import OAI.MathematicalPhysics.Transonic.Shooting.RegularGlobalRange import OAI.MathematicalPhysics.Transonic.Shooting.RegularHighEndpoint namespace OAI section namespace SepticProfile.RegularHighData open Set IntegerPolynomial theorem range_0 {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u (startTime R step0) ∈ Icc ((head step0.lo:ℝ)/Q) ((head step0.hi:ℝ)/Q)) : ∀ t ∈ Icc (startTime R step0) (stopTime R step15), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have ht0 := ht.1 by_cases hstop0 : t ≤ stopTime R step0 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert0 hud hu t ⟨ht0,hstop0⟩ have h0 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert0 hud hu have hu1 := join_bound Q_pos join1 h0 have ht1 : startTime R step1 ≤ t := by rw [start_eq_stop join1.1] exact (lt_of_not_ge hstop0).le by_cases hstop1 : t ≤ stopTime R step1 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert1 hud hu1 t ⟨ht1,hstop1⟩ have h1 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert1 hud hu1 have hu2 := join_bound Q_pos join2 h1 have ht2 : startTime R step2 ≤ t := by rw [start_eq_stop join2.1] exact (lt_of_not_ge hstop1).le by_cases hstop2 : t ≤ stopTime R step2 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert2 hud hu2 t ⟨ht2,hstop2⟩ have h2 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert2 hud hu2 have hu3 := join_bound Q_pos join3 h2 have ht3 : startTime R step3 ≤ t := by rw [start_eq_stop join3.1] exact (lt_of_not_ge hstop2).le by_cases hstop3 : t ≤ stopTime R step3 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert3 hud hu3 t ⟨ht3,hstop3⟩ have h3 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert3 hud hu3 have hu4 := join_bound Q_pos join4 h3 have ht4 : startTime R step4 ≤ t := by rw [start_eq_stop join4.1] exact (lt_of_not_ge hstop3).le by_cases hstop4 : t ≤ stopTime R step4 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert4 hud hu4 t ⟨ht4,hstop4⟩ have h4 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert4 hud hu4 have hu5 := join_bound Q_pos join5 h4 have ht5 : startTime R step5 ≤ t := by rw [start_eq_stop join5.1] exact (lt_of_not_ge hstop4).le by_cases hstop5 : t ≤ stopTime R step5 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert5 hud hu5 t ⟨ht5,hstop5⟩ have h5 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert5 hud hu5 have hu6 := join_bound Q_pos join6 h5 have ht6 : startTime R step6 ≤ t := by rw [start_eq_stop join6.1] exact (lt_of_not_ge hstop5).le by_cases hstop6 : t ≤ stopTime R step6 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert6 hud hu6 t ⟨ht6,hstop6⟩ have h6 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert6 hud hu6 have hu7 := join_bound Q_pos join7 h6 have ht7 : startTime R step7 ≤ t := by rw [start_eq_stop join7.1] exact (lt_of_not_ge hstop6).le by_cases hstop7 : t ≤ stopTime R step7 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert7 hud hu7 t ⟨ht7,hstop7⟩ have h7 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert7 hud hu7 have hu8 := join_bound Q_pos join8 h7 have ht8 : startTime R step8 ≤ t := by rw [start_eq_stop join8.1] exact (lt_of_not_ge hstop7).le by_cases hstop8 : t ≤ stopTime R step8 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert8 hud hu8 t ⟨ht8,hstop8⟩ have h8 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert8 hud hu8 have hu9 := join_bound Q_pos join9 h8 have ht9 : startTime R step9 ≤ t := by rw [start_eq_stop join9.1] exact (lt_of_not_ge hstop8).le by_cases hstop9 : t ≤ stopTime R step9 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert9 hud hu9 t ⟨ht9,hstop9⟩ have h9 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert9 hud hu9 have hu10 := join_bound Q_pos join10 h9 have ht10 : startTime R step10 ≤ t := by rw [start_eq_stop join10.1] exact (lt_of_not_ge hstop9).le by_cases hstop10 : t ≤ stopTime R step10 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert10 hud hu10 t ⟨ht10,hstop10⟩ have h10 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert10 hud hu10 have hu11 := join_bound Q_pos join11 h10 have ht11 : startTime R step11 ≤ t := by rw [start_eq_stop join11.1] exact (lt_of_not_ge hstop10).le by_cases hstop11 : t ≤ stopTime R step11 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert11 hud hu11 t ⟨ht11,hstop11⟩ have h11 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert11 hud hu11 have hu12 := join_bound Q_pos join12 h11 have ht12 : startTime R step12 ≤ t := by rw [start_eq_stop join12.1] exact (lt_of_not_ge hstop11).le by_cases hstop12 : t ≤ stopTime R step12 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert12 hud hu12 t ⟨ht12,hstop12⟩ have h12 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert12 hud hu12 have hu13 := join_bound Q_pos join13 h12 have ht13 : startTime R step13 ≤ t := by rw [start_eq_stop join13.1] exact (lt_of_not_ge hstop12).le by_cases hstop13 : t ≤ stopTime R step13 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert13 hud hu13 t ⟨ht13,hstop13⟩ have h13 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert13 hud hu13 have hu14 := join_bound Q_pos join14 h13 have ht14 : startTime R step14 ≤ t := by rw [start_eq_stop join14.1] exact (lt_of_not_ge hstop13).le by_cases hstop14 : t ≤ stopTime R step14 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert14 hud hu14 t ⟨ht14,hstop14⟩ have h14 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert14 hud hu14 have hu15 := join_bound Q_pos join15 h14 have ht15 : startTime R step15 ≤ t := by rw [start_eq_stop join15.1] exact (lt_of_not_ge hstop14).le exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert15 hud hu15 t ⟨ht15,ht.2⟩ theorem range_16 {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u (startTime R step16) ∈ Icc ((head step16.lo:ℝ)/Q) ((head step16.hi:ℝ)/Q)) : ∀ t ∈ Icc (startTime R step16) (stopTime R step31), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have ht16 := ht.1 by_cases hstop16 : t ≤ stopTime R step16 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert16 hud hu t ⟨ht16,hstop16⟩ have h16 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert16 hud hu have hu17 := join_bound Q_pos join17 h16 have ht17 : startTime R step17 ≤ t := by rw [start_eq_stop join17.1] exact (lt_of_not_ge hstop16).le by_cases hstop17 : t ≤ stopTime R step17 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert17 hud hu17 t ⟨ht17,hstop17⟩ have h17 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert17 hud hu17 have hu18 := join_bound Q_pos join18 h17 have ht18 : startTime R step18 ≤ t := by rