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
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|---|---|---|---|---|---|---|---|---|---|---|
coe_mk (z : ℂ) (hz : abs z < 1) : (mk z hz : ℂ) = z | rfl | lemma | complex.unit_disc.coe_mk | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (z : 𝔻) (hz : abs (z : ℂ) < 1 := z.abs_lt_one) :
mk z hz = z | subtype.eta _ _ | lemma | complex.unit_disc.mk_coe | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_neg (z : ℂ) (hz : abs (-z) < 1) :
mk (-z) hz = -mk z (abs.map_neg z ▸ hz) | rfl | lemma | complex.unit_disc.mk_neg | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : 𝔻) : ℂ) = 0 | rfl | lemma | complex.unit_disc.coe_zero | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_zero {z : 𝔻} : (z : ℂ) = 0 ↔ z = 0 | coe_injective.eq_iff' coe_zero | lemma | complex.unit_disc.coe_eq_zero | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
circle_action : mul_action circle 𝔻 | mul_action_sphere_ball | instance | complex.unit_disc.circle_action | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"circle",
"mul_action",
"mul_action_sphere_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_circle_circle : is_scalar_tower circle circle 𝔻 | is_scalar_tower_sphere_sphere_ball | instance | complex.unit_disc.is_scalar_tower_circle_circle | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"circle",
"is_scalar_tower",
"is_scalar_tower_sphere_sphere_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_circle : is_scalar_tower circle 𝔻 𝔻 | is_scalar_tower_sphere_ball_ball | instance | complex.unit_disc.is_scalar_tower_circle | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"circle",
"is_scalar_tower",
"is_scalar_tower_sphere_ball_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_circle : smul_comm_class circle 𝔻 𝔻 | smul_comm_class_sphere_ball_ball | instance | complex.unit_disc.smul_comm_class_circle | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"circle",
"smul_comm_class",
"smul_comm_class_sphere_ball_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_circle' : smul_comm_class 𝔻 circle 𝔻 | smul_comm_class.symm _ _ _ | instance | complex.unit_disc.smul_comm_class_circle' | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"circle",
"smul_comm_class",
"smul_comm_class.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul_circle (z : circle) (w : 𝔻) : ↑(z • w) = (z * w : ℂ) | rfl | lemma | complex.unit_disc.coe_smul_circle | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_action : mul_action (closed_ball (0 : ℂ) 1) 𝔻 | mul_action_closed_ball_ball | instance | complex.unit_disc.closed_ball_action | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"mul_action",
"mul_action_closed_ball_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_closed_ball_closed_ball :
is_scalar_tower (closed_ball (0 : ℂ) 1) (closed_ball (0 : ℂ) 1) 𝔻 | is_scalar_tower_closed_ball_closed_ball_ball | instance | complex.unit_disc.is_scalar_tower_closed_ball_closed_ball | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"is_scalar_tower",
"is_scalar_tower_closed_ball_closed_ball_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_closed_ball : is_scalar_tower (closed_ball (0 : ℂ) 1) 𝔻 𝔻 | is_scalar_tower_closed_ball_ball_ball | instance | complex.unit_disc.is_scalar_tower_closed_ball | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"is_scalar_tower",
"is_scalar_tower_closed_ball_ball_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_closed_ball : smul_comm_class (closed_ball (0 : ℂ) 1) 𝔻 𝔻 | ⟨λ a b c, subtype.ext $ mul_left_comm _ _ _⟩ | instance | complex.unit_disc.smul_comm_class_closed_ball | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"mul_left_comm",
"smul_comm_class",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_closed_ball' : smul_comm_class 𝔻 (closed_ball (0 : ℂ) 1) 𝔻 | smul_comm_class.symm _ _ _ | instance | complex.unit_disc.smul_comm_class_closed_ball' | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"smul_comm_class",
