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(* AUTOMATICALLY GENERATED, DO NOT EDIT! *)
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theory HOL4Real imports HOL4Base begin
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;setup_theory realax
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lemma HREAL_RDISTRIB: "ALL (x::hreal) (y::hreal) z::hreal.
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hreal_mul (hreal_add x y) z = hreal_add (hreal_mul x z) (hreal_mul y z)"
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by (import realax HREAL_RDISTRIB)
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lemma HREAL_EQ_ADDL: "ALL (x::hreal) y::hreal. x ~= hreal_add x y"
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by (import realax HREAL_EQ_ADDL)
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lemma HREAL_EQ_LADD: "ALL (x::hreal) (y::hreal) z::hreal.
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(hreal_add x y = hreal_add x z) = (y = z)"
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by (import realax HREAL_EQ_LADD)
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lemma HREAL_LT_REFL: "ALL x::hreal. ~ hreal_lt x x"
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by (import realax HREAL_LT_REFL)
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lemma HREAL_LT_ADDL: "ALL (x::hreal) y::hreal. hreal_lt x (hreal_add x y)"
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by (import realax HREAL_LT_ADDL)
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lemma HREAL_LT_NE: "ALL (x::hreal) y::hreal. hreal_lt x y --> x ~= y"
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by (import realax HREAL_LT_NE)
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lemma HREAL_LT_ADDR: "ALL (x::hreal) y::hreal. ~ hreal_lt (hreal_add x y) x"
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by (import realax HREAL_LT_ADDR)
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lemma HREAL_LT_GT: "ALL (x::hreal) y::hreal. hreal_lt x y --> ~ hreal_lt y x"
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by (import realax HREAL_LT_GT)
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lemma HREAL_LT_ADD2: "ALL (x1::hreal) (x2::hreal) (y1::hreal) y2::hreal.
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hreal_lt x1 y1 & hreal_lt x2 y2 -->
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hreal_lt (hreal_add x1 x2) (hreal_add y1 y2)"
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by (import realax HREAL_LT_ADD2)
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lemma HREAL_LT_LADD: "ALL (x::hreal) (y::hreal) z::hreal.
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hreal_lt (hreal_add x y) (hreal_add x z) = hreal_lt y z"
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by (import realax HREAL_LT_LADD)
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constdefs
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treal_0 :: "hreal * hreal"
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"treal_0 == (hreal_1, hreal_1)"
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lemma treal_0: "treal_0 = (hreal_1, hreal_1)"
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by (import realax treal_0)
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constdefs
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treal_1 :: "hreal * hreal"
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"treal_1 == (hreal_add hreal_1 hreal_1, hreal_1)"
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lemma treal_1: "treal_1 = (hreal_add hreal_1 hreal_1, hreal_1)"
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by (import realax treal_1)
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constdefs
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treal_neg :: "hreal * hreal => hreal * hreal"
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"treal_neg == %(x::hreal, y::hreal). (y, x)"
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lemma treal_neg: "ALL (x::hreal) y::hreal. treal_neg (x, y) = (y, x)"
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by (import realax treal_neg)
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constdefs
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treal_add :: "hreal * hreal => hreal * hreal => hreal * hreal"
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"treal_add ==
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%(x1::hreal, y1::hreal) (x2::hreal, y2::hreal).
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(hreal_add x1 x2, hreal_add y1 y2)"
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lemma treal_add: "ALL (x1::hreal) (y1::hreal) (x2::hreal) y2::hreal.
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treal_add (x1, y1) (x2, y2) = (hreal_add x1 x2, hreal_add y1 y2)"
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by (import realax treal_add)
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constdefs
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treal_mul :: "hreal * hreal => hreal * hreal => hreal * hreal"
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"treal_mul ==
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%(x1::hreal, y1::hreal) (x2::hreal, y2::hreal).
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(hreal_add (hreal_mul x1 x2) (hreal_mul y1 y2),
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hreal_add (hreal_mul x1 y2) (hreal_mul y1 x2))"
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lemma treal_mul: "ALL (x1::hreal) (y1::hreal) (x2::hreal) y2::hreal.
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treal_mul (x1, y1) (x2, y2) =
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(hreal_add (hreal_mul x1 x2) (hreal_mul y1 y2),
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hreal_add (hreal_mul x1 y2) (hreal_mul y1 x2))"
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by (import realax treal_mul)
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constdefs
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treal_lt :: "hreal * hreal => hreal * hreal => bool"
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"treal_lt ==
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%(x1::hreal, y1::hreal) (x2::hreal, y2::hreal).
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hreal_lt (hreal_add x1 y2) (hreal_add x2 y1)"
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lemma treal_lt: "ALL (x1::hreal) (y1::hreal) (x2::hreal) y2::hreal.
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treal_lt (x1, y1) (x2, y2) = hreal_lt (hreal_add x1 y2) (hreal_add x2 y1)"
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by (import realax treal_lt)
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constdefs
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treal_inv :: "hreal * hreal => hreal * hreal"
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"treal_inv ==
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%(x::hreal, y::hreal).
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if x = y then treal_0
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else if hreal_lt y x
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then (hreal_add (hreal_inv (hreal_sub x y)) hreal_1, hreal_1)
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else (hreal_1, hreal_add (hreal_inv (hreal_sub y x)) hreal_1)"
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lemma treal_inv: "ALL (x::hreal) y::hreal.
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treal_inv (x, y) =
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(if x = y then treal_0
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else if hreal_lt y x
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then (hreal_add (hreal_inv (hreal_sub x y)) hreal_1, hreal_1)
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else (hreal_1, hreal_add (hreal_inv (hreal_sub y x)) hreal_1))"
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by (import realax treal_inv)
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constdefs
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treal_eq :: "hreal * hreal => hreal * hreal => bool"
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"treal_eq ==
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%(x1::hreal, y1::hreal) (x2::hreal, y2::hreal).
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hreal_add x1 y2 = hreal_add x2 y1"
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lemma treal_eq: "ALL (x1::hreal) (y1::hreal) (x2::hreal) y2::hreal.
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treal_eq (x1, y1) (x2, y2) = (hreal_add x1 y2 = hreal_add x2 y1)"
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by (import realax treal_eq)
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lemma TREAL_EQ_REFL: "ALL x::hreal * hreal. treal_eq x x"
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by (import realax TREAL_EQ_REFL)
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lemma TREAL_EQ_SYM: "ALL (x::hreal * hreal) y::hreal * hreal. treal_eq x y = treal_eq y x"
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by (import realax TREAL_EQ_SYM)
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lemma TREAL_EQ_TRANS: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
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treal_eq x y & treal_eq y z --> treal_eq x z"
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by (import realax TREAL_EQ_TRANS)
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lemma TREAL_EQ_EQUIV: "ALL (p::hreal * hreal) q::hreal * hreal.
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treal_eq p q = (treal_eq p = treal_eq q)"
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by (import realax TREAL_EQ_EQUIV)
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lemma TREAL_EQ_AP: "ALL (p::hreal * hreal) q::hreal * hreal. p = q --> treal_eq p q"
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by (import realax TREAL_EQ_AP)
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lemma TREAL_10: "~ treal_eq treal_1 treal_0"
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by (import realax TREAL_10)
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lemma TREAL_ADD_SYM: "ALL (x::hreal * hreal) y::hreal * hreal. treal_add x y = treal_add y x"
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by (import realax TREAL_ADD_SYM)
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lemma TREAL_MUL_SYM: "ALL (x::hreal * hreal) y::hreal * hreal. treal_mul x y = treal_mul y x"
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by (import realax TREAL_MUL_SYM)
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lemma TREAL_ADD_ASSOC: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
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treal_add x (treal_add y z) = treal_add (treal_add x y) z"
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by (import realax TREAL_ADD_ASSOC)
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lemma TREAL_MUL_ASSOC: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
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treal_mul x (treal_mul y z) = treal_mul (treal_mul x y) z"
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by (import realax TREAL_MUL_ASSOC)
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lemma TREAL_LDISTRIB: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
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treal_mul x (treal_add y z) = treal_add (treal_mul x y) (treal_mul x z)"
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by (import realax TREAL_LDISTRIB)
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lemma TREAL_ADD_LID: "ALL x::hreal * hreal. treal_eq (treal_add treal_0 x) x"
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by (import realax TREAL_ADD_LID)
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lemma TREAL_MUL_LID: "ALL x::hreal * hreal. treal_eq (treal_mul treal_1 x) x"
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by (import realax TREAL_MUL_LID)
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lemma TREAL_ADD_LINV: "ALL x::hreal * hreal. treal_eq (treal_add (treal_neg x) x) treal_0"
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by (import realax TREAL_ADD_LINV)
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lemma TREAL_INV_0: "treal_eq (treal_inv treal_0) treal_0"
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by (import realax TREAL_INV_0)
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lemma TREAL_MUL_LINV: "ALL x::hreal * hreal.
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~ treal_eq x treal_0 --> treal_eq (treal_mul (treal_inv x) x) treal_1"
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by (import realax TREAL_MUL_LINV)
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lemma TREAL_LT_TOTAL: "ALL (x::hreal * hreal) y::hreal * hreal.
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treal_eq x y | treal_lt x y | treal_lt y x"
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by (import realax TREAL_LT_TOTAL)
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lemma TREAL_LT_REFL: "ALL x::hreal * hreal. ~ treal_lt x x"
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by (import realax TREAL_LT_REFL)
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lemma TREAL_LT_TRANS: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
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treal_lt x y & treal_lt y z --> treal_lt x z"
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by (import realax TREAL_LT_TRANS)
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lemma TREAL_LT_ADD: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
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treal_lt y z --> treal_lt (treal_add x y) (treal_add x z)"
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by (import realax TREAL_LT_ADD)
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lemma TREAL_LT_MUL: "ALL (x::hreal * hreal) y::hreal * hreal.
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treal_lt treal_0 x & treal_lt treal_0 y -->
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treal_lt treal_0 (treal_mul x y)"
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by (import realax TREAL_LT_MUL)
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constdefs
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treal_of_hreal :: "hreal => hreal * hreal"
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"treal_of_hreal == %x::hreal. (hreal_add x hreal_1, hreal_1)"
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lemma treal_of_hreal: "ALL x::hreal. treal_of_hreal x = (hreal_add x hreal_1, hreal_1)"
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by (import realax treal_of_hreal)
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constdefs
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hreal_of_treal :: "hreal * hreal => hreal"
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"hreal_of_treal == %(x::hreal, y::hreal). SOME d::hreal. x = hreal_add y d"
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lemma hreal_of_treal: "ALL (x::hreal) y::hreal.
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hreal_of_treal (x, y) = (SOME d::hreal. x = hreal_add y d)"
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by (import realax hreal_of_treal)
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lemma TREAL_BIJ: "(ALL h::hreal. hreal_of_treal (treal_of_hreal h) = h) &
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(ALL r::hreal * hreal.
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treal_lt treal_0 r = treal_eq (treal_of_hreal (hreal_of_treal r)) r)"
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by (import realax TREAL_BIJ)
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lemma TREAL_ISO: "ALL (h::hreal) i::hreal.
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hreal_lt h i --> treal_lt (treal_of_hreal h) (treal_of_hreal i)"
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by (import realax TREAL_ISO)
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lemma TREAL_BIJ_WELLDEF: "ALL (h::hreal * hreal) i::hreal * hreal.
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treal_eq h i --> hreal_of_treal h = hreal_of_treal i"
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by (import realax TREAL_BIJ_WELLDEF)
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lemma TREAL_NEG_WELLDEF: "ALL (x1::hreal * hreal) x2::hreal * hreal.
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treal_eq x1 x2 --> treal_eq (treal_neg x1) (treal_neg x2)"
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by (import realax TREAL_NEG_WELLDEF)
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lemma TREAL_ADD_WELLDEFR: "ALL (x1::hreal * hreal) (x2::hreal * hreal) y::hreal * hreal.
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treal_eq x1 x2 --> treal_eq (treal_add x1 y) (treal_add x2 y)"
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by (import realax TREAL_ADD_WELLDEFR)
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lemma TREAL_ADD_WELLDEF: "ALL (x1::hreal * hreal) (x2::hreal * hreal) (y1::hreal * hreal)
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y2::hreal * hreal.
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treal_eq x1 x2 & treal_eq y1 y2 -->
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treal_eq (treal_add x1 y1) (treal_add x2 y2)"
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by (import realax TREAL_ADD_WELLDEF)
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lemma TREAL_MUL_WELLDEFR: "ALL (x1::hreal * hreal) (x2::hreal * hreal) y::hreal * hreal.
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treal_eq x1 x2 --> treal_eq (treal_mul x1 y) (treal_mul x2 y)"
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by (import realax TREAL_MUL_WELLDEFR)
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lemma TREAL_MUL_WELLDEF: "ALL (x1::hreal * hreal) (x2::hreal * hreal) (y1::hreal * hreal)
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y2::hreal * hreal.
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treal_eq x1 x2 & treal_eq y1 y2 -->
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treal_eq (treal_mul x1 y1) (treal_mul x2 y2)"
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by (import realax TREAL_MUL_WELLDEF)
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lemma TREAL_LT_WELLDEFR: "ALL (x1::hreal * hreal) (x2::hreal * hreal) y::hreal * hreal.
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treal_eq x1 x2 --> treal_lt x1 y = treal_lt x2 y"
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by (import realax TREAL_LT_WELLDEFR)
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lemma TREAL_LT_WELLDEFL: "ALL (x::hreal * hreal) (y1::hreal * hreal) y2::hreal * hreal.
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treal_eq y1 y2 --> treal_lt x y1 = treal_lt x y2"
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by (import realax TREAL_LT_WELLDEFL)
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lemma TREAL_LT_WELLDEF: "ALL (x1::hreal * hreal) (x2::hreal * hreal) (y1::hreal * hreal)
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y2::hreal * hreal.
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treal_eq x1 x2 & treal_eq y1 y2 --> treal_lt x1 y1 = treal_lt x2 y2"
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by (import realax TREAL_LT_WELLDEF)
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lemma TREAL_INV_WELLDEF: "ALL (x1::hreal * hreal) x2::hreal * hreal.
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treal_eq x1 x2 --> treal_eq (treal_inv x1) (treal_inv x2)"
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by (import realax TREAL_INV_WELLDEF)
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;end_setup
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;setup_theory real
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lemma REAL_0: "(op =::real => real => bool) (0::real) (0::real)"
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by (import real REAL_0)
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lemma REAL_1: "(op =::real => real => bool) (1::real) (1::real)"
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by (import real REAL_1)
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lemma REAL_ADD_LID_UNIQ: "ALL (x::real) y::real. (x + y = y) = (x = 0)"
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by (import real REAL_ADD_LID_UNIQ)
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lemma REAL_ADD_RID_UNIQ: "ALL (x::real) y::real. (x + y = x) = (y = 0)"
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by (import real REAL_ADD_RID_UNIQ)
|
|
281 |
|
17652
|
282 |
lemma REAL_LNEG_UNIQ: "ALL (x::real) y::real. (x + y = 0) = (x = - y)"
|
14516
|
283 |
by (import real REAL_LNEG_UNIQ)
|
|
284 |
|
|
285 |
lemma REAL_LT_ANTISYM: "ALL (x::real) y::real. ~ (x < y & y < x)"
|
|
286 |
by (import real REAL_LT_ANTISYM)
|
|
287 |
|
|
288 |
lemma REAL_LTE_TOTAL: "ALL (x::real) y::real. x < y | y <= x"
|
|
289 |
by (import real REAL_LTE_TOTAL)
|
|
290 |
|
|
291 |
lemma REAL_LET_ANTISYM: "ALL (x::real) y::real. ~ (x < y & y <= x)"
|
|
292 |
by (import real REAL_LET_ANTISYM)
|
|
293 |
|
|
294 |
lemma REAL_LTE_ANTSYM: "ALL (x::real) y::real. ~ (x <= y & y < x)"
|
|
295 |
by (import real REAL_LTE_ANTSYM)
|
|
296 |
|
17652
|
297 |
lemma REAL_LT_NEGTOTAL: "ALL x::real. x = 0 | 0 < x | 0 < - x"
|
14516
|
298 |
by (import real REAL_LT_NEGTOTAL)
|
|
299 |
|
17652
|
300 |
lemma REAL_LE_NEGTOTAL: "ALL x::real. 0 <= x | 0 <= - x"
|
14516
|
301 |
by (import real REAL_LE_NEGTOTAL)
|
|
302 |
|
|
303 |
lemma REAL_LT_ADDNEG: "ALL (x::real) (y::real) z::real. (y < x + - z) = (y + z < x)"
|
|
304 |
by (import real REAL_LT_ADDNEG)
|
|
305 |
|
|
306 |
lemma REAL_LT_ADDNEG2: "ALL (x::real) (y::real) z::real. (x + - y < z) = (x < z + y)"
|
|
307 |
by (import real REAL_LT_ADDNEG2)
|
|
308 |
|
17652
|
309 |
lemma REAL_LT_ADD1: "ALL (x::real) y::real. x <= y --> x < y + 1"
|
14516
|
310 |
by (import real REAL_LT_ADD1)
|
|
311 |
|
14796
|
312 |
lemma REAL_SUB_ADD2: "ALL (x::real) y::real. y + (x - y) = x"
|
|
313 |
by (import real REAL_SUB_ADD2)
|
|
314 |
|
17652
|
315 |
lemma REAL_SUB_LT: "ALL (x::real) y::real. (0 < x - y) = (y < x)"
|
15647
|
316 |
by (import real REAL_SUB_LT)
|
|
317 |
|
17652
|
318 |
lemma REAL_SUB_LE: "ALL (x::real) y::real. (0 <= x - y) = (y <= x)"
|
15647
|
319 |
by (import real REAL_SUB_LE)
|
|
320 |
|
14796
|
321 |
lemma REAL_ADD_SUB: "ALL (x::real) y::real. x + y - x = y"
|
|
322 |
by (import real REAL_ADD_SUB)
|
14516
|
323 |
|
|
324 |
lemma REAL_NEG_EQ: "ALL (x::real) y::real. (- x = y) = (x = - y)"
|
|
325 |
by (import real REAL_NEG_EQ)
|
|
326 |
|
17652
|
327 |
lemma REAL_NEG_MINUS1: "ALL x::real. - x = - 1 * x"
|
14516
|
328 |
by (import real REAL_NEG_MINUS1)
|
|
329 |
|
17652
|
330 |
lemma REAL_LT_LMUL_0: "ALL (x::real) y::real. 0 < x --> (0 < x * y) = (0 < y)"
|
14516
|
331 |
by (import real REAL_LT_LMUL_0)
|
|
332 |
|
17652
|
333 |
lemma REAL_LT_RMUL_0: "ALL (x::real) y::real. 0 < y --> (0 < x * y) = (0 < x)"
|
14516
|
334 |
by (import real REAL_LT_RMUL_0)
|
|
335 |
|
17652
|
336 |
lemma REAL_LT_LMUL: "ALL (x::real) (y::real) z::real. 0 < x --> (x * y < x * z) = (y < z)"
|
14516
|
337 |
by (import real REAL_LT_LMUL)
|
|
338 |
|
17652
|
339 |
lemma REAL_LINV_UNIQ: "ALL (x::real) y::real. x * y = 1 --> x = inverse y"
|
14516
|
340 |
by (import real REAL_LINV_UNIQ)
|
|
341 |
|
17652
|
342 |
lemma REAL_LE_INV: "(All::(real => bool) => bool)
|
|
343 |
(%x::real.
|
|
344 |
(op -->::bool => bool => bool)
|
|
345 |
((op <=::real => real => bool) (0::real) x)
|
|
346 |
((op <=::real => real => bool) (0::real) ((inverse::real => real) x)))"
|
14516
|
347 |
by (import real REAL_LE_INV)
|
|
348 |
|
17652
|
349 |
lemma REAL_LE_ADDR: "ALL (x::real) y::real. (x <= x + y) = (0 <= y)"
|
14516
|
350 |
by (import real REAL_LE_ADDR)
|
|
351 |
|
17652
|
352 |
lemma REAL_LE_ADDL: "ALL (x::real) y::real. (y <= x + y) = (0 <= x)"
|
14516
|
353 |
by (import real REAL_LE_ADDL)
|
|
354 |
|
17652
|
355 |
lemma REAL_LT_ADDR: "ALL (x::real) y::real. (x < x + y) = (0 < y)"
|
14516
|
356 |
by (import real REAL_LT_ADDR)
|
|
357 |
|
17652
|
358 |
lemma REAL_LT_ADDL: "ALL (x::real) y::real. (y < x + y) = (0 < x)"
|
14516
|
359 |
by (import real REAL_LT_ADDL)
|
|
360 |
|
17652
|
361 |
lemma REAL_LT_NZ: "ALL n::nat. (real n ~= 0) = (0 < real n)"
|
14516
|
362 |
by (import real REAL_LT_NZ)
|
|
363 |
|
17652
|
364 |
lemma REAL_NZ_IMP_LT: "ALL n::nat. n ~= 0 --> 0 < real n"
|
14516
|
365 |
by (import real REAL_NZ_IMP_LT)
|
|
366 |
|
17652
|
367 |
lemma REAL_LT_RDIV_0: "ALL (y::real) z::real. 0 < z --> (0 < y / z) = (0 < y)"
|
14516
|
368 |
by (import real REAL_LT_RDIV_0)
|
|
369 |
|
17652
|
370 |
lemma REAL_LT_RDIV: "ALL (x::real) (y::real) z::real. 0 < z --> (x / z < y / z) = (x < y)"
|
14516
|
371 |
by (import real REAL_LT_RDIV)
|
|
372 |
|
17652
|
373 |
lemma REAL_LT_FRACTION_0: "ALL (n::nat) d::real. n ~= 0 --> (0 < d / real n) = (0 < d)"
|
14516
|
374 |
by (import real REAL_LT_FRACTION_0)
|
|
375 |
|
17652
|
376 |
lemma REAL_LT_MULTIPLE: "ALL (x::nat) xa::real. 1 < x --> (xa < real x * xa) = (0 < xa)"
|
14516
|
377 |
by (import real REAL_LT_MULTIPLE)
|
|
378 |
|
17652
|
379 |
lemma REAL_LT_FRACTION: "ALL (n::nat) d::real. 1 < n --> (d / real n < d) = (0 < d)"
|
14516
|
380 |
by (import real REAL_LT_FRACTION)
|
|
381 |
|
17652
|
382 |
lemma REAL_LT_HALF2: "ALL d::real. (d / 2 < d) = (0 < d)"
|
14516
|
383 |
by (import real REAL_LT_HALF2)
|
|
384 |
|
17652
|
385 |
lemma REAL_DIV_LMUL: "ALL (x::real) y::real. y ~= 0 --> y * (x / y) = x"
|
14516
|
386 |
by (import real REAL_DIV_LMUL)
|
|
387 |
|
17652
|
388 |
lemma REAL_DIV_RMUL: "ALL (x::real) y::real. y ~= 0 --> x / y * y = x"
|
14516
|
389 |
by (import real REAL_DIV_RMUL)
|
|
390 |
|
17652
|
391 |
lemma REAL_DOWN: "(All::(real => bool) => bool)
|
|
392 |
(%x::real.
|
|
393 |
(op -->::bool => bool => bool)
|
|
394 |
((op <::real => real => bool) (0::real) x)
|
|
395 |
((Ex::(real => bool) => bool)
|
|
396 |
(%xa::real.
|
|
397 |
(op &::bool => bool => bool)
|
|
398 |
((op <::real => real => bool) (0::real) xa)
|
|
399 |
((op <::real => real => bool) xa x))))"
|
14516
|
400 |
by (import real REAL_DOWN)
|
|
401 |
|
|
402 |
lemma REAL_SUB_SUB: "ALL (x::real) y::real. x - y - x = - y"
|
|
403 |
by (import real REAL_SUB_SUB)
|
|
404 |
|
|
405 |
lemma REAL_ADD2_SUB2: "ALL (a::real) (b::real) (c::real) d::real. a + b - (c + d) = a - c + (b - d)"
|
|
406 |
by (import real REAL_ADD2_SUB2)
|
|
407 |
|
|
408 |
lemma REAL_SUB_LNEG: "ALL (x::real) y::real. - x - y = - (x + y)"
|
|
409 |
by (import real REAL_SUB_LNEG)
|
|
410 |
|
|
411 |
lemma REAL_SUB_NEG2: "ALL (x::real) y::real. - x - - y = y - x"
|
|
412 |
by (import real REAL_SUB_NEG2)
|
|
413 |
|
|
414 |
lemma REAL_SUB_TRIANGLE: "ALL (a::real) (b::real) c::real. a - b + (b - c) = a - c"
|
|
415 |
by (import real REAL_SUB_TRIANGLE)
|
|
416 |
|
|
417 |
lemma REAL_INV_MUL: "ALL (x::real) y::real.
|
17652
|
418 |
x ~= 0 & y ~= 0 --> inverse (x * y) = inverse x * inverse y"
|
14516
|
419 |
by (import real REAL_INV_MUL)
|
|
420 |
|
|
421 |
lemma REAL_SUB_INV2: "ALL (x::real) y::real.
|
17652
|
422 |
x ~= 0 & y ~= 0 --> inverse x - inverse y = (y - x) / (x * y)"
|
14516
|
423 |
by (import real REAL_SUB_INV2)
|
|
424 |
|
|
425 |
lemma REAL_SUB_SUB2: "ALL (x::real) y::real. x - (x - y) = y"
|
|
426 |
by (import real REAL_SUB_SUB2)
|
|
427 |
|
|
428 |
lemma REAL_ADD_SUB2: "ALL (x::real) y::real. x - (x + y) = - y"
|
|
429 |
by (import real REAL_ADD_SUB2)
|
|
430 |
|
|
431 |
lemma REAL_LE_MUL2: "ALL (x1::real) (x2::real) (y1::real) y2::real.
|
17652
|
432 |
0 <= x1 & 0 <= y1 & x1 <= x2 & y1 <= y2 --> x1 * y1 <= x2 * y2"
|
14516
|
433 |
by (import real REAL_LE_MUL2)
|
|
434 |
|
17652
|
435 |
lemma REAL_LE_DIV: "ALL (x::real) xa::real. 0 <= x & 0 <= xa --> 0 <= x / xa"
|
14516
|
436 |
by (import real REAL_LE_DIV)
|
|
437 |
|
17652
|
438 |
lemma REAL_LT_1: "ALL (x::real) y::real. 0 <= x & x < y --> x / y < 1"
|
14516
|
439 |
by (import real REAL_LT_1)
|
|
440 |
|
17652
|
441 |
lemma REAL_POS_NZ: "(All::(real => bool) => bool)
|
|
442 |
(%x::real.
|
|
443 |
(op -->::bool => bool => bool)
|
|
444 |
((op <::real => real => bool) (0::real) x)
|
|
445 |
((Not::bool => bool) ((op =::real => real => bool) x (0::real))))"
|
14516
|
446 |
by (import real REAL_POS_NZ)
|
|
447 |
|
17652
|
448 |
lemma REAL_EQ_LMUL_IMP: "ALL (x::real) (xa::real) xb::real. x ~= 0 & x * xa = x * xb --> xa = xb"
|
14516
|
449 |
by (import real REAL_EQ_LMUL_IMP)
|
|
450 |
|
17652
|
451 |
lemma REAL_FACT_NZ: "ALL n::nat. real (FACT n) ~= 0"
|
14516
|
452 |
by (import real REAL_FACT_NZ)
|
|
453 |
|
|
454 |
lemma REAL_DIFFSQ: "ALL (x::real) y::real. (x + y) * (x - y) = x * x - y * y"
|
|
455 |
by (import real REAL_DIFFSQ)
|
|
456 |
|
17652
|
457 |
lemma REAL_POASQ: "ALL x::real. (0 < x * x) = (x ~= 0)"
|
14516
|
458 |
by (import real REAL_POASQ)
|
|
459 |
|
17652
|
460 |
lemma REAL_SUMSQ: "ALL (x::real) y::real. (x * x + y * y = 0) = (x = 0 & y = 0)"
|
14516
|
461 |
by (import real REAL_SUMSQ)
|
|
462 |
|
17188
|
463 |
lemma REAL_DIV_MUL2: "ALL (x::real) z::real.
|
17652
|
464 |
x ~= 0 & z ~= 0 --> (ALL y::real. y / z = x * y / (x * z))"
|
17188
|
465 |
by (import real REAL_DIV_MUL2)
|
|
466 |
|
17652
|
467 |
lemma REAL_MIDDLE1: "ALL (a::real) b::real. a <= b --> a <= (a + b) / 2"
|
14516
|
468 |
by (import real REAL_MIDDLE1)
|
|
469 |
|
17652
|
470 |
lemma REAL_MIDDLE2: "ALL (a::real) b::real. a <= b --> (a + b) / 2 <= b"
|
14516
|
471 |
by (import real REAL_MIDDLE2)
|
|
472 |
|
|
473 |
lemma ABS_LT_MUL2: "ALL (w::real) (x::real) (y::real) z::real.
|
|
474 |
abs w < y & abs x < z --> abs (w * x) < y * z"
|
|
475 |
by (import real ABS_LT_MUL2)
|
|
476 |
|
17652
|
477 |
lemma ABS_REFL: "ALL x::real. (abs x = x) = (0 <= x)"
|
14516
|
478 |
by (import real ABS_REFL)
|
|
479 |
|
|
480 |
lemma ABS_BETWEEN: "ALL (x::real) (y::real) d::real.
|
17652
|
481 |
(0 < d & x - d < y & y < x + d) = (abs (y - x) < d)"
|
14516
|
482 |
by (import real ABS_BETWEEN)
|
|
483 |
|
|
484 |
lemma ABS_BOUND: "ALL (x::real) (y::real) d::real. abs (x - y) < d --> y < x + d"
|
|
485 |
by (import real ABS_BOUND)
|
|
486 |
|
17652
|
487 |
lemma ABS_STILLNZ: "ALL (x::real) y::real. abs (x - y) < abs y --> x ~= 0"
|
14516
|
488 |
by (import real ABS_STILLNZ)
|
|
489 |
|
17652
|
490 |
lemma ABS_CASES: "ALL x::real. x = 0 | 0 < abs x"
|
14516
|
491 |
by (import real ABS_CASES)
|
|
492 |
|
|
493 |
lemma ABS_BETWEEN1: "ALL (x::real) (y::real) z::real. x < z & abs (y - x) < z - x --> y < z"
|
|
494 |
by (import real ABS_BETWEEN1)
|
|
495 |
|
17652
|
496 |
lemma ABS_SIGN: "ALL (x::real) y::real. abs (x - y) < y --> 0 < x"
|
14516
|
497 |
by (import real ABS_SIGN)
|
|
498 |
|
17652
|
499 |
lemma ABS_SIGN2: "ALL (x::real) y::real. abs (x - y) < - y --> x < 0"
|
14516
|
500 |
by (import real ABS_SIGN2)
|
|
501 |
|
|
502 |
lemma ABS_CIRCLE: "ALL (x::real) (y::real) h::real.
|
|
503 |
abs h < abs y - abs x --> abs (x + h) < abs y"
|
|
504 |
by (import real ABS_CIRCLE)
|
|
505 |
|
|
506 |
lemma ABS_BETWEEN2: "ALL (x0::real) (x::real) (y0::real) y::real.
|
17652
|
507 |
x0 < y0 & abs (x - x0) < (y0 - x0) / 2 & abs (y - y0) < (y0 - x0) / 2 -->
|
14516
|
508 |
x < y"
|
|
509 |
by (import real ABS_BETWEEN2)
|
|
510 |
|
17652
|
511 |
lemma POW_PLUS1: "ALL e>0. ALL n::nat. 1 + real n * e <= (1 + e) ^ n"
|
14516
|
512 |
by (import real POW_PLUS1)
|
|
513 |
|
17652
|
514 |
lemma POW_M1: "(All::(nat => bool) => bool)
|
|
515 |
(%n::nat.
|
|
516 |
(op =::real => real => bool)
|
|
517 |
((abs::real => real)
|
|
518 |
((op ^::real => nat => real) ((uminus::real => real) (1::real)) n))
|
|
519 |
(1::real))"
|
14516
|
520 |
by (import real POW_M1)
|
|
521 |
|
17652
|
522 |
lemma REAL_LE1_POW2: "(All::(real => bool) => bool)
|
|
523 |
(%x::real.
