rw [to_Ico_div_sub_eq_to_Ico_div_add, zero_add] | lemma | to_Ico_div_eq_sub | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_sub_eq_to_Ico_div_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_eq_sub (a b : α) : to_Ioc_div hp a b = to_Ioc_div hp 0 (b - a) | by rw [to_Ioc_div_sub_eq_to_Ioc_div_add, zero_add] | lemma | to_Ioc_div_eq_sub | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_div_sub_eq_to_Ioc_div_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_eq_sub (a b : α) : to_Ico_mod hp a b = to_Ico_mod hp 0 (b - a) + a | by rw [to_Ico_mod_sub_eq_sub, zero_add, sub_add_cancel] | lemma | to_Ico_mod_eq_sub | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_sub_eq_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_eq_sub (a b : α) : to_Ioc_mod hp a b = to_Ioc_mod hp 0 (b - a) + a | by rw [to_Ioc_mod_sub_eq_sub, zero_add, sub_add_cancel] | lemma | to_Ioc_mod_eq_sub | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_mod",
"to_Ioc_mod_sub_eq_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_add_to_Ioc_mod_zero (a b : α) :
to_Ico_mod hp 0 (a - b) + to_Ioc_mod hp 0 (b - a) = p | by rw [to_Ico_mod_zero_sub_comm, sub_add_cancel] | lemma | to_Ico_mod_add_to_Ioc_mod_zero | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_zero_sub_comm",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_add_to_Ico_mod_zero (a b : α) :
to_Ioc_mod hp 0 (a - b) + to_Ico_mod hp 0 (b - a) = p | by rw [add_comm, to_Ico_mod_add_to_Ioc_mod_zero] | lemma | to_Ioc_mod_add_to_Ico_mod_zero | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_add_to_Ioc_mod_zero",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_add_group.equiv_Ico_mod (a : α) :
(α ⧸ add_subgroup.zmultiples p) ≃ set.Ico a (a + p) | { to_fun := λ b, ⟨(to_Ico_mod_periodic hp a).lift b,
quotient_add_group.induction_on' b $ to_Ico_mod_mem_Ico hp a⟩,
inv_fun := coe,
right_inv := λ b, subtype.ext $ (to_Ico_mod_eq_self hp).mpr b.prop,
left_inv := λ b, begin
induction b using quotient_add_group.induction_on',
dsimp,
rw [quotient_add... | def | quotient_add_group.equiv_Ico_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"add_subgroup.zmultiples",
"inv_fun",
"lift",
"set.Ico",
"subtype.ext",
"to_Ico_mod_eq_self",
"to_Ico_mod_mem_Ico",
"to_Ico_mod_periodic",
"to_Ico_mod_sub_self"
] | `to_Ico_mod` as an equiv from the quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_add_group.equiv_Ico_mod_coe (a b : α) :
quotient_add_group.equiv_Ico_mod hp a ↑b = ⟨to_Ico_mod hp a b, to_Ico_mod_mem_Ico hp a _⟩ | rfl | lemma | quotient_add_group.equiv_Ico_mod_coe | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"quotient_add_group.equiv_Ico_mod",
"to_Ico_mod_mem_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_add_group.equiv_Ico_mod_zero (a : α) :
quotient_add_group.equiv_Ico_mod hp a 0 = ⟨to_Ico_mod hp a 0, to_Ico_mod_mem_Ico hp a _⟩ | rfl | lemma | quotient_add_group.equiv_Ico_mod_zero | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"quotient_add_group.equiv_Ico_mod",
"to_Ico_mod_mem_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_add_group.equiv_Ioc_mod (a : α) :
(α ⧸ add_subgroup.zmultiples p) ≃ set.Ioc a (a + p) | { to_fun := λ b, ⟨(to_Ioc_mod_periodic hp a).lift b,