"smul_comm_class.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_circle_closed_ball : smul_comm_class circle (closed_ball (0 : ℂ) 1) 𝔻 | smul_comm_class_sphere_closed_ball_ball | instance | complex.unit_disc.smul_comm_class_circle_closed_ball | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"circle",
"smul_comm_class",
"smul_comm_class_sphere_closed_ball_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class_closed_ball_circle : smul_comm_class (closed_ball (0 : ℂ) 1) circle 𝔻 | smul_comm_class.symm _ _ _ | instance | complex.unit_disc.smul_comm_class_closed_ball_circle | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"circle",
"smul_comm_class",
"smul_comm_class.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul_closed_ball (z : closed_ball (0 : ℂ) 1) (w : 𝔻) : ↑(z • w) = (z * w : ℂ) | rfl | lemma | complex.unit_disc.coe_smul_closed_ball | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
re (z : 𝔻) : ℝ | re z | def | complex.unit_disc.re | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | Real part of a point of the unit disc. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
im (z : 𝔻) : ℝ | im z | def | complex.unit_disc.im | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | Imaginary part of a point of the unit disc. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
re_coe (z : 𝔻) : (z : ℂ).re = z.re | rfl | lemma | complex.unit_disc.re_coe | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_coe (z : 𝔻) : (z : ℂ).im = z.im | rfl | lemma | complex.unit_disc.im_coe | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
re_neg (z : 𝔻) : (-z).re = -z.re | rfl | lemma | complex.unit_disc.re_neg | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_neg (z : 𝔻) : (-z).im = -z.im | rfl | lemma | complex.unit_disc.im_neg | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj (z : 𝔻) : 𝔻 | mk (conj' ↑z) $ (abs_conj z).symm ▸ z.abs_lt_one | def | complex.unit_disc.conj | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | Conjugate point of the unit disc. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_conj (z : 𝔻) : (z.conj : ℂ) = conj' ↑z | rfl | lemma | complex.unit_disc.coe_conj | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_zero : conj 0 = 0 | coe_injective (map_zero conj') | lemma | complex.unit_disc.conj_zero | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_conj (z : 𝔻) : conj (conj z) = z | coe_injective $ complex.conj_conj z | lemma | complex.unit_disc.conj_conj | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_neg (z : 𝔻) : (-z).conj = -z.conj | rfl | lemma | complex.unit_disc.conj_neg | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
re_conj (z : 𝔻) : z.conj.re = z.re | rfl | lemma | complex.unit_disc.re_conj | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_conj (z : 𝔻) : z.conj.im = -z.im | rfl | lemma | complex.unit_disc.im_conj | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_mul (z w : 𝔻) : (z * w).conj = z.conj * w.conj | subtype.ext $ map_mul _ _ _ | lemma | complex.unit_disc.conj_mul | analysis.complex.unit_disc | src/analysis/complex/unit_disc/basic.lean | [
"analysis.complex.circle",
"analysis.normed_space.ball_action"
] | [
"conj_mul",
"map_mul",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_half_plane | {point : ℂ // 0 < point.im} | def | upper_half_plane | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | The open upper half plane | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
can_lift : can_lift ℂ ℍ coe (λ z, 0 < z.im) | subtype.can_lift (λ z, 0 < z.im) | instance | upper_half_plane.can_lift | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"can_lift",
"subtype.can_lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im (z : ℍ) | (z : ℂ).im | def | upper_half_plane.im | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | Imaginary part | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