|
|
524 |
(op -->::bool => bool => bool)
|
|
525 |
((op <=::real => real => bool) (1::real) x)
|
|
526 |
((op <=::real => real => bool) (1::real)
|
|
527 |
((op ^::real => nat => real) x
|
20485
|
528 |
((number_of \<Colon> int => nat)
|
|
529 |
((op BIT \<Colon> int => bit => int)
|
|
530 |
((op BIT \<Colon> int => bit => int) (Numeral.Pls \<Colon> int) (bit.B1::bit))
|
17652
|
531 |
(bit.B0::bit))))))"
|
14516
|
532 |
by (import real REAL_LE1_POW2)
|
|
533 |
|
17652
|
534 |
lemma REAL_LT1_POW2: "(All::(real => bool) => bool)
|
|
535 |
(%x::real.
|
|
536 |
(op -->::bool => bool => bool)
|
|
537 |
((op <::real => real => bool) (1::real) x)
|
|
538 |
((op <::real => real => bool) (1::real)
|
|
539 |
((op ^::real => nat => real) x
|
20485
|
540 |
((number_of \<Colon> int => nat)
|
|
541 |
((op BIT \<Colon> int => bit => int)
|
|
542 |
((op BIT \<Colon> int => bit => int) (Numeral.Pls \<Colon> int) (bit.B1::bit))
|
17652
|
543 |
(bit.B0::bit))))))"
|
14516
|
544 |
by (import real REAL_LT1_POW2)
|
|
545 |
|
17652
|
546 |
lemma POW_POS_LT: "ALL (x::real) n::nat. 0 < x --> 0 < x ^ Suc n"
|
14516
|
547 |
by (import real POW_POS_LT)
|
|
548 |
|
17652
|
549 |
lemma POW_LT: "ALL (n::nat) (x::real) y::real. 0 <= x & x < y --> x ^ Suc n < y ^ Suc n"
|
14516
|
550 |
by (import real POW_LT)
|
|
551 |
|
17652
|
552 |
lemma POW_ZERO_EQ: "ALL (n::nat) x::real. (x ^ Suc n = 0) = (x = 0)"
|
14516
|
553 |
by (import real POW_ZERO_EQ)
|
|
554 |
|
17652
|
555 |
lemma REAL_POW_LT2: "ALL (n::nat) (x::real) y::real. n ~= 0 & 0 <= x & x < y --> x ^ n < y ^ n"
|
14516
|
556 |
by (import real REAL_POW_LT2)
|
|
557 |
|
17652
|
558 |
lemma REAL_POW_MONO_LT: "ALL (m::nat) (n::nat) x::real. 1 < x & m < n --> x ^ m < x ^ n"
|
17188
|
559 |
by (import real REAL_POW_MONO_LT)
|
|
560 |
|
14516
|
561 |
lemma REAL_SUP_SOMEPOS: "ALL P::real => bool.
|
17652
|
562 |
(EX x::real. P x & 0 < x) & (EX z::real. ALL x::real. P x --> x < z) -->
|
14516
|
563 |
(EX s::real. ALL y::real. (EX x::real. P x & y < x) = (y < s))"
|
|
564 |
by (import real REAL_SUP_SOMEPOS)
|
|
565 |
|
|
566 |
lemma SUP_LEMMA1: "ALL (P::real => bool) (s::real) d::real.
|
|
567 |
(ALL y::real. (EX x::real. P (x + d) & y < x) = (y < s)) -->
|
|
568 |
(ALL y::real. (EX x::real. P x & y < x) = (y < s + d))"
|
|
569 |
by (import real SUP_LEMMA1)
|
|
570 |
|
17652
|
571 |
lemma SUP_LEMMA2: "ALL P::real => bool. Ex P --> (EX (d::real) x::real. P (x + d) & 0 < x)"
|
14516
|
572 |
by (import real SUP_LEMMA2)
|
|
573 |
|
|
574 |
lemma SUP_LEMMA3: "ALL d::real.
|
|
575 |
(EX z::real. ALL x::real. (P::real => bool) x --> x < z) -->
|
|
576 |
(EX x::real. ALL xa::real. P (xa + d) --> xa < x)"
|
|
577 |
by (import real SUP_LEMMA3)
|
|
578 |
|
|
579 |
lemma REAL_SUP_EXISTS: "ALL P::real => bool.
|
|
580 |
Ex P & (EX z::real. ALL x::real. P x --> x < z) -->
|
|
581 |
(EX x::real. ALL y::real. (EX x::real. P x & y < x) = (y < x))"
|
|
582 |
by (import real REAL_SUP_EXISTS)
|
|
583 |
|
|
584 |
constdefs
|
|
585 |
sup :: "(real => bool) => real"
|
17644
|
586 |
"sup ==
|
|
587 |
%P::real => bool.
|
|
588 |
SOME s::real. ALL y::real. (EX x::real. P x & y < x) = (y < s)"
|
|
589 |
|
|
590 |
lemma sup: "ALL P::real => bool.
|
|
591 |
sup P = (SOME s::real. ALL y::real. (EX x::real. P x & y < x) = (y < s))"
|
14516
|
592 |
by (import real sup)
|
|
593 |
|
17644
|
594 |
lemma REAL_SUP: "ALL P::real => bool.
|
|
595 |
Ex P & (EX z::real. ALL x::real. P x --> x < z) -->
|
|
596 |
(ALL y::real. (EX x::real. P x & y < x) = (y < sup P))"
|
14516
|
597 |
by (import real REAL_SUP)
|
|
598 |
|
17644
|
599 |
lemma REAL_SUP_UBOUND: "ALL P::real => bool.
|
|
600 |
Ex P & (EX z::real. ALL x::real. P x --> x < z) -->
|
|
601 |
(ALL y::real. P y --> y <= sup P)"
|
14516
|
602 |
by (import real REAL_SUP_UBOUND)
|
|
603 |
|
|
604 |
lemma SETOK_LE_LT: "ALL P::real => bool.
|
|
605 |
(Ex P & (EX z::real. ALL x::real. P x --> x <= z)) =
|
|
606 |
(Ex P & (EX z::real. ALL x::real. P x --> x < z))"
|
|
607 |
by (import real SETOK_LE_LT)
|
|
608 |
|
17644
|
609 |
lemma REAL_SUP_LE: "ALL P::real => bool.
|
|
610 |
Ex P & (EX z::real. ALL x::real. P x --> x <= z) -->
|
|
611 |
(ALL y::real. (EX x::real. P x & y < x) = (y < sup P))"
|
14516
|
612 |
by (import real REAL_SUP_LE)
|
|
613 |
|
17644
|
614 |
lemma REAL_SUP_UBOUND_LE: "ALL P::real => bool.
|
|
615 |
Ex P & (EX z::real. ALL x::real. P x --> x <= z) -->
|
|
616 |
(ALL y::real. P y --> y <= sup P)"
|
14516
|
617 |
by (import real REAL_SUP_UBOUND_LE)
|
|
618 |
|
17652
|
619 |
lemma REAL_ARCH_LEAST: "ALL y>0. ALL x>=0. EX n::nat. real n * y <= x & x < real (Suc n) * y"
|
14516
|
620 |
by (import real REAL_ARCH_LEAST)
|
|
621 |
|
|
622 |
consts
|
|
623 |
sumc :: "nat => nat => (nat => real) => real"
|
|
624 |
|
17652
|
625 |
specification (sumc) sumc: "(ALL (n::nat) f::nat => real. sumc n 0 f = 0) &
|
17644
|
626 |
(ALL (n::nat) (m::nat) f::nat => real.
|
|
627 |
sumc n (Suc m) f = sumc n m f + f (n + m))"
|
14516
|
628 |
by (import real sumc)
|
|
629 |
|
14694
|
630 |
consts
|
14516
|
631 |
sum :: "nat * nat => (nat => real) => real"
|
14694
|
632 |
|
|
633 |
defs
|
15647
|
634 |
sum_def: "(op ==::(nat * nat => (nat => real) => real)
|
|
635 |
=> (nat * nat => (nat => real) => real) => prop)
|
|
636 |
(real.sum::nat * nat => (nat => real) => real)
|
|
637 |
((split::(nat => nat => (nat => real) => real)
|
|
638 |
=> nat * nat => (nat => real) => real)
|
|
639 |
(sumc::nat => nat => (nat => real) => real))"
|
14516
|
640 |
|
17644
|
641 |
lemma SUM_DEF: "ALL (m::nat) (n::nat) f::nat => real. real.sum (m, n) f = sumc m n f"
|
14516
|
642 |
by (import real SUM_DEF)
|
|
643 |
|
17644
|
644 |
lemma sum: "ALL (x::nat => real) (xa::nat) xb::nat.
|
17652
|
645 |
real.sum (xa, 0) x = 0 &
|
14516
|
646 |
real.sum (xa, Suc xb) x = real.sum (xa, xb) x + x (xa + xb)"
|
|
647 |
by (import real sum)
|
|
648 |
|
17644
|
649 |
lemma SUM_TWO: "ALL (f::nat => real) (n::nat) p::nat.
|
17652
|
650 |
real.sum (0, n) f + real.sum (n, p) f = real.sum (0, n + p) f"
|
14516
|
651 |
by (import real SUM_TWO)
|
|
652 |
|
17644
|
653 |
lemma SUM_DIFF: "ALL (f::nat => real) (m::nat) n::nat.
|
17652
|
654 |
real.sum (m, n) f = real.sum (0, m + n) f - real.sum (0, m) f"
|
14516
|
655 |
by (import real SUM_DIFF)
|
|
656 |
|
17644
|
657 |
lemma ABS_SUM: "ALL (f::nat => real) (m::nat) n::nat.
|
|
658 |
abs (real.sum (m, n) f) <= real.sum (m, n) (%n::nat. abs (f n))"
|
14516
|
659 |
by (import real ABS_SUM)
|
|
660 |
|
17644
|
661 |
lemma SUM_LE: "ALL (f::nat => real) (g::nat => real) (m::nat) n::nat.
|
|
662 |
(ALL r::nat. m <= r & r < n + m --> f r <= g r) -->
|
14516
|
663 |
real.sum (m, n) f <= real.sum (m, n) g"
|
|
664 |
by (import real SUM_LE)
|
|
665 |
|
17644
|
666 |
lemma SUM_EQ: "ALL (f::nat => real) (g::nat => real) (m::nat) n::nat.
|
|
667 |
(ALL r::nat. m <= r & r < n + m --> f r = g r) -->
|
14516
|
668 |
real.sum (m, n) f = real.sum (m, n) g"
|
|
669 |
by (import real SUM_EQ)
|
|
670 |
|
17644
|
671 |
lemma SUM_POS: "ALL f::nat => real.
|
17652
|
672 |
(ALL n::nat. 0 <= f n) --> (ALL (m::nat) n::nat. 0 <= real.sum (m, n) f)"
|
14516
|
673 |
by (import real SUM_POS)
|
|
674 |
|
17644
|
675 |
lemma SUM_POS_GEN: "ALL (f::nat => real) m::nat.
|
17652
|
676 |
(ALL n::nat. m <= n --> 0 <= f n) -->
|
|
677 |
(ALL n::nat. 0 <= real.sum (m, n) f)"
|
14516
|
678 |
by (import real SUM_POS_GEN)
|
|
679 |
|
17644
|
680 |
lemma SUM_ABS: "ALL (f::nat => real) (m::nat) x::nat.
|
|
681 |
abs (real.sum (m, x) (%m::nat. abs (f m))) =
|
|
682 |
real.sum (m, x) (%m::nat. abs (f m))"
|
14516
|
683 |
by (import real SUM_ABS)
|
|
684 |
|
17644
|
685 |
lemma SUM_ABS_LE: "ALL (f::nat => real) (m::nat) n::nat.
|
|
686 |
abs (real.sum (m, n) f) <= real.sum (m, n) (%n::nat. abs (f n))"
|
14516
|
687 |
by (import real SUM_ABS_LE)
|
|
688 |
|
17644
|
689 |
lemma SUM_ZERO: "ALL (f::nat => real) N::nat.
|
17652
|
690 |
(ALL n::nat. N <= n --> f n = 0) -->
|
|
691 |
(ALL (m::nat) n::nat. N <= m --> real.sum (m, n) f = 0)"
|
14516
|
692 |
by (import real SUM_ZERO)
|
|
693 |
|
17644
|
694 |
lemma SUM_ADD: "ALL (f::nat => real) (g::nat => real) (m::nat) n::nat.
|
|
695 |
real.sum (m, n) (%n::nat. f n + g n) =
|
|
696 |
real.sum (m, n) f + real.sum (m, n) g"
|
14516
|
697 |
by (import real SUM_ADD)
|
|
698 |
|
17644
|
699 |
lemma SUM_CMUL: "ALL (f::nat => real) (c::real) (m::nat) n::nat.
|
|
700 |
real.sum (m, n) (%n::nat. c * f n) = c * real.sum (m, n) f"
|
14516
|
701 |
by (import real SUM_CMUL)
|
|
702 |
|
17644
|
703 |
lemma SUM_NEG: "ALL (f::nat => real) (n::nat) d::nat.
|
|
704 |
real.sum (n, d) (%n::nat. - f n) = - real.sum (n, d) f"
|
14516
|
705 |
by (import real SUM_NEG)
|
|
706 |
|
17644
|
707 |
lemma SUM_SUB: "ALL (f::nat => real) (g::nat => real) (m::nat) n::nat.
|
|
708 |
real.sum (m, n) (%x::nat. f x - g x) =
|
|
709 |
real.sum (m, n) f - real.sum (m, n) g"
|
14516
|
710 |
by (import real SUM_SUB)
|
|
711 |
|
17644
|
712 |
lemma SUM_SUBST: "ALL (f::nat => real) (g::nat => real) (m::nat) n::nat.
|
|
713 |
(ALL p::nat. m <= p & p < m + n --> f p = g p) -->
|
14516
|
714 |
real.sum (m, n) f = real.sum (m, n) g"
|
|
715 |
by (import real SUM_SUBST)
|
|
716 |
|
17644
|
717 |
lemma SUM_NSUB: "ALL (n::nat) (f::nat => real) c::real.
|
17652
|
718 |
real.sum (0, n) f - real n * c = real.sum (0, n) (%p::nat. f p - c)"
|
14516
|
719 |
by (import real SUM_NSUB)
|
|
720 |
|
17644
|
721 |
lemma SUM_BOUND: "ALL (f::nat => real) (k::real) (m::nat) n::nat.
|
|
722 |
(ALL p::nat. m <= p & p < m + n --> f p <= k) -->
|
14516
|
723 |
real.sum (m, n) f <= real n * k"
|
|
724 |
by (import real SUM_BOUND)
|
|
725 |
|
17644
|
726 |
lemma SUM_GROUP: "ALL (n::nat) (k::nat) f::nat => real.
|
17652
|
727 |
real.sum (0, n) (%m::nat. real.sum (m * k, k) f) = real.sum (0, n * k) f"
|
14516
|
728 |
by (import real SUM_GROUP)
|
|
729 |
|
17652
|
730 |
lemma SUM_1: "ALL (f::nat => real) n::nat. real.sum (n, 1) f = f n"
|
14516
|
731 |
by (import real SUM_1)
|
|
732 |
|
17652
|
733 |
lemma SUM_2: "ALL (f::nat => real) n::nat. real.sum (n, 2) f = f n + f (n + 1)"
|
14516
|
734 |
by (import real SUM_2)
|
|
735 |
|
17644
|
736 |
lemma SUM_OFFSET: "ALL (f::nat => real) (n::nat) k::nat.
|
17652
|
737 |
real.sum (0, n) (%m::nat. f (m + k)) =
|
|
738 |
real.sum (0, n + k) f - real.sum (0, k) f"
|
14516
|
739 |
by (import real SUM_OFFSET)
|
|
740 |
|
17644
|
741 |
lemma SUM_REINDEX: "ALL (f::nat => real) (m::nat) (k::nat) n::nat.
|
|
742 |
real.sum (m + k, n) f = real.sum (m, n) (%r::nat. f (r + k))"
|
14516
|
743 |
by (import real SUM_REINDEX)
|
|
744 |
|
17652
|
745 |
lemma SUM_0: "ALL (m::nat) n::nat. real.sum (m, n) (%r::nat. 0) = 0"
|
14516
|
746 |
by (import real SUM_0)
|
|
747 |
|
14847
|
748 |
lemma SUM_PERMUTE_0: "(All::(nat => bool) => bool)
|
|
749 |
(%n::nat.
|
|
750 |
(All::((nat => nat) => bool) => bool)
|
|
751 |
(%p::nat => nat.
|
|
752 |
(op -->::bool => bool => bool)
|
|
753 |
((All::(nat => bool) => bool)
|
|
754 |
(%y::nat.
|
|
755 |
(op -->::bool => bool => bool)
|
|
756 |
((op <::nat => nat => bool) y n)
|
|
757 |
((Ex1::(nat => bool) => bool)
|
|
758 |
(%x::nat.
|
|
759 |
(op &::bool => bool => bool)
|
|
760 |
((op <::nat => nat => bool) x n)
|
|
761 |
((op =::nat => nat => bool) (p x) y)))))
|
|
762 |
((All::((nat => real) => bool) => bool)
|
|
763 |
(%f::nat => real.
|
|
764 |
(op =::real => real => bool)
|
|
765 |
((real.sum::nat * nat => (nat => real) => real)
|
|
766 |
((Pair::nat => nat => nat * nat) (0::nat) n)
|
|
767 |
(%n::nat. f (p n)))
|
|
768 |
((real.sum::nat * nat => (nat => real) => real)
|
|
769 |
((Pair::nat => nat => nat * nat) (0::nat) n) f)))))"
|
14516
|
770 |
by (import real SUM_PERMUTE_0)
|
|
771 |
|
17644
|
772 |
lemma SUM_CANCEL: "ALL (f::nat => real) (n::nat) d::nat.
|
|
773 |
real.sum (n, d) (%n::nat. f (Suc n) - f n) = f (n + d) - f n"
|
14516
|
774 |
by (import real SUM_CANCEL)
|
|
775 |
|
17652
|
776 |
lemma REAL_EQ_RDIV_EQ: "ALL (x::real) (xa::real) xb::real. 0 < xb --> (x = xa / xb) = (x * xb = xa)"
|
14516
|
777 |
by (import real REAL_EQ_RDIV_EQ)
|
|
778 |
|
17652
|
779 |
lemma REAL_EQ_LDIV_EQ: "ALL (x::real) (xa::real) xb::real. 0 < xb --> (x / xb = xa) = (x = xa * xb)"
|
14516
|
780 |
by (import real REAL_EQ_LDIV_EQ)
|
|
781 |
|
|
782 |
;end_setup
|
|
783 |
|
|
784 |
;setup_theory topology
|
|
785 |
|
|
786 |
constdefs
|
17652
|
787 |
re_Union :: "(('a => bool) => bool) => 'a => bool"
|
17644
|
788 |
"re_Union ==
|
|
789 |
%(P::('a::type => bool) => bool) x::'a::type.
|
|
790 |
EX s::'a::type => bool. P s & s x"
|
|
791 |
|
|
792 |
lemma re_Union: "ALL P::('a::type => bool) => bool.
|
|
793 |
re_Union P = (%x::'a::type. EX s::'a::type => bool. P s & s x)"
|
14516
|
794 |
by (import topology re_Union)
|
|
795 |
|
|
796 |
constdefs
|
17652
|
797 |
re_union :: "('a => bool) => ('a => bool) => 'a => bool"
|
17644
|
798 |
"re_union ==
|
|
799 |
%(P::'a::type => bool) (Q::'a::type => bool) x::'a::type. P x | Q x"
|
|
800 |
|
|
801 |
lemma re_union: "ALL (P::'a::type => bool) Q::'a::type => bool.
|
|
802 |
re_union P Q = (%x::'a::type. P x | Q x)"
|
14516
|
803 |
by (import topology re_union)
|
|
804 |
|
|
805 |
constdefs
|
17652
|
806 |
re_intersect :: "('a => bool) => ('a => bool) => 'a => bool"
|
17644
|
807 |
"re_intersect ==
|
|
808 |
%(P::'a::type => bool) (Q::'a::type => bool) x::'a::type. P x & Q x"
|
|
809 |
|
|
810 |
lemma re_intersect: "ALL (P::'a::type => bool) Q::'a::type => bool.
|
|
811 |
re_intersect P Q = (%x::'a::type. P x & Q x)"
|
14516
|
812 |
by (import topology re_intersect)
|
|
813 |
|
|
814 |
constdefs
|
17652
|
815 |
re_null :: "'a => bool"
|
17644
|
816 |
"re_null == %x::'a::type. False"
|
|
817 |
|
|
818 |
lemma re_null: "re_null = (%x::'a::type. False)"
|
14516
|
819 |
by (import topology re_null)
|
|
820 |
|
|
821 |
constdefs
|
17652
|
822 |
re_universe :: "'a => bool"
|
17644
|
823 |
"re_universe == %x::'a::type. True"
|
|
824 |
|
|
825 |
lemma re_universe: "re_universe = (%x::'a::type. True)"
|
14516
|
826 |
by (import topology re_universe)
|
|
827 |
|
|
828 |
constdefs
|
17652
|
829 |
re_subset :: "('a => bool) => ('a => bool) => bool"
|
17644
|
830 |
"re_subset ==
|
|
831 |
%(P::'a::type => bool) Q::'a::type => bool. ALL x::'a::type. P x --> Q x"
|
|
832 |
|
|
833 |
lemma re_subset: "ALL (P::'a::type => bool) Q::'a::type => bool.
|
|
834 |
re_subset P Q = (ALL x::'a::type. P x --> Q x)"
|
14516
|
835 |
by (import topology re_subset)
|
|
836 |
|
|
837 |
constdefs
|
17652
|
838 |
re_compl :: "('a => bool) => 'a => bool"
|
17644
|
839 |
"re_compl == %(P::'a::type => bool) x::'a::type. ~ P x"
|
|
840 |
|
|
841 |
lemma re_compl: "ALL P::'a::type => bool. re_compl P = (%x::'a::type. ~ P x)"
|
14516
|
842 |
by (import topology re_compl)
|
|
843 |
|
17644
|
844 |
lemma SUBSET_REFL: "ALL P::'a::type => bool. re_subset P P"
|
14516
|
845 |
by (import topology SUBSET_REFL)
|
|
846 |
|
17644
|
847 |
lemma COMPL_MEM: "ALL (P::'a::type => bool) x::'a::type. P x = (~ re_compl P x)"
|
14516
|
848 |
by (import topology COMPL_MEM)
|
|
849 |
|
17644
|
850 |
lemma SUBSET_ANTISYM: "ALL (P::'a::type => bool) Q::'a::type => bool.
|
|
851 |
(re_subset P Q & re_subset Q P) = (P = Q)"
|
14516
|
852 |
by (import topology SUBSET_ANTISYM)
|
|
853 |
|
17644
|
854 |
lemma SUBSET_TRANS: "ALL (P::'a::type => bool) (Q::'a::type => bool) R::'a::type => bool.
|
|
855 |
re_subset P Q & re_subset Q R --> re_subset P R"
|
14516
|
856 |
by (import topology SUBSET_TRANS)
|
|
857 |
|
|
858 |
constdefs
|
17652
|
859 |
istopology :: "(('a => bool) => bool) => bool"
|
14516
|
860 |
"istopology ==
|
17644
|
861 |
%L::('a::type => bool) => bool.
|
|
862 |
L re_null &
|
|
863 |
L re_universe &
|
|
864 |
(ALL (a::'a::type => bool) b::'a::type => bool.
|
|
865 |
L a & L b --> L (re_intersect a b)) &
|
|
866 |
(ALL P::('a::type => bool) => bool. re_subset P L --> L (re_Union P))"
|
|
867 |
|
|
868 |
lemma istopology: "ALL L::('a::type => bool) => bool.
|
14516
|
869 |
istopology L =
|
|
870 |
(L re_null &
|
|
871 |
L re_universe &
|
17644
|
872 |
(ALL (a::'a::type => bool) b::'a::type => bool.
|
|
873 |
L a & L b --> L (re_intersect a b)) &
|
|
874 |
(ALL P::('a::type => bool) => bool. re_subset P L --> L (re_Union P)))"
|
14516
|
875 |
by (import topology istopology)
|
|
876 |
|
17644
|
877 |
typedef (open) ('a) topology = "(Collect::((('a::type => bool) => bool) => bool)
|
|
878 |
=> (('a::type => bool) => bool) set)
|
|
879 |
(istopology::(('a::type => bool) => bool) => bool)"
|
14516
|
880 |
by (rule typedef_helper,import topology topology_TY_DEF)
|
|
881 |
|
|
882 |
lemmas topology_TY_DEF = typedef_hol2hol4 [OF type_definition_topology]
|
|
883 |
|
|
884 |
consts
|
17652
|
885 |
topology :: "(('a => bool) => bool) => 'a topology"
|
|
886 |
"open" :: "'a topology => ('a => bool) => bool"
|
17644
|
887 |
|
|
888 |
specification ("open" topology) topology_tybij: "(ALL a::'a::type topology. topology (open a) = a) &
|
|
889 |
(ALL r::('a::type => bool) => bool. istopology r = (open (topology r) = r))"
|
14516
|
890 |
by (import topology topology_tybij)
|
|
891 |
|
17644
|
892 |
lemma TOPOLOGY: "ALL L::'a::type topology.
|
14516
|
893 |
open L re_null &
|
|
894 |
open L re_universe &
|
17644
|
895 |
(ALL (a::'a::type => bool) b::'a::type => bool.
|
|
896 |
open L a & open L b --> open L (re_intersect a b)) &
|
|
897 |
(ALL P::('a::type => bool) => bool.
|
|
898 |
re_subset P (open L) --> open L (re_Union P))"
|
14516
|
899 |
by (import topology TOPOLOGY)
|
|
900 |
|
17644
|
901 |
lemma TOPOLOGY_UNION: "ALL (x::'a::type topology) xa::('a::type => bool) => bool.
|
|
902 |
re_subset xa (open x) --> open x (re_Union xa)"
|
14516
|
903 |
by (import topology TOPOLOGY_UNION)
|
|
904 |
|
|
905 |
constdefs
|
17652
|
906 |
neigh :: "'a topology => ('a => bool) * 'a => bool"
|
17644
|
907 |
"neigh ==
|
|
908 |
%(top::'a::type topology) (N::'a::type => bool, x::'a::type).
|
|
909 |
EX P::'a::type => bool. open top P & re_subset P N & P x"
|
|
910 |
|
|
911 |
lemma neigh: "ALL (top::'a::type topology) (N::'a::type => bool) x::'a::type.
|
|
912 |
neigh top (N, x) =
|
|
913 |
(EX P::'a::type => bool. open top P & re_subset P N & P x)"
|
14516
|
914 |
by (import topology neigh)
|
|
915 |
|
17644
|
916 |
lemma OPEN_OWN_NEIGH: "ALL (S'::'a::type => bool) (top::'a::type topology) x::'a::type.
|
|
917 |
open top S' & S' x --> neigh top (S', x)"
|
14516
|
918 |
by (import topology OPEN_OWN_NEIGH)
|
|
919 |
|
17644
|
920 |
lemma OPEN_UNOPEN: "ALL (S'::'a::type => bool) top::'a::type topology.
|
|
921 |
open top S' =
|
|
922 |
(re_Union (%P::'a::type => bool. open top P & re_subset P S') = S')"
|
14516
|
923 |
by (import topology OPEN_UNOPEN)
|
|
924 |
|
17644
|
925 |
lemma OPEN_SUBOPEN: "ALL (S'::'a::type => bool) top::'a::type topology.
|
|
926 |
open top S' =
|
|
927 |
(ALL x::'a::type.
|
|
928 |
S' x --> (EX P::'a::type => bool. P x & open top P & re_subset P S'))"
|
14516
|
929 |
by (import topology OPEN_SUBOPEN)
|
|
930 |
|
17644
|
931 |
lemma OPEN_NEIGH: "ALL (S'::'a::type => bool) top::'a::type topology.
|
|
932 |
open top S' =
|
|
933 |
(ALL x::'a::type.
|
|
934 |
S' x --> (EX N::'a::type => bool. neigh top (N, x) & re_subset N S'))"
|
14516
|
935 |
by (import topology OPEN_NEIGH)
|
|
936 |
|
|
937 |
constdefs
|
17652
|
938 |
closed :: "'a topology => ('a => bool) => bool"
|
17644
|
939 |
"closed == %(L::'a::type topology) S'::'a::type => bool. open L (re_compl S')"
|
|
940 |
|
|
941 |
lemma closed: "ALL (L::'a::type topology) S'::'a::type => bool.
|
|
942 |
closed L S' = open L (re_compl S')"
|
14516
|
943 |
by (import topology closed)
|
|
944 |
|
|
945 |
constdefs
|
17652
|
946 |
limpt :: "'a topology => 'a => ('a => bool) => bool"
|
17644
|
947 |
"limpt ==
|
|
948 |
%(top::'a::type topology) (x::'a::type) S'::'a::type => bool.
|
|
949 |
ALL N::'a::type => bool.
|
|
950 |
neigh top (N, x) --> (EX y::'a::type. x ~= y & S' y & N y)"
|
|
951 |
|
|
952 |
lemma limpt: "ALL (top::'a::type topology) (x::'a::type) S'::'a::type => bool.
|
14516
|
953 |
limpt top x S' =
|
17644
|
954 |
(ALL N::'a::type => bool.
|
|
955 |
neigh top (N, x) --> (EX y::'a::type. x ~= y & S' y & N y))"
|
14516
|
956 |
by (import topology limpt)
|
|
957 |
|
17644
|
958 |
lemma CLOSED_LIMPT: "ALL (top::'a::type topology) S'::'a::type => bool.
|
|
959 |
closed top S' = (ALL x::'a::type. limpt top x S' --> S' x)"
|
14516
|
960 |
by (import topology CLOSED_LIMPT)
|
|
961 |
|
|
962 |
constdefs
|
17652
|
963 |
ismet :: "('a * 'a => real) => bool"
|
14516
|
964 |
"ismet ==
|
17644
|
965 |
%m::'a::type * 'a::type => real.
|
17652
|
966 |
(ALL (x::'a::type) y::'a::type. (m (x, y) = 0) = (x = y)) &
|
17644
|
967 |
(ALL (x::'a::type) (y::'a::type) z::'a::type.
|
|
968 |
m (y, z) <= m (x, y) + m (x, z))"
|
|
969 |
|
|
970 |
lemma ismet: "ALL m::'a::type * 'a::type => real.
|
14516
|
971 |
ismet m =
|
17652
|
972 |
((ALL (x::'a::type) y::'a::type. (m (x, y) = 0) = (x = y)) &
|
17644
|
973 |
(ALL (x::'a::type) (y::'a::type) z::'a::type.
|
|
974 |
m (y, z) <= m (x, y) + m (x, z)))"
|
14516
|
975 |
by (import topology ismet)
|
|
976 |
|
17644
|
977 |
typedef (open) ('a) metric = "(Collect::(('a::type * 'a::type => real) => bool)
|
|
978 |
=> ('a::type * 'a::type => real) set)
|
|
979 |
(ismet::('a::type * 'a::type => real) => bool)"
|
14516
|
980 |
by (rule typedef_helper,import topology metric_TY_DEF)
|
|
981 |
|
|
982 |
lemmas metric_TY_DEF = typedef_hol2hol4 [OF type_definition_metric]
|
|
983 |
|
|
984 |
consts
|
17652
|
985 |
metric :: "('a * 'a => real) => 'a metric"
|
|
986 |
dist :: "'a metric => 'a * 'a => real"
|
17644
|
987 |
|
|
988 |
specification (dist metric) metric_tybij: "(ALL a::'a::type metric. metric (dist a) = a) &
|
|
989 |
(ALL r::'a::type * 'a::type => real. ismet r = (dist (metric r) = r))"
|
14516
|
990 |
by (import topology metric_tybij)
|
|
991 |
|
17644
|
992 |
lemma METRIC_ISMET: "ALL m::'a::type metric. ismet (dist m)"
|
14516
|
993 |
by (import topology METRIC_ISMET)
|
|
994 |
|
17644
|
995 |
lemma METRIC_ZERO: "ALL (m::'a::type metric) (x::'a::type) y::'a::type.
|
17652
|
996 |
(dist m (x, y) = 0) = (x = y)"
|
14516
|
997 |
by (import topology METRIC_ZERO)
|
|
998 |
|
17652
|
999 |
lemma METRIC_SAME: "ALL (m::'a::type metric) x::'a::type. dist m (x, x) = 0"
|
14516
|
1000 |
by (import topology METRIC_SAME)
|
|
1001 |
|
17652
|
1002 |
lemma METRIC_POS: "ALL (m::'a::type metric) (x::'a::type) y::'a::type. 0 <= dist m (x, y)"
|
14516
|
1003 |
by (import topology METRIC_POS)
|
|
1004 |
|
17644
|
1005 |
lemma METRIC_SYM: "ALL (m::'a::type metric) (x::'a::type) y::'a::type.