quotient_add_group.induction_on' b $ to_Ioc_mod_mem_Ioc hp a⟩,
inv_fun := coe,
right_inv := λ b, subtype.ext $ (to_Ioc_mod_eq_self hp).mpr b.prop,
left_inv := λ b, begin
induction b using quotient_add_group.induction_on',
dsimp,
rw [quotient_add... | def | quotient_add_group.equiv_Ioc_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"add_subgroup.zmultiples",
"inv_fun",
"lift",
"set.Ioc",
"subtype.ext",
"to_Ioc_mod_eq_self",
"to_Ioc_mod_mem_Ioc",
"to_Ioc_mod_periodic",
"to_Ioc_mod_sub_self"
] | `to_Ioc_mod` as an equiv from the quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_add_group.equiv_Ioc_mod_coe (a b : α) :
quotient_add_group.equiv_Ioc_mod hp a ↑b = ⟨to_Ioc_mod hp a b, to_Ioc_mod_mem_Ioc hp a _⟩ | rfl | lemma | quotient_add_group.equiv_Ioc_mod_coe | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"quotient_add_group.equiv_Ioc_mod",
"to_Ioc_mod_mem_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_add_group.equiv_Ioc_mod_zero (a : α) :
quotient_add_group.equiv_Ioc_mod hp a 0 = ⟨to_Ioc_mod hp a 0, to_Ioc_mod_mem_Ioc hp a _⟩ | rfl | lemma | quotient_add_group.equiv_Ioc_mod_zero | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"quotient_add_group.equiv_Ioc_mod",
"to_Ioc_mod_mem_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ixx_mod_iff (x₁ x₂ x₃ : α) :
to_Ico_mod hp x₁ x₂ ≤ to_Ioc_mod hp x₁ x₃ ↔
to_Ico_mod hp 0 (x₂ - x₁) + to_Ico_mod hp 0 (x₁ - x₃) ≤ p | by rw [to_Ico_mod_eq_sub, to_Ioc_mod_eq_sub _ x₁, add_le_add_iff_right, ←neg_sub x₁ x₃,
to_Ioc_mod_neg, neg_zero, le_sub_iff_add_le] | lemma | to_Ixx_mod_iff | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_eq_sub",
"to_Ioc_mod",
"to_Ioc_mod_eq_sub",
"to_Ioc_mod_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ixx_mod_cyclic_left {x₁ x₂ x₃ : α}
(h : to_Ico_mod hp x₁ x₂ ≤ to_Ioc_mod hp x₁ x₃) :
to_Ico_mod hp x₂ x₃ ≤ to_Ioc_mod hp x₂ x₁ | begin
let x₂' := to_Ico_mod hp x₁ x₂,
let x₃' := to_Ico_mod hp x₂' x₃,
have h : x₂' ≤ to_Ioc_mod hp x₁ x₃' := by simpa,
have h₂₁ : x₂' < x₁ + p := to_Ico_mod_lt_right _ _ _,
have h₃₂ : x₃' - p < x₂' := sub_lt_iff_lt_add.2 (to_Ico_mod_lt_right _ _ _),
suffices hequiv : x₃' ≤ to_Ioc_mod hp x₂' x₁,
{ obtain... | lemma | to_Ixx_mod_cyclic_left | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"left_le_to_Ico_mod",
"to_Ico_mod",
"to_Ico_mod_eq_iff",
"to_Ico_mod_lt_right",
"to_Ioc_mod",
"to_Ioc_mod_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ixx_mod_antisymm (h₁₂₃ : to_Ico_mod hp a b ≤ to_Ioc_mod hp a c)
(h₁₃₂ : to_Ico_mod hp a c ≤ to_Ioc_mod hp a b) :
b ≡ a [PMOD p] ∨ c ≡ b [PMOD p] ∨ a ≡ c [PMOD p] | begin
by_contra' h,
rw modeq_comm at h,
rw ←(not_modeq_iff_to_Ico_mod_eq_to_Ioc_mod hp).mp h.2.2 at h₁₂₃,
rw ←(not_modeq_iff_to_Ico_mod_eq_to_Ioc_mod hp).mp h.1 at h₁₃₂,
exact h.2.1 ((to_Ico_mod_inj _).1 $ h₁₃₂.antisymm h₁₂₃),
end | lemma | to_Ixx_mod_antisymm | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_inj",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ixx_mod_total' (a b c : α) :
to_Ico_mod hp b a ≤ to_Ioc_mod hp b c ∨ to_Ico_mod hp b c ≤ to_Ioc_mod hp b a | begin
/- an essential ingredient is the lemma sabing {a-b} + {b-a} = period if a ≠ b (and = 0 if a = b).