re (z : ℍ) | (z : ℂ).re | def | upper_half_plane.re | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | Real part | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk (z : ℂ) (h : 0 < z.im) : ℍ | ⟨z, h⟩ | def | upper_half_plane.mk | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | Constructor for `upper_half_plane`. It is useful if `⟨z, h⟩` makes Lean use a wrong
typeclass instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_im (z : ℍ) : (z : ℂ).im = z.im | rfl | lemma | upper_half_plane.coe_im | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_re (z : ℍ) : (z : ℂ).re = z.re | rfl | lemma | upper_half_plane.coe_re | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_re (z : ℂ) (h : 0 < z.im) : (mk z h).re = z.re | rfl | lemma | upper_half_plane.mk_re | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_im (z : ℂ) (h : 0 < z.im) : (mk z h).im = z.im | rfl | lemma | upper_half_plane.mk_im | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (z : ℂ) (h : 0 < z.im) : (mk z h : ℂ) = z | rfl | lemma | upper_half_plane.coe_mk | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (z : ℍ) (h : 0 < (z : ℂ).im := z.2) : mk z h = z | subtype.eta z h | lemma | upper_half_plane.mk_coe | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
re_add_im (z : ℍ) : (z.re + z.im * complex.I : ℂ) = z | complex.re_add_im z | lemma | upper_half_plane.re_add_im | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"complex.I",
"complex.re_add_im"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_pos (z : ℍ) : 0 < z.im | z.2 | lemma | upper_half_plane.im_pos | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_ne_zero (z : ℍ) : z.im ≠ 0 | z.im_pos.ne' | lemma | upper_half_plane.im_ne_zero | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero (z : ℍ) : (z : ℂ) ≠ 0 | mt (congr_arg complex.im) z.im_ne_zero | lemma | upper_half_plane.ne_zero | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_pos (z : ℍ) : 0 < complex.norm_sq (z : ℂ) | by { rw complex.norm_sq_pos, exact z.ne_zero } | lemma | upper_half_plane.norm_sq_pos | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"complex.norm_sq",
"complex.norm_sq_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_ne_zero (z : ℍ) : complex.norm_sq (z : ℂ) ≠ 0 | (norm_sq_pos z).ne' | lemma | upper_half_plane.norm_sq_ne_zero | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"complex.norm_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_inv_neg_coe_pos (z : ℍ) : 0 < ((-z : ℂ)⁻¹).im | by simpa using div_pos z.property (norm_sq_pos z) | lemma | upper_half_plane.im_inv_neg_coe_pos | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"div_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num (g : GL(2, ℝ)⁺) (z : ℍ) : ℂ | (↑ₘg 0 0 : ℝ) * z + (↑ₘg 0 1 : ℝ) | def | upper_half_plane.num | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"num"
] | Numerator of the formula for a fractional linear transformation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
denom (g : GL(2, ℝ)⁺) (z : ℍ) : ℂ | (↑ₘg 1 0 : ℝ) * z + (↑ₘg 1 1 : ℝ) | def | upper_half_plane.denom | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | Denominator of the formula for a fractional linear transformation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_ne_zero (cd : fin 2 → ℝ) (z : ℍ) (h : cd ≠ 0) : (cd 0 : ℂ) * z + cd 1 ≠ 0 | begin
contrapose! h,
have : cd 0 = 0, -- we will need this twice
{ apply_fun complex.im at h,
simpa only [z.im_ne_zero, complex.add_im, add_zero, coe_im, zero_mul, or_false,
complex.of_real_im, complex.zero_im, complex.mul_im, mul_eq_zero] using h, },
simp only [this, zero_mul, complex.of_real_zero, z... | lemma | upper_half_plane.linear_ne_zero | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"complex.add_im",
"complex.mul_im",
"complex.of_real_eq_zero",
"complex.of_real_im",
"complex.of_real_zero",
"complex.zero_im",
"mul_eq_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
denom_ne_zero (g : GL(2, ℝ)⁺) (z : ℍ) : denom g z ≠ 0 | begin
intro H,
have DET := (mem_GL_pos _).1 g.prop,
have hz := z.prop,
simp only [general_linear_group.coe_det_apply] at DET,