|
|
1006 |
dist m (x, y) = dist m (y, x)"
|
14516
|
1007 |
by (import topology METRIC_SYM)
|
|
1008 |
|
17644
|
1009 |
lemma METRIC_TRIANGLE: "ALL (m::'a::type metric) (x::'a::type) (y::'a::type) z::'a::type.
|
|
1010 |
dist m (x, z) <= dist m (x, y) + dist m (y, z)"
|
14516
|
1011 |
by (import topology METRIC_TRIANGLE)
|
|
1012 |
|
17644
|
1013 |
lemma METRIC_NZ: "ALL (m::'a::type metric) (x::'a::type) y::'a::type.
|
17652
|
1014 |
x ~= y --> 0 < dist m (x, y)"
|
14516
|
1015 |
by (import topology METRIC_NZ)
|
|
1016 |
|
|
1017 |
constdefs
|
17652
|
1018 |
mtop :: "'a metric => 'a topology"
|
14516
|
1019 |
"mtop ==
|
17644
|
1020 |
%m::'a::type metric.
|
|
1021 |
topology
|
|
1022 |
(%S'::'a::type => bool.
|
|
1023 |
ALL x::'a::type.
|
17652
|
1024 |
S' x --> (EX e>0. ALL y::'a::type. dist m (x, y) < e --> S' y))"
|
17644
|
1025 |
|
|
1026 |
lemma mtop: "ALL m::'a::type metric.
|
14516
|
1027 |
mtop m =
|
|
1028 |
topology
|
17644
|
1029 |
(%S'::'a::type => bool.
|
|
1030 |
ALL x::'a::type.
|
17652
|
1031 |
S' x --> (EX e>0. ALL y::'a::type. dist m (x, y) < e --> S' y))"
|
14516
|
1032 |
by (import topology mtop)
|
|
1033 |
|
17644
|
1034 |
lemma mtop_istopology: "ALL m::'a::type metric.
|
14516
|
1035 |
istopology
|
17644
|
1036 |
(%S'::'a::type => bool.
|
|
1037 |
ALL x::'a::type.
|
17652
|
1038 |
S' x --> (EX e>0. ALL y::'a::type. dist m (x, y) < e --> S' y))"
|
14516
|
1039 |
by (import topology mtop_istopology)
|
|
1040 |
|
17644
|
1041 |
lemma MTOP_OPEN: "ALL (S'::'a::type => bool) x::'a::type metric.
|
14516
|
1042 |
open (mtop x) S' =
|
17644
|
1043 |
(ALL xa::'a::type.
|
17652
|
1044 |
S' xa --> (EX e>0. ALL y::'a::type. dist x (xa, y) < e --> S' y))"
|
14516
|
1045 |
by (import topology MTOP_OPEN)
|
|
1046 |
|
|
1047 |
constdefs
|
17652
|
1048 |
B :: "'a metric => 'a * real => 'a => bool"
|
17644
|
1049 |
"B ==
|
|
1050 |
%(m::'a::type metric) (x::'a::type, e::real) y::'a::type. dist m (x, y) < e"
|
|
1051 |
|
|
1052 |
lemma ball: "ALL (m::'a::type metric) (x::'a::type) e::real.
|
|
1053 |
B m (x, e) = (%y::'a::type. dist m (x, y) < e)"
|
14516
|
1054 |
by (import topology ball)
|
|
1055 |
|
17644
|
1056 |
lemma BALL_OPEN: "ALL (m::'a::type metric) (x::'a::type) e::real.
|
17652
|
1057 |
0 < e --> open (mtop m) (B m (x, e))"
|
14516
|
1058 |
by (import topology BALL_OPEN)
|
|
1059 |
|
17644
|
1060 |
lemma BALL_NEIGH: "ALL (m::'a::type metric) (x::'a::type) e::real.
|
17652
|
1061 |
0 < e --> neigh (mtop m) (B m (x, e), x)"
|
14516
|
1062 |
by (import topology BALL_NEIGH)
|
|
1063 |
|
17644
|
1064 |
lemma MTOP_LIMPT: "ALL (m::'a::type metric) (x::'a::type) S'::'a::type => bool.
|
|
1065 |
limpt (mtop m) x S' =
|
17652
|
1066 |
(ALL e>0. EX y::'a::type. x ~= y & S' y & dist m (x, y) < e)"
|
14516
|
1067 |
by (import topology MTOP_LIMPT)
|
|
1068 |
|
17644
|
1069 |
lemma ISMET_R1: "ismet (%(x::real, y::real). abs (y - x))"
|
14516
|
1070 |
by (import topology ISMET_R1)
|
|
1071 |
|
|
1072 |
constdefs
|
|
1073 |
mr1 :: "real metric"
|
17644
|
1074 |
"mr1 == metric (%(x::real, y::real). abs (y - x))"
|
|
1075 |
|
|
1076 |
lemma mr1: "mr1 = metric (%(x::real, y::real). abs (y - x))"
|
14516
|
1077 |
by (import topology mr1)
|
|
1078 |
|
17644
|
1079 |
lemma MR1_DEF: "ALL (x::real) y::real. dist mr1 (x, y) = abs (y - x)"
|
14516
|
1080 |
by (import topology MR1_DEF)
|
|
1081 |
|
17644
|
1082 |
lemma MR1_ADD: "ALL (x::real) d::real. dist mr1 (x, x + d) = abs d"
|
14516
|
1083 |
by (import topology MR1_ADD)
|
|
1084 |
|
17644
|
1085 |
lemma MR1_SUB: "ALL (x::real) d::real. dist mr1 (x, x - d) = abs d"
|
14516
|
1086 |
by (import topology MR1_SUB)
|
|
1087 |
|
17652
|
1088 |
lemma MR1_ADD_POS: "ALL (x::real) d::real. 0 <= d --> dist mr1 (x, x + d) = d"
|
14516
|
1089 |
by (import topology MR1_ADD_POS)
|
|
1090 |
|
17652
|
1091 |
lemma MR1_SUB_LE: "ALL (x::real) d::real. 0 <= d --> dist mr1 (x, x - d) = d"
|
14516
|
1092 |
by (import topology MR1_SUB_LE)
|
|
1093 |
|
17652
|
1094 |
lemma MR1_ADD_LT: "ALL (x::real) d::real. 0 < d --> dist mr1 (x, x + d) = d"
|
14516
|
1095 |
by (import topology MR1_ADD_LT)
|
|
1096 |
|
17652
|
1097 |
lemma MR1_SUB_LT: "ALL (x::real) d::real. 0 < d --> dist mr1 (x, x - d) = d"
|
14516
|
1098 |
by (import topology MR1_SUB_LT)
|
|
1099 |
|
17644
|
1100 |
lemma MR1_BETWEEN1: "ALL (x::real) (y::real) z::real. x < z & dist mr1 (x, y) < z - x --> y < z"
|
14516
|
1101 |
by (import topology MR1_BETWEEN1)
|
|
1102 |
|
17644
|
1103 |
lemma MR1_LIMPT: "ALL x::real. limpt (mtop mr1) x re_universe"
|
14516
|
1104 |
by (import topology MR1_LIMPT)
|
|
1105 |
|
|
1106 |
;end_setup
|
|
1107 |
|
|
1108 |
;setup_theory nets
|
|
1109 |
|
|
1110 |
constdefs
|
17652
|
1111 |
dorder :: "('a => 'a => bool) => bool"
|
14516
|
1112 |
"dorder ==
|
17644
|
1113 |
%g::'a::type => 'a::type => bool.
|
|
1114 |
ALL (x::'a::type) y::'a::type.
|
|
1115 |
g x x & g y y -->
|
|
1116 |
(EX z::'a::type. g z z & (ALL w::'a::type. g w z --> g w x & g w y))"
|
|
1117 |
|
|
1118 |
lemma dorder: "ALL g::'a::type => 'a::type => bool.
|
14516
|
1119 |
dorder g =
|
17644
|
1120 |
(ALL (x::'a::type) y::'a::type.
|
|
1121 |
g x x & g y y -->
|
|
1122 |
(EX z::'a::type. g z z & (ALL w::'a::type. g w z --> g w x & g w y)))"
|
14516
|
1123 |
by (import nets dorder)
|
|
1124 |
|
|
1125 |
constdefs
|
17652
|
1126 |
tends :: "('b => 'a) => 'a => 'a topology * ('b => 'b => bool) => bool"
|
14516
|
1127 |
"tends ==
|
17644
|
1128 |
%(s::'b::type => 'a::type) (l::'a::type) (top::'a::type topology,
|
|
1129 |
g::'b::type => 'b::type => bool).
|
|
1130 |
ALL N::'a::type => bool.
|
14516
|
1131 |
neigh top (N, l) -->
|
17644
|
1132 |
(EX n::'b::type. g n n & (ALL m::'b::type. g m n --> N (s m)))"
|
|
1133 |
|
|
1134 |
lemma tends: "ALL (s::'b::type => 'a::type) (l::'a::type) (top::'a::type topology)
|
|
1135 |
g::'b::type => 'b::type => bool.
|
14516
|
1136 |
tends s l (top, g) =
|
17644
|
1137 |
(ALL N::'a::type => bool.
|
14516
|
1138 |
neigh top (N, l) -->
|
17644
|
1139 |
(EX n::'b::type. g n n & (ALL m::'b::type. g m n --> N (s m))))"
|
14516
|
1140 |
by (import nets tends)
|
|
1141 |
|
|
1142 |
constdefs
|
17652
|
1143 |
bounded :: "'a metric * ('b => 'b => bool) => ('b => 'a) => bool"
|
14516
|
1144 |
"bounded ==
|
17644
|
1145 |
%(m::'a::type metric, g::'b::type => 'b::type => bool)
|
|
1146 |
f::'b::type => 'a::type.
|
|
1147 |
EX (k::real) (x::'a::type) N::'b::type.
|
|
1148 |
g N N & (ALL n::'b::type. g n N --> dist m (f n, x) < k)"
|
|
1149 |
|
|
1150 |
lemma bounded: "ALL (m::'a::type metric) (g::'b::type => 'b::type => bool)
|
|
1151 |
f::'b::type => 'a::type.
|
14516
|
1152 |
bounded (m, g) f =
|
17644
|
1153 |
(EX (k::real) (x::'a::type) N::'b::type.
|
|
1154 |
g N N & (ALL n::'b::type. g n N --> dist m (f n, x) < k))"
|
14516
|
1155 |
by (import nets bounded)
|
|
1156 |
|
|
1157 |
constdefs
|
17652
|
1158 |
tendsto :: "'a metric * 'a => 'a => 'a => bool"
|
17644
|
1159 |
"tendsto ==
|
|
1160 |
%(m::'a::type metric, x::'a::type) (y::'a::type) z::'a::type.
|
17652
|
1161 |
0 < dist m (x, y) & dist m (x, y) <= dist m (x, z)"
|
17644
|
1162 |
|
|
1163 |
lemma tendsto: "ALL (m::'a::type metric) (x::'a::type) (y::'a::type) z::'a::type.
|
17652
|
1164 |
tendsto (m, x) y z = (0 < dist m (x, y) & dist m (x, y) <= dist m (x, z))"
|
14516
|
1165 |
by (import nets tendsto)
|
|
1166 |
|
17644
|
1167 |
lemma DORDER_LEMMA: "ALL g::'a::type => 'a::type => bool.
|
14516
|
1168 |
dorder g -->
|
17644
|
1169 |
(ALL (P::'a::type => bool) Q::'a::type => bool.
|
|
1170 |
(EX n::'a::type. g n n & (ALL m::'a::type. g m n --> P m)) &
|
|
1171 |
(EX n::'a::type. g n n & (ALL m::'a::type. g m n --> Q m)) -->
|
|
1172 |
(EX n::'a::type. g n n & (ALL m::'a::type. g m n --> P m & Q m)))"
|
14516
|
1173 |
by (import nets DORDER_LEMMA)
|
|
1174 |
|
|
1175 |
lemma DORDER_NGE: "dorder nat_ge"
|
|
1176 |
by (import nets DORDER_NGE)
|
|
1177 |
|
17644
|
1178 |
lemma DORDER_TENDSTO: "ALL (m::'a::type metric) x::'a::type. dorder (tendsto (m, x))"
|
14516
|
1179 |
by (import nets DORDER_TENDSTO)
|
|
1180 |
|
17644
|
1181 |
lemma MTOP_TENDS: "ALL (d::'a::type metric) (g::'b::type => 'b::type => bool)
|
|
1182 |
(x::'b::type => 'a::type) x0::'a::type.
|
14516
|
1183 |
tends x x0 (mtop d, g) =
|
17652
|
1184 |
(ALL e>0.
|
17644
|
1185 |
EX n::'b::type.
|
|
1186 |
g n n & (ALL m::'b::type. g m n --> dist d (x m, x0) < e))"
|
14516
|
1187 |
by (import nets MTOP_TENDS)
|
|
1188 |
|
17644
|
1189 |
lemma MTOP_TENDS_UNIQ: "ALL (g::'b::type => 'b::type => bool) d::'a::type metric.
|
14516
|
1190 |
dorder g -->
|
17644
|
1191 |
tends (x::'b::type => 'a::type) (x0::'a::type) (mtop d, g) &
|
|
1192 |
tends x (x1::'a::type) (mtop d, g) -->
|
14516
|
1193 |
x0 = x1"
|
|
1194 |
by (import nets MTOP_TENDS_UNIQ)
|
|
1195 |
|
17644
|
1196 |
lemma SEQ_TENDS: "ALL (d::'a::type metric) (x::nat => 'a::type) x0::'a::type.
|
14516
|
1197 |
tends x x0 (mtop d, nat_ge) =
|
17652
|
1198 |
(ALL xa>0. EX xb::nat. ALL xc::nat. xb <= xc --> dist d (x xc, x0) < xa)"
|
14516
|
1199 |
by (import nets SEQ_TENDS)
|
|
1200 |
|
17644
|
1201 |
lemma LIM_TENDS: "ALL (m1::'a::type metric) (m2::'b::type metric) (f::'a::type => 'b::type)
|
|
1202 |
(x0::'a::type) y0::'b::type.
|
14516
|
1203 |
limpt (mtop m1) x0 re_universe -->
|
|
1204 |
tends f y0 (mtop m2, tendsto (m1, x0)) =
|
17652
|
1205 |
(ALL e>0.
|
|
1206 |
EX d>0.
|
17644
|
1207 |
ALL x::'a::type.
|
17652
|
1208 |
0 < dist m1 (x, x0) & dist m1 (x, x0) <= d -->
|
15647
|
1209 |
dist m2 (f x, y0) < e)"
|
14516
|
1210 |
by (import nets LIM_TENDS)
|
|
1211 |
|
17644
|
1212 |
lemma LIM_TENDS2: "ALL (m1::'a::type metric) (m2::'b::type metric) (f::'a::type => 'b::type)
|
|
1213 |
(x0::'a::type) y0::'b::type.
|
14516
|
1214 |
limpt (mtop m1) x0 re_universe -->
|
|
1215 |
tends f y0 (mtop m2, tendsto (m1, x0)) =
|
17652
|
1216 |
(ALL e>0.
|
|
1217 |
EX d>0.
|
17644
|
1218 |
ALL x::'a::type.
|
17652
|
1219 |
0 < dist m1 (x, x0) & dist m1 (x, x0) < d -->
|
15647
|
1220 |
dist m2 (f x, y0) < e)"
|
14516
|
1221 |
by (import nets LIM_TENDS2)
|
|
1222 |
|
17644
|
1223 |
lemma MR1_BOUNDED: "ALL (g::'a::type => 'a::type => bool) f::'a::type => real.
|
|
1224 |
bounded (mr1, g) f =
|
|
1225 |
(EX (k::real) N::'a::type.
|
|
1226 |
g N N & (ALL n::'a::type. g n N --> abs (f n) < k))"
|
14516
|
1227 |
by (import nets MR1_BOUNDED)
|
|
1228 |
|
17644
|
1229 |
lemma NET_NULL: "ALL (g::'a::type => 'a::type => bool) (x::'a::type => real) x0::real.
|
17652
|
1230 |
tends x x0 (mtop mr1, g) = tends (%n::'a::type. x n - x0) 0 (mtop mr1, g)"
|
14516
|
1231 |
by (import nets NET_NULL)
|
|
1232 |
|
17644
|
1233 |
lemma NET_CONV_BOUNDED: "ALL (g::'a::type => 'a::type => bool) (x::'a::type => real) x0::real.
|
|
1234 |
tends x x0 (mtop mr1, g) --> bounded (mr1, g) x"
|
14516
|
1235 |
by (import nets NET_CONV_BOUNDED)
|
|
1236 |
|
17644
|
1237 |
lemma NET_CONV_NZ: "ALL (g::'a::type => 'a::type => bool) (x::'a::type => real) x0::real.
|
17652
|
1238 |
tends x x0 (mtop mr1, g) & x0 ~= 0 -->
|
|
1239 |
(EX N::'a::type. g N N & (ALL n::'a::type. g n N --> x n ~= 0))"
|
14516
|
1240 |
by (import nets NET_CONV_NZ)
|
|
1241 |
|
17644
|
1242 |
lemma NET_CONV_IBOUNDED: "ALL (g::'a::type => 'a::type => bool) (x::'a::type => real) x0::real.
|
17652
|
1243 |
tends x x0 (mtop mr1, g) & x0 ~= 0 -->
|
17644
|
1244 |
bounded (mr1, g) (%n::'a::type. inverse (x n))"
|
14516
|
1245 |
by (import nets NET_CONV_IBOUNDED)
|
|
1246 |
|
17644
|
1247 |
lemma NET_NULL_ADD: "ALL g::'a::type => 'a::type => bool.
|
14516
|
1248 |
dorder g -->
|
17644
|
1249 |
(ALL (x::'a::type => real) y::'a::type => real.
|
17652
|
1250 |
tends x 0 (mtop mr1, g) & tends y 0 (mtop mr1, g) -->
|
|
1251 |
tends (%n::'a::type. x n + y n) 0 (mtop mr1, g))"
|
14516
|
1252 |
by (import nets NET_NULL_ADD)
|
|
1253 |
|
17644
|
1254 |
lemma NET_NULL_MUL: "ALL g::'a::type => 'a::type => bool.
|
14516
|
1255 |
dorder g -->
|
17644
|
1256 |
(ALL (x::'a::type => real) y::'a::type => real.
|
17652
|
1257 |
bounded (mr1, g) x & tends y 0 (mtop mr1, g) -->
|
|
1258 |
tends (%n::'a::type. x n * y n) 0 (mtop mr1, g))"
|
14516
|
1259 |
by (import nets NET_NULL_MUL)
|
|
1260 |
|
17644
|
1261 |
lemma NET_NULL_CMUL: "ALL (g::'a::type => 'a::type => bool) (k::real) x::'a::type => real.
|
17652
|
1262 |
tends x 0 (mtop mr1, g) --> tends (%n::'a::type. k * x n) 0 (mtop mr1, g)"
|
14516
|
1263 |
by (import nets NET_NULL_CMUL)
|
|
1264 |
|
17644
|
1265 |
lemma NET_ADD: "ALL g::'a::type => 'a::type => bool.
|
14516
|
1266 |
dorder g -->
|
17644
|
1267 |
(ALL (x::'a::type => real) (x0::real) (y::'a::type => real) y0::real.
|
14516
|
1268 |
tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) -->
|
17644
|
1269 |
tends (%n::'a::type. x n + y n) (x0 + y0) (mtop mr1, g))"
|
14516
|
1270 |
by (import nets NET_ADD)
|
|
1271 |
|
17644
|
1272 |
lemma NET_NEG: "ALL g::'a::type => 'a::type => bool.
|
14516
|
1273 |
dorder g -->
|
17644
|
1274 |
(ALL (x::'a::type => real) x0::real.
|
|
1275 |
tends x x0 (mtop mr1, g) =
|
|
1276 |
tends (%n::'a::type. - x n) (- x0) (mtop mr1, g))"
|
14516
|
1277 |
by (import nets NET_NEG)
|
|
1278 |
|
17644
|
1279 |
lemma NET_SUB: "ALL g::'a::type => 'a::type => bool.
|
14516
|
1280 |
dorder g -->
|
17644
|
1281 |
(ALL (x::'a::type => real) (x0::real) (y::'a::type => real) y0::real.
|
14516
|
1282 |
tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) -->
|
17644
|
1283 |
tends (%xa::'a::type. x xa - y xa) (x0 - y0) (mtop mr1, g))"
|
14516
|
1284 |
by (import nets NET_SUB)
|
|
1285 |
|
17644
|
1286 |
lemma NET_MUL: "ALL g::'a::type => 'a::type => bool.
|
14516
|
1287 |
dorder g -->
|
17644
|
1288 |
(ALL (x::'a::type => real) (y::'a::type => real) (x0::real) y0::real.
|
14516
|
1289 |
tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) -->
|
17644
|
1290 |
tends (%n::'a::type. x n * y n) (x0 * y0) (mtop mr1, g))"
|
14516
|
1291 |
by (import nets NET_MUL)
|
|
1292 |
|
17644
|
1293 |
lemma NET_INV: "ALL g::'a::type => 'a::type => bool.
|
14516
|
1294 |
dorder g -->
|
17644
|
1295 |
(ALL (x::'a::type => real) x0::real.
|
17652
|
1296 |
tends x x0 (mtop mr1, g) & x0 ~= 0 -->
|
17644
|
1297 |
tends (%n::'a::type. inverse (x n)) (inverse x0) (mtop mr1, g))"
|
14516
|
1298 |
by (import nets NET_INV)
|
|
1299 |
|
17644
|
1300 |
lemma NET_DIV: "ALL g::'a::type => 'a::type => bool.
|
14516
|
1301 |
dorder g -->
|
17644
|
1302 |
(ALL (x::'a::type => real) (x0::real) (y::'a::type => real) y0::real.
|
17652
|
1303 |
tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) & y0 ~= 0 -->
|
17644
|
1304 |
tends (%xa::'a::type. x xa / y xa) (x0 / y0) (mtop mr1, g))"
|
14516
|
1305 |
by (import nets NET_DIV)
|
|
1306 |
|
17644
|
1307 |
lemma NET_ABS: "ALL (g::'a::type => 'a::type => bool) (x::'a::type => real) x0::real.
|
|
1308 |
tends x x0 (mtop mr1, g) -->
|
|
1309 |
tends (%n::'a::type. abs (x n)) (abs x0) (mtop mr1, g)"
|
14516
|
1310 |
by (import nets NET_ABS)
|
|
1311 |
|
17644
|
1312 |
lemma NET_LE: "ALL g::'a::type => 'a::type => bool.
|
14516
|
1313 |
dorder g -->
|
17644
|
1314 |
(ALL (x::'a::type => real) (x0::real) (y::'a::type => real) y0::real.
|
14516
|
1315 |
tends x x0 (mtop mr1, g) &
|
|
1316 |
tends y y0 (mtop mr1, g) &
|
17644
|
1317 |
(EX N::'a::type. g N N & (ALL n::'a::type. g n N --> x n <= y n)) -->
|
14516
|
1318 |
x0 <= y0)"
|
|
1319 |
by (import nets NET_LE)
|
|
1320 |
|
|
1321 |
;end_setup
|
|
1322 |
|
|
1323 |
;setup_theory seq
|
|
1324 |
|
14694
|
1325 |
consts
|
17694
|
1326 |
"hol4-->" :: "(nat => real) => real => bool" ("hol4-->")
|
14694
|
1327 |
|
|
1328 |
defs
|
17694
|
1329 |
"hol4-->_def": "hol4--> == %(x::nat => real) x0::real. tends x x0 (mtop mr1, nat_ge)"
|
|
1330 |
|
|
1331 |
lemma tends_num_real: "ALL (x::nat => real) x0::real. hol4--> x x0 = tends x x0 (mtop mr1, nat_ge)"
|
14516
|
1332 |
by (import seq tends_num_real)
|
|
1333 |
|
17694
|
1334 |
lemma SEQ: "ALL (x::nat => real) x0::real.
|
|
1335 |
hol4--> x x0 =
|
|
1336 |
(ALL e>0. EX N::nat. ALL n::nat. N <= n --> abs (x n - x0) < e)"
|
14516
|
1337 |
by (import seq SEQ)
|
|
1338 |
|
17694
|
1339 |
lemma SEQ_CONST: "ALL k::real. hol4--> (%x::nat. k) k"
|
14516
|
1340 |
by (import seq SEQ_CONST)
|
|
1341 |
|
17694
|
1342 |
lemma SEQ_ADD: "ALL (x::nat => real) (x0::real) (y::nat => real) y0::real.
|
|
1343 |
hol4--> x x0 & hol4--> y y0 --> hol4--> (%n::nat. x n + y n) (x0 + y0)"
|
14516
|
1344 |
by (import seq SEQ_ADD)
|
|
1345 |
|
17694
|
1346 |
lemma SEQ_MUL: "ALL (x::nat => real) (x0::real) (y::nat => real) y0::real.
|
|
1347 |
hol4--> x x0 & hol4--> y y0 --> hol4--> (%n::nat. x n * y n) (x0 * y0)"
|
14516
|
1348 |
by (import seq SEQ_MUL)
|
|
1349 |
|
17694
|
1350 |
lemma SEQ_NEG: "ALL (x::nat => real) x0::real.
|
|
1351 |
hol4--> x x0 = hol4--> (%n::nat. - x n) (- x0)"
|
14516
|
1352 |
by (import seq SEQ_NEG)
|
|
1353 |
|
17694
|
1354 |
lemma SEQ_INV: "ALL (x::nat => real) x0::real.
|
|
1355 |
hol4--> x x0 & x0 ~= 0 --> hol4--> (%n::nat. inverse (x n)) (inverse x0)"
|
14516
|
1356 |
by (import seq SEQ_INV)
|
|
1357 |
|
17694
|
1358 |
lemma SEQ_SUB: "ALL (x::nat => real) (x0::real) (y::nat => real) y0::real.
|
|
1359 |
hol4--> x x0 & hol4--> y y0 --> hol4--> (%n::nat. x n - y n) (x0 - y0)"
|
14516
|
1360 |
by (import seq SEQ_SUB)
|
|
1361 |
|
17694
|
1362 |
lemma SEQ_DIV: "ALL (x::nat => real) (x0::real) (y::nat => real) y0::real.
|
|
1363 |
hol4--> x x0 & hol4--> y y0 & y0 ~= 0 -->
|
|
1364 |
hol4--> (%n::nat. x n / y n) (x0 / y0)"
|
14516
|
1365 |
by (import seq SEQ_DIV)
|
|
1366 |
|
17694
|
1367 |
lemma SEQ_UNIQ: "ALL (x::nat => real) (x1::real) x2::real.
|
|
1368 |
hol4--> x x1 & hol4--> x x2 --> x1 = x2"
|
14516
|
1369 |
by (import seq SEQ_UNIQ)
|
|
1370 |
|
|
1371 |
constdefs
|
|
1372 |
convergent :: "(nat => real) => bool"
|
17694
|
1373 |
"convergent == %f::nat => real. Ex (hol4--> f)"
|
|
1374 |
|
|
1375 |
lemma convergent: "ALL f::nat => real. convergent f = Ex (hol4--> f)"
|
14516
|
1376 |
by (import seq convergent)
|
|
1377 |
|
|
1378 |
constdefs
|
|
1379 |
cauchy :: "(nat => real) => bool"
|
17694
|
1380 |
"cauchy ==
|
|
1381 |
%f::nat => real.
|
|
1382 |
ALL e>0.
|
|
1383 |
EX N::nat.
|
|
1384 |
ALL (m::nat) n::nat. N <= m & N <= n --> abs (f m - f n) < e"
|
|
1385 |
|
|
1386 |
lemma cauchy: "ALL f::nat => real.
|
|
1387 |
cauchy f =
|
|
1388 |
(ALL e>0.
|
|
1389 |
EX N::nat.
|
|
1390 |
ALL (m::nat) n::nat. N <= m & N <= n --> abs (f m - f n) < e)"
|
14516
|
1391 |
by (import seq cauchy)
|
|
1392 |
|
|
1393 |
constdefs
|
|
1394 |
lim :: "(nat => real) => real"
|
17694
|
1395 |
"lim == %f::nat => real. Eps (hol4--> f)"
|
|
1396 |
|
|
1397 |
lemma lim: "ALL f::nat => real. lim f = Eps (hol4--> f)"
|
14516
|
1398 |
by (import seq lim)
|
|
1399 |
|
17694
|
1400 |
lemma SEQ_LIM: "ALL f::nat => real. convergent f = hol4--> f (lim f)"
|
14516
|
1401 |
by (import seq SEQ_LIM)
|
|
1402 |
|
|
1403 |
constdefs
|
|
1404 |
subseq :: "(nat => nat) => bool"
|
17694
|
1405 |
"subseq == %f::nat => nat. ALL (m::nat) n::nat. m < n --> f m < f n"
|
|
1406 |
|
|
1407 |
lemma subseq: "ALL f::nat => nat. subseq f = (ALL (m::nat) n::nat. m < n --> f m < f n)"
|
14516
|
1408 |
by (import seq subseq)
|
|
1409 |
|
17644
|
1410 |
lemma SUBSEQ_SUC: "ALL f::nat => nat. subseq f = (ALL n::nat. f n < f (Suc n))"
|
14516
|
1411 |
by (import seq SUBSEQ_SUC)
|
|
1412 |
|
14694
|
1413 |
consts
|
14516
|
1414 |
mono :: "(nat => real) => bool"
|
14694
|
1415 |
|
|
1416 |
defs
|
17694
|
1417 |
mono_def: "seq.mono ==
|
|
1418 |
%f::nat => real.
|
|
1419 |
(ALL (m::nat) n::nat. m <= n --> f m <= f n) |
|
|
1420 |
(ALL (m::nat) n::nat. m <= n --> f n <= f m)"
|
|
1421 |
|
|
1422 |
lemma mono: "ALL f::nat => real.
|
|
1423 |
seq.mono f =
|
|
1424 |
((ALL (m::nat) n::nat. m <= n --> f m <= f n) |
|
|
1425 |
(ALL (m::nat) n::nat. m <= n --> f n <= f m))"
|
14516
|
1426 |
by (import seq mono)
|
|
1427 |
|
17644
|
1428 |
lemma MONO_SUC: "ALL f::nat => real.
|
|
1429 |
seq.mono f =
|
|
1430 |
((ALL x::nat. f x <= f (Suc x)) | (ALL n::nat. f (Suc n) <= f n))"
|
14516
|
1431 |
by (import seq MONO_SUC)
|
|
1432 |
|
14847
|
1433 |
lemma MAX_LEMMA: "(All::((nat => real) => bool) => bool)
|
|
1434 |
(%s::nat => real.
|
|
1435 |
(All::(nat => bool) => bool)
|
|
1436 |
(%N::nat.
|
|
1437 |
(Ex::(real => bool) => bool)
|
|
1438 |
(%k::real.
|
|
1439 |
(All::(nat => bool) => bool)
|
|
1440 |
(%n::nat.
|
|
1441 |
(op -->::bool => bool => bool)
|
|
1442 |
((op <::nat => nat => bool) n N)
|
|
1443 |
((op <::real => real => bool)
|
|
1444 |
((abs::real => real) (s n)) k)))))"
|
14516
|
1445 |
by (import seq MAX_LEMMA)
|
|
1446 |
|
17644
|
1447 |
lemma SEQ_BOUNDED: "ALL s::nat => real.
|
|
1448 |
bounded (mr1, nat_ge) s = (EX k::real. ALL n::nat. abs (s n) < k)"
|
14516
|
1449 |
by (import seq SEQ_BOUNDED)
|
|
1450 |
|
17694
|
1451 |
lemma SEQ_BOUNDED_2: "ALL (f::nat => real) (k::real) k'::real.
|
|
1452 |
(ALL n::nat. k <= f n & f n <= k') --> bounded (mr1, nat_ge) f"
|
14516
|
1453 |
by (import seq SEQ_BOUNDED_2)
|
|
1454 |
|
17694
|
1455 |
lemma SEQ_CBOUNDED: "ALL f::nat => real. cauchy f --> bounded (mr1, nat_ge) f"
|
14516
|
1456 |
by (import seq SEQ_CBOUNDED)
|
|
1457 |
|
17694
|
1458 |
lemma SEQ_ICONV: "ALL f::nat => real.