Thus if a ≠ b and b ≠ c then ({a-b} + {b-c}) + ({c-b} + {b-a}) = 2 * period, so one of
`{a-b} + {b-c}` and `{c-b} + {b-a}` must be `≤ period` -/
have := congr_arg2 (+)
(to_Ico_mod_add_to_Ioc_mod_zero hp ... | lemma | to_Ixx_mod_total' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"congr_arg2",
"min_le_iff",
"to_Ico_mod",
"to_Ico_mod_add_to_Ioc_mod_zero",
"to_Ico_mod_le_to_Ioc_mod",
"to_Ioc_mod",
"to_Ixx_mod_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ixx_mod_total (a b c : α) :
to_Ico_mod hp a b ≤ to_Ioc_mod hp a c ∨ to_Ico_mod hp c b ≤ to_Ioc_mod hp c a | (to_Ixx_mod_total' _ _ _ _).imp_right $ to_Ixx_mod_cyclic_left _ | lemma | to_Ixx_mod_total | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ioc_mod",
"to_Ixx_mod_cyclic_left",
"to_Ixx_mod_total'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ixx_mod_trans {x₁ x₂ x₃ x₄ : α}
(h₁₂₃ : to_Ico_mod hp x₁ x₂ ≤ to_Ioc_mod hp x₁ x₃
∧ ¬to_Ico_mod hp x₃ x₂ ≤ to_Ioc_mod hp x₃ x₁)
(h₂₃₄ : to_Ico_mod hp x₂ x₄ ≤ to_Ioc_mod hp x₂ x₃
∧ ¬to_Ico_mod hp x₃ x₄ ≤ to_Ioc_mod hp x₃ x₂) :
to_Ico_mod hp x₁ x₄ ≤ to_Ioc_mod hp x₁ x₃
∧ ¬to_Ico_mod hp x₃ x₄ ≤ to_Ioc... | begin
split,
{ suffices h : ¬x₃ ≡ x₂ [PMOD p],
{ have h₁₂₃' := to_Ixx_mod_cyclic_left _ (to_Ixx_mod_cyclic_left _ h₁₂₃.1),
have h₂₃₄' := to_Ixx_mod_cyclic_left _ (to_Ixx_mod_cyclic_left _ h₂₃₄.1),
rw [(not_modeq_iff_to_Ico_mod_eq_to_Ioc_mod hp).1 h] at h₂₃₄',
exact to_Ixx_mod_cyclic_left _ (h₁... | lemma | to_Ixx_mod_trans | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"by_contra",
"left_lt_to_Ioc_mod",
"to_Ico_mod",
"to_Ico_mod_le_to_Ioc_mod",
"to_Ioc_mod",
"to_Ixx_mod_cyclic_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
btw_coe_iff' {x₁ x₂ x₃ : α} :
has_btw.btw (x₁ : α ⧸ add_subgroup.zmultiples p) x₂ x₃ ↔
to_Ico_mod hp'.out 0 (x₂ - x₁) ≤ to_Ioc_mod hp'.out 0 (x₃ - x₁) | iff.rfl | lemma | quotient_add_group.btw_coe_iff' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"add_subgroup.zmultiples",
"to_Ico_mod",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
btw_coe_iff {x₁ x₂ x₃ : α} :
has_btw.btw (x₁ : α ⧸ add_subgroup.zmultiples p) x₂ x₃ ↔
to_Ico_mod hp'.out x₁ x₂ ≤ to_Ioc_mod hp'.out x₁ x₃ | by rw [btw_coe_iff', to_Ioc_mod_sub_eq_sub, to_Ico_mod_sub_eq_sub, zero_add, sub_le_sub_iff_right] | lemma | quotient_add_group.btw_coe_iff | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"add_subgroup.zmultiples",
"to_Ico_mod",
"to_Ico_mod_sub_eq_sub",
"to_Ioc_mod",
"to_Ioc_mod_sub_eq_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
circular_preorder : circular_preorder (α ⧸ add_subgroup.zmultiples p) | { btw_refl := λ x, show _ ≤ _, by simp [sub_self, hp'.out.le],
btw_cyclic_left := λ x₁ x₂ x₃ h, begin
induction x₁ using quotient_add_group.induction_on',
induction x₂ using quotient_add_group.induction_on',
induction x₃ using quotient_add_group.induction_on',
simp_rw [btw_coe_iff] at h ⊢,
apply t... | instance | quotient_add_group.circular_preorder | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"add_subgroup.zmultiples",
"circular_preorder",
"sbtw",
"sbtw_iff_btw_not_btw",
"to_Ixx_mod_cyclic_left",
"to_Ixx_mod_trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
circular_order : circular_order (α ⧸ add_subgroup.zmultiples p) | { btw_antisymm := λ x₁ x₂ x₃ h₁₂₃ h₃₂₁, begin
induction x₁ using quotient_add_group.induction_on',
induction x₂ using quotient_add_group.induction_on',
induction x₃ using quotient_add_group.induction_on',
rw btw_cyclic at h₃₂₁,
simp_rw [btw_coe_iff] at h₁₂₃ h₃₂₁,
simp_rw ←modeq_iff_eq_mod_zmulti... | instance | quotient_add_group.circular_order | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"add_subgroup.zmultiples",