have H1 : (↑ₘg 1 0 : ℝ) = 0 ∨ z.im = 0, by simpa using congr_arg complex.im H,
cases H1,
{ simp only [H1, complex.of_real_zero, denom, coe_fn_eq_coe, zero_mul, zero_add,
comple... | lemma | upper_half_plane.denom_ne_zero | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"coe_coe",
"complex.of_real_eq_zero",
"complex.of_real_zero",
"lt_self_iff_false",
"matrix",
"matrix.det_fin_two",
"mul_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_denom_pos (g : GL(2, ℝ)⁺) (z : ℍ) : 0 < complex.norm_sq (denom g z) | complex.norm_sq_pos.mpr (denom_ne_zero g z) | lemma | upper_half_plane.norm_sq_denom_pos | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"complex.norm_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_denom_ne_zero (g : GL(2, ℝ)⁺) (z : ℍ) : complex.norm_sq (denom g z) ≠ 0 | ne_of_gt (norm_sq_denom_pos g z) | lemma | upper_half_plane.norm_sq_denom_ne_zero | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"complex.norm_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_aux' (g : GL(2, ℝ)⁺) (z : ℍ) : ℂ | num g z / denom g z | def | upper_half_plane.smul_aux' | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"num"
] | Fractional linear transformation, also known as the Moebius transformation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_aux'_im (g : GL(2, ℝ)⁺) (z : ℍ) :
(smul_aux' g z).im = ((det ↑ₘg) * z.im) / (denom g z).norm_sq | begin
rw [smul_aux', complex.div_im],
set NsqBot := (denom g z).norm_sq,
have : NsqBot ≠ 0,
{ simp only [denom_ne_zero g z, map_eq_zero, ne.def, not_false_iff], },
field_simp [smul_aux', -coe_coe],
rw (matrix.det_fin_two (↑ₘg)),
ring,
end | lemma | upper_half_plane.smul_aux'_im | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"coe_coe",
"complex.div_im",
"map_eq_zero",
"matrix.det_fin_two",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_aux (g : GL(2, ℝ)⁺) (z : ℍ) : ℍ | ⟨smul_aux' g z, begin
rw smul_aux'_im,
convert (mul_pos ((mem_GL_pos _).1 g.prop)
(div_pos z.im_pos (complex.norm_sq_pos.mpr (denom_ne_zero g z)))),
simp only [general_linear_group.coe_det_apply, coe_coe],
ring
end⟩ | def | upper_half_plane.smul_aux | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"coe_coe",
"div_pos",
"ring"
] | Fractional linear transformation, also known as the Moebius transformation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
denom_cocycle (x y : GL(2, ℝ)⁺) (z : ℍ) :
denom (x * y) z = denom x (smul_aux y z) * denom y z | begin
change _ = (_ * (_ / _) + _) * _,
field_simp [denom_ne_zero, -denom, -num],
simp only [matrix.mul, dot_product, fin.sum_univ_succ, denom, num, coe_coe, subgroup.coe_mul,
general_linear_group.coe_mul, fintype.univ_of_subsingleton, fin.mk_zero,
finset.sum_singleton, fin.succ_zero_eq_one, complex.of_re... | lemma | upper_half_plane.denom_cocycle | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"coe_coe",
"complex.of_real_add",
"complex.of_real_mul",
"fin.mk_zero",
"fin.succ_zero_eq_one",
"fintype.univ_of_subsingleton",
"matrix.mul",
"num",
"ring",
"subgroup.coe_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_smul' (x y : GL(2, ℝ)⁺) (z : ℍ) :
smul_aux (x * y) z = smul_aux x (smul_aux y z) | begin
ext1,
change _ / _ = (_ * (_ / _) + _) * _,
rw denom_cocycle,
field_simp [denom_ne_zero, -denom, -num],
simp only [matrix.mul, dot_product, fin.sum_univ_succ, num, denom, coe_coe, subgroup.coe_mul,
general_linear_group.coe_mul, fintype.univ_of_subsingleton, fin.mk_zero,
finset.sum_singleton, fin... | lemma | upper_half_plane.mul_smul' | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"coe_coe",
"complex.of_real_add",
"complex.of_real_mul",
"fin.mk_zero",
"fin.succ_zero_eq_one",
"fintype.univ_of_subsingleton",
"matrix.mul",
"num",
"ring",
"subgroup.coe_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
SL_action {R : Type*} [comm_ring R] [algebra R ℝ] : mul_action SL(2, R) ℍ | mul_action.comp_hom ℍ $ (special_linear_group.to_GL_pos).comp $ map (algebra_map R ℝ) | instance | upper_half_plane.SL_action | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"algebra",