|
|
1459 |
bounded (mr1, nat_ge) f &
|
|
1460 |
(ALL (m::nat) n::nat. n <= m --> f n <= f m) -->
|
|
1461 |
convergent f"
|
14516
|
1462 |
by (import seq SEQ_ICONV)
|
|
1463 |
|
17644
|
1464 |
lemma SEQ_NEG_CONV: "ALL f::nat => real. convergent f = convergent (%n::nat. - f n)"
|
14516
|
1465 |
by (import seq SEQ_NEG_CONV)
|
|
1466 |
|
17644
|
1467 |
lemma SEQ_NEG_BOUNDED: "ALL f::nat => real.
|
|
1468 |
bounded (mr1, nat_ge) (%n::nat. - f n) = bounded (mr1, nat_ge) f"
|
14516
|
1469 |
by (import seq SEQ_NEG_BOUNDED)
|
|
1470 |
|
17694
|
1471 |
lemma SEQ_BCONV: "ALL f::nat => real. bounded (mr1, nat_ge) f & seq.mono f --> convergent f"
|
14516
|
1472 |
by (import seq SEQ_BCONV)
|
|
1473 |
|
17644
|
1474 |
lemma SEQ_MONOSUB: "ALL s::nat => real. EX f::nat => nat. subseq f & seq.mono (%n::nat. s (f n))"
|
14516
|
1475 |
by (import seq SEQ_MONOSUB)
|
|
1476 |
|
17694
|
1477 |
lemma SEQ_SBOUNDED: "ALL (s::nat => real) f::nat => nat.
|
|
1478 |
bounded (mr1, nat_ge) s --> bounded (mr1, nat_ge) (%n::nat. s (f n))"
|
14516
|
1479 |
by (import seq SEQ_SBOUNDED)
|
|
1480 |
|
17694
|
1481 |
lemma SEQ_SUBLE: "ALL f::nat => nat. subseq f --> (ALL n::nat. n <= f n)"
|
14516
|
1482 |
by (import seq SEQ_SUBLE)
|
|
1483 |
|
17694
|
1484 |
lemma SEQ_DIRECT: "ALL f::nat => nat.
|
|
1485 |
subseq f --> (ALL (N1::nat) N2::nat. EX x::nat. N1 <= x & N2 <= f x)"
|
14516
|
1486 |
by (import seq SEQ_DIRECT)
|
|
1487 |
|
17644
|
1488 |
lemma SEQ_CAUCHY: "ALL f::nat => real. cauchy f = convergent f"
|
14516
|
1489 |
by (import seq SEQ_CAUCHY)
|
|
1490 |
|
17694
|
1491 |
lemma SEQ_LE: "ALL (f::nat => real) (g::nat => real) (l::real) m::real.
|
|
1492 |
hol4--> f l &
|
|
1493 |
hol4--> g m & (EX x::nat. ALL xa::nat. x <= xa --> f xa <= g xa) -->
|
|
1494 |
l <= m"
|
14516
|
1495 |
by (import seq SEQ_LE)
|
|
1496 |
|
17694
|
1497 |
lemma SEQ_SUC: "ALL (f::nat => real) l::real. hol4--> f l = hol4--> (%n::nat. f (Suc n)) l"
|
14516
|
1498 |
by (import seq SEQ_SUC)
|
|
1499 |
|
17694
|
1500 |
lemma SEQ_ABS: "ALL f::nat => real. hol4--> (%n::nat. abs (f n)) 0 = hol4--> f 0"
|
14516
|
1501 |
by (import seq SEQ_ABS)
|
|
1502 |
|
17694
|
1503 |
lemma SEQ_ABS_IMP: "ALL (f::nat => real) l::real.
|
|
1504 |
hol4--> f l --> hol4--> (%n::nat. abs (f n)) (abs l)"
|
14516
|
1505 |
by (import seq SEQ_ABS_IMP)
|
|
1506 |
|
17694
|
1507 |
lemma SEQ_INV0: "ALL f::nat => real.
|
|
1508 |
(ALL y::real. EX N::nat. ALL n::nat. N <= n --> y < f n) -->
|
|
1509 |
hol4--> (%n::nat. inverse (f n)) 0"
|
14516
|
1510 |
by (import seq SEQ_INV0)
|
|
1511 |
|
17694
|
1512 |
lemma SEQ_POWER_ABS: "ALL c::real. abs c < 1 --> hol4--> (op ^ (abs c)) 0"
|
14516
|
1513 |
by (import seq SEQ_POWER_ABS)
|
|
1514 |
|
17694
|
1515 |
lemma SEQ_POWER: "ALL c::real. abs c < 1 --> hol4--> (op ^ c) 0"
|
14516
|
1516 |
by (import seq SEQ_POWER)
|
|
1517 |
|
17694
|
1518 |
lemma NEST_LEMMA: "ALL (f::nat => real) g::nat => real.
|
|
1519 |
(ALL n::nat. f n <= f (Suc n)) &
|
|
1520 |
(ALL n::nat. g (Suc n) <= g n) & (ALL n::nat. f n <= g n) -->
|
|
1521 |
(EX (l::real) m::real.
|
|
1522 |
l <= m &
|
|
1523 |
((ALL n::nat. f n <= l) & hol4--> f l) &
|
|
1524 |
(ALL n::nat. m <= g n) & hol4--> g m)"
|
14516
|
1525 |
by (import seq NEST_LEMMA)
|
|
1526 |
|
17694
|
1527 |
lemma NEST_LEMMA_UNIQ: "ALL (f::nat => real) g::nat => real.
|
|
1528 |
(ALL n::nat. f n <= f (Suc n)) &
|
|
1529 |
(ALL n::nat. g (Suc n) <= g n) &
|
|
1530 |
(ALL n::nat. f n <= g n) & hol4--> (%n::nat. f n - g n) 0 -->
|
|
1531 |
(EX x::real.
|
|
1532 |
((ALL n::nat. f n <= x) & hol4--> f x) &
|
|
1533 |
(ALL n::nat. x <= g n) & hol4--> g x)"
|
14516
|
1534 |
by (import seq NEST_LEMMA_UNIQ)
|
|
1535 |
|
17694
|
1536 |
lemma BOLZANO_LEMMA: "ALL P::real * real => bool.
|
|
1537 |
(ALL (a::real) (b::real) c::real.
|
|
1538 |
a <= b & b <= c & P (a, b) & P (b, c) --> P (a, c)) &
|
|
1539 |
(ALL x::real.
|
|
1540 |
EX d>0.
|
|
1541 |
ALL (a::real) b::real.
|
|
1542 |
a <= x & x <= b & b - a < d --> P (a, b)) -->
|
|
1543 |
(ALL (a::real) b::real. a <= b --> P (a, b))"
|
14516
|
1544 |
by (import seq BOLZANO_LEMMA)
|
|
1545 |
|
|
1546 |
constdefs
|
|
1547 |
sums :: "(nat => real) => real => bool"
|
17694
|
1548 |
"sums == %f::nat => real. hol4--> (%n::nat. real.sum (0, n) f)"
|
|
1549 |
|
|
1550 |
lemma sums: "ALL (f::nat => real) s::real.
|
|
1551 |
sums f s = hol4--> (%n::nat. real.sum (0, n) f) s"
|
14516
|
1552 |
by (import seq sums)
|
|
1553 |
|
|
1554 |
constdefs
|
|
1555 |
summable :: "(nat => real) => bool"
|
17644
|
1556 |
"summable == %f::nat => real. Ex (sums f)"
|
|
1557 |
|
|
1558 |
lemma summable: "ALL f::nat => real. summable f = Ex (sums f)"
|
14516
|
1559 |
by (import seq summable)
|
|
1560 |
|
|
1561 |
constdefs
|
|
1562 |
suminf :: "(nat => real) => real"
|
17644
|
1563 |
"suminf == %f::nat => real. Eps (sums f)"
|
|
1564 |
|
|
1565 |
lemma suminf: "ALL f::nat => real. suminf f = Eps (sums f)"
|
14516
|
1566 |
by (import seq suminf)
|
|
1567 |
|
17694
|
1568 |
lemma SUM_SUMMABLE: "ALL (f::nat => real) l::real. sums f l --> summable f"
|
14516
|
1569 |
by (import seq SUM_SUMMABLE)
|
|
1570 |
|
17694
|
1571 |
lemma SUMMABLE_SUM: "ALL f::nat => real. summable f --> sums f (suminf f)"
|
14516
|
1572 |
by (import seq SUMMABLE_SUM)
|
|
1573 |
|
17694
|
1574 |
lemma SUM_UNIQ: "ALL (f::nat => real) x::real. sums f x --> x = suminf f"
|
14516
|
1575 |
by (import seq SUM_UNIQ)
|
|
1576 |
|
17694
|
1577 |
lemma SER_0: "ALL (f::nat => real) n::nat.
|
|
1578 |
(ALL m::nat. n <= m --> f m = 0) --> sums f (real.sum (0, n) f)"
|
14516
|
1579 |
by (import seq SER_0)
|
|
1580 |
|
17694
|
1581 |
lemma SER_POS_LE: "ALL (f::nat => real) n::nat.
|
|
1582 |
summable f & (ALL m::nat. n <= m --> 0 <= f m) -->
|
|
1583 |
real.sum (0, n) f <= suminf f"
|
14516
|
1584 |
by (import seq SER_POS_LE)
|
|
1585 |
|
17694
|
1586 |
lemma SER_POS_LT: "ALL (f::nat => real) n::nat.
|
|
1587 |
summable f & (ALL m::nat. n <= m --> 0 < f m) -->
|
|
1588 |
real.sum (0, n) f < suminf f"
|
14516
|
1589 |
by (import seq SER_POS_LT)
|
|
1590 |
|
17694
|
1591 |
lemma SER_GROUP: "ALL (f::nat => real) k::nat.
|
|
1592 |
summable f & 0 < k --> sums (%n::nat. real.sum (n * k, k) f) (suminf f)"
|
14516
|
1593 |
by (import seq SER_GROUP)
|
|
1594 |
|
17694
|
1595 |
lemma SER_PAIR: "ALL f::nat => real.
|
|
1596 |
summable f --> sums (%n::nat. real.sum (2 * n, 2) f) (suminf f)"
|
14516
|
1597 |
by (import seq SER_PAIR)
|
|
1598 |
|
17694
|
1599 |
lemma SER_OFFSET: "ALL f::nat => real.
|
|
1600 |
summable f -->
|
|
1601 |
(ALL k::nat. sums (%n::nat. f (n + k)) (suminf f - real.sum (0, k) f))"
|
14516
|
1602 |
by (import seq SER_OFFSET)
|
|
1603 |
|
17694
|
1604 |
lemma SER_POS_LT_PAIR: "ALL (f::nat => real) n::nat.
|
|
1605 |
summable f & (ALL d::nat. 0 < f (n + 2 * d) + f (n + (2 * d + 1))) -->
|
|
1606 |
real.sum (0, n) f < suminf f"
|
14516
|
1607 |
by (import seq SER_POS_LT_PAIR)
|
|
1608 |
|
17694
|
1609 |
lemma SER_ADD: "ALL (x::nat => real) (x0::real) (y::nat => real) y0::real.
|
|
1610 |
sums x x0 & sums y y0 --> sums (%n::nat. x n + y n) (x0 + y0)"
|
14516
|
1611 |
by (import seq SER_ADD)
|
|
1612 |
|
17694
|
1613 |
lemma SER_CMUL: "ALL (x::nat => real) (x0::real) c::real.
|
|
1614 |
sums x x0 --> sums (%n::nat. c * x n) (c * x0)"
|
14516
|
1615 |
by (import seq SER_CMUL)
|
|
1616 |
|
17694
|
1617 |
lemma SER_NEG: "ALL (x::nat => real) x0::real. sums x x0 --> sums (%xa::nat. - x xa) (- x0)"
|
14516
|
1618 |
by (import seq SER_NEG)
|
|
1619 |
|
17694
|
1620 |
lemma SER_SUB: "ALL (x::nat => real) (x0::real) (y::nat => real) y0::real.
|
|
1621 |
sums x x0 & sums y y0 --> sums (%xa::nat. x xa - y xa) (x0 - y0)"
|
14516
|
1622 |
by (import seq SER_SUB)
|
|
1623 |
|
17694
|
1624 |
lemma SER_CDIV: "ALL (x::nat => real) (x0::real) c::real.
|
|
1625 |
sums x x0 --> sums (%xa::nat. x xa / c) (x0 / c)"
|
14516
|
1626 |
by (import seq SER_CDIV)
|
|
1627 |
|
17694
|
1628 |
lemma SER_CAUCHY: "ALL f::nat => real.
|
|
1629 |
summable f =
|
|
1630 |
(ALL e>0.
|
|
1631 |
EX N::nat.
|
|
1632 |
ALL (m::nat) n::nat. N <= m --> abs (real.sum (m, n) f) < e)"
|
14516
|
1633 |
by (import seq SER_CAUCHY)
|
|
1634 |
|
17694
|
1635 |
lemma SER_ZERO: "ALL f::nat => real. summable f --> hol4--> f 0"
|
14516
|
1636 |
by (import seq SER_ZERO)
|
|
1637 |
|
17694
|
1638 |
lemma SER_COMPAR: "ALL (f::nat => real) g::nat => real.
|
|
1639 |
(EX x::nat. ALL xa::nat. x <= xa --> abs (f xa) <= g xa) & summable g -->
|
|
1640 |
summable f"
|
14516
|
1641 |
by (import seq SER_COMPAR)
|
|
1642 |
|
17694
|
1643 |
lemma SER_COMPARA: "ALL (f::nat => real) g::nat => real.
|
|
1644 |
(EX x::nat. ALL xa::nat. x <= xa --> abs (f xa) <= g xa) & summable g -->
|
|
1645 |
summable (%k::nat. abs (f k))"
|
14516
|
1646 |
by (import seq SER_COMPARA)
|
|
1647 |
|
17694
|
1648 |
lemma SER_LE: "ALL (f::nat => real) g::nat => real.
|
|
1649 |
(ALL n::nat. f n <= g n) & summable f & summable g -->
|
|
1650 |
suminf f <= suminf g"
|
14516
|
1651 |
by (import seq SER_LE)
|
|
1652 |
|
17694
|
1653 |
lemma SER_LE2: "ALL (f::nat => real) g::nat => real.
|
|
1654 |
(ALL n::nat. abs (f n) <= g n) & summable g -->
|
|
1655 |
summable f & suminf f <= suminf g"
|
14516
|
1656 |
by (import seq SER_LE2)
|
|
1657 |
|
17694
|
1658 |
lemma SER_ACONV: "ALL f::nat => real. summable (%n::nat. abs (f n)) --> summable f"
|
14516
|
1659 |
by (import seq SER_ACONV)
|
|
1660 |
|
17694
|
1661 |
lemma SER_ABS: "ALL f::nat => real.
|
|
1662 |
summable (%n::nat. abs (f n)) -->
|
|
1663 |
abs (suminf f) <= suminf (%n::nat. abs (f n))"
|
14516
|
1664 |
by (import seq SER_ABS)
|
|
1665 |
|
17694
|
1666 |
lemma GP_FINITE: "ALL x::real.
|
|
1667 |
x ~= 1 --> (ALL n::nat. real.sum (0, n) (op ^ x) = (x ^ n - 1) / (x - 1))"
|
14516
|
1668 |
by (import seq GP_FINITE)
|
|
1669 |
|
17694
|
1670 |
lemma GP: "ALL x::real. abs x < 1 --> sums (op ^ x) (inverse (1 - x))"
|
14516
|
1671 |
by (import seq GP)
|
|
1672 |
|
|
1673 |
lemma ABS_NEG_LEMMA: "(All::(real => bool) => bool)
|
|
1674 |
(%c::real.
|
|
1675 |
(op -->::bool => bool => bool)
|
|
1676 |
((op <=::real => real => bool) c (0::real))
|
|
1677 |
((All::(real => bool) => bool)
|
|
1678 |
(%x::real.
|
|
1679 |
(All::(real => bool) => bool)
|
|
1680 |
(%y::real.
|
|
1681 |
(op -->::bool => bool => bool)
|
|
1682 |
((op <=::real => real => bool) ((abs::real => real) x)
|
|
1683 |
((op *::real => real => real) c
|
|
1684 |
((abs::real => real) y)))
|
|
1685 |
((op =::real => real => bool) x (0::real))))))"
|
|
1686 |
by (import seq ABS_NEG_LEMMA)
|
|
1687 |
|
17694
|
1688 |
lemma SER_RATIO: "ALL (f::nat => real) (c::real) N::nat.
|
|
1689 |
c < 1 & (ALL n::nat. N <= n --> abs (f (Suc n)) <= c * abs (f n)) -->
|
|
1690 |
summable f"
|
14516
|
1691 |
by (import seq SER_RATIO)
|
|
1692 |
|
|
1693 |
;end_setup
|
|
1694 |
|
|
1695 |
;setup_theory lim
|
|
1696 |
|
|
1697 |
constdefs
|
|
1698 |
tends_real_real :: "(real => real) => real => real => bool"
|
17644
|
1699 |
"tends_real_real ==
|
|
1700 |
%(f::real => real) (l::real) x0::real.
|
|
1701 |
tends f l (mtop mr1, tendsto (mr1, x0))"
|
|
1702 |
|
|
1703 |
lemma tends_real_real: "ALL (f::real => real) (l::real) x0::real.
|
|
1704 |
tends_real_real f l x0 = tends f l (mtop mr1, tendsto (mr1, x0))"
|
14516
|
1705 |
by (import lim tends_real_real)
|
|
1706 |
|
17694
|
1707 |
lemma LIM: "ALL (f::real => real) (y0::real) x0::real.
|
|
1708 |
tends_real_real f y0 x0 =
|
|
1709 |
(ALL e>0.
|
|
1710 |
EX d>0.
|
|
1711 |
ALL x::real.
|
|
1712 |
0 < abs (x - x0) & abs (x - x0) < d --> abs (f x - y0) < e)"
|
14516
|
1713 |
by (import lim LIM)
|
|
1714 |
|
17644
|
1715 |
lemma LIM_CONST: "ALL k::real. All (tends_real_real (%x::real. k) k)"
|
14516
|
1716 |
by (import lim LIM_CONST)
|
|
1717 |
|
17694
|
1718 |
lemma LIM_ADD: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
|
|
1719 |
tends_real_real f l x & tends_real_real g m x -->
|
|
1720 |
tends_real_real (%x::real. f x + g x) (l + m) x"
|
14516
|
1721 |
by (import lim LIM_ADD)
|
|
1722 |
|
17694
|
1723 |
lemma LIM_MUL: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
|
|
1724 |
tends_real_real f l x & tends_real_real g m x -->
|
|
1725 |
tends_real_real (%x::real. f x * g x) (l * m) x"
|
14516
|
1726 |
by (import lim LIM_MUL)
|
|
1727 |
|
17644
|
1728 |
lemma LIM_NEG: "ALL (f::real => real) (l::real) x::real.
|
|
1729 |
tends_real_real f l x = tends_real_real (%x::real. - f x) (- l) x"
|
14516
|
1730 |
by (import lim LIM_NEG)
|
|
1731 |
|
17694
|
1732 |
lemma LIM_INV: "ALL (f::real => real) (l::real) x::real.
|
|
1733 |
tends_real_real f l x & l ~= 0 -->
|
|
1734 |
tends_real_real (%x::real. inverse (f x)) (inverse l) x"
|
14516
|
1735 |
by (import lim LIM_INV)
|
|
1736 |
|
17694
|
1737 |
lemma LIM_SUB: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
|
|
1738 |
tends_real_real f l x & tends_real_real g m x -->
|
|
1739 |
tends_real_real (%x::real. f x - g x) (l - m) x"
|
14516
|
1740 |
by (import lim LIM_SUB)
|
|
1741 |
|
17694
|
1742 |
lemma LIM_DIV: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
|
|
1743 |
tends_real_real f l x & tends_real_real g m x & m ~= 0 -->
|
|
1744 |
tends_real_real (%x::real. f x / g x) (l / m) x"
|
14516
|
1745 |
by (import lim LIM_DIV)
|
|
1746 |
|
17644
|
1747 |
lemma LIM_NULL: "ALL (f::real => real) (l::real) x::real.
|
17652
|
1748 |
tends_real_real f l x = tends_real_real (%x::real. f x - l) 0 x"
|
14516
|
1749 |
by (import lim LIM_NULL)
|
|
1750 |
|
17644
|
1751 |
lemma LIM_X: "ALL x0::real. tends_real_real (%x::real. x) x0 x0"
|
14516
|
1752 |
by (import lim LIM_X)
|
|
1753 |
|
17694
|
1754 |
lemma LIM_UNIQ: "ALL (f::real => real) (l::real) (m::real) x::real.
|
|
1755 |
tends_real_real f l x & tends_real_real f m x --> l = m"
|
14516
|
1756 |
by (import lim LIM_UNIQ)
|
|
1757 |
|
17694
|
1758 |
lemma LIM_EQUAL: "ALL (f::real => real) (g::real => real) (l::real) x0::real.
|
|
1759 |
(ALL x::real. x ~= x0 --> f x = g x) -->
|
|
1760 |
tends_real_real f l x0 = tends_real_real g l x0"
|
14516
|
1761 |
by (import lim LIM_EQUAL)
|
|
1762 |
|
17694
|
1763 |
lemma LIM_TRANSFORM: "ALL (f::real => real) (g::real => real) (x0::real) l::real.
|
|
1764 |
tends_real_real (%x::real. f x - g x) 0 x0 & tends_real_real g l x0 -->
|
|
1765 |
tends_real_real f l x0"
|
14516
|
1766 |
by (import lim LIM_TRANSFORM)
|
|
1767 |
|
|
1768 |
constdefs
|
|
1769 |
diffl :: "(real => real) => real => real => bool"
|
17644
|
1770 |
"diffl ==
|
|
1771 |
%(f::real => real) (l::real) x::real.
|
17652
|
1772 |
tends_real_real (%h::real. (f (x + h) - f x) / h) l 0"
|
17644
|
1773 |
|
|
1774 |
lemma diffl: "ALL (f::real => real) (l::real) x::real.
|
17652
|
1775 |
diffl f l x = tends_real_real (%h::real. (f (x + h) - f x) / h) l 0"
|
14516
|
1776 |
by (import lim diffl)
|
|
1777 |
|
|
1778 |
constdefs
|
|
1779 |
contl :: "(real => real) => real => bool"
|
17644
|
1780 |
"contl ==
|
17652
|
1781 |
%(f::real => real) x::real. tends_real_real (%h::real. f (x + h)) (f x) 0"
|
17644
|
1782 |
|
|
1783 |
lemma contl: "ALL (f::real => real) x::real.
|
17652
|
1784 |
contl f x = tends_real_real (%h::real. f (x + h)) (f x) 0"
|
14516
|
1785 |
by (import lim contl)
|
|
1786 |
|
|
1787 |
constdefs
|
|
1788 |
differentiable :: "(real => real) => real => bool"
|
17644
|
1789 |
"differentiable == %(f::real => real) x::real. EX l::real. diffl f l x"
|
|
1790 |
|
|
1791 |
lemma differentiable: "ALL (f::real => real) x::real.
|
|
1792 |
differentiable f x = (EX l::real. diffl f l x)"
|
14516
|
1793 |
by (import lim differentiable)
|
|
1794 |
|
17694
|
1795 |
lemma DIFF_UNIQ: "ALL (f::real => real) (l::real) (m::real) x::real.
|
|
1796 |
diffl f l x & diffl f m x --> l = m"
|
14516
|
1797 |
by (import lim DIFF_UNIQ)
|
|
1798 |
|
17694
|
1799 |
lemma DIFF_CONT: "ALL (f::real => real) (l::real) x::real. diffl f l x --> contl f x"
|
14516
|
1800 |
by (import lim DIFF_CONT)
|
|
1801 |
|
17644
|
1802 |
lemma CONTL_LIM: "ALL (f::real => real) x::real. contl f x = tends_real_real f (f x) x"
|
14516
|
1803 |
by (import lim CONTL_LIM)
|
|
1804 |
|
17644
|
1805 |
lemma DIFF_CARAT: "ALL (f::real => real) (l::real) x::real.
|
14516
|
1806 |
diffl f l x =
|
17644
|
1807 |
(EX g::real => real.
|
|
1808 |
(ALL z::real. f z - f x = g z * (z - x)) & contl g x & g x = l)"
|
14516
|
1809 |
by (import lim DIFF_CARAT)
|
|
1810 |
|
17644
|
1811 |
lemma CONT_CONST: "ALL k::real. All (contl (%x::real. k))"
|
14516
|
1812 |
by (import lim CONT_CONST)
|
|
1813 |
|
17694
|
1814 |
lemma CONT_ADD: "ALL (f::real => real) (g::real => real) x::real.
|
|
1815 |
contl f x & contl g x --> contl (%x::real. f x + g x) x"
|
14516
|
1816 |
by (import lim CONT_ADD)
|
|
1817 |
|
17694
|
1818 |
lemma CONT_MUL: "ALL (f::real => real) (g::real => real) x::real.
|
|
1819 |
contl f x & contl g x --> contl (%x::real. f x * g x) x"
|
14516
|
1820 |
by (import lim CONT_MUL)
|
|
1821 |
|
17694
|
1822 |
lemma CONT_NEG: "ALL (f::real => real) x::real. contl f x --> contl (%x::real. - f x) x"
|
14516
|
1823 |
by (import lim CONT_NEG)
|
|
1824 |
|
17694
|
1825 |
lemma CONT_INV: "ALL (f::real => real) x::real.
|
|
1826 |
contl f x & f x ~= 0 --> contl (%x::real. inverse (f x)) x"
|
14516
|
1827 |
by (import lim CONT_INV)
|
|
1828 |
|
17694
|
1829 |
lemma CONT_SUB: "ALL (f::real => real) (g::real => real) x::real.
|
|
1830 |
contl f x & contl g x --> contl (%x::real. f x - g x) x"
|
14516
|
1831 |
by (import lim CONT_SUB)
|
|
1832 |
|
17694
|
1833 |
lemma CONT_DIV: "ALL (f::real => real) (g::real => real) x::real.
|
|
1834 |
contl f x & contl g x & g x ~= 0 --> contl (%x::real. f x / g x) x"
|
14516
|
1835 |
by (import lim CONT_DIV)
|
|
1836 |
|
17694
|
1837 |
lemma CONT_COMPOSE: "ALL (f::real => real) (g::real => real) x::real.
|
|
1838 |
contl f x & contl g (f x) --> contl (%x::real. g (f x)) x"
|
14516
|
1839 |
by (import lim CONT_COMPOSE)
|
|
1840 |
|
17694
|
1841 |
lemma IVT: "ALL (f::real => real) (a::real) (b::real) y::real.
|
|
1842 |
a <= b &
|
|
1843 |
(f a <= y & y <= f b) & (ALL x::real. a <= x & x <= b --> contl f x) -->
|
|
1844 |
(EX x::real. a <= x & x <= b & f x = y)"
|
14516
|
1845 |
by (import lim IVT)
|
|
1846 |
|
17694
|
1847 |
lemma IVT2: "ALL (f::real => real) (a::real) (b::real) y::real.
|
|
1848 |
a <= b &
|
|
1849 |
(f b <= y & y <= f a) & (ALL x::real. a <= x & x <= b --> contl f x) -->
|
|
1850 |
(EX x::real. a <= x & x <= b & f x = y)"
|
14516
|
1851 |
by (import lim IVT2)
|
|
1852 |
|
17652
|
1853 |
lemma DIFF_CONST: "ALL k::real. All (diffl (%x::real. k) 0)"
|
14516
|
1854 |
by (import lim DIFF_CONST)
|
|
1855 |
|
17694
|
1856 |
lemma DIFF_ADD: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
|
|
1857 |
diffl f l x & diffl g m x --> diffl (%x::real. f x + g x) (l + m) x"
|
14516
|
1858 |
by (import lim DIFF_ADD)
|
|
1859 |
|
17694
|
1860 |
lemma DIFF_MUL: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
|
|
1861 |
diffl f l x & diffl g m x -->
|
|
1862 |
diffl (%x::real. f x * g x) (l * g x + m * f x) x"
|
14516
|
1863 |
by (import lim DIFF_MUL)
|
|
1864 |
|
17694
|
1865 |
lemma DIFF_CMUL: "ALL (f::real => real) (c::real) (l::real) x::real.
|
|
1866 |
diffl f l x --> diffl (%x::real. c * f x) (c * l) x"
|
14516
|
1867 |
by (import lim DIFF_CMUL)
|
|
1868 |
|
17694
|
1869 |
lemma DIFF_NEG: "ALL (f::real => real) (l::real) x::real.
|
|
1870 |
diffl f l x --> diffl (%x::real. - f x) (- l) x"
|
14516
|
1871 |
by (import lim DIFF_NEG)
|
|
1872 |
|
17694
|
1873 |
lemma DIFF_SUB: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
|
|
1874 |
diffl f l x & diffl g m x --> diffl (%x::real. f x - g x) (l - m) x"
|
14516
|
1875 |
by (import lim DIFF_SUB)
|
|
1876 |
|
17694
|
1877 |
lemma DIFF_CHAIN: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
|
|
1878 |
diffl f l (g x) & diffl g m x --> diffl (%x::real. f (g x)) (l * m) x"
|
14516
|
1879 |
by (import lim DIFF_CHAIN)
|
|
1880 |
|
17652
|
1881 |
lemma DIFF_X: "All (diffl (%x::real. x) 1)"
|
14516
|
1882 |
by (import lim DIFF_X)
|
|
1883 |
|
17652
|
1884 |
lemma DIFF_POW: "ALL (n::nat) x::real. diffl (%x::real. x ^ n) (real n * x ^ (n - 1)) x"
|
14516
|
1885 |
by (import lim DIFF_POW)
|
|
1886 |
|
17694
|
1887 |
lemma DIFF_XM1: "ALL x::real. x ~= 0 --> diffl inverse (- (inverse x ^ 2)) x"
|
14516
|
1888 |
by (import lim DIFF_XM1)
|
|
1889 |
|
17694
|
1890 |
lemma DIFF_INV: "ALL (f::real => real) (l::real) x::real.
|
|
1891 |
diffl f l x & f x ~= 0 -->
|
|
1892 |
diffl (%x::real. inverse (f x)) (- (l / f x ^ 2)) x"
|
14516
|
1893 |
by (import lim DIFF_INV)
|
|
1894 |
|
17694
|
1895 |
lemma DIFF_DIV: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
|
|
1896 |
diffl f l x & diffl g m x & g x ~= 0 -->
|
|
1897 |
diffl (%x::real. f x / g x) ((l * g x - m * f x) / g x ^ 2) x"
|
14516
|
1898 |
by (import lim DIFF_DIV)
|
|
1899 |
|
17694
|
1900 |
lemma DIFF_SUM: "ALL (f::nat => real => real) (f'::nat => real => real) (m::nat) (n::nat)
|
|
1901 |
x::real.
|
|
1902 |
(ALL r::nat. m <= r & r < m + n --> diffl (f r) (f' r x) x) -->
|
|
1903 |
diffl (%x::real. real.sum (m, n) (%n::nat. f n x))
|
|
1904 |
(real.sum (m, n) (%r::nat. f' r x)) x"
|
14516
|
1905 |
by (import lim DIFF_SUM)
|
|
1906 |
|
17694
|
1907 |
lemma CONT_BOUNDED: "ALL (f::real => real) (a::real) b::real.
|
|
1908 |
a <= b & (ALL x::real. a <= x & x <= b --> contl f x) -->
|
|
1909 |
(EX M::real. ALL x::real. a <= x & x <= b --> f x <= M)"
|
14516
|
1910 |
by (import lim CONT_BOUNDED)
|
|
1911 |
|
|
1912 |
lemma CONT_HASSUP: "(All::((real => real) => bool) => bool)
|
|
1913 |
(%f::real => real.
|
|
1914 |
(All::(real => bool) => bool)
|
|
1915 |
(%a::real.
|
|
1916 |
(All::(real => bool) => bool)
|
|
1917 |
(%b::real.
|
|
1918 |
(op -->::bool => bool => bool)
|
|
1919 |
((op &::bool => bool => bool)
|
|
1920 |
((op <=::real => real => bool) a b)
|
|
1921 |
((All::(real => bool) => bool)
|
|
1922 |
(%x::real.
|
|
1923 |
(op -->::bool => bool => bool)
|
|
1924 |
((op &::bool => bool => bool)
|
|
1925 |
((op <=::real => real => bool) a x)
|
|
1926 |
((op <=::real => real => bool) x b))
|
|
1927 |
((contl::(real => real) => real => bool) f x))))
|
|
1928 |
((Ex::(real => bool) => bool)
|
|
1929 |
(%M::real.
|
|
1930 |
(op &::bool => bool => bool)
|
|
1931 |
((All::(real => bool) => bool)
|
|
1932 |
(%x::real.