"btw_cyclic",
"circular_order",
"quotient_add_group.circular_preorder",
"to_Ixx_mod_antisymm",
"to_Ixx_mod_total"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_eq_floor (a b : α) : to_Ico_div hp a b = ⌊(b - a) / p⌋ | begin
refine to_Ico_div_eq_of_sub_zsmul_mem_Ico hp _,
rw [set.mem_Ico, zsmul_eq_mul, ←sub_nonneg, add_comm, sub_right_comm, ←sub_lt_iff_lt_add,
sub_right_comm _ _ a],
exact ⟨int.sub_floor_div_mul_nonneg _ hp, int.sub_floor_div_mul_lt _ hp⟩,
end | lemma | to_Ico_div_eq_floor | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"int.sub_floor_div_mul_lt",
"set.mem_Ico",
"to_Ico_div",
"to_Ico_div_eq_of_sub_zsmul_mem_Ico",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_eq_neg_floor (a b : α) : to_Ioc_div hp a b = -⌊(a + p - b) / p⌋ | begin
refine to_Ioc_div_eq_of_sub_zsmul_mem_Ioc hp _,
rw [set.mem_Ioc, zsmul_eq_mul, int.cast_neg, neg_mul, sub_neg_eq_add, ←sub_nonneg,
sub_add_eq_sub_sub],
refine ⟨_, int.sub_floor_div_mul_nonneg _ hp⟩,
rw [←add_lt_add_iff_right p, add_assoc, add_comm b, ←sub_lt_iff_lt_add, add_comm (_ * _),
←sub_lt... | lemma | to_Ioc_div_eq_neg_floor | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"int.cast_neg",
"int.sub_floor_div_mul_lt",
"int.sub_floor_div_mul_nonneg",
"neg_mul",
"set.mem_Ioc",
"to_Ioc_div",
"to_Ioc_div_eq_of_sub_zsmul_mem_Ioc",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_zero_one (b : α) : to_Ico_div (zero_lt_one' α) 0 b = ⌊b⌋ | by simp [to_Ico_div_eq_floor] | lemma | to_Ico_div_zero_one | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_eq_floor",
"zero_lt_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_eq_add_fract_mul (a b : α) : to_Ico_mod hp a b = a + int.fract ((b - a) / p) * p | begin
rw [to_Ico_mod, to_Ico_div_eq_floor, int.fract],
field_simp [hp.ne.symm],
ring
end | lemma | to_Ico_mod_eq_add_fract_mul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"int.fract",
"ring",
"to_Ico_div_eq_floor",
"to_Ico_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_eq_fract_mul (b : α) : to_Ico_mod hp 0 b = int.fract (b / p) * p | by simp [to_Ico_mod_eq_add_fract_mul] | lemma | to_Ico_mod_eq_fract_mul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"int.fract",
"to_Ico_mod",
"to_Ico_mod_eq_add_fract_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_eq_sub_fract_mul (a b : α) :
to_Ioc_mod hp a b = a + p - int.fract ((a + p - b) / p) * p | begin
rw [to_Ioc_mod, to_Ioc_div_eq_neg_floor, int.fract],
field_simp [hp.ne.symm],
ring
end | lemma | to_Ioc_mod_eq_sub_fract_mul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"int.fract",
"ring",
"to_Ioc_div_eq_neg_floor",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_zero_one (b : α) : to_Ico_mod (zero_lt_one' α) 0 b = int.fract b | by simp [to_Ico_mod_eq_add_fract_mul] | lemma | to_Ico_mod_zero_one | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"int.fract",
"to_Ico_mod",
"to_Ico_mod_eq_add_fract_mul",
"zero_lt_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_Ioc_add_zsmul : (⋃ (n : ℤ), Ioc (a + n • p) (a + (n + 1) • p)) = univ | begin
refine eq_univ_iff_forall.mpr (λ b, mem_Union.mpr _),
rcases sub_to_Ioc_div_zsmul_mem_Ioc hp a b with ⟨hl, hr⟩,
refine ⟨to_Ioc_div hp a b, ⟨lt_sub_iff_add_lt.mp hl, _⟩⟩,
rw [add_smul, one_smul, ←add_assoc],
convert sub_le_iff_le_add.mp hr using 1, abel,
end | lemma | Union_Ioc_add_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"add_smul",
"one_smul",
"sub_to_Ioc_div_zsmul_mem_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_Ico_add_zsmul : (⋃ (n : ℤ), Ico (a + n • p) (a + (n + 1) • p)) = univ | begin