"algebra_map",
"comm_ring",
"mul_action",
"mul_action.comp_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
SL_on_GL_pos : has_smul SL(2,ℤ) (GL(2, ℝ)⁺) | ⟨λ s g, s * g⟩ | instance | upper_half_plane.SL_on_GL_pos | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
SL_on_GL_pos_smul_apply (s : SL(2,ℤ)) (g : (GL(2, ℝ)⁺)) (z : ℍ) :
(s • g) • z = ( (s : GL(2, ℝ)⁺) * g) • z | rfl | lemma | upper_half_plane.SL_on_GL_pos_smul_apply | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
SL_to_GL_tower : is_scalar_tower SL(2,ℤ) (GL(2, ℝ)⁺) ℍ | { smul_assoc := by {intros s g z, simp only [SL_on_GL_pos_smul_apply, coe_coe], apply mul_smul',},} | instance | upper_half_plane.SL_to_GL_tower | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"coe_coe",
"is_scalar_tower",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup_GL_pos : has_smul Γ (GL(2, ℝ)⁺) | ⟨λ s g, s * g⟩ | instance | upper_half_plane.subgroup_GL_pos | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup_on_GL_pos_smul_apply (s : Γ) (g : (GL(2, ℝ)⁺)) (z : ℍ) :
(s • g) • z = ( (s : GL(2, ℝ)⁺) * g) • z | rfl | lemma | upper_half_plane.subgroup_on_GL_pos_smul_apply | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup_on_GL_pos : is_scalar_tower Γ (GL(2, ℝ)⁺) ℍ | { smul_assoc :=
by {intros s g z, simp only [subgroup_on_GL_pos_smul_apply, coe_coe], apply mul_smul',},} | instance | upper_half_plane.subgroup_on_GL_pos | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"coe_coe",
"is_scalar_tower",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup_SL : has_smul Γ SL(2,ℤ) | ⟨λ s g, s * g⟩ | instance | upper_half_plane.subgroup_SL | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup_on_SL_apply (s : Γ) (g : SL(2,ℤ) ) (z : ℍ) :
(s • g) • z = ( (s : SL(2, ℤ)) * g) • z | rfl | lemma | upper_half_plane.subgroup_on_SL_apply | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup_to_SL_tower : is_scalar_tower Γ SL(2,ℤ) ℍ | { smul_assoc := λ s g z, by { rw subgroup_on_SL_apply, apply mul_action.mul_smul } } | instance | upper_half_plane.subgroup_to_SL_tower | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"is_scalar_tower",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
special_linear_group_apply {R : Type*} [comm_ring R] [algebra R ℝ] (g : SL(2, R)) (z : ℍ) :
g • z = mk ((((↑(↑ₘ[R] g 0 0) : ℝ) : ℂ) * z + ((↑(↑ₘ[R] g 0 1) : ℝ) : ℂ)) /
(((↑(↑ₘ[R] g 1 0) : ℝ) : ℂ) * z + ((↑(↑ₘ[R] g 1 1) : ℝ) : ℂ))) (g • z).property | rfl | lemma | upper_half_plane.special_linear_group_apply | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"algebra",
"comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (g : GL(2, ℝ)⁺) (z : ℍ) : ↑(g • z) = num g z / denom g z | rfl | lemma | upper_half_plane.coe_smul | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
re_smul (g : GL(2, ℝ)⁺) (z : ℍ) : (g • z).re = (num g z / denom g z).re | rfl | lemma | upper_half_plane.re_smul | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_smul (g : GL(2, ℝ)⁺) (z : ℍ) : (g • z).im = (num g z / denom g z).im | rfl | lemma | upper_half_plane.im_smul | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_smul_eq_div_norm_sq (g : GL(2, ℝ)⁺) (z : ℍ) :
(g • z).im = (det ↑ₘg * z.im) / (complex.norm_sq (denom g z)) | smul_aux'_im g z | lemma | upper_half_plane.im_smul_eq_div_norm_sq | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"complex.norm_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_smul (g : GL(2, ℝ)⁺) (z : ℍ) : -g • z = g • z | begin
ext1,
change _ / _ = _ / _,
field_simp [denom_ne_zero, -denom, -num],
simp only [num, denom, coe_coe, complex.of_real_neg, neg_mul, GL_pos.coe_neg_GL, units.coe_neg,
pi.neg_apply],
ring_nf,
end | lemma | upper_half_plane.neg_smul | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"coe_coe",
"complex.of_real_neg",
"neg_mul",
"neg_smul",
"num",