|
|
1933 |
(op -->::bool => bool => bool)
|
|
1934 |
((op &::bool => bool => bool)
|
|
1935 |
((op <=::real => real => bool) a x)
|
|
1936 |
((op <=::real => real => bool) x b))
|
|
1937 |
((op <=::real => real => bool) (f x) M)))
|
|
1938 |
((All::(real => bool) => bool)
|
|
1939 |
(%N::real.
|
|
1940 |
(op -->::bool => bool => bool)
|
|
1941 |
((op <::real => real => bool) N M)
|
|
1942 |
((Ex::(real => bool) => bool)
|
|
1943 |
(%x::real.
|
|
1944 |
(op &::bool => bool => bool)
|
|
1945 |
((op <=::real => real => bool) a x)
|
|
1946 |
((op &::bool => bool => bool)
|
|
1947 |
((op <=::real => real => bool) x b)
|
|
1948 |
((op <::real => real => bool) N (f x))))))))))))"
|
|
1949 |
by (import lim CONT_HASSUP)
|
|
1950 |
|
17694
|
1951 |
lemma CONT_ATTAINS: "ALL (f::real => real) (a::real) b::real.
|
|
1952 |
a <= b & (ALL x::real. a <= x & x <= b --> contl f x) -->
|
|
1953 |
(EX x::real.
|
|
1954 |
(ALL xa::real. a <= xa & xa <= b --> f xa <= x) &
|
|
1955 |
(EX xa::real. a <= xa & xa <= b & f xa = x))"
|
14516
|
1956 |
by (import lim CONT_ATTAINS)
|
|
1957 |
|
17694
|
1958 |
lemma CONT_ATTAINS2: "ALL (f::real => real) (a::real) b::real.
|
|
1959 |
a <= b & (ALL x::real. a <= x & x <= b --> contl f x) -->
|
|
1960 |
(EX x::real.
|
|
1961 |
(ALL xa::real. a <= xa & xa <= b --> x <= f xa) &
|
|
1962 |
(EX xa::real. a <= xa & xa <= b & f xa = x))"
|
14516
|
1963 |
by (import lim CONT_ATTAINS2)
|
|
1964 |
|
17694
|
1965 |
lemma CONT_ATTAINS_ALL: "ALL (f::real => real) (a::real) b::real.
|
|
1966 |
a <= b & (ALL x::real. a <= x & x <= b --> contl f x) -->
|
|
1967 |
(EX (x::real) M::real.
|
|
1968 |
x <= M &
|
|
1969 |
(ALL y::real.
|
|
1970 |
x <= y & y <= M --> (EX x::real. a <= x & x <= b & f x = y)) &
|
|
1971 |
(ALL xa::real. a <= xa & xa <= b --> x <= f xa & f xa <= M))"
|
14516
|
1972 |
by (import lim CONT_ATTAINS_ALL)
|
|
1973 |
|
17694
|
1974 |
lemma DIFF_LINC: "ALL (f::real => real) (x::real) l::real.
|
|
1975 |
diffl f l x & 0 < l -->
|
|
1976 |
(EX d>0. ALL h::real. 0 < h & h < d --> f x < f (x + h))"
|
14516
|
1977 |
by (import lim DIFF_LINC)
|
|
1978 |
|
17694
|
1979 |
lemma DIFF_LDEC: "ALL (f::real => real) (x::real) l::real.
|
|
1980 |
diffl f l x & l < 0 -->
|
|
1981 |
(EX d>0. ALL h::real. 0 < h & h < d --> f x < f (x - h))"
|
14516
|
1982 |
by (import lim DIFF_LDEC)
|
|
1983 |
|
17694
|
1984 |
lemma DIFF_LMAX: "ALL (f::real => real) (x::real) l::real.
|
|
1985 |
diffl f l x & (EX d>0. ALL y::real. abs (x - y) < d --> f y <= f x) -->
|
|
1986 |
l = 0"
|
14516
|
1987 |
by (import lim DIFF_LMAX)
|
|
1988 |
|
17694
|
1989 |
lemma DIFF_LMIN: "ALL (f::real => real) (x::real) l::real.
|
|
1990 |
diffl f l x & (EX d>0. ALL y::real. abs (x - y) < d --> f x <= f y) -->
|
|
1991 |
l = 0"
|
14516
|
1992 |
by (import lim DIFF_LMIN)
|
|
1993 |
|
17694
|
1994 |
lemma DIFF_LCONST: "ALL (f::real => real) (x::real) l::real.
|
|
1995 |
diffl f l x & (EX d>0. ALL y::real. abs (x - y) < d --> f y = f x) -->
|
|
1996 |
l = 0"
|
14516
|
1997 |
by (import lim DIFF_LCONST)
|
|
1998 |
|
17694
|
1999 |
lemma INTERVAL_LEMMA: "ALL (a::real) (b::real) x::real.
|
|
2000 |
a < x & x < b -->
|
|
2001 |
(EX d>0. ALL y::real. abs (x - y) < d --> a <= y & y <= b)"
|
14516
|
2002 |
by (import lim INTERVAL_LEMMA)
|
|
2003 |
|
17694
|
2004 |
lemma ROLLE: "ALL (f::real => real) (a::real) b::real.
|
|
2005 |
a < b &
|
|
2006 |
f a = f b &
|
|
2007 |
(ALL x::real. a <= x & x <= b --> contl f x) &
|
|
2008 |
(ALL x::real. a < x & x < b --> differentiable f x) -->
|
|
2009 |
(EX z::real. a < z & z < b & diffl f 0 z)"
|
14516
|
2010 |
by (import lim ROLLE)
|
|
2011 |
|
|
2012 |
lemma MVT_LEMMA: "ALL (f::real => real) (a::real) b::real.
|
|
2013 |
f a - (f b - f a) / (b - a) * a = f b - (f b - f a) / (b - a) * b"
|
|
2014 |
by (import lim MVT_LEMMA)
|
|
2015 |
|
17694
|
2016 |
lemma MVT: "ALL (f::real => real) (a::real) b::real.
|
|
2017 |
a < b &
|
|
2018 |
(ALL x::real. a <= x & x <= b --> contl f x) &
|
|
2019 |
(ALL x::real. a < x & x < b --> differentiable f x) -->
|
|
2020 |
(EX (l::real) z::real.
|
|
2021 |
a < z & z < b & diffl f l z & f b - f a = (b - a) * l)"
|
14516
|
2022 |
by (import lim MVT)
|
|
2023 |
|
17694
|
2024 |
lemma DIFF_ISCONST_END: "ALL (f::real => real) (a::real) b::real.
|
|
2025 |
a < b &
|
|
2026 |
(ALL x::real. a <= x & x <= b --> contl f x) &
|
|
2027 |
(ALL x::real. a < x & x < b --> diffl f 0 x) -->
|
|
2028 |
f b = f a"
|
14516
|
2029 |
by (import lim DIFF_ISCONST_END)
|
|
2030 |
|
17694
|
2031 |
lemma DIFF_ISCONST: "ALL (f::real => real) (a::real) b::real.
|
|
2032 |
a < b &
|
|
2033 |
(ALL x::real. a <= x & x <= b --> contl f x) &
|
|
2034 |
(ALL x::real. a < x & x < b --> diffl f 0 x) -->
|
|
2035 |
(ALL x::real. a <= x & x <= b --> f x = f a)"
|
14516
|
2036 |
by (import lim DIFF_ISCONST)
|
|
2037 |
|
17694
|
2038 |
lemma DIFF_ISCONST_ALL: "ALL f::real => real. All (diffl f 0) --> (ALL (x::real) y::real. f x = f y)"
|
14516
|
2039 |
by (import lim DIFF_ISCONST_ALL)
|
|
2040 |
|
|
2041 |
lemma INTERVAL_ABS: "ALL (x::real) (z::real) d::real.
|
|
2042 |
(x - d <= z & z <= x + d) = (abs (z - x) <= d)"
|
|
2043 |
by (import lim INTERVAL_ABS)
|
|
2044 |
|
17694
|
2045 |
lemma CONT_INJ_LEMMA: "ALL (f::real => real) (g::real => real) (x::real) d::real.
|
|
2046 |
0 < d &
|
|
2047 |
(ALL z::real. abs (z - x) <= d --> g (f z) = z) &
|
|
2048 |
(ALL z::real. abs (z - x) <= d --> contl f z) -->
|
|
2049 |
~ (ALL z::real. abs (z - x) <= d --> f z <= f x)"
|
14516
|
2050 |
by (import lim CONT_INJ_LEMMA)
|
|
2051 |
|
17694
|
2052 |
lemma CONT_INJ_LEMMA2: "ALL (f::real => real) (g::real => real) (x::real) d::real.
|
|
2053 |
0 < d &
|
|
2054 |
(ALL z::real. abs (z - x) <= d --> g (f z) = z) &
|
|
2055 |
(ALL z::real. abs (z - x) <= d --> contl f z) -->
|
|
2056 |
~ (ALL z::real. abs (z - x) <= d --> f x <= f z)"
|
14516
|
2057 |
by (import lim CONT_INJ_LEMMA2)
|
|
2058 |
|
17694
|
2059 |
lemma CONT_INJ_RANGE: "ALL (f::real => real) (g::real => real) (x::real) d::real.
|
|
2060 |
0 < d &
|
|
2061 |
(ALL z::real. abs (z - x) <= d --> g (f z) = z) &
|
|
2062 |
(ALL z::real. abs (z - x) <= d --> contl f z) -->
|
|
2063 |
(EX e>0.
|
|
2064 |
ALL y::real.
|
|
2065 |
abs (y - f x) <= e --> (EX z::real. abs (z - x) <= d & f z = y))"
|
14516
|
2066 |
by (import lim CONT_INJ_RANGE)
|
|
2067 |
|
17694
|
2068 |
lemma CONT_INVERSE: "ALL (f::real => real) (g::real => real) (x::real) d::real.
|
|
2069 |
0 < d &
|
|
2070 |
(ALL z::real. abs (z - x) <= d --> g (f z) = z) &
|
|
2071 |
(ALL z::real. abs (z - x) <= d --> contl f z) -->
|
|
2072 |
contl g (f x)"
|
14516
|
2073 |
by (import lim CONT_INVERSE)
|
|
2074 |
|
17694
|
2075 |
lemma DIFF_INVERSE: "ALL (f::real => real) (g::real => real) (l::real) (x::real) d::real.
|
|
2076 |
0 < d &
|
|
2077 |
(ALL z::real. abs (z - x) <= d --> g (f z) = z) &
|
|
2078 |
(ALL z::real. abs (z - x) <= d --> contl f z) & diffl f l x & l ~= 0 -->
|
|
2079 |
diffl g (inverse l) (f x)"
|
14516
|
2080 |
by (import lim DIFF_INVERSE)
|
|
2081 |
|
17694
|
2082 |
lemma DIFF_INVERSE_LT: "ALL (f::real => real) (g::real => real) (l::real) (x::real) d::real.
|
|
2083 |
0 < d &
|
|
2084 |
(ALL z::real. abs (z - x) < d --> g (f z) = z) &
|
|
2085 |
(ALL z::real. abs (z - x) < d --> contl f z) & diffl f l x & l ~= 0 -->
|
|
2086 |
diffl g (inverse l) (f x)"
|
14516
|
2087 |
by (import lim DIFF_INVERSE_LT)
|
|
2088 |
|
17694
|
2089 |
lemma INTERVAL_CLEMMA: "ALL (a::real) (b::real) x::real.
|
|
2090 |
a < x & x < b -->
|
|
2091 |
(EX d>0. ALL y::real. abs (y - x) <= d --> a < y & y < b)"
|
14516
|
2092 |
by (import lim INTERVAL_CLEMMA)
|
|
2093 |
|
17694
|
2094 |
lemma DIFF_INVERSE_OPEN: "ALL (f::real => real) (g::real => real) (l::real) (a::real) (x::real)
|
|
2095 |
b::real.
|
|
2096 |
a < x &
|
|
2097 |
x < b &
|
|
2098 |
(ALL z::real. a < z & z < b --> g (f z) = z & contl f z) &
|
|
2099 |
diffl f l x & l ~= 0 -->
|
|
2100 |
diffl g (inverse l) (f x)"
|
14516
|
2101 |
by (import lim DIFF_INVERSE_OPEN)
|
|
2102 |
|
|
2103 |
;end_setup
|
|
2104 |
|
|
2105 |
;setup_theory powser
|
|
2106 |
|
17644
|
2107 |
lemma POWDIFF_LEMMA: "ALL (n::nat) (x::real) y::real.
|
17652
|
2108 |
real.sum (0, Suc n) (%p::nat. x ^ p * y ^ (Suc n - p)) =
|
|
2109 |
y * real.sum (0, Suc n) (%p::nat. x ^ p * y ^ (n - p))"
|
14516
|
2110 |
by (import powser POWDIFF_LEMMA)
|
|
2111 |
|
17644
|
2112 |
lemma POWDIFF: "ALL (n::nat) (x::real) y::real.
|
14516
|
2113 |
x ^ Suc n - y ^ Suc n =
|
17652
|
2114 |
(x - y) * real.sum (0, Suc n) (%p::nat. x ^ p * y ^ (n - p))"
|
14516
|
2115 |
by (import powser POWDIFF)
|
|
2116 |
|
17644
|
2117 |
lemma POWREV: "ALL (n::nat) (x::real) y::real.
|
17652
|
2118 |
real.sum (0, Suc n) (%xa::nat. x ^ xa * y ^ (n - xa)) =
|
|
2119 |
real.sum (0, Suc n) (%xa::nat. x ^ (n - xa) * y ^ xa)"
|
14516
|
2120 |
by (import powser POWREV)
|
|
2121 |
|
17694
|
2122 |
lemma POWSER_INSIDEA: "ALL (f::nat => real) (x::real) z::real.
|
|
2123 |
summable (%n::nat. f n * x ^ n) & abs z < abs x -->
|
|
2124 |
summable (%n::nat. abs (f n) * z ^ n)"
|
14516
|
2125 |
by (import powser POWSER_INSIDEA)
|
|
2126 |
|
17694
|
2127 |
lemma POWSER_INSIDE: "ALL (f::nat => real) (x::real) z::real.
|
|
2128 |
summable (%n::nat. f n * x ^ n) & abs z < abs x -->
|
|
2129 |
summable (%n::nat. f n * z ^ n)"
|
14516
|
2130 |
by (import powser POWSER_INSIDE)
|
|
2131 |
|
|
2132 |
constdefs
|
|
2133 |
diffs :: "(nat => real) => nat => real"
|
17644
|
2134 |
"diffs == %(c::nat => real) n::nat. real (Suc n) * c (Suc n)"
|
|
2135 |
|
|
2136 |
lemma diffs: "ALL c::nat => real. diffs c = (%n::nat. real (Suc n) * c (Suc n))"
|
14516
|
2137 |
by (import powser diffs)
|
|
2138 |
|
17644
|
2139 |
lemma DIFFS_NEG: "ALL c::nat => real. diffs (%n::nat. - c n) = (%x::nat. - diffs c x)"
|
14516
|
2140 |
by (import powser DIFFS_NEG)
|
|
2141 |
|
17644
|
2142 |
lemma DIFFS_LEMMA: "ALL (n::nat) (c::nat => real) x::real.
|
17652
|
2143 |
real.sum (0, n) (%n::nat. diffs c n * x ^ n) =
|
|
2144 |
real.sum (0, n) (%n::nat. real n * (c n * x ^ (n - 1))) +
|
|
2145 |
real n * (c n * x ^ (n - 1))"
|
14516
|
2146 |
by (import powser DIFFS_LEMMA)
|
|
2147 |
|
17644
|
2148 |
lemma DIFFS_LEMMA2: "ALL (n::nat) (c::nat => real) x::real.
|
17652
|
2149 |
real.sum (0, n) (%n::nat. real n * (c n * x ^ (n - 1))) =
|
|
2150 |
real.sum (0, n) (%n::nat. diffs c n * x ^ n) -
|
|
2151 |
real n * (c n * x ^ (n - 1))"
|
14516
|
2152 |
by (import powser DIFFS_LEMMA2)
|
|
2153 |
|
17694
|
2154 |
lemma DIFFS_EQUIV: "ALL (c::nat => real) x::real.
|
|
2155 |
summable (%n::nat. diffs c n * x ^ n) -->
|
|
2156 |
sums (%n::nat. real n * (c n * x ^ (n - 1)))
|
|
2157 |
(suminf (%n::nat. diffs c n * x ^ n))"
|
14516
|
2158 |
by (import powser DIFFS_EQUIV)
|
|
2159 |
|
17644
|
2160 |
lemma TERMDIFF_LEMMA1: "ALL (m::nat) (z::real) h::real.
|
17652
|
2161 |
real.sum (0, m) (%p::nat. (z + h) ^ (m - p) * z ^ p - z ^ m) =
|
|
2162 |
real.sum (0, m) (%p::nat. z ^ p * ((z + h) ^ (m - p) - z ^ (m - p)))"
|
14516
|
2163 |
by (import powser TERMDIFF_LEMMA1)
|
|
2164 |
|
17694
|
2165 |
lemma TERMDIFF_LEMMA2: "ALL (z::real) (h::real) n::nat.
|
|
2166 |
h ~= 0 -->
|
|
2167 |
((z + h) ^ n - z ^ n) / h - real n * z ^ (n - 1) =
|
|
2168 |
h *
|
|
2169 |
real.sum (0, n - 1)
|
|
2170 |
(%p::nat.
|
|
2171 |
z ^ p *
|
|
2172 |
real.sum (0, n - 1 - p)
|
|
2173 |
(%q::nat. (z + h) ^ q * z ^ (n - 2 - p - q)))"
|
14516
|
2174 |
by (import powser TERMDIFF_LEMMA2)
|
|
2175 |
|
17694
|
2176 |
lemma TERMDIFF_LEMMA3: "ALL (z::real) (h::real) (n::nat) k'::real.
|
|
2177 |
h ~= 0 & abs z <= k' & abs (z + h) <= k' -->
|
|
2178 |
abs (((z + h) ^ n - z ^ n) / h - real n * z ^ (n - 1))
|
|
2179 |
<= real n * (real (n - 1) * (k' ^ (n - 2) * abs h))"
|
14516
|
2180 |
by (import powser TERMDIFF_LEMMA3)
|
|
2181 |
|
17694
|
2182 |
lemma TERMDIFF_LEMMA4: "ALL (f::real => real) (k'::real) k::real.
|
|
2183 |
0 < k &
|
|
2184 |
(ALL h::real. 0 < abs h & abs h < k --> abs (f h) <= k' * abs h) -->
|
|
2185 |
tends_real_real f 0 0"
|
14516
|
2186 |
by (import powser TERMDIFF_LEMMA4)
|
|
2187 |
|
17694
|
2188 |
lemma TERMDIFF_LEMMA5: "ALL (f::nat => real) (g::real => nat => real) k::real.
|
|
2189 |
0 < k &
|
|
2190 |
summable f &
|
|
2191 |
(ALL h::real.
|
|
2192 |
0 < abs h & abs h < k -->
|
|
2193 |
(ALL n::nat. abs (g h n) <= f n * abs h)) -->
|
|
2194 |
tends_real_real (%h::real. suminf (g h)) 0 0"
|
14516
|
2195 |
by (import powser TERMDIFF_LEMMA5)
|
|
2196 |
|
17694
|
2197 |
lemma TERMDIFF: "ALL (c::nat => real) (k'::real) x::real.
|
|
2198 |
summable (%n::nat. c n * k' ^ n) &
|
|
2199 |
summable (%n::nat. diffs c n * k' ^ n) &
|
|
2200 |
summable (%n::nat. diffs (diffs c) n * k' ^ n) & abs x < abs k' -->
|
|
2201 |
diffl (%x::real. suminf (%n::nat. c n * x ^ n))
|
|
2202 |
(suminf (%n::nat. diffs c n * x ^ n)) x"
|
14516
|
2203 |
by (import powser TERMDIFF)
|
|
2204 |
|
|
2205 |
;end_setup
|
|
2206 |
|
|
2207 |
;setup_theory transc
|
|
2208 |
|
|
2209 |
constdefs
|
|
2210 |
exp :: "real => real"
|
17644
|
2211 |
"exp == %x::real. suminf (%n::nat. inverse (real (FACT n)) * x ^ n)"
|
|
2212 |
|
|
2213 |
lemma exp: "ALL x::real. exp x = suminf (%n::nat. inverse (real (FACT n)) * x ^ n)"
|
14516
|
2214 |
by (import transc exp)
|
|
2215 |
|
|
2216 |
constdefs
|
|
2217 |
cos :: "real => real"
|
17652
|
2218 |
"cos ==
|
|
2219 |
%x::real.
|
|
2220 |
suminf
|
|
2221 |
(%n::nat.
|
|
2222 |
(if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) * x ^ n)"
|
|
2223 |
|
|
2224 |
lemma cos: "ALL x::real.
|
|
2225 |
cos x =
|
|
2226 |
suminf
|
|
2227 |
(%n::nat.
|
|
2228 |
(if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) * x ^ n)"
|
14516
|
2229 |
by (import transc cos)
|
|
2230 |
|
|
2231 |
constdefs
|
|
2232 |
sin :: "real => real"
|
|
2233 |
"sin ==
|
17644
|
2234 |
%x::real.
|
|
2235 |
suminf
|
|
2236 |
(%n::nat.
|
17652
|
2237 |
(if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
|
17644
|
2238 |
x ^ n)"
|
|
2239 |
|
|
2240 |
lemma sin: "ALL x::real.
|
14516
|
2241 |
sin x =
|
|
2242 |
suminf
|
17644
|
2243 |
(%n::nat.
|
17652
|
2244 |
(if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
|
17644
|
2245 |
x ^ n)"
|
14516
|
2246 |
by (import transc sin)
|
|
2247 |
|
17644
|
2248 |
lemma EXP_CONVERGES: "ALL x::real. sums (%n::nat. inverse (real (FACT n)) * x ^ n) (exp x)"
|
14516
|
2249 |
by (import transc EXP_CONVERGES)
|
|
2250 |
|
17644
|
2251 |
lemma SIN_CONVERGES: "ALL x::real.
|
14516
|
2252 |
sums
|
17644
|
2253 |
(%n::nat.
|
17652
|
2254 |
(if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
|
17644
|
2255 |
x ^ n)
|
14516
|
2256 |
(sin x)"
|
|
2257 |
by (import transc SIN_CONVERGES)
|
|
2258 |
|
17652
|
2259 |
lemma COS_CONVERGES: "ALL x::real.
|
|
2260 |
sums
|
|
2261 |
(%n::nat.
|
|
2262 |
(if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) * x ^ n)
|
|
2263 |
(cos x)"
|
14516
|
2264 |
by (import transc COS_CONVERGES)
|
|
2265 |
|
17644
|
2266 |
lemma EXP_FDIFF: "diffs (%n::nat. inverse (real (FACT n))) =
|
|
2267 |
(%n::nat. inverse (real (FACT n)))"
|
14516
|
2268 |
by (import transc EXP_FDIFF)
|
|
2269 |
|
17652
|
2270 |
lemma SIN_FDIFF: "diffs
|
|
2271 |
(%n::nat. if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) =
|
|
2272 |
(%n::nat. if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0)"
|
14516
|
2273 |
by (import transc SIN_FDIFF)
|
|
2274 |
|
17652
|
2275 |
lemma COS_FDIFF: "diffs (%n::nat. if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) =
|
|
2276 |
(%n::nat. - (if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)))"
|
14516
|
2277 |
by (import transc COS_FDIFF)
|
|
2278 |
|
17644
|
2279 |
lemma SIN_NEGLEMMA: "ALL x::real.
|
14516
|
2280 |
- sin x =
|
|
2281 |
suminf
|
17644
|
2282 |
(%n::nat.
|
17652
|
2283 |
- ((if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
|
17644
|
2284 |
x ^ n))"
|
14516
|
2285 |
by (import transc SIN_NEGLEMMA)
|
|
2286 |
|
17644
|
2287 |
lemma DIFF_EXP: "ALL x::real. diffl exp (exp x) x"
|
14516
|
2288 |
by (import transc DIFF_EXP)
|
|
2289 |
|
17644
|
2290 |
lemma DIFF_SIN: "ALL x::real. diffl sin (cos x) x"
|
14516
|
2291 |
by (import transc DIFF_SIN)
|
|
2292 |
|
17644
|
2293 |
lemma DIFF_COS: "ALL x::real. diffl cos (- sin x) x"
|
14516
|
2294 |
by (import transc DIFF_COS)
|
|
2295 |
|
17694
|
2296 |
lemma DIFF_COMPOSITE: "(diffl (f::real => real) (l::real) (x::real) & f x ~= 0 -->
|
|
2297 |
diffl (%x::real. inverse (f x)) (- (l / f x ^ 2)) x) &
|
|
2298 |
(diffl f l x & diffl (g::real => real) (m::real) x & g x ~= 0 -->
|
|
2299 |
diffl (%x::real. f x / g x) ((l * g x - m * f x) / g x ^ 2) x) &
|
|
2300 |
(diffl f l x & diffl g m x --> diffl (%x::real. f x + g x) (l + m) x) &
|
|
2301 |
(diffl f l x & diffl g m x -->
|
|
2302 |
diffl (%x::real. f x * g x) (l * g x + m * f x) x) &
|
|
2303 |
(diffl f l x & diffl g m x --> diffl (%x::real. f x - g x) (l - m) x) &
|
|
2304 |
(diffl f l x --> diffl (%x::real. - f x) (- l) x) &
|
|
2305 |
(diffl g m x -->
|
|
2306 |
diffl (%x::real. g x ^ (n::nat)) (real n * g x ^ (n - 1) * m) x) &
|
|
2307 |
(diffl g m x --> diffl (%x::real. exp (g x)) (exp (g x) * m) x) &
|
|
2308 |
(diffl g m x --> diffl (%x::real. sin (g x)) (cos (g x) * m) x) &
|
|
2309 |
(diffl g m x --> diffl (%x::real. cos (g x)) (- sin (g x) * m) x)"
|
14516
|
2310 |
by (import transc DIFF_COMPOSITE)
|
|
2311 |
|
17652
|
2312 |
lemma EXP_0: "exp 0 = 1"
|
14516
|
2313 |
by (import transc EXP_0)
|
|
2314 |
|
17652
|
2315 |
lemma EXP_LE_X: "ALL x>=0. 1 + x <= exp x"
|
14516
|
2316 |
by (import transc EXP_LE_X)
|
|
2317 |
|
17652
|
2318 |
lemma EXP_LT_1: "ALL x>0. 1 < exp x"
|
14516
|
2319 |
by (import transc EXP_LT_1)
|
|
2320 |
|
17644
|
2321 |
lemma EXP_ADD_MUL: "ALL (x::real) y::real. exp (x + y) * exp (- x) = exp y"
|
14516
|
2322 |
by (import transc EXP_ADD_MUL)
|
|
2323 |
|
17652
|
2324 |
lemma EXP_NEG_MUL: "ALL x::real. exp x * exp (- x) = 1"
|
14516
|
2325 |
by (import transc EXP_NEG_MUL)
|
|
2326 |
|
17652
|
2327 |
lemma EXP_NEG_MUL2: "ALL x::real. exp (- x) * exp x = 1"
|
14516
|
2328 |
by (import transc EXP_NEG_MUL2)
|
|
2329 |
|
17644
|
2330 |
lemma EXP_NEG: "ALL x::real. exp (- x) = inverse (exp x)"
|
14516
|
2331 |
by (import transc EXP_NEG)
|
|
2332 |
|
17644
|
2333 |
lemma EXP_ADD: "ALL (x::real) y::real. exp (x + y) = exp x * exp y"
|
14516
|
2334 |
by (import transc EXP_ADD)
|
|
2335 |
|
17652
|
2336 |
lemma EXP_POS_LE: "ALL x::real. 0 <= exp x"
|
14516
|
2337 |
by (import transc EXP_POS_LE)
|
|
2338 |
|
17652
|
2339 |
lemma EXP_NZ: "ALL x::real. exp x ~= 0"
|
14516
|
2340 |
by (import transc EXP_NZ)
|
|
2341 |
|
17652
|
2342 |
lemma EXP_POS_LT: "ALL x::real. 0 < exp x"
|
14516
|
2343 |
by (import transc EXP_POS_LT)
|
|
2344 |
|
17644
|
2345 |
lemma EXP_N: "ALL (n::nat) x::real. exp (real n * x) = exp x ^ n"
|
14516
|
2346 |
by (import transc EXP_N)
|
|
2347 |
|
17644
|
2348 |
lemma EXP_SUB: "ALL (x::real) y::real. exp (x - y) = exp x / exp y"
|
14516
|
2349 |
by (import transc EXP_SUB)
|
|
2350 |
|
17694
|
2351 |
lemma EXP_MONO_IMP: "ALL (x::real) y::real. x < y --> exp x < exp y"
|
14516
|
2352 |
by (import transc EXP_MONO_IMP)
|
|
2353 |
|
17644
|
2354 |
lemma EXP_MONO_LT: "ALL (x::real) y::real. (exp x < exp y) = (x < y)"
|
14516
|
2355 |
by (import transc EXP_MONO_LT)
|
|
2356 |
|
17644
|
2357 |
lemma EXP_MONO_LE: "ALL (x::real) y::real. (exp x <= exp y) = (x <= y)"
|
14516
|
2358 |
by (import transc EXP_MONO_LE)
|
|
2359 |
|
17644
|
2360 |
lemma EXP_INJ: "ALL (x::real) y::real. (exp x = exp y) = (x = y)"
|
14516
|
2361 |
by (import transc EXP_INJ)
|
|
2362 |
|
17652
|
2363 |
lemma EXP_TOTAL_LEMMA: "ALL y>=1. EX x>=0. x <= y - 1 & exp x = y"
|
14516
|
2364 |
by (import transc EXP_TOTAL_LEMMA)
|
|
2365 |
|
17652
|
2366 |
lemma EXP_TOTAL: "ALL y>0. EX x::real. exp x = y"
|
14516
|
2367 |
by (import transc EXP_TOTAL)
|
|
2368 |
|
|
2369 |
constdefs
|
|
2370 |
ln :: "real => real"
|
17644
|
2371 |
"ln == %x::real. SOME u::real. exp u = x"
|
|
2372 |
|
|
2373 |
lemma ln: "ALL x::real. ln x = (SOME u::real. exp u = x)"
|
14516
|
2374 |
by (import transc ln)
|
|
2375 |
|
17644
|
2376 |
lemma LN_EXP: "ALL x::real. ln (exp x) = x"
|
14516
|
2377 |
by (import transc LN_EXP)
|
|
2378 |
|
17652
|
2379 |
lemma EXP_LN: "ALL x::real. (exp (ln x) = x) = (0 < x)"
|
14516
|
2380 |
by (import transc EXP_LN)
|
|
2381 |
|
17694
|
2382 |
lemma LN_MUL: "ALL (x::real) y::real. 0 < x & 0 < y --> ln (x * y) = ln x + ln y"
|
14516
|
2383 |
by (import transc LN_MUL)
|
|
2384 |
|
17694
|
2385 |
lemma LN_INJ: "ALL (x::real) y::real. 0 < x & 0 < y --> (ln x = ln y) = (x = y)"
|
14516
|
2386 |
by (import transc LN_INJ)
|
|
2387 |
|
17652
|
2388 |
lemma LN_1: "ln 1 = 0"
|
14516
|
2389 |
by (import transc LN_1)
|
|
2390 |
|
17652
|
2391 |
lemma LN_INV: "ALL x>0. ln (inverse x) = - ln x"
|
14516
|
2392 |
by (import transc LN_INV)
|
|
2393 |
|
17694
|
2394 |
lemma LN_DIV: "ALL (x::real) y::real. 0 < x & 0 < y --> ln (x / y) = ln x - ln y"
|
14516
|
2395 |
by (import transc LN_DIV)
|
|
2396 |
|
17694
|
2397 |
lemma LN_MONO_LT: "ALL (x::real) y::real. 0 < x & 0 < y --> (ln x < ln y) = (x < y)"
|
14516
|
2398 |
by (import transc LN_MONO_LT)
|
|
2399 |
|
17694
|
2400 |
lemma LN_MONO_LE: "ALL (x::real) y::real. 0 < x & 0 < y --> (ln x <= ln y) = (x <= y)"
|
14516
|
2401 |
by (import transc LN_MONO_LE)
|
|
2402 |
|
17694
|
2403 |
lemma LN_POW: "ALL (n::nat) x::real. 0 < x --> ln (x ^ n) = real n * ln x"
|
14516
|
2404 |
by (import transc LN_POW)
|
|
2405 |
|
17652
|
2406 |
lemma LN_LE: "ALL x>=0. ln (1 + x) <= x"
|
14516
|
2407 |
by (import transc LN_LE)
|
|
2408 |
|
17652
|
2409 |
lemma LN_LT_X: "ALL x>0. ln x < x"
|
14516
|
2410 |
by (import transc LN_LT_X)
|
|
2411 |
|
17652
|
2412 |
lemma LN_POS: "ALL x>=1. 0 <= ln x"
|
14516
|
2413 |
by (import transc LN_POS)
|
|
2414 |
|
|
2415 |
constdefs
|
|
2416 |
root :: "nat => real => real"
|
17694
|
2417 |
"root == %(n::nat) x::real. SOME u::real. (0 < x --> 0 < u) & u ^ n = x"
|
|
2418 |
|
|
2419 |
lemma root: "ALL (n::nat) x::real.