refine eq_univ_iff_forall.mpr (λ b, mem_Union.mpr _),
rcases sub_to_Ico_div_zsmul_mem_Ico hp a b with ⟨hl, hr⟩,
refine ⟨to_Ico_div hp a b, ⟨le_sub_iff_add_le.mp hl, _⟩⟩,
rw [add_smul, one_smul, ←add_assoc],
convert sub_lt_iff_lt_add.mp hr using 1, abel,
end | lemma | Union_Ico_add_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"add_smul",
"one_smul",
"sub_to_Ico_div_zsmul_mem_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_Icc_add_zsmul : (⋃ (n : ℤ), Icc (a + n • p) (a + (n + 1) • p)) = univ | by simpa only [Union_Ioc_add_zsmul hp a, univ_subset_iff] using
Union_mono (λ n : ℤ, (Ioc_subset_Icc_self : Ioc (a + n • p) (a + (n + 1) • p) ⊆ Icc _ _)) | lemma | Union_Icc_add_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"Union_Ioc_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_Ioc_zsmul : (⋃ (n : ℤ), Ioc (n • p) ((n + 1) • p)) = univ | by simpa only [zero_add] using Union_Ioc_add_zsmul hp 0 | lemma | Union_Ioc_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"Union_Ioc_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_Ico_zsmul : (⋃ (n : ℤ), Ico (n • p) ((n + 1) • p)) = univ | by simpa only [zero_add] using Union_Ico_add_zsmul hp 0 | lemma | Union_Ico_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"Union_Ico_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_Icc_zsmul : (⋃ (n : ℤ), Icc (n • p) ((n + 1) • p)) = univ | by simpa only [zero_add] using Union_Icc_add_zsmul hp 0 | lemma | Union_Icc_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"Union_Icc_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_Ioc_add_int_cast : (⋃ (n : ℤ), Ioc (a + n) (a + n + 1)) = set.univ | by simpa only [zsmul_one, int.cast_add, int.cast_one, ←add_assoc]
using Union_Ioc_add_zsmul zero_lt_one a | lemma | Union_Ioc_add_int_cast | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"Union_Ioc_add_zsmul",
"int.cast_add",
"int.cast_one",
"zero_lt_one",
"zsmul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_Ico_add_int_cast : (⋃ (n : ℤ), Ico (a + n) (a + n + 1)) = set.univ | by simpa only [zsmul_one, int.cast_add, int.cast_one, ←add_assoc]
using Union_Ico_add_zsmul zero_lt_one a | lemma | Union_Ico_add_int_cast | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"Union_Ico_add_zsmul",
"int.cast_add",
"int.cast_one",
"zero_lt_one",
"zsmul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_Icc_add_int_cast : (⋃ (n : ℤ), Icc (a + n) (a + n + 1)) = set.univ | by simpa only [zsmul_one, int.cast_add, int.cast_one, ←add_assoc]
using Union_Icc_add_zsmul zero_lt_one a | lemma | Union_Icc_add_int_cast | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"Union_Icc_add_zsmul",
"int.cast_add",
"int.cast_one",
"zero_lt_one",
"zsmul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_Ioc_int_cast : (⋃ (n : ℤ), Ioc (n : α) (n + 1)) = set.univ | by simpa only [zero_add] using Union_Ioc_add_int_cast (0 : α) | lemma | Union_Ioc_int_cast | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"Union_Ioc_add_int_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_Ico_int_cast : (⋃ (n : ℤ), Ico (n : α) (n + 1)) = set.univ | by simpa only [zero_add] using Union_Ico_add_int_cast (0 : α) | lemma | Union_Ico_int_cast | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"Union_Ico_add_int_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_Icc_int_cast : (⋃ (n : ℤ), Icc (n : α) (n + 1)) = set.univ | by simpa only [zero_add] using Union_Icc_add_int_cast (0 : α) | lemma | Union_Icc_int_cast | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"Union_Icc_add_int_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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