"units.coe_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sl_moeb (A : SL(2,ℤ)) (z : ℍ) : A • z = (A : (GL(2, ℝ)⁺)) • z | rfl | lemma | upper_half_plane.sl_moeb | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup_moeb (A : Γ) (z : ℍ) : A • z = (A : (GL(2, ℝ)⁺)) • z | rfl | lemma | upper_half_plane.subgroup_moeb | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subgroup_to_sl_moeb (A : Γ) (z : ℍ) : A • z = (A : SL(2,ℤ)) • z | rfl | lemma | upper_half_plane.subgroup_to_sl_moeb | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
SL_neg_smul (g : SL(2,ℤ)) (z : ℍ) : -g • z = g • z | begin
simp only [coe_GL_pos_neg, sl_moeb, coe_coe, coe_int_neg, neg_smul],
end | lemma | upper_half_plane.SL_neg_smul | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"coe_coe",
"neg_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
c_mul_im_sq_le_norm_sq_denom (z : ℍ) (g : SL(2, ℝ)) :
((↑ₘg 1 0 : ℝ) * (z.im))^2 ≤ complex.norm_sq (denom g z) | begin
let c := (↑ₘg 1 0 : ℝ),
let d := (↑ₘg 1 1 : ℝ),
calc (c * z.im)^2 ≤ (c * z.im)^2 + (c * z.re + d)^2 : by nlinarith
... = complex.norm_sq (denom g z) : by simp [complex.norm_sq]; ring,
end | lemma | upper_half_plane.c_mul_im_sq_le_norm_sq_denom | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"complex.norm_sq",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
special_linear_group.im_smul_eq_div_norm_sq :
(g • z).im = z.im / (complex.norm_sq (denom g z)) | begin
convert (im_smul_eq_div_norm_sq g z),
simp only [coe_coe, general_linear_group.coe_det_apply,coe_GL_pos_coe_GL_coe_matrix,
int.coe_cast_ring_hom,(g : SL(2,ℝ)).prop, one_mul],
end | lemma | upper_half_plane.special_linear_group.im_smul_eq_div_norm_sq | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"coe_coe",
"complex.norm_sq",
"int.coe_cast_ring_hom",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
denom_apply (g : SL(2, ℤ)) (z : ℍ) : denom g z = (↑g : matrix (fin 2) (fin 2) ℤ) 1 0 * z +
(↑g : matrix (fin 2) (fin 2) ℤ) 1 1 | by simp | lemma | upper_half_plane.denom_apply | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_real_action : mul_action {x : ℝ // 0 < x} ℍ | { smul := λ x z, mk ((x : ℝ) • z) $ by simpa using mul_pos x.2 z.2,
one_smul := λ z, subtype.ext $ one_smul _ _,
mul_smul := λ x y z, subtype.ext $ mul_smul (x : ℝ) y (z : ℂ) } | instance | upper_half_plane.pos_real_action | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"mul_action",
"one_smul",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pos_real_smul : ↑(x • z) = (x : ℝ) • (z : ℂ) | rfl | lemma | upper_half_plane.coe_pos_real_smul | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_real_im : (x • z).im = x * z.im | complex.smul_im _ _ | lemma | upper_half_plane.pos_real_im | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"complex.smul_im"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_real_re : (x • z).re = x * z.re | complex.smul_re _ _ | lemma | upper_half_plane.pos_real_re | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"complex.smul_re"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_vadd : ↑(x +ᵥ z) = (x + z : ℂ) | rfl | lemma | upper_half_plane.coe_vadd | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vadd_re : (x +ᵥ z).re = x + z.re | rfl | lemma | upper_half_plane.vadd_re | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vadd_im : (x +ᵥ z).im = z.im | zero_add _ | lemma | upper_half_plane.vadd_im | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modular_S_smul (z : ℍ) : modular_group.S • z = mk (-z : ℂ)⁻¹ z.im_inv_neg_coe_pos | by { rw special_linear_group_apply, simp [modular_group.S, neg_div, inv_neg], } | lemma | upper_half_plane.modular_S_smul | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"inv_neg",
"modular_group.S",
"neg_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modular_T_zpow_smul (z : ℍ) (n : ℤ) : modular_group.T ^ n • z = (n : ℝ) +ᵥ z | begin
rw [←subtype.coe_inj, coe_vadd, add_comm, special_linear_group_apply, coe_mk,
modular_group.coe_T_zpow],
simp only [of_apply, cons_val_zero, algebra_map.coe_one, complex.of_real_one, one_mul,
cons_val_one, head_cons, algebra_map.coe_zero, zero_mul, zero_add, div_one],