|
|
2420 |
root n x = (SOME u::real. (0 < x --> 0 < u) & u ^ n = x)"
|
14516
|
2421 |
by (import transc root)
|
|
2422 |
|
|
2423 |
constdefs
|
|
2424 |
sqrt :: "real => real"
|
17652
|
2425 |
"sqrt == root 2"
|
|
2426 |
|
|
2427 |
lemma sqrt: "ALL x::real. sqrt x = root 2 x"
|
14516
|
2428 |
by (import transc sqrt)
|
|
2429 |
|
17694
|
2430 |
lemma ROOT_LT_LEMMA: "ALL (n::nat) x::real. 0 < x --> exp (ln x / real (Suc n)) ^ Suc n = x"
|
14516
|
2431 |
by (import transc ROOT_LT_LEMMA)
|
|
2432 |
|
17694
|
2433 |
lemma ROOT_LN: "ALL (n::nat) x::real. 0 < x --> root (Suc n) x = exp (ln x / real (Suc n))"
|
14516
|
2434 |
by (import transc ROOT_LN)
|
|
2435 |
|
17652
|
2436 |
lemma ROOT_0: "ALL n::nat. root (Suc n) 0 = 0"
|
14516
|
2437 |
by (import transc ROOT_0)
|
|
2438 |
|
17652
|
2439 |
lemma ROOT_1: "ALL n::nat. root (Suc n) 1 = 1"
|
14516
|
2440 |
by (import transc ROOT_1)
|
|
2441 |
|
17694
|
2442 |
lemma ROOT_POS_LT: "ALL (n::nat) x::real. 0 < x --> 0 < root (Suc n) x"
|
14516
|
2443 |
by (import transc ROOT_POS_LT)
|
|
2444 |
|
17694
|
2445 |
lemma ROOT_POW_POS: "ALL (n::nat) x::real. 0 <= x --> root (Suc n) x ^ Suc n = x"
|
14516
|
2446 |
by (import transc ROOT_POW_POS)
|
|
2447 |
|
17694
|
2448 |
lemma POW_ROOT_POS: "ALL (n::nat) x::real. 0 <= x --> root (Suc n) (x ^ Suc n) = x"
|
14516
|
2449 |
by (import transc POW_ROOT_POS)
|
|
2450 |
|
17694
|
2451 |
lemma ROOT_POS: "ALL (n::nat) x::real. 0 <= x --> 0 <= root (Suc n) x"
|
14516
|
2452 |
by (import transc ROOT_POS)
|
|
2453 |
|
17694
|
2454 |
lemma ROOT_POS_UNIQ: "ALL (n::nat) (x::real) y::real.
|
|
2455 |
0 <= x & 0 <= y & y ^ Suc n = x --> root (Suc n) x = y"
|
14516
|
2456 |
by (import transc ROOT_POS_UNIQ)
|
|
2457 |
|
17694
|
2458 |
lemma ROOT_MUL: "ALL (n::nat) (x::real) y::real.
|
|
2459 |
0 <= x & 0 <= y -->
|
|
2460 |
root (Suc n) (x * y) = root (Suc n) x * root (Suc n) y"
|
14516
|
2461 |
by (import transc ROOT_MUL)
|
|
2462 |
|
17694
|
2463 |
lemma ROOT_INV: "ALL (n::nat) x::real.
|
|
2464 |
0 <= x --> root (Suc n) (inverse x) = inverse (root (Suc n) x)"
|
14516
|
2465 |
by (import transc ROOT_INV)
|
|
2466 |
|
17694
|
2467 |
lemma ROOT_DIV: "ALL (x::nat) (xa::real) xb::real.
|
|
2468 |
0 <= xa & 0 <= xb -->
|
|
2469 |
root (Suc x) (xa / xb) = root (Suc x) xa / root (Suc x) xb"
|
14516
|
2470 |
by (import transc ROOT_DIV)
|
|
2471 |
|
17694
|
2472 |
lemma ROOT_MONO_LE: "ALL (x::real) y::real.
|
|
2473 |
0 <= x & x <= y --> root (Suc (n::nat)) x <= root (Suc n) y"
|
14516
|
2474 |
by (import transc ROOT_MONO_LE)
|
|
2475 |
|
17652
|
2476 |
lemma SQRT_0: "sqrt 0 = 0"
|
14516
|
2477 |
by (import transc SQRT_0)
|
|
2478 |
|
17652
|
2479 |
lemma SQRT_1: "sqrt 1 = 1"
|
14516
|
2480 |
by (import transc SQRT_1)
|
|
2481 |
|
17652
|
2482 |
lemma SQRT_POS_LT: "ALL x>0. 0 < sqrt x"
|
14516
|
2483 |
by (import transc SQRT_POS_LT)
|
|
2484 |
|
17652
|
2485 |
lemma SQRT_POS_LE: "ALL x>=0. 0 <= sqrt x"
|
14516
|
2486 |
by (import transc SQRT_POS_LE)
|
|
2487 |
|
17652
|
2488 |
lemma SQRT_POW2: "ALL x::real. (sqrt x ^ 2 = x) = (0 <= x)"
|
14516
|
2489 |
by (import transc SQRT_POW2)
|
|
2490 |
|
17652
|
2491 |
lemma SQRT_POW_2: "ALL x>=0. sqrt x ^ 2 = x"
|
14516
|
2492 |
by (import transc SQRT_POW_2)
|
|
2493 |
|
17694
|
2494 |
lemma POW_2_SQRT: "0 <= (x::real) --> sqrt (x ^ 2) = x"
|
14516
|
2495 |
by (import transc POW_2_SQRT)
|
|
2496 |
|
17694
|
2497 |
lemma SQRT_POS_UNIQ: "ALL (x::real) xa::real. 0 <= x & 0 <= xa & xa ^ 2 = x --> sqrt x = xa"
|
14516
|
2498 |
by (import transc SQRT_POS_UNIQ)
|
|
2499 |
|
17694
|
2500 |
lemma SQRT_MUL: "ALL (x::real) xa::real.
|
|
2501 |
0 <= x & 0 <= xa --> sqrt (x * xa) = sqrt x * sqrt xa"
|
14516
|
2502 |
by (import transc SQRT_MUL)
|
|
2503 |
|
17652
|
2504 |
lemma SQRT_INV: "ALL x>=0. sqrt (inverse x) = inverse (sqrt x)"
|
14516
|
2505 |
by (import transc SQRT_INV)
|
|
2506 |
|
17694
|
2507 |
lemma SQRT_DIV: "ALL (x::real) xa::real.
|
|
2508 |
0 <= x & 0 <= xa --> sqrt (x / xa) = sqrt x / sqrt xa"
|
14516
|
2509 |
by (import transc SQRT_DIV)
|
|
2510 |
|
17694
|
2511 |
lemma SQRT_MONO_LE: "ALL (x::real) xa::real. 0 <= x & x <= xa --> sqrt x <= sqrt xa"
|
14516
|
2512 |
by (import transc SQRT_MONO_LE)
|
|
2513 |
|
17694
|
2514 |
lemma SQRT_EVEN_POW2: "ALL n::nat. EVEN n --> sqrt (2 ^ n) = 2 ^ (n div 2)"
|
14516
|
2515 |
by (import transc SQRT_EVEN_POW2)
|
|
2516 |
|
17652
|
2517 |
lemma REAL_DIV_SQRT: "ALL x>=0. x / sqrt x = sqrt x"
|
14516
|
2518 |
by (import transc REAL_DIV_SQRT)
|
|
2519 |
|
17694
|
2520 |
lemma SQRT_EQ: "ALL (x::real) y::real. x ^ 2 = y & 0 <= x --> x = sqrt y"
|
14516
|
2521 |
by (import transc SQRT_EQ)
|
|
2522 |
|
17652
|
2523 |
lemma SIN_0: "sin 0 = 0"
|
14516
|
2524 |
by (import transc SIN_0)
|
|
2525 |
|
17652
|
2526 |
lemma COS_0: "cos 0 = 1"
|
14516
|
2527 |
by (import transc COS_0)
|
|
2528 |
|
17652
|
2529 |
lemma SIN_CIRCLE: "ALL x::real. sin x ^ 2 + cos x ^ 2 = 1"
|
14516
|
2530 |
by (import transc SIN_CIRCLE)
|
|
2531 |
|
17652
|
2532 |
lemma SIN_BOUND: "ALL x::real. abs (sin x) <= 1"
|
14516
|
2533 |
by (import transc SIN_BOUND)
|
|
2534 |
|
17652
|
2535 |
lemma SIN_BOUNDS: "ALL x::real. - 1 <= sin x & sin x <= 1"
|
14516
|
2536 |
by (import transc SIN_BOUNDS)
|
|
2537 |
|
17652
|
2538 |
lemma COS_BOUND: "ALL x::real. abs (cos x) <= 1"
|
14516
|
2539 |
by (import transc COS_BOUND)
|
|
2540 |
|
17652
|
2541 |
lemma COS_BOUNDS: "ALL x::real. - 1 <= cos x & cos x <= 1"
|
14516
|
2542 |
by (import transc COS_BOUNDS)
|
|
2543 |
|
17644
|
2544 |
lemma SIN_COS_ADD: "ALL (x::real) y::real.
|
14516
|
2545 |
(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
|
|
2546 |
(cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 =
|
17652
|
2547 |
0"
|
14516
|
2548 |
by (import transc SIN_COS_ADD)
|
|
2549 |
|
17652
|
2550 |
lemma SIN_COS_NEG: "ALL x::real. (sin (- x) + sin x) ^ 2 + (cos (- x) - cos x) ^ 2 = 0"
|
14516
|
2551 |
by (import transc SIN_COS_NEG)
|
|
2552 |
|
17644
|
2553 |
lemma SIN_ADD: "ALL (x::real) y::real. sin (x + y) = sin x * cos y + cos x * sin y"
|
14516
|
2554 |
by (import transc SIN_ADD)
|
|
2555 |
|
17644
|
2556 |
lemma COS_ADD: "ALL (x::real) y::real. cos (x + y) = cos x * cos y - sin x * sin y"
|
14516
|
2557 |
by (import transc COS_ADD)
|
|
2558 |
|
17644
|
2559 |
lemma SIN_NEG: "ALL x::real. sin (- x) = - sin x"
|
14516
|
2560 |
by (import transc SIN_NEG)
|
|
2561 |
|
17644
|
2562 |
lemma COS_NEG: "ALL x::real. cos (- x) = cos x"
|
14516
|
2563 |
by (import transc COS_NEG)
|
|
2564 |
|
17652
|
2565 |
lemma SIN_DOUBLE: "ALL x::real. sin (2 * x) = 2 * (sin x * cos x)"
|
14516
|
2566 |
by (import transc SIN_DOUBLE)
|
|
2567 |
|
17652
|
2568 |
lemma COS_DOUBLE: "ALL x::real. cos (2 * x) = cos x ^ 2 - sin x ^ 2"
|
14516
|
2569 |
by (import transc COS_DOUBLE)
|
|
2570 |
|
17644
|
2571 |
lemma SIN_PAIRED: "ALL x::real.
|
17652
|
2572 |
sums (%n::nat. (- 1) ^ n / real (FACT (2 * n + 1)) * x ^ (2 * n + 1))
|
17644
|
2573 |
(sin x)"
|
14516
|
2574 |
by (import transc SIN_PAIRED)
|
|
2575 |
|
17694
|
2576 |
lemma SIN_POS: "ALL x::real. 0 < x & x < 2 --> 0 < sin x"
|
14516
|
2577 |
by (import transc SIN_POS)
|
|
2578 |
|
17644
|
2579 |
lemma COS_PAIRED: "ALL x::real.
|
17652
|
2580 |
sums (%n::nat. (- 1) ^ n / real (FACT (2 * n)) * x ^ (2 * n)) (cos x)"
|
14516
|
2581 |
by (import transc COS_PAIRED)
|
|
2582 |
|
17652
|
2583 |
lemma COS_2: "cos 2 < 0"
|
14516
|
2584 |
by (import transc COS_2)
|
|
2585 |
|
17652
|
2586 |
lemma COS_ISZERO: "EX! x::real. 0 <= x & x <= 2 & cos x = 0"
|
14516
|
2587 |
by (import transc COS_ISZERO)
|
|
2588 |
|
|
2589 |
constdefs
|
|
2590 |
pi :: "real"
|
17652
|
2591 |
"pi == 2 * (SOME x::real. 0 <= x & x <= 2 & cos x = 0)"
|
|
2592 |
|
|
2593 |
lemma pi: "pi = 2 * (SOME x::real. 0 <= x & x <= 2 & cos x = 0)"
|
14516
|
2594 |
by (import transc pi)
|
|
2595 |
|
17652
|
2596 |
lemma PI2: "pi / 2 = (SOME x::real. 0 <= x & x <= 2 & cos x = 0)"
|
14516
|
2597 |
by (import transc PI2)
|
|
2598 |
|
17652
|
2599 |
lemma COS_PI2: "cos (pi / 2) = 0"
|
14516
|
2600 |
by (import transc COS_PI2)
|
|
2601 |
|
17652
|
2602 |
lemma PI2_BOUNDS: "0 < pi / 2 & pi / 2 < 2"
|
14516
|
2603 |
by (import transc PI2_BOUNDS)
|
|
2604 |
|
17652
|
2605 |
lemma PI_POS: "0 < pi"
|
14516
|
2606 |
by (import transc PI_POS)
|
|
2607 |
|
17652
|
2608 |
lemma SIN_PI2: "sin (pi / 2) = 1"
|
14516
|
2609 |
by (import transc SIN_PI2)
|
|
2610 |
|
17652
|
2611 |
lemma COS_PI: "cos pi = - 1"
|
14516
|
2612 |
by (import transc COS_PI)
|
|
2613 |
|
17652
|
2614 |
lemma SIN_PI: "sin pi = 0"
|
14516
|
2615 |
by (import transc SIN_PI)
|
|
2616 |
|
17652
|
2617 |
lemma SIN_COS: "ALL x::real. sin x = cos (pi / 2 - x)"
|
14516
|
2618 |
by (import transc SIN_COS)
|
|
2619 |
|
17652
|
2620 |
lemma COS_SIN: "ALL x::real. cos x = sin (pi / 2 - x)"
|
14516
|
2621 |
by (import transc COS_SIN)
|
|
2622 |
|
17644
|
2623 |
lemma SIN_PERIODIC_PI: "ALL x::real. sin (x + pi) = - sin x"
|
14516
|
2624 |
by (import transc SIN_PERIODIC_PI)
|
|
2625 |
|
17644
|
2626 |
lemma COS_PERIODIC_PI: "ALL x::real. cos (x + pi) = - cos x"
|
14516
|
2627 |
by (import transc COS_PERIODIC_PI)
|
|
2628 |
|
17652
|
2629 |
lemma SIN_PERIODIC: "ALL x::real. sin (x + 2 * pi) = sin x"
|
14516
|
2630 |
by (import transc SIN_PERIODIC)
|
|
2631 |
|
17652
|
2632 |
lemma COS_PERIODIC: "ALL x::real. cos (x + 2 * pi) = cos x"
|
14516
|
2633 |
by (import transc COS_PERIODIC)
|
|
2634 |
|
17652
|
2635 |
lemma COS_NPI: "ALL n::nat. cos (real n * pi) = (- 1) ^ n"
|
14516
|
2636 |
by (import transc COS_NPI)
|
|
2637 |
|
17652
|
2638 |
lemma SIN_NPI: "ALL n::nat. sin (real n * pi) = 0"
|
14516
|
2639 |
by (import transc SIN_NPI)
|
|
2640 |
|
17694
|
2641 |
lemma SIN_POS_PI2: "ALL x::real. 0 < x & x < pi / 2 --> 0 < sin x"
|
14516
|
2642 |
by (import transc SIN_POS_PI2)
|
|
2643 |
|
17694
|
2644 |
lemma COS_POS_PI2: "ALL x::real. 0 < x & x < pi / 2 --> 0 < cos x"
|
14516
|
2645 |
by (import transc COS_POS_PI2)
|
|
2646 |
|
17694
|
2647 |
lemma COS_POS_PI: "ALL x::real. - (pi / 2) < x & x < pi / 2 --> 0 < cos x"
|
14516
|
2648 |
by (import transc COS_POS_PI)
|
|
2649 |
|
17694
|
2650 |
lemma SIN_POS_PI: "ALL x::real. 0 < x & x < pi --> 0 < sin x"
|
14516
|
2651 |
by (import transc SIN_POS_PI)
|
|
2652 |
|
17694
|
2653 |
lemma COS_POS_PI2_LE: "ALL x::real. 0 <= x & x <= pi / 2 --> 0 <= cos x"
|
14516
|
2654 |
by (import transc COS_POS_PI2_LE)
|
|
2655 |
|
17694
|
2656 |
lemma COS_POS_PI_LE: "ALL x::real. - (pi / 2) <= x & x <= pi / 2 --> 0 <= cos x"
|
14516
|
2657 |
by (import transc COS_POS_PI_LE)
|
|
2658 |
|
17694
|
2659 |
lemma SIN_POS_PI2_LE: "ALL x::real. 0 <= x & x <= pi / 2 --> 0 <= sin x"
|
14516
|
2660 |
by (import transc SIN_POS_PI2_LE)
|
|
2661 |
|
17694
|
2662 |
lemma SIN_POS_PI_LE: "ALL x::real. 0 <= x & x <= pi --> 0 <= sin x"
|
14516
|
2663 |
by (import transc SIN_POS_PI_LE)
|
|
2664 |
|
17694
|
2665 |
lemma COS_TOTAL: "ALL y::real.
|
|
2666 |
- 1 <= y & y <= 1 --> (EX! x::real. 0 <= x & x <= pi & cos x = y)"
|
14516
|
2667 |
by (import transc COS_TOTAL)
|
|
2668 |
|
17694
|
2669 |
lemma SIN_TOTAL: "ALL y::real.
|
|
2670 |
- 1 <= y & y <= 1 -->
|
|
2671 |
(EX! x::real. - (pi / 2) <= x & x <= pi / 2 & sin x = y)"
|
14516
|
2672 |
by (import transc SIN_TOTAL)
|
|
2673 |
|
17694
|
2674 |
lemma COS_ZERO_LEMMA: "ALL x::real.
|
|
2675 |
0 <= x & cos x = 0 --> (EX n::nat. ~ EVEN n & x = real n * (pi / 2))"
|
14516
|
2676 |
by (import transc COS_ZERO_LEMMA)
|
|
2677 |
|
17694
|
2678 |
lemma SIN_ZERO_LEMMA: "ALL x::real.
|
|
2679 |
0 <= x & sin x = 0 --> (EX n::nat. EVEN n & x = real n * (pi / 2))"
|
14516
|
2680 |
by (import transc SIN_ZERO_LEMMA)
|
|
2681 |
|
17644
|
2682 |
lemma COS_ZERO: "ALL x::real.
|
17652
|
2683 |
(cos x = 0) =
|
|
2684 |
((EX n::nat. ~ EVEN n & x = real n * (pi / 2)) |
|
|
2685 |
(EX n::nat. ~ EVEN n & x = - (real n * (pi / 2))))"
|
14516
|
2686 |
by (import transc COS_ZERO)
|
|
2687 |
|
17644
|
2688 |
lemma SIN_ZERO: "ALL x::real.
|
17652
|
2689 |
(sin x = 0) =
|
|
2690 |
((EX n::nat. EVEN n & x = real n * (pi / 2)) |
|
|
2691 |
(EX n::nat. EVEN n & x = - (real n * (pi / 2))))"
|
14516
|
2692 |
by (import transc SIN_ZERO)
|
|
2693 |
|
|
2694 |
constdefs
|
|
2695 |
tan :: "real => real"
|
17644
|
2696 |
"tan == %x::real. sin x / cos x"
|
|
2697 |
|
|
2698 |
lemma tan: "ALL x::real. tan x = sin x / cos x"
|
14516
|
2699 |
by (import transc tan)
|
|
2700 |
|
17652
|
2701 |
lemma TAN_0: "tan 0 = 0"
|
14516
|
2702 |
by (import transc TAN_0)
|
|
2703 |
|
17652
|
2704 |
lemma TAN_PI: "tan pi = 0"
|
14516
|
2705 |
by (import transc TAN_PI)
|
|
2706 |
|
17652
|
2707 |
lemma TAN_NPI: "ALL n::nat. tan (real n * pi) = 0"
|
14516
|
2708 |
by (import transc TAN_NPI)
|
|
2709 |
|
17644
|
2710 |
lemma TAN_NEG: "ALL x::real. tan (- x) = - tan x"
|
14516
|
2711 |
by (import transc TAN_NEG)
|
|
2712 |
|
17652
|
2713 |
lemma TAN_PERIODIC: "ALL x::real. tan (x + 2 * pi) = tan x"
|
14516
|
2714 |
by (import transc TAN_PERIODIC)
|
|
2715 |
|
17694
|
2716 |
lemma TAN_ADD: "ALL (x::real) y::real.
|
|
2717 |
cos x ~= 0 & cos y ~= 0 & cos (x + y) ~= 0 -->
|
|
2718 |
tan (x + y) = (tan x + tan y) / (1 - tan x * tan y)"
|
14516
|
2719 |
by (import transc TAN_ADD)
|
|
2720 |
|
17694
|
2721 |
lemma TAN_DOUBLE: "ALL x::real.
|
|
2722 |
cos x ~= 0 & cos (2 * x) ~= 0 -->
|
|
2723 |
tan (2 * x) = 2 * tan x / (1 - tan x ^ 2)"
|
14516
|
2724 |
by (import transc TAN_DOUBLE)
|
|
2725 |
|
17694
|
2726 |
lemma TAN_POS_PI2: "ALL x::real. 0 < x & x < pi / 2 --> 0 < tan x"
|
14516
|
2727 |
by (import transc TAN_POS_PI2)
|
|
2728 |
|
17694
|
2729 |
lemma DIFF_TAN: "ALL x::real. cos x ~= 0 --> diffl tan (inverse (cos x ^ 2)) x"
|
14516
|
2730 |
by (import transc DIFF_TAN)
|
|
2731 |
|
17652
|
2732 |
lemma TAN_TOTAL_LEMMA: "ALL y>0. EX x>0. x < pi / 2 & y < tan x"
|
14516
|
2733 |
by (import transc TAN_TOTAL_LEMMA)
|
|
2734 |
|
17652
|
2735 |
lemma TAN_TOTAL_POS: "ALL y>=0. EX x>=0. x < pi / 2 & tan x = y"
|
14516
|
2736 |
by (import transc TAN_TOTAL_POS)
|
|
2737 |
|
17652
|
2738 |
lemma TAN_TOTAL: "ALL y::real. EX! x::real. - (pi / 2) < x & x < pi / 2 & tan x = y"
|
14516
|
2739 |
by (import transc TAN_TOTAL)
|
|
2740 |
|
|
2741 |
constdefs
|
|
2742 |
asn :: "real => real"
|
17652
|
2743 |
"asn == %y::real. SOME x::real. - (pi / 2) <= x & x <= pi / 2 & sin x = y"
|
17644
|
2744 |
|
|
2745 |
lemma asn: "ALL y::real.
|
17652
|
2746 |
asn y = (SOME x::real. - (pi / 2) <= x & x <= pi / 2 & sin x = y)"
|
14516
|
2747 |
by (import transc asn)
|
|
2748 |
|
|
2749 |
constdefs
|
|
2750 |
acs :: "real => real"
|
17652
|
2751 |
"acs == %y::real. SOME x::real. 0 <= x & x <= pi & cos x = y"
|
|
2752 |
|
|
2753 |
lemma acs: "ALL y::real. acs y = (SOME x::real. 0 <= x & x <= pi & cos x = y)"
|
14516
|
2754 |
by (import transc acs)
|
|
2755 |
|
|
2756 |
constdefs
|
|
2757 |
atn :: "real => real"
|
17652
|
2758 |
"atn == %y::real. SOME x::real. - (pi / 2) < x & x < pi / 2 & tan x = y"
|
|
2759 |
|
|
2760 |
lemma atn: "ALL y::real. atn y = (SOME x::real. - (pi / 2) < x & x < pi / 2 & tan x = y)"
|
14516
|
2761 |
by (import transc atn)
|
|
2762 |
|
17694
|
2763 |
lemma ASN: "ALL y::real.
|
|
2764 |
- 1 <= y & y <= 1 -->
|
|
2765 |
- (pi / 2) <= asn y & asn y <= pi / 2 & sin (asn y) = y"
|
14516
|
2766 |
by (import transc ASN)
|
|
2767 |
|
17694
|
2768 |
lemma ASN_SIN: "ALL y::real. - 1 <= y & y <= 1 --> sin (asn y) = y"
|
14516
|
2769 |
by (import transc ASN_SIN)
|
|
2770 |
|
17694
|
2771 |
lemma ASN_BOUNDS: "ALL y::real. - 1 <= y & y <= 1 --> - (pi / 2) <= asn y & asn y <= pi / 2"
|
14516
|
2772 |
by (import transc ASN_BOUNDS)
|
|
2773 |
|
17694
|
2774 |
lemma ASN_BOUNDS_LT: "ALL y::real. - 1 < y & y < 1 --> - (pi / 2) < asn y & asn y < pi / 2"
|
14516
|
2775 |
by (import transc ASN_BOUNDS_LT)
|
|
2776 |
|
17694
|
2777 |
lemma SIN_ASN: "ALL x::real. - (pi / 2) <= x & x <= pi / 2 --> asn (sin x) = x"
|
14516
|
2778 |
by (import transc SIN_ASN)
|
|
2779 |
|
17694
|
2780 |
lemma ACS: "ALL y::real.
|
|
2781 |
- 1 <= y & y <= 1 --> 0 <= acs y & acs y <= pi & cos (acs y) = y"
|
14516
|
2782 |
by (import transc ACS)
|
|
2783 |
|
17694
|
2784 |
lemma ACS_COS: "ALL y::real. - 1 <= y & y <= 1 --> cos (acs y) = y"
|
14516
|
2785 |
by (import transc ACS_COS)
|
|
2786 |
|
17694
|
2787 |
lemma ACS_BOUNDS: "ALL y::real. - 1 <= y & y <= 1 --> 0 <= acs y & acs y <= pi"
|
14516
|
2788 |
by (import transc ACS_BOUNDS)
|
|
2789 |
|
17694
|
2790 |
lemma ACS_BOUNDS_LT: "ALL y::real. - 1 < y & y < 1 --> 0 < acs y & acs y < pi"
|
14516
|
2791 |
by (import transc ACS_BOUNDS_LT)
|
|
2792 |
|
17694
|
2793 |
lemma COS_ACS: "ALL x::real. 0 <= x & x <= pi --> acs (cos x) = x"
|
14516
|
2794 |
by (import transc COS_ACS)
|
|
2795 |
|
17652
|
2796 |
lemma ATN: "ALL y::real. - (pi / 2) < atn y & atn y < pi / 2 & tan (atn y) = y"
|
14516
|
2797 |
by (import transc ATN)
|
|
2798 |
|
17644
|
2799 |
lemma ATN_TAN: "ALL x::real. tan (atn x) = x"
|
14516
|
2800 |
by (import transc ATN_TAN)
|
|
2801 |
|
17652
|
2802 |
lemma ATN_BOUNDS: "ALL x::real. - (pi / 2) < atn x & atn x < pi / 2"
|
14516
|
2803 |
by (import transc ATN_BOUNDS)
|
|
2804 |
|
17694
|
2805 |
lemma TAN_ATN: "ALL x::real. - (pi / 2) < x & x < pi / 2 --> atn (tan x) = x"
|
14516
|
2806 |
by (import transc TAN_ATN)
|
|
2807 |
|
17694
|
2808 |
lemma TAN_SEC: "ALL x::real. cos x ~= 0 --> 1 + tan x ^ 2 = inverse (cos x) ^ 2"
|
14516
|
2809 |
by (import transc TAN_SEC)
|
|
2810 |
|
17694
|
2811 |
lemma SIN_COS_SQ: "ALL x::real. 0 <= x & x <= pi --> sin x = sqrt (1 - cos x ^ 2)"
|
14516
|
2812 |
by (import transc SIN_COS_SQ)
|
|
2813 |
|
17694
|
2814 |
lemma COS_SIN_SQ: "ALL x::real. - (pi / 2) <= x & x <= pi / 2 --> cos x = sqrt (1 - sin x ^ 2)"
|
14516
|
2815 |
by (import transc COS_SIN_SQ)
|
|
2816 |
|
17652
|
2817 |
lemma COS_ATN_NZ: "ALL x::real. cos (atn x) ~= 0"
|
14516
|
2818 |
by (import transc COS_ATN_NZ)
|
|
2819 |
|
17694
|
2820 |
lemma COS_ASN_NZ: "ALL x::real. - 1 < x & x < 1 --> cos (asn x) ~= 0"
|
14516
|
2821 |
by (import transc COS_ASN_NZ)
|
|
2822 |
|
17694
|
2823 |
lemma SIN_ACS_NZ: "ALL x::real. - 1 < x & x < 1 --> sin (acs x) ~= 0"
|
14516
|
2824 |
by (import transc SIN_ACS_NZ)
|
|
2825 |
|
17694
|
2826 |
lemma COS_SIN_SQRT: "ALL x::real. 0 <= cos x --> cos x = sqrt (1 - sin x ^ 2)"
|
14516
|
2827 |
by (import transc COS_SIN_SQRT)
|
|
2828 |
|
17694
|
2829 |
lemma SIN_COS_SQRT: "ALL x::real. 0 <= sin x --> sin x = sqrt (1 - cos x ^ 2)"
|
14516
|
2830 |
by (import transc SIN_COS_SQRT)
|
|
2831 |
|
17652
|
2832 |
lemma DIFF_LN: "ALL x>0. diffl ln (inverse x) x"
|
14516
|
2833 |
by (import transc DIFF_LN)
|
|
2834 |
|
17694
|
2835 |
lemma DIFF_ASN_LEMMA: "ALL x::real. - 1 < x & x < 1 --> diffl asn (inverse (cos (asn x))) x"
|
14516
|
2836 |
by (import transc DIFF_ASN_LEMMA)
|
|
2837 |
|
17694
|
2838 |
lemma DIFF_ASN: "ALL x::real. - 1 < x & x < 1 --> diffl asn (inverse (sqrt (1 - x ^ 2))) x"
|
14516
|
2839 |
by (import transc DIFF_ASN)
|
|
2840 |
|
17694
|
2841 |
lemma DIFF_ACS_LEMMA: "ALL x::real. - 1 < x & x < 1 --> diffl acs (inverse (- sin (acs x))) x"
|
14516
|
2842 |
by (import transc DIFF_ACS_LEMMA)
|
|
2843 |
|
17694
|
2844 |
lemma DIFF_ACS: "ALL x::real. - 1 < x & x < 1 --> diffl acs (- inverse (sqrt (1 - x ^ 2))) x"
|
14516
|
2845 |
by (import transc DIFF_ACS)
|
|
2846 |
|
17652
|
2847 |
lemma DIFF_ATN: "ALL x::real. diffl atn (inverse (1 + x ^ 2)) x"
|
14516
|
2848 |
by (import transc DIFF_ATN)
|
|
2849 |
|
|
2850 |
constdefs
|
|
2851 |
division :: "real * real => (nat => real) => bool"
|
|
2852 |
"(op ==::(real * real => (nat => real) => bool)
|
|
2853 |
=> (real * real => (nat => real) => bool) => prop)
|
|
2854 |
(division::real * real => (nat => real) => bool)
|
|
2855 |
((split::(real => real => (nat => real) => bool)
|
|
2856 |
=> real * real => (nat => real) => bool)
|
|
2857 |
(%(a::real) (b::real) D::nat => real.
|
|
2858 |
(op &::bool => bool => bool)
|
|
2859 |
((op =::real => real => bool) (D (0::nat)) a)
|
|
2860 |
((Ex::(nat => bool) => bool)
|
|
2861 |
(%N::nat.
|
|
2862 |
(op &::bool => bool => bool)
|
|
2863 |
((All::(nat => bool) => bool)
|
|
2864 |
(%n::nat.
|
|
2865 |
(op -->::bool => bool => bool)
|
|
2866 |
((op <::nat => nat => bool) n N)
|
|
2867 |
((op <::real => real => bool) (D n)
|
|
2868 |
(D ((Suc::nat => nat) n)))))
|
|
2869 |
((All::(nat => bool) => bool)
|
|
2870 |
(%n::nat.
|
|
2871 |
(op -->::bool => bool => bool)
|
|
2872 |
((op <=::nat => nat => bool) N n)
|
|
2873 |
((op =::real => real => bool) (D n) b)))))))"
|
|
2874 |
|
|
2875 |
lemma division: "(All::(real => bool) => bool)
|
|
2876 |
(%a::real.
|
|
2877 |
(All::(real => bool) => bool)
|
|
2878 |
(%b::real.
|
|
2879 |
(All::((nat => real) => bool) => bool)
|
|
2880 |
(%D::nat => real.
|
|
2881 |
(op =::bool => bool => bool)
|
|
2882 |
((division::real * real => (nat => real) => bool)
|
|
2883 |
((Pair::real => real => real * real) a b) D)
|
|
2884 |
((op &::bool => bool => bool)
|
|
2885 |
((op =::real => real => bool) (D (0::nat)) a)
|
|
2886 |
((Ex::(nat => bool) => bool)
|
|
2887 |
(%N::nat.
|
|
2888 |
(op &::bool => bool => bool)
|
|
2889 |
((All::(nat => bool) => bool)
|
|
2890 |
(%n::nat.
|
|
2891 |
(op -->::bool => bool => bool)
|
|
2892 |
((op <::nat => nat => bool) n N)
|
|
2893 |
((op <::real => real => bool) (D n)
|
|
2894 |
(D ((Suc::nat => nat) n)))))
|
|
2895 |
((All::(nat => bool) => bool)
|
|
2896 |
(%n::nat.
|
|
2897 |
(op -->::bool => bool => bool)
|
|
2898 |
((op <=::nat => nat => bool) N n)
|
|
2899 |
((op =::real => real => bool) (D n)
|
|
2900 |
b)))))))))"
|
|
2901 |
by (import transc division)
|
|
2902 |
|
|
2903 |
constdefs
|
|
2904 |
dsize :: "(nat => real) => nat"
|
|
2905 |
"(op ==::((nat => real) => nat) => ((nat => real) => nat) => prop)
|
|
2906 |
(dsize::(nat => real) => nat)
|
|
2907 |
(%D::nat => real.
|
|
2908 |
(Eps::(nat => bool) => nat)
|
|
2909 |
(%N::nat.
|
|
2910 |
(op &::bool => bool => bool)
|
|
2911 |
((All::(nat => bool) => bool)
|
|
2912 |
(%n::nat.
|
|
2913 |
(op -->::bool => bool => bool)
|
|
2914 |
((op <::nat => nat => bool) n N)
|
|
2915 |
((op <::real => real => bool) (D n)
|
|
2916 |
(D ((Suc::nat => nat) n)))))
|
|
2917 |
((All::(nat => bool) => bool)
|
|
2918 |
(%n::nat.
|
|
2919 |
(op -->::bool => bool => bool)
|
|
2920 |
((op <=::nat => nat => bool) N n)
|
|
2921 |
((op =::real => real => bool) (D n) (D N))))))"
|
|
2922 |
|
|
2923 |
lemma dsize: "(All::((nat => real) => bool) => bool)
|
|
2924 |
(%D::nat => real.
|
|
2925 |
(op =::nat => nat => bool) ((dsize::(nat => real) => nat) D)
|
|
2926 |
((Eps::(nat => bool) => nat)
|
|
2927 |
(%N::nat.
|
|
2928 |
(op &::bool => bool => bool)
|
|
2929 |
((All::(nat => bool) => bool)
|
|
2930 |
(%n::nat.
|
|
2931 |
(op -->::bool => bool => bool)
|
|
2932 |
((op <::nat => nat => bool) n N)
|
|
2933 |
((op <::real => real => bool) (D n)
|
|
2934 |
(D ((Suc::nat => nat) n)))))
|
|
2935 |
((All::(nat => bool) => bool)
|
|
2936 |
(%n::nat.
|
|
2937 |
(op -->::bool => bool => bool)
|
|
2938 |
((op <=::nat => nat => bool) N n)
|
|
2939 |
((op =::real => real => bool) (D n) (D N)))))))"
|
|
2940 |
by (import transc dsize)
|
|
2941 |
|
|
2942 |
constdefs
|
|
2943 |
tdiv :: "real * real => (nat => real) * (nat => real) => bool"
|
|
2944 |
"tdiv ==
|
17644
|
2945 |
%(a::real, b::real) (D::nat => real, p::nat => real).
|
|
2946 |
division (a, b) D & (ALL n::nat. D n <= p n & p n <= D (Suc n))"
|
|
2947 |
|
|
2948 |
lemma tdiv: "ALL (a::real) (b::real) (D::nat => real) p::nat => real.
|
14516
|
2949 |
tdiv (a, b) (D, p) =
|
17644
|
2950 |
(division (a, b) D & (ALL n::nat. D n <= p n & p n <= D (Suc n)))"
|
14516
|
2951 |
by (import transc tdiv)
|
|
2952 |
|
|
2953 |
constdefs
|
|
2954 |
gauge :: "(real => bool) => (real => real) => bool"
|
17694
|
2955 |
"gauge == %(E::real => bool) g::real => real. ALL x::real. E x --> 0 < g x"
|
|
2956 |
|
|
2957 |
lemma gauge: "ALL (E::real => bool) g::real => real.
|
|
2958 |
gauge E g = (ALL x::real. E x --> 0 < g x)"
|
14516
|
2959 |
by (import transc gauge)
|
|
2960 |
|
|
2961 |
constdefs
|
|
2962 |
fine :: "(real => real) => (nat => real) * (nat => real) => bool"
|
14847
|
2963 |
"(op ==::((real => real) => (nat => real) * (nat => real) => bool)
|
|
2964 |
=> ((real => real) => (nat => real) * (nat => real) => bool)
|
|
2965 |
=> prop)
|
|
2966 |
(fine::(real => real) => (nat => real) * (nat => real) => bool)
|
|
2967 |
(%g::real => real.
|
|
2968 |
(split::((nat => real) => (nat => real) => bool)
|
|
2969 |
=> (nat => real) * (nat => real) => bool)
|
|
2970 |
(%(D::nat => real) p::nat => real.
|
|
2971 |
(All::(nat => bool) => bool)
|
|
2972 |
(%n::nat.
|
|
2973 |
(op -->::bool => bool => bool)
|
|
2974 |
((op <::nat => nat => bool) n
|
|
2975 |
((dsize::(nat => real) => nat) D))
|
|
2976 |
((op <::real => real => bool)
|
|
2977 |
((op -::real => real => real) (D ((Suc::nat => nat) n))
|
|
2978 |
(D n))
|
|
2979 |
(g (p n))))))"
|
|
2980 |
|
|
2981 |
lemma fine: "(All::((real => real) => bool) => bool)
|
|
2982 |
(%g::real => real.
|
|
2983 |
(All::((nat => real) => bool) => bool)
|
|
2984 |
(%D::nat => real.
|
|
2985 |
(All::((nat => real) => bool) => bool)
|
|
2986 |
(%p::nat => real.
|
|
2987 |
(op =::bool => bool => bool)
|
|
2988 |
((fine::(real => real)
|
|
2989 |
=> (nat => real) * (nat => real) => bool)
|
|
2990 |
g ((Pair::(nat => real)
|
|
2991 |
=> (nat => real)
|
|
2992 |
=> (nat => real) * (nat => real))
|
|
2993 |
D p))
|
|
2994 |
((All::(nat => bool) => bool)
|
|
2995 |
(%n::nat.
|
|
2996 |
(op -->::bool => bool => bool)
|
|
2997 |
((op <::nat => nat => bool) n
|
|
2998 |
((dsize::(nat => real) => nat) D))
|
|
2999 |
((op <::real => real => bool)
|
|
3000 |
((op -::real => real => real)
|
|
3001 |
(D ((Suc::nat => nat) n)) (D n))
|
|
3002 |
(g (p n))))))))"
|
14516
|
3003 |
by (import transc fine)
|
|
3004 |
|
|
3005 |
constdefs
|
|
3006 |
rsum :: "(nat => real) * (nat => real) => (real => real) => real"
|
17644
|
3007 |
"rsum ==
|
|
3008 |
%(D::nat => real, p::nat => real) f::real => real.
|
17652
|
3009 |
real.sum (0, dsize D) (%n::nat. f (p n) * (D (Suc n) - D n))"
|
17644
|
3010 |
|
|
3011 |
lemma rsum: "ALL (D::nat => real) (p::nat => real) f::real => real.
|
|
3012 |
rsum (D, p) f =
|
17652
|
3013 |
real.sum (0, dsize D) (%n::nat. f (p n) * (D (Suc n) - D n))"
|
14516
|
3014 |
by (import transc rsum)
|
|
3015 |
|
|
3016 |
constdefs
|
|
3017 |
Dint :: "real * real => (real => real) => real => bool"
|
17694
|
3018 |
"Dint ==
|
|
3019 |
%(a::real, b::real) (f::real => real) k::real.
|
|
3020 |
ALL e>0.
|
|
3021 |
EX g::real => real.
|
|
3022 |
gauge (%x::real. a <= x & x <= b) g &
|
|
3023 |
(ALL (D::nat => real) p::nat => real.
|
|
3024 |
tdiv (a, b) (D, p) & fine g (D, p) -->
|
|
3025 |
abs (rsum (D, p) f - k) < e)"
|
|
3026 |
|
|
3027 |
lemma Dint: "ALL (a::real) (b::real) (f::real => real) k::real.
|
|
3028 |
Dint (a, b) f k =
|
|
3029 |
(ALL e>0.
|
|
3030 |
EX g::real => real.
|
|
3031 |
gauge (%x::real. a <= x & x <= b) g &
|
|
3032 |
(ALL (D::nat => real) p::nat => real.
|
|
3033 |
tdiv (a, b) (D, p) & fine g (D, p) -->
|
|
3034 |
abs (rsum (D, p) f - k) < e))"
|
14516
|
3035 |
by (import transc Dint)
|
|
3036 |
|
17694
|
3037 |
lemma DIVISION_0: "ALL (a::real) b::real. a = b --> dsize (%n::nat. if n = 0 then a else b) = 0"
|
14516
|
3038 |
by (import transc DIVISION_0)
|
|
3039 |
|
17694
|
3040 |
lemma DIVISION_1: "ALL (a::real) b::real. a < b --> dsize (%n::nat. if n = 0 then a else b) = 1"
|
14516
|
3041 |
by (import transc DIVISION_1)
|
|
3042 |
|
17694
|
3043 |
lemma DIVISION_SINGLE: "ALL (a::real) b::real.
|
|
3044 |
a <= b --> division (a, b) (%n::nat. if n = 0 then a else b)"
|
14516
|
3045 |
by (import transc DIVISION_SINGLE)
|
|
3046 |
|
17694
|
3047 |
lemma DIVISION_LHS: "ALL (D::nat => real) (a::real) b::real. division (a, b) D --> D 0 = a"
|
14516
|
3048 |
by (import transc DIVISION_LHS)
|
|
3049 |
|
|
3050 |
lemma DIVISION_THM: "(All::((nat => real) => bool) => bool)
|
|
3051 |
(%D::nat => real.
|
|
3052 |
(All::(real => bool) => bool)
|
|
3053 |
(%a::real.
|
|
3054 |
(All::(real => bool) => bool)
|
|
3055 |
(%b::real.
|
|
3056 |
(op =::bool => bool => bool)
|
|
3057 |
((division::real * real => (nat => real) => bool)
|
|
3058 |
((Pair::real => real => real * real) a b) D)
|
|
3059 |
((op &::bool => bool => bool)
|
|
3060 |
((op =::real => real => bool) (D (0::nat)) a)
|
|
3061 |
((op &::bool => bool => bool)
|
|
3062 |
((All::(nat => bool) => bool)
|
|
3063 |
(%n::nat.
|
|
3064 |
(op -->::bool => bool => bool)
|
|
3065 |
((op <::nat => nat => bool) n
|
|
3066 |
((dsize::(nat => real) => nat) D))
|
|
3067 |
((op <::real => real => bool) (D n)
|
|
3068 |
(D ((Suc::nat => nat) n)))))
|
|
3069 |
((All::(nat => bool) => bool)
|
|
3070 |
(%n::nat.
|
|
3071 |
(op -->::bool => bool => bool)
|
|
3072 |
((op <=::nat => nat => bool)
|
|
3073 |
((dsize::(nat => real) => nat) D) n)
|
|
3074 |
((op =::real => real => bool) (D n) b))))))))"
|
|
3075 |
by (import transc DIVISION_THM)
|
|
3076 |
|
17694
|
3077 |
lemma DIVISION_RHS: "ALL (D::nat => real) (a::real) b::real.
|
|
3078 |
division (a, b) D --> D (dsize D) = b"
|
14516
|
3079 |
by (import transc DIVISION_RHS)
|
|
3080 |
|
17694
|
3081 |
lemma DIVISION_LT_GEN: "ALL (D::nat => real) (a::real) (b::real) (m::nat) n::nat.
|
|
3082 |
division (a, b) D & m < n & n <= dsize D --> D m < D n"
|
14516
|
3083 |
by (import transc DIVISION_LT_GEN)
|
|
3084 |
|
|
3085 |
lemma DIVISION_LT: "(All::((nat => real) => bool) => bool)
|
|
3086 |
(%D::nat => real.
|
|
3087 |
(All::(real => bool) => bool)
|
|
3088 |
(%a::real.
|
|
3089 |
(All::(real => bool) => bool)
|
|
3090 |
(%b::real.
|
|
3091 |
(op -->::bool => bool => bool)
|
|
3092 |
((division::real * real => (nat => real) => bool)
|
|
3093 |
((Pair::real => real => real * real) a b) D)
|
|
3094 |
((All::(nat => bool) => bool)
|
|
3095 |
(%n::nat.
|
|
3096 |
(op -->::bool => bool => bool)
|
|
3097 |
((op <::nat => nat => bool) n
|
|
3098 |
((dsize::(nat => real) => nat) D))
|
|
3099 |
((op <::real => real => bool) (D (0::nat))
|
|
3100 |
(D ((Suc::nat => nat) n))))))))"
|
|
3101 |
by (import transc DIVISION_LT)
|
|
3102 |
|
17694
|
3103 |
lemma DIVISION_LE: "ALL (D::nat => real) (a::real) b::real. division (a, b) D --> a <= b"
|
14516
|
3104 |
by (import transc DIVISION_LE)
|
|
3105 |
|
|
3106 |
lemma DIVISION_GT: "(All::((nat => real) => bool) => bool)
|
|
3107 |
(%D::nat => real.
|
|
3108 |
(All::(real => bool) => bool)
|
|
3109 |
(%a::real.
|
|
3110 |
(All::(real => bool) => bool)
|
|
3111 |
(%b::real.
|
|
3112 |
(op -->::bool => bool => bool)
|
|
3113 |
((division::real * real => (nat => real) => bool)
|
|
3114 |
((Pair::real => real => real * real) a b) D)
|
|
3115 |
((All::(nat => bool) => bool)
|
|
3116 |
(%n::nat.
|
|
3117 |
(op -->::bool => bool => bool)
|
|
3118 |
((op <::nat => nat => bool) n
|
|
3119 |
((dsize::(nat => real) => nat) D))
|
|
3120 |
((op <::real => real => bool) (D n)
|
|
3121 |
(D ((dsize::(nat => real) => nat) D))))))))"
|
|
3122 |
by (import transc DIVISION_GT)
|
|
3123 |
|
17694
|
3124 |
lemma DIVISION_EQ: "ALL (D::nat => real) (a::real) b::real.
|
|
3125 |
division (a, b) D --> (a = b) = (dsize D = 0)"
|
14516
|
3126 |
by (import transc DIVISION_EQ)
|
|
3127 |
|
17694
|
3128 |
lemma DIVISION_LBOUND: "ALL (D::nat => real) (a::real) b::real.
|
|
3129 |
division (a, b) D --> (ALL r::nat. a <= D r)"
|
14516
|
3130 |
by (import transc DIVISION_LBOUND)
|
|
3131 |
|
17694
|
3132 |
lemma DIVISION_LBOUND_LT: "ALL (D::nat => real) (a::real) b::real.
|
|
3133 |
division (a, b) D & dsize D ~= 0 --> (ALL n::nat. a < D (Suc n))"
|
14516
|
3134 |
by (import transc DIVISION_LBOUND_LT)
|
|
3135 |
|
17694
|
3136 |
lemma DIVISION_UBOUND: "ALL (D::nat => real) (a::real) b::real.
|
|
3137 |
division (a, b) D --> (ALL r::nat. D r <= b)"
|
14516
|
3138 |
by (import transc DIVISION_UBOUND)
|
|
3139 |
|
17694
|
3140 |
lemma DIVISION_UBOUND_LT: "ALL (D::nat => real) (a::real) (b::real) n::nat.
|
|
3141 |
division (a, b) D & n < dsize D --> D n < b"
|
14516
|
3142 |
by (import transc DIVISION_UBOUND_LT)
|
|
3143 |
|
17694
|
3144 |
lemma DIVISION_APPEND: "ALL (a::real) (b::real) c::real.
|
|
3145 |
(EX (D1::nat => real) p1::nat => real.
|
|
3146 |
tdiv (a, b) (D1, p1) & fine (g::real => real) (D1, p1)) &
|
|
3147 |
(EX (D2::nat => real) p2::nat => real.
|
|
3148 |
tdiv (b, c) (D2, p2) & fine g (D2, p2)) -->
|
|
3149 |
(EX (x::nat => real) p::nat => real. tdiv (a, c) (x, p) & fine g (x, p))"
|
14516
|
3150 |
by (import transc DIVISION_APPEND)
|
|
3151 |
|
17694
|
3152 |
lemma DIVISION_EXISTS: "ALL (a::real) (b::real) g::real => real.
|
|
3153 |
a <= b & gauge (%x::real. a <= x & x <= b) g -->
|
|
3154 |
(EX (D::nat => real) p::nat => real. tdiv (a, b) (D, p) & fine g (D, p))"
|
14516
|
3155 |
by (import transc DIVISION_EXISTS)
|
|
3156 |
|
17694
|
3157 |
lemma GAUGE_MIN: "ALL (E::real => bool) (g1::real => real) g2::real => real.
|
|
3158 |
gauge E g1 & gauge E g2 -->
|
|
3159 |
gauge E (%x::real. if g1 x < g2 x then g1 x else g2 x)"
|
14516
|
3160 |
by (import transc GAUGE_MIN)
|
|
3161 |
|
17694
|
3162 |
lemma FINE_MIN: "ALL (g1::real => real) (g2::real => real) (D::nat => real) p::nat => real.
|
|
3163 |
fine (%x::real. if g1 x < g2 x then g1 x else g2 x) (D, p) -->
|
|
3164 |
fine g1 (D, p) & fine g2 (D, p)"
|
14516
|
3165 |
by (import transc FINE_MIN)
|
|
3166 |
|
17694
|
3167 |
lemma DINT_UNIQ: "ALL (a::real) (b::real) (f::real => real) (k1::real) k2::real.
|
|
3168 |
a <= b & Dint (a, b) f k1 & Dint (a, b) f k2 --> k1 = k2"
|
14516
|
3169 |
by (import transc DINT_UNIQ)
|
|
3170 |
|
17652
|
3171 |
lemma INTEGRAL_NULL: "ALL (f::real => real) a::real. Dint (a, a) f 0"
|
14516
|
3172 |
by (import transc INTEGRAL_NULL)
|
|
3173 |
|
17694
|
3174 |
lemma FTC1: "ALL (f::real => real) (f'::real => real) (a::real) b::real.
|
|
3175 |
a <= b & (ALL x::real. a <= x & x <= b --> diffl f (f' x) x) -->
|
|
3176 |
Dint (a, b) f' (f b - f a)"
|
14516
|
3177 |
by (import transc FTC1)
|
|
3178 |
|
17694
|
3179 |
lemma MCLAURIN: "ALL (f::real => real) (diff::nat => real => real) (h::real) n::nat.
|
|
3180 |
0 < h &
|
|
3181 |
0 < n &
|
|
3182 |
diff 0 = f &
|
|
3183 |
(ALL (m::nat) t::real.
|
|
3184 |
m < n & 0 <= t & t <= h --> diffl (diff m) (diff (Suc m) t) t) -->
|
|
3185 |
(EX t>0.
|
|
3186 |
t < h &
|
|
3187 |
f h =
|
|
3188 |
real.sum (0, n) (%m::nat. diff m 0 / real (FACT m) * h ^ m) +
|
|
3189 |
diff n t / real (FACT n) * h ^ n)"
|
14516
|
3190 |
by (import transc MCLAURIN)
|
|
3191 |
|
17694
|
3192 |
lemma MCLAURIN_NEG: "ALL (f::real => real) (diff::nat => real => real) (h::real) n::nat.
|
|
3193 |
h < 0 &
|
|
3194 |
0 < n &
|
|
3195 |
diff 0 = f &
|
|
3196 |
(ALL (m::nat) t::real.
|
|
3197 |
m < n & h <= t & t <= 0 --> diffl (diff m) (diff (Suc m) t) t) -->
|
|
3198 |
(EX t::real.
|
|
3199 |
h < t &
|
|
3200 |
t < 0 &
|
|
3201 |
f h =
|
|
3202 |
real.sum (0, n) (%m::nat. diff m 0 / real (FACT m) * h ^ m) +
|
|
3203 |
diff n t / real (FACT n) * h ^ n)"
|
14516
|
3204 |
by (import transc MCLAURIN_NEG)
|
|
3205 |
|
17694
|
3206 |
lemma MCLAURIN_ALL_LT: "ALL (f::real => real) diff::nat => real => real.
|
|
3207 |
diff 0 = f &
|
|
3208 |
(ALL (m::nat) x::real. diffl (diff m) (diff (Suc m) x) x) -->
|
|
3209 |
(ALL (x::real) n::nat.
|
|
3210 |
x ~= 0 & 0 < n -->
|
|
3211 |
(EX t::real.
|
|
3212 |
0 < abs t &
|
|
3213 |
abs t < abs x &
|
|
3214 |
f x =
|
|
3215 |
real.sum (0, n) (%m::nat. diff m 0 / real (FACT m) * x ^ m) +
|
|
3216 |
diff n t / real (FACT n) * x ^ n))"
|
14516
|
3217 |
by (import transc MCLAURIN_ALL_LT)
|
|
3218 |
|
17694
|
3219 |
lemma MCLAURIN_ZERO: "ALL (diff::nat => real => real) (n::nat) x::real.
|
|
3220 |
x = 0 & 0 < n -->
|
|
3221 |
real.sum (0, n) (%m::nat. diff m 0 / real (FACT m) * x ^ m) = diff 0 0"
|
14516
|
3222 |
by (import transc MCLAURIN_ZERO)
|
|
3223 |
|
17694
|
3224 |
lemma MCLAURIN_ALL_LE: "ALL (f::real => real) diff::nat => real => real.
|
|
3225 |
diff 0 = f &
|
|
3226 |
(ALL (m::nat) x::real. diffl (diff m) (diff (Suc m) x) x) -->
|
|
3227 |
(ALL (x::real) n::nat.
|
|
3228 |
EX t::real.
|
|
3229 |
abs t <= abs x &
|
|
3230 |
f x =
|
|
3231 |
real.sum (0, n) (%m::nat. diff m 0 / real (FACT m) * x ^ m) +
|
|
3232 |
diff n t / real (FACT n) * x ^ n)"
|
14516
|
3233 |
by (import transc MCLAURIN_ALL_LE)
|
|
3234 |
|
17694
|
3235 |
lemma MCLAURIN_EXP_LT: "ALL (x::real) n::nat.
|
|
3236 |
x ~= 0 & 0 < n -->
|
|
3237 |
(EX xa::real.
|
|
3238 |
0 < abs xa &
|
|
3239 |
abs xa < abs x &
|
|
3240 |
exp x =
|
|
3241 |
real.sum (0, n) (%m::nat. x ^ m / real (FACT m)) +
|
|
3242 |
exp xa / real (FACT n) * x ^ n)"
|
14516
|
3243 |
by (import transc MCLAURIN_EXP_LT)
|
|
3244 |
|
17644
|
3245 |
lemma MCLAURIN_EXP_LE: "ALL (x::real) n::nat.
|
|
3246 |
EX xa::real.
|
14516
|
3247 |
abs xa <= abs x &
|
|
3248 |
exp x =
|
17652
|
3249 |
real.sum (0, n) (%m::nat. x ^ m / real (FACT m)) +
|
14516
|
3250 |
exp xa / real (FACT n) * x ^ n"
|
|
3251 |
by (import transc MCLAURIN_EXP_LE)
|
|
3252 |
|
17694
|
3253 |
lemma DIFF_LN_COMPOSITE: "ALL (g::real => real) (m::real) x::real.
|
|
3254 |
diffl g m x & 0 < g x -->
|
|
3255 |
diffl (%x::real. ln (g x)) (inverse (g x) * m) x"
|
14516
|
3256 |
by (import transc DIFF_LN_COMPOSITE)
|
|
3257 |
|
|
3258 |
;end_setup
|
|
3259 |
|
|
3260 |
;setup_theory poly
|
|
3261 |
|
|
3262 |
consts
|
|
3263 |
poly :: "real list => real => real"
|
|
3264 |
|
17652
|
3265 |
specification (poly_primdef: poly) poly_def: "(ALL x::real. poly [] x = 0) &
|
17644
|
3266 |
(ALL (h::real) (t::real list) x::real. poly (h # t) x = h + x * poly t x)"
|
14516
|
3267 |
by (import poly poly_def)
|
|
3268 |
|
|
3269 |
consts
|
|
3270 |
poly_add :: "real list => real list => real list"
|
|
3271 |
|
17644
|
3272 |
specification (poly_add_primdef: poly_add) poly_add_def: "(ALL l2::real list. poly_add [] l2 = l2) &
|
|
3273 |
(ALL (h::real) (t::real list) l2::real list.
|
14516
|
3274 |
poly_add (h # t) l2 =
|
|
3275 |
(if l2 = [] then h # t else (h + hd l2) # poly_add t (tl l2)))"
|
|
3276 |
by (import poly poly_add_def)
|
|
3277 |
|
|
3278 |
consts
|
|
3279 |
"##" :: "real => real list => real list" ("##")
|
|
3280 |
|
17644
|
3281 |
specification ("##") poly_cmul_def: "(ALL c::real. ## c [] = []) &
|
|
3282 |
(ALL (c::real) (h::real) t::real list. ## c (h # t) = c * h # ## c t)"
|
14516
|
3283 |
by (import poly poly_cmul_def)
|
|
3284 |
|
|
3285 |
consts
|
|
3286 |
poly_neg :: "real list => real list"
|
|
3287 |
|
|
3288 |
defs
|
17652
|
3289 |
poly_neg_primdef: "poly_neg == ## (- 1)"
|
|
3290 |
|
|
3291 |
lemma poly_neg_def: "poly_neg = ## (- 1)"
|
14516
|
3292 |
by (import poly poly_neg_def)
|
|
3293 |
|
|
3294 |
consts
|
|
3295 |
poly_mul :: "real list => real list => real list"
|
|
3296 |
|
17644
|
3297 |
specification (poly_mul_primdef: poly_mul) poly_mul_def: "(ALL l2::real list. poly_mul [] l2 = []) &
|
|
3298 |
(ALL (h::real) (t::real list) l2::real list.
|
14516
|
3299 |
poly_mul (h # t) l2 =
|
17652
|
3300 |
(if t = [] then ## h l2 else poly_add (## h l2) (0 # poly_mul t l2)))"
|
14516
|
3301 |
by (import poly poly_mul_def)
|
|
3302 |
|
|
3303 |
consts
|
|
3304 |
poly_exp :: "real list => nat => real list"
|
|
3305 |
|
17652
|
3306 |
specification (poly_exp_primdef: poly_exp) poly_exp_def: "(ALL p::real list. poly_exp p 0 = [1]) &
|
17644
|
3307 |
(ALL (p::real list) n::nat. poly_exp p (Suc n) = poly_mul p (poly_exp p n))"
|
14516
|
3308 |
by (import poly poly_exp_def)
|
|
3309 |
|
|
3310 |
consts
|
|
3311 |
poly_diff_aux :: "nat => real list => real list"
|
|
3312 |
|
17644
|
3313 |
specification (poly_diff_aux_primdef: poly_diff_aux) poly_diff_aux_def: "(ALL n::nat. poly_diff_aux n [] = []) &
|
|
3314 |
(ALL (n::nat) (h::real) t::real list.
|
|
3315 |
poly_diff_aux n (h # t) = real n * h # poly_diff_aux (Suc n) t)"
|
14516
|
3316 |
by (import poly poly_diff_aux_def)
|
|
3317 |
|
|
3318 |
constdefs
|
|
3319 |
diff :: "real list => real list"
|
17652
|
3320 |
"diff == %l::real list. if l = [] then [] else poly_diff_aux 1 (tl l)"
|
|
3321 |
|
|
3322 |
lemma poly_diff_def: "ALL l::real list. diff l = (if l = [] then [] else poly_diff_aux 1 (tl l))"
|
14516
|
3323 |
by (import poly poly_diff_def)
|
|
3324 |
|
17644
|
3325 |
lemma POLY_ADD_CLAUSES: "poly_add [] (p2::real list) = p2 &
|
|
3326 |
poly_add (p1::real list) [] = p1 &
|
|
3327 |
poly_add ((h1::real) # (t1::real list)) ((h2::real) # (t2::real list)) =
|
|
3328 |
(h1 + h2) # poly_add t1 t2"
|
14516
|
3329 |
by (import poly POLY_ADD_CLAUSES)
|
|
3330 |
|
17644
|
3331 |
lemma POLY_CMUL_CLAUSES: "## (c::real) [] = [] & ## c ((h::real) # (t::real list)) = c * h # ## c t"
|
14516
|
3332 |
by (import poly POLY_CMUL_CLAUSES)
|
|
3333 |
|
17644
|
3334 |
lemma POLY_NEG_CLAUSES: "poly_neg [] = [] & poly_neg ((h::real) # (t::real list)) = - h # poly_neg t"
|
14516
|
3335 |
by (import poly POLY_NEG_CLAUSES)
|
|
3336 |
|
17644
|
3337 |
lemma POLY_MUL_CLAUSES: "poly_mul [] (p2::real list) = [] &
|
|
3338 |
poly_mul [h1::real] p2 = ## h1 p2 &
|
|
3339 |
poly_mul (h1 # (k1::real) # (t1::real list)) p2 =
|
17652
|
3340 |
poly_add (## h1 p2) (0 # poly_mul (k1 # t1) p2)"
|
14516
|
3341 |
by (import poly POLY_MUL_CLAUSES)
|
|
3342 |
|
17644
|
3343 |
lemma POLY_DIFF_CLAUSES: "diff [] = [] &
|
17652
|
3344 |
diff [c::real] = [] & diff ((h::real) # (t::real list)) = poly_diff_aux 1 t"
|
14516
|
3345 |
by (import poly POLY_DIFF_CLAUSES)
|
|
3346 |
|
17644
|
3347 |
lemma POLY_ADD: "ALL (t::real list) (p2::real list) x::real.
|
|
3348 |
poly (poly_add t p2) x = poly t x + poly p2 x"
|
14516
|
3349 |
by (import poly POLY_ADD)
|
|
3350 |
|
17644
|
3351 |
lemma POLY_CMUL: "ALL (t::real list) (c::real) x::real. poly (## c t) x = c * poly t x"
|
14516
|
3352 |
by (import poly POLY_CMUL)
|
|
3353 |
|
17644
|
3354 |
lemma POLY_NEG: "ALL (x::real list) xa::real. poly (poly_neg x) xa = - poly x xa"
|
14516
|
3355 |
by (import poly POLY_NEG)
|
|
3356 |
|
17644
|
3357 |
lemma POLY_MUL: "ALL (x::real) (t::real list) p2::real list.