end | lemma | upper_half_plane.modular_T_zpow_smul | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"algebra_map.coe_one",
"algebra_map.coe_zero",
"complex.of_real_one",
"div_one",
"modular_group.T",
"modular_group.coe_T_zpow",
"one_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modular_T_smul (z : ℍ) : modular_group.T • z = (1 : ℝ) +ᵥ z | by simpa only [algebra_map.coe_one] using modular_T_zpow_smul z 1 | lemma | upper_half_plane.modular_T_smul | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"algebra_map.coe_one",
"modular_group.T"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_SL2_smul_eq_of_apply_zero_one_eq_zero (g : SL(2, ℝ)) (hc : ↑ₘ[ℝ] g 1 0 = 0) :
∃ (u : {x : ℝ // 0 < x}) (v : ℝ),
((•) g : ℍ → ℍ) = (λ z, v +ᵥ z) ∘ (λ z, u • z) | begin
obtain ⟨a, b, ha, rfl⟩ := g.fin_two_exists_eq_mk_of_apply_zero_one_eq_zero hc,
refine ⟨⟨_, mul_self_pos.mpr ha⟩, b * a, _⟩,
ext1 ⟨z, hz⟩, ext1,
suffices : ↑a * z * a + b * a = b * a + a * a * z,
{ rw special_linear_group_apply, simpa [add_mul], },
ring,
end | lemma | upper_half_plane.exists_SL2_smul_eq_of_apply_zero_one_eq_zero | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_SL2_smul_eq_of_apply_zero_one_ne_zero (g : SL(2, ℝ)) (hc : ↑ₘ[ℝ] g 1 0 ≠ 0) :
∃ (u : {x : ℝ // 0 < x}) (v w : ℝ),
((•) g : ℍ → ℍ) = ((+ᵥ) w : ℍ → ℍ) ∘ ((•) modular_group.S : ℍ → ℍ)
∘ ((+ᵥ) v : ℍ → ℍ) ∘ ((•) u : ℍ → ℍ) | begin
have h_denom := denom_ne_zero g,
induction g using matrix.special_linear_group.fin_two_induction with a b c d h,
replace hc : c ≠ 0, { simpa using hc, },
refine ⟨⟨_, mul_self_pos.mpr hc⟩, c * d, a / c, _⟩,
ext1 ⟨z, hz⟩, ext1,
suffices : (↑a * z + b) / (↑c * z + d) = a / c - (c * d + ↑c * ↑c * z)⁻¹,
... | lemma | upper_half_plane.exists_SL2_smul_eq_of_apply_zero_one_ne_zero | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/basic.lean | [
"data.fintype.parity",
"linear_algebra.matrix.special_linear_group",
"analysis.complex.basic",
"group_theory.group_action.defs",
"linear_algebra.matrix.general_linear_group",
"tactic.linear_combination"
] | [
"complex.of_real_div",
"complex.of_real_mul",
"complex.real_smul",
"inv_neg",
"matrix.special_linear_group.fin_two_induction",
"modular_group.S",
"mul_assoc",
"mul_ne_zero",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
at_im_infty | filter.at_top.comap upper_half_plane.im | def | upper_half_plane.at_im_infty | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/functions_bounded_at_infty.lean | [
"algebra.module.submodule.basic",
"analysis.complex.upper_half_plane.basic",
"order.filter.zero_and_bounded_at_filter"
] | [
"upper_half_plane.im"
] | Filter for approaching `i∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
at_im_infty_basis : (at_im_infty).has_basis (λ _, true) (λ (i : ℝ), im ⁻¹' set.Ici i) | filter.has_basis.comap upper_half_plane.im filter.at_top_basis | lemma | upper_half_plane.at_im_infty_basis | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/functions_bounded_at_infty.lean | [
"algebra.module.submodule.basic",
"analysis.complex.upper_half_plane.basic",
"order.filter.zero_and_bounded_at_filter"
] | [
"filter.at_top_basis",
"filter.has_basis.comap",
"set.Ici",
"upper_half_plane.im"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
at_im_infty_mem (S : set ℍ) : S ∈ at_im_infty ↔ (∃ A : ℝ, ∀ z : ℍ, A ≤ im z → z ∈ S) | begin
simp only [at_im_infty, filter.mem_comap', filter.mem_at_top_sets, ge_iff_le, set.mem_set_of_eq,
upper_half_plane.coe_im],
refine ⟨λ ⟨a, h⟩, ⟨a, (λ z hz, h (im z) hz rfl)⟩, _⟩,
rintro ⟨A, h⟩,
refine ⟨A, λ b hb x hx, h x _⟩,
rwa hx,
end | lemma | upper_half_plane.at_im_infty_mem | analysis.complex.upper_half_plane | src/analysis/complex/upper_half_plane/functions_bounded_at_infty.lean | [
"algebra.module.submodule.basic",
"analysis.complex.upper_half_plane.basic",
"order.filter.zero_and_bounded_at_filter"
] | [
"filter.mem_at_top_sets",
"filter.mem_comap'",
"ge_iff_le",
"upper_half_plane.coe_im"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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