|
|
3358 |
poly (poly_mul t p2) x = poly t x * poly p2 x"
|
14516
|
3359 |
by (import poly POLY_MUL)
|
|
3360 |
|
17644
|
3361 |
lemma POLY_EXP: "ALL (p::real list) (n::nat) x::real. poly (poly_exp p n) x = poly p x ^ n"
|
14516
|
3362 |
by (import poly POLY_EXP)
|
|
3363 |
|
17644
|
3364 |
lemma POLY_DIFF_LEMMA: "ALL (t::real list) (n::nat) x::real.
|
|
3365 |
diffl (%x::real. x ^ Suc n * poly t x)
|
14516
|
3366 |
(x ^ n * poly (poly_diff_aux (Suc n) t) x) x"
|
|
3367 |
by (import poly POLY_DIFF_LEMMA)
|
|
3368 |
|
17644
|
3369 |
lemma POLY_DIFF: "ALL (t::real list) x::real. diffl (poly t) (poly (diff t) x) x"
|
14516
|
3370 |
by (import poly POLY_DIFF)
|
|
3371 |
|
17644
|
3372 |
lemma POLY_DIFFERENTIABLE: "ALL l::real list. All (differentiable (poly l))"
|
14516
|
3373 |
by (import poly POLY_DIFFERENTIABLE)
|
|
3374 |
|
17644
|
3375 |
lemma POLY_CONT: "ALL l::real list. All (contl (poly l))"
|
14516
|
3376 |
by (import poly POLY_CONT)
|
|
3377 |
|
17694
|
3378 |
lemma POLY_IVT_POS: "ALL (x::real list) (xa::real) xb::real.
|
|
3379 |
xa < xb & poly x xa < 0 & 0 < poly x xb -->
|
|
3380 |
(EX xc::real. xa < xc & xc < xb & poly x xc = 0)"
|
14516
|
3381 |
by (import poly POLY_IVT_POS)
|
|
3382 |
|
17694
|
3383 |
lemma POLY_IVT_NEG: "ALL (p::real list) (a::real) b::real.
|
|
3384 |
a < b & 0 < poly p a & poly p b < 0 -->
|
|
3385 |
(EX x::real. a < x & x < b & poly p x = 0)"
|
14516
|
3386 |
by (import poly POLY_IVT_NEG)
|
|
3387 |
|
17694
|
3388 |
lemma POLY_MVT: "ALL (p::real list) (a::real) b::real.
|
|
3389 |
a < b -->
|
|
3390 |
(EX x::real.
|
|
3391 |
a < x & x < b & poly p b - poly p a = (b - a) * poly (diff p) x)"
|
14516
|
3392 |
by (import poly POLY_MVT)
|
|
3393 |
|
17644
|
3394 |
lemma POLY_ADD_RZERO: "ALL x::real list. poly (poly_add x []) = poly x"
|
14516
|
3395 |
by (import poly POLY_ADD_RZERO)
|
|
3396 |
|
17644
|
3397 |
lemma POLY_MUL_ASSOC: "ALL (x::real list) (xa::real list) xb::real list.
|
14516
|
3398 |
poly (poly_mul x (poly_mul xa xb)) = poly (poly_mul (poly_mul x xa) xb)"
|
|
3399 |
by (import poly POLY_MUL_ASSOC)
|
|
3400 |
|
17644
|
3401 |
lemma POLY_EXP_ADD: "ALL (x::nat) (xa::nat) xb::real list.
|
14516
|
3402 |
poly (poly_exp xb (xa + x)) =
|
|
3403 |
poly (poly_mul (poly_exp xb xa) (poly_exp xb x))"
|
|
3404 |
by (import poly POLY_EXP_ADD)
|
|
3405 |
|
17644
|
3406 |
lemma POLY_DIFF_AUX_ADD: "ALL (t::real list) (p2::real list) n::nat.
|
14516
|
3407 |
poly (poly_diff_aux n (poly_add t p2)) =
|
|
3408 |
poly (poly_add (poly_diff_aux n t) (poly_diff_aux n p2))"
|
|
3409 |
by (import poly POLY_DIFF_AUX_ADD)
|
|
3410 |
|
17644
|
3411 |
lemma POLY_DIFF_AUX_CMUL: "ALL (t::real list) (c::real) n::nat.
|
|
3412 |
poly (poly_diff_aux n (## c t)) = poly (## c (poly_diff_aux n t))"
|
14516
|
3413 |
by (import poly POLY_DIFF_AUX_CMUL)
|
|
3414 |
|
17644
|
3415 |
lemma POLY_DIFF_AUX_NEG: "ALL (x::real list) xa::nat.
|
14516
|
3416 |
poly (poly_diff_aux xa (poly_neg x)) =
|
|
3417 |
poly (poly_neg (poly_diff_aux xa x))"
|
|
3418 |
by (import poly POLY_DIFF_AUX_NEG)
|
|
3419 |
|
17644
|
3420 |
lemma POLY_DIFF_AUX_MUL_LEMMA: "ALL (t::real list) n::nat.
|
14516
|
3421 |
poly (poly_diff_aux (Suc n) t) = poly (poly_add (poly_diff_aux n t) t)"
|
|
3422 |
by (import poly POLY_DIFF_AUX_MUL_LEMMA)
|
|
3423 |
|
17644
|
3424 |
lemma POLY_DIFF_ADD: "ALL (t::real list) p2::real list.
|
|
3425 |
poly (diff (poly_add t p2)) = poly (poly_add (diff t) (diff p2))"
|
14516
|
3426 |
by (import poly POLY_DIFF_ADD)
|
|
3427 |
|
17644
|
3428 |
lemma POLY_DIFF_CMUL: "ALL (t::real list) c::real. poly (diff (## c t)) = poly (## c (diff t))"
|
14516
|
3429 |
by (import poly POLY_DIFF_CMUL)
|
|
3430 |
|
17644
|
3431 |
lemma POLY_DIFF_NEG: "ALL x::real list. poly (diff (poly_neg x)) = poly (poly_neg (diff x))"
|
14516
|
3432 |
by (import poly POLY_DIFF_NEG)
|
|
3433 |
|
17644
|
3434 |
lemma POLY_DIFF_MUL_LEMMA: "ALL (x::real list) xa::real.
|
17652
|
3435 |
poly (diff (xa # x)) = poly (poly_add (0 # diff x) x)"
|
14516
|
3436 |
by (import poly POLY_DIFF_MUL_LEMMA)
|
|
3437 |
|
17644
|
3438 |
lemma POLY_DIFF_MUL: "ALL (t::real list) p2::real list.
|
14516
|
3439 |
poly (diff (poly_mul t p2)) =
|
|
3440 |
poly (poly_add (poly_mul t (diff p2)) (poly_mul (diff t) p2))"
|
|
3441 |
by (import poly POLY_DIFF_MUL)
|
|
3442 |
|
17644
|
3443 |
lemma POLY_DIFF_EXP: "ALL (p::real list) n::nat.
|
14516
|
3444 |
poly (diff (poly_exp p (Suc n))) =
|
|
3445 |
poly (poly_mul (## (real (Suc n)) (poly_exp p n)) (diff p))"
|
|
3446 |
by (import poly POLY_DIFF_EXP)
|
|
3447 |
|
17644
|
3448 |
lemma POLY_DIFF_EXP_PRIME: "ALL (n::nat) a::real.
|
17652
|
3449 |
poly (diff (poly_exp [- a, 1] (Suc n))) =
|
|
3450 |
poly (## (real (Suc n)) (poly_exp [- a, 1] n))"
|
14516
|
3451 |
by (import poly POLY_DIFF_EXP_PRIME)
|
|
3452 |
|
17644
|
3453 |
lemma POLY_LINEAR_REM: "ALL (t::real list) h::real.
|
|
3454 |
EX (q::real list) r::real.
|
17652
|
3455 |
h # t = poly_add [r] (poly_mul [- (a::real), 1] q)"
|
14516
|
3456 |
by (import poly POLY_LINEAR_REM)
|
|
3457 |
|
17644
|
3458 |
lemma POLY_LINEAR_DIVIDES: "ALL (a::real) t::real list.
|
17652
|
3459 |
(poly t a = 0) = (t = [] | (EX q::real list. t = poly_mul [- a, 1] q))"
|
14516
|
3460 |
by (import poly POLY_LINEAR_DIVIDES)
|
|
3461 |
|
17652
|
3462 |
lemma POLY_LENGTH_MUL: "ALL x::real list. length (poly_mul [- (a::real), 1] x) = Suc (length x)"
|
14516
|
3463 |
by (import poly POLY_LENGTH_MUL)
|
|
3464 |
|
|
3465 |
lemma POLY_ROOTS_INDEX_LEMMA: "(All::(nat => bool) => bool)
|
|
3466 |
(%n::nat.
|
|
3467 |
(All::(real list => bool) => bool)
|
|
3468 |
(%p::real list.
|
|
3469 |
(op -->::bool => bool => bool)
|
|
3470 |
((op &::bool => bool => bool)
|
|
3471 |
((Not::bool => bool)
|
|
3472 |
((op =::(real => real) => (real => real) => bool)
|
|
3473 |
((poly::real list => real => real) p)
|
|
3474 |
((poly::real list => real => real) ([]::real list))))
|
|
3475 |
((op =::nat => nat => bool) ((size::real list => nat) p) n))
|
|
3476 |
((Ex::((nat => real) => bool) => bool)
|
|
3477 |
(%i::nat => real.
|
|
3478 |
(All::(real => bool) => bool)
|
|
3479 |
(%x::real.
|
|
3480 |
(op -->::bool => bool => bool)
|
|
3481 |
((op =::real => real => bool)
|
|
3482 |
((poly::real list => real => real) p x) (0::real))
|
|
3483 |
((Ex::(nat => bool) => bool)
|
|
3484 |
(%m::nat.
|
|
3485 |
(op &::bool => bool => bool)
|
|
3486 |
((op <=::nat => nat => bool) m n)
|
|
3487 |
((op =::real => real => bool) x (i m)))))))))"
|
|
3488 |
by (import poly POLY_ROOTS_INDEX_LEMMA)
|
|
3489 |
|
|
3490 |
lemma POLY_ROOTS_INDEX_LENGTH: "(All::(real list => bool) => bool)
|
|
3491 |
(%p::real list.
|
|
3492 |
(op -->::bool => bool => bool)
|
|
3493 |
((Not::bool => bool)
|
|
3494 |
((op =::(real => real) => (real => real) => bool)
|
|
3495 |
((poly::real list => real => real) p)
|
|
3496 |
((poly::real list => real => real) ([]::real list))))
|
|
3497 |
((Ex::((nat => real) => bool) => bool)
|
|
3498 |
(%i::nat => real.
|
|
3499 |
(All::(real => bool) => bool)
|
|
3500 |
(%x::real.
|
|
3501 |
(op -->::bool => bool => bool)
|
|
3502 |
((op =::real => real => bool)
|
|
3503 |
((poly::real list => real => real) p x) (0::real))
|
|
3504 |
((Ex::(nat => bool) => bool)
|
|
3505 |
(%n::nat.
|
|
3506 |
(op &::bool => bool => bool)
|
|
3507 |
((op <=::nat => nat => bool) n
|
|
3508 |
((size::real list => nat) p))
|
|
3509 |
((op =::real => real => bool) x (i n))))))))"
|
|
3510 |
by (import poly POLY_ROOTS_INDEX_LENGTH)
|
|
3511 |
|
|
3512 |
lemma POLY_ROOTS_FINITE_LEMMA: "(All::(real list => bool) => bool)
|
|
3513 |
(%p::real list.
|
|
3514 |
(op -->::bool => bool => bool)
|
|
3515 |
((Not::bool => bool)
|
|
3516 |
((op =::(real => real) => (real => real) => bool)
|
|
3517 |
((poly::real list => real => real) p)
|
|
3518 |
((poly::real list => real => real) ([]::real list))))
|
|
3519 |
((Ex::(nat => bool) => bool)
|
|
3520 |
(%N::nat.
|
|
3521 |
(Ex::((nat => real) => bool) => bool)
|
|
3522 |
(%i::nat => real.
|
|
3523 |
(All::(real => bool) => bool)
|
|
3524 |
(%x::real.
|
|
3525 |
(op -->::bool => bool => bool)
|
|
3526 |
((op =::real => real => bool)
|
|
3527 |
((poly::real list => real => real) p x) (0::real))
|
|
3528 |
((Ex::(nat => bool) => bool)
|
|
3529 |
(%n::nat.
|
|
3530 |
(op &::bool => bool => bool)
|
|
3531 |
((op <::nat => nat => bool) n N)
|
|
3532 |
((op =::real => real => bool) x (i n)))))))))"
|
|
3533 |
by (import poly POLY_ROOTS_FINITE_LEMMA)
|
|
3534 |
|
|
3535 |
lemma FINITE_LEMMA: "(All::((nat => real) => bool) => bool)
|
|
3536 |
(%i::nat => real.
|
|
3537 |
(All::(nat => bool) => bool)
|
|
3538 |
(%x::nat.
|
|
3539 |
(All::((real => bool) => bool) => bool)
|
|
3540 |
(%xa::real => bool.
|
|
3541 |
(op -->::bool => bool => bool)
|
|
3542 |
((All::(real => bool) => bool)
|
|
3543 |
(%xb::real.
|
|
3544 |
(op -->::bool => bool => bool) (xa xb)
|
|
3545 |
((Ex::(nat => bool) => bool)
|
|
3546 |
(%n::nat.
|
|
3547 |
(op &::bool => bool => bool)
|
|
3548 |
((op <::nat => nat => bool) n x)
|
|
3549 |
((op =::real => real => bool) xb (i n))))))
|
|
3550 |
((Ex::(real => bool) => bool)
|
|
3551 |
(%a::real.
|
|
3552 |
(All::(real => bool) => bool)
|
|
3553 |
(%x::real.
|
|
3554 |
(op -->::bool => bool => bool) (xa x)
|
|
3555 |
((op <::real => real => bool) x a)))))))"
|
|
3556 |
by (import poly FINITE_LEMMA)
|
|
3557 |
|
|
3558 |
lemma POLY_ROOTS_FINITE: "(All::(real list => bool) => bool)
|
|
3559 |
(%p::real list.
|
|
3560 |
(op =::bool => bool => bool)
|
|
3561 |
((Not::bool => bool)
|
|
3562 |
((op =::(real => real) => (real => real) => bool)
|
|
3563 |
((poly::real list => real => real) p)
|
|
3564 |
((poly::real list => real => real) ([]::real list))))
|
|
3565 |
((Ex::(nat => bool) => bool)
|
|
3566 |
(%N::nat.
|
|
3567 |
(Ex::((nat => real) => bool) => bool)
|
|
3568 |
(%i::nat => real.
|
|
3569 |
(All::(real => bool) => bool)
|
|
3570 |
(%x::real.
|
|
3571 |
(op -->::bool => bool => bool)
|
|
3572 |
((op =::real => real => bool)
|
|
3573 |
((poly::real list => real => real) p x) (0::real))
|
|
3574 |
((Ex::(nat => bool) => bool)
|
|
3575 |
(%n::nat.
|
|
3576 |
(op &::bool => bool => bool)
|
|
3577 |
((op <::nat => nat => bool) n N)
|
|
3578 |
((op =::real => real => bool) x (i n)))))))))"
|
|
3579 |
by (import poly POLY_ROOTS_FINITE)
|
|
3580 |
|
17694
|
3581 |
lemma POLY_ENTIRE_LEMMA: "ALL (p::real list) q::real list.
|
|
3582 |
poly p ~= poly [] & poly q ~= poly [] --> poly (poly_mul p q) ~= poly []"
|
14516
|
3583 |
by (import poly POLY_ENTIRE_LEMMA)
|
|
3584 |
|
17644
|
3585 |
lemma POLY_ENTIRE: "ALL (p::real list) q::real list.
|
14516
|
3586 |
(poly (poly_mul p q) = poly []) = (poly p = poly [] | poly q = poly [])"
|
|
3587 |
by (import poly POLY_ENTIRE)
|
|
3588 |
|
17644
|
3589 |
lemma POLY_MUL_LCANCEL: "ALL (x::real list) (xa::real list) xb::real list.
|
14516
|
3590 |
(poly (poly_mul x xa) = poly (poly_mul x xb)) =
|
|
3591 |
(poly x = poly [] | poly xa = poly xb)"
|
|
3592 |
by (import poly POLY_MUL_LCANCEL)
|
|
3593 |
|
17644
|
3594 |
lemma POLY_EXP_EQ_0: "ALL (p::real list) n::nat.
|
17652
|
3595 |
(poly (poly_exp p n) = poly []) = (poly p = poly [] & n ~= 0)"
|
14516
|
3596 |
by (import poly POLY_EXP_EQ_0)
|
|
3597 |
|
17652
|
3598 |
lemma POLY_PRIME_EQ_0: "ALL a::real. poly [a, 1] ~= poly []"
|
14516
|
3599 |
by (import poly POLY_PRIME_EQ_0)
|
|
3600 |
|
17652
|
3601 |
lemma POLY_EXP_PRIME_EQ_0: "ALL (a::real) n::nat. poly (poly_exp [a, 1] n) ~= poly []"
|
14516
|
3602 |
by (import poly POLY_EXP_PRIME_EQ_0)
|
|
3603 |
|
17694
|
3604 |
lemma POLY_ZERO_LEMMA: "ALL (h::real) t::real list.
|
|
3605 |
poly (h # t) = poly [] --> h = 0 & poly t = poly []"
|
14516
|
3606 |
by (import poly POLY_ZERO_LEMMA)
|
|
3607 |
|
17652
|
3608 |
lemma POLY_ZERO: "ALL t::real list. (poly t = poly []) = list_all (%c::real. c = 0) t"
|
14516
|
3609 |
by (import poly POLY_ZERO)
|
|
3610 |
|
17644
|
3611 |
lemma POLY_DIFF_AUX_ISZERO: "ALL (t::real list) n::nat.
|
17652
|
3612 |
list_all (%c::real. c = 0) (poly_diff_aux (Suc n) t) =
|
|
3613 |
list_all (%c::real. c = 0) t"
|
14516
|
3614 |
by (import poly POLY_DIFF_AUX_ISZERO)
|
|
3615 |
|
17694
|
3616 |
lemma POLY_DIFF_ISZERO: "ALL x::real list.
|
|
3617 |
poly (diff x) = poly [] --> (EX h::real. poly x = poly [h])"
|
14516
|
3618 |
by (import poly POLY_DIFF_ISZERO)
|
|
3619 |
|
17694
|
3620 |
lemma POLY_DIFF_ZERO: "ALL x::real list. poly x = poly [] --> poly (diff x) = poly []"
|
14516
|
3621 |
by (import poly POLY_DIFF_ZERO)
|
|
3622 |
|
17694
|
3623 |
lemma POLY_DIFF_WELLDEF: "ALL (p::real list) q::real list.
|
|
3624 |
poly p = poly q --> poly (diff p) = poly (diff q)"
|
14516
|
3625 |
by (import poly POLY_DIFF_WELLDEF)
|
|
3626 |
|
|
3627 |
constdefs
|
|
3628 |
poly_divides :: "real list => real list => bool"
|
17644
|
3629 |
"poly_divides ==
|
|
3630 |
%(p1::real list) p2::real list.
|
|
3631 |
EX q::real list. poly p2 = poly (poly_mul p1 q)"
|
|
3632 |
|
|
3633 |
lemma poly_divides: "ALL (p1::real list) p2::real list.
|
|
3634 |
poly_divides p1 p2 = (EX q::real list. poly p2 = poly (poly_mul p1 q))"
|
14516
|
3635 |
by (import poly poly_divides)
|
|
3636 |
|
17644
|
3637 |
lemma POLY_PRIMES: "ALL (a::real) (p::real list) q::real list.
|
17652
|
3638 |
poly_divides [a, 1] (poly_mul p q) =
|
|
3639 |
(poly_divides [a, 1] p | poly_divides [a, 1] q)"
|
14516
|
3640 |
by (import poly POLY_PRIMES)
|
|
3641 |
|
17644
|
3642 |
lemma POLY_DIVIDES_REFL: "ALL p::real list. poly_divides p p"
|
14516
|
3643 |
by (import poly POLY_DIVIDES_REFL)
|
|
3644 |
|
17694
|
3645 |
lemma POLY_DIVIDES_TRANS: "ALL (p::real list) (q::real list) r::real list.
|
|
3646 |
poly_divides p q & poly_divides q r --> poly_divides p r"
|
14516
|
3647 |
by (import poly POLY_DIVIDES_TRANS)
|
|
3648 |
|
17694
|
3649 |
lemma POLY_DIVIDES_EXP: "ALL (p::real list) (m::nat) n::nat.
|
|
3650 |
m <= n --> poly_divides (poly_exp p m) (poly_exp p n)"
|
14516
|
3651 |
by (import poly POLY_DIVIDES_EXP)
|
|
3652 |
|
17694
|
3653 |
lemma POLY_EXP_DIVIDES: "ALL (p::real list) (q::real list) (m::nat) n::nat.
|
|
3654 |
poly_divides (poly_exp p n) q & m <= n --> poly_divides (poly_exp p m) q"
|
14516
|
3655 |
by (import poly POLY_EXP_DIVIDES)
|
|
3656 |
|
17694
|
3657 |
lemma POLY_DIVIDES_ADD: "ALL (p::real list) (q::real list) r::real list.
|
|
3658 |
poly_divides p q & poly_divides p r --> poly_divides p (poly_add q r)"
|
14516
|
3659 |
by (import poly POLY_DIVIDES_ADD)
|
|
3660 |
|
17694
|
3661 |
lemma POLY_DIVIDES_SUB: "ALL (p::real list) (q::real list) r::real list.
|
|
3662 |
poly_divides p q & poly_divides p (poly_add q r) --> poly_divides p r"
|
14516
|
3663 |
by (import poly POLY_DIVIDES_SUB)
|
|
3664 |
|
17694
|
3665 |
lemma POLY_DIVIDES_SUB2: "ALL (p::real list) (q::real list) r::real list.
|
|
3666 |
poly_divides p r & poly_divides p (poly_add q r) --> poly_divides p q"
|
14516
|
3667 |
by (import poly POLY_DIVIDES_SUB2)
|
|
3668 |
|
17694
|
3669 |
lemma POLY_DIVIDES_ZERO: "ALL (p::real list) q::real list. poly p = poly [] --> poly_divides q p"
|
14516
|
3670 |
by (import poly POLY_DIVIDES_ZERO)
|
|
3671 |
|
17694
|
3672 |
lemma POLY_ORDER_EXISTS: "ALL (a::real) (d::nat) p::real list.
|
|
3673 |
length p = d & poly p ~= poly [] -->
|
|
3674 |
(EX x::nat.
|
|
3675 |
poly_divides (poly_exp [- a, 1] x) p &
|
|
3676 |
~ poly_divides (poly_exp [- a, 1] (Suc x)) p)"
|
14516
|
3677 |
by (import poly POLY_ORDER_EXISTS)
|
|
3678 |
|
17694
|
3679 |
lemma POLY_ORDER: "ALL (p::real list) a::real.
|
|
3680 |
poly p ~= poly [] -->
|
|
3681 |
(EX! n::nat.
|
|
3682 |
poly_divides (poly_exp [- a, 1] n) p &
|
|
3683 |
~ poly_divides (poly_exp [- a, 1] (Suc n)) p)"
|
14516
|
3684 |
by (import poly POLY_ORDER)
|
|
3685 |
|
|
3686 |
constdefs
|
|
3687 |
poly_order :: "real => real list => nat"
|
|
3688 |
"poly_order ==
|
17644
|
3689 |
%(a::real) p::real list.
|
|
3690 |
SOME n::nat.
|
17652
|
3691 |
poly_divides (poly_exp [- a, 1] n) p &
|
|
3692 |
~ poly_divides (poly_exp [- a, 1] (Suc n)) p"
|
17644
|
3693 |
|
|
3694 |
lemma poly_order: "ALL (a::real) p::real list.
|
14516
|
3695 |
poly_order a p =
|
17644
|
3696 |
(SOME n::nat.
|
17652
|
3697 |
poly_divides (poly_exp [- a, 1] n) p &
|
|
3698 |
~ poly_divides (poly_exp [- a, 1] (Suc n)) p)"
|
14516
|
3699 |
by (import poly poly_order)
|
|
3700 |
|
17644
|
3701 |
lemma ORDER: "ALL (p::real list) (a::real) n::nat.
|
17652
|
3702 |
(poly_divides (poly_exp [- a, 1] n) p &
|
|
3703 |
~ poly_divides (poly_exp [- a, 1] (Suc n)) p) =
|
14516
|
3704 |
(n = poly_order a p & poly p ~= poly [])"
|
|
3705 |
by (import poly ORDER)
|
|
3706 |
|
17694
|
3707 |
lemma ORDER_THM: "ALL (p::real list) a::real.
|
|
3708 |
poly p ~= poly [] -->
|
|
3709 |
poly_divides (poly_exp [- a, 1] (poly_order a p)) p &
|
|
3710 |
~ poly_divides (poly_exp [- a, 1] (Suc (poly_order a p))) p"
|
14516
|
3711 |
by (import poly ORDER_THM)
|
|
3712 |
|
17694
|
3713 |
lemma ORDER_UNIQUE: "ALL (p::real list) (a::real) n::nat.
|
|
3714 |
poly p ~= poly [] &
|
|
3715 |
poly_divides (poly_exp [- a, 1] n) p &
|
|
3716 |
~ poly_divides (poly_exp [- a, 1] (Suc n)) p -->
|
|
3717 |
n = poly_order a p"
|
14516
|
3718 |
by (import poly ORDER_UNIQUE)
|
|
3719 |
|
17694
|
3720 |
lemma ORDER_POLY: "ALL (p::real list) (q::real list) a::real.
|
|
3721 |
poly p = poly q --> poly_order a p = poly_order a q"
|
14516
|
3722 |
by (import poly ORDER_POLY)
|
|
3723 |
|
17644
|
3724 |
lemma ORDER_ROOT: "ALL (p::real list) a::real.
|
17652
|
3725 |
(poly p a = 0) = (poly p = poly [] | poly_order a p ~= 0)"
|
14516
|
3726 |
by (import poly ORDER_ROOT)
|
|
3727 |
|
17644
|
3728 |
lemma ORDER_DIVIDES: "ALL (p::real list) (a::real) n::nat.
|
17652
|
3729 |
poly_divides (poly_exp [- a, 1] n) p =
|
14516
|
3730 |
(poly p = poly [] | n <= poly_order a p)"
|
|
3731 |
by (import poly ORDER_DIVIDES)
|
|
3732 |
|
17694
|
3733 |
lemma ORDER_DECOMP: "ALL (p::real list) a::real.
|
|
3734 |
poly p ~= poly [] -->
|
|
3735 |
(EX x::real list.
|
|
3736 |
poly p = poly (poly_mul (poly_exp [- a, 1] (poly_order a p)) x) &
|
|
3737 |
~ poly_divides [- a, 1] x)"
|
14516
|
3738 |
by (import poly ORDER_DECOMP)
|
|
3739 |
|
17694
|
3740 |
lemma ORDER_MUL: "ALL (a::real) (p::real list) q::real list.
|
|
3741 |
poly (poly_mul p q) ~= poly [] -->
|
|
3742 |
poly_order a (poly_mul p q) = poly_order a p + poly_order a q"
|
14516
|
3743 |
by (import poly ORDER_MUL)
|
|
3744 |
|
17694
|
3745 |
lemma ORDER_DIFF: "ALL (p::real list) a::real.
|
|
3746 |
poly (diff p) ~= poly [] & poly_order a p ~= 0 -->
|
|
3747 |
poly_order a p = Suc (poly_order a (diff p))"
|
14516
|
3748 |
by (import poly ORDER_DIFF)
|
|
3749 |
|
17694
|
3750 |
lemma POLY_SQUAREFREE_DECOMP_ORDER: "ALL (p::real list) (q::real list) (d::real list) (e::real list)
|
|
3751 |
(r::real list) s::real list.
|
|
3752 |
poly (diff p) ~= poly [] &
|
|
3753 |
poly p = poly (poly_mul q d) &
|
|
3754 |
poly (diff p) = poly (poly_mul e d) &
|
|
3755 |
poly d = poly (poly_add (poly_mul r p) (poly_mul s (diff p))) -->
|
|
3756 |
(ALL a::real. poly_order a q = (if poly_order a p = 0 then 0 else 1))"
|
14516
|
3757 |
by (import poly POLY_SQUAREFREE_DECOMP_ORDER)
|
|
3758 |
|
|
3759 |
constdefs
|
|
3760 |
rsquarefree :: "real list => bool"
|
|
3761 |
"rsquarefree ==
|
17644
|
3762 |
%p::real list.
|
|
3763 |
poly p ~= poly [] &
|
17652
|
3764 |
(ALL a::real. poly_order a p = 0 | poly_order a p = 1)"
|
17644
|
3765 |
|
|
3766 |
lemma rsquarefree: "ALL p::real list.
|
14516
|
3767 |
rsquarefree p =
|
17644
|
3768 |
(poly p ~= poly [] &
|
17652
|
3769 |
(ALL a::real. poly_order a p = 0 | poly_order a p = 1))"
|
14516
|
3770 |
by (import poly rsquarefree)
|
|
3771 |
|
17644
|
3772 |
lemma RSQUAREFREE_ROOTS: "ALL p::real list.
|
17652
|
3773 |
rsquarefree p = (ALL a::real. ~ (poly p a = 0 & poly (diff p) a = 0))"
|
14516
|
3774 |
by (import poly RSQUAREFREE_ROOTS)
|
|
3775 |
|
17694
|
3776 |
lemma RSQUAREFREE_DECOMP: "ALL (p::real list) a::real.
|
|
3777 |
rsquarefree p & poly p a = 0 -->
|
|
3778 |
(EX q::real list. poly p = poly (poly_mul [- a, 1] q) & poly q a ~= 0)"
|
14516
|
3779 |
by (import poly RSQUAREFREE_DECOMP)
|
|
3780 |
|
17694
|
3781 |
lemma POLY_SQUAREFREE_DECOMP: "ALL (p::real list) (q::real list) (d::real list) (e::real list)
|
|
3782 |
(r::real list) s::real list.
|
|
3783 |
poly (diff p) ~= poly [] &
|
|
3784 |
poly p = poly (poly_mul q d) &
|
|
3785 |
poly (diff p) = poly (poly_mul e d) &
|
|
3786 |
poly d = poly (poly_add (poly_mul r p) (poly_mul s (diff p))) -->
|
|
3787 |
rsquarefree q & (ALL x::real. (poly q x = 0) = (poly p x = 0))"
|
14516
|
3788 |
by (import poly POLY_SQUAREFREE_DECOMP)
|
|
3789 |
|
|
3790 |
consts
|
|
3791 |
normalize :: "real list => real list"
|
|
3792 |
|
|
3793 |
specification (normalize) normalize: "normalize [] = [] &
|
17644
|
3794 |
(ALL (h::real) t::real list.
|
14516
|
3795 |
normalize (h # t) =
|
17652
|
3796 |
(if normalize t = [] then if h = 0 then [] else [h]
|
14516
|
3797 |
else h # normalize t))"
|
|
3798 |
by (import poly normalize)
|
|
3799 |
|
17644
|
3800 |
lemma POLY_NORMALIZE: "ALL t::real list. poly (normalize t) = poly t"
|
14516
|
3801 |
by (import poly POLY_NORMALIZE)
|
|
3802 |
|
|
3803 |
constdefs
|
|
3804 |
degree :: "real list => nat"
|
17644
|
3805 |
"degree == %p::real list. PRE (length (normalize p))"
|
|
3806 |
|
|
3807 |
lemma degree: "ALL p::real list. degree p = PRE (length (normalize p))"
|
14516
|
3808 |
by (import poly degree)
|
|
3809 |
|
17694
|
3810 |
lemma DEGREE_ZERO: "ALL p::real list. poly p = poly [] --> degree p = 0"
|
14516
|
3811 |
by (import poly DEGREE_ZERO)
|
|
3812 |
|
17694
|
3813 |
lemma POLY_ROOTS_FINITE_SET: "ALL p::real list.
|
|
3814 |
poly p ~= poly [] --> FINITE (GSPEC (%x::real. (x, poly p x = 0)))"
|
14516
|
3815 |
by (import poly POLY_ROOTS_FINITE_SET)
|
|
3816 |
|
17694
|
3817 |
lemma POLY_MONO: "ALL (x::real) (k::real) xa::real list.
|
|
3818 |
abs x <= k --> abs (poly xa x) <= poly (map abs xa) k"
|
14516
|
3819 |
by (import poly POLY_MONO)
|
|
3820 |
|
|
3821 |
;end_setup
|
|
3822 |
|
|
3823 |
end
|
|
3824 |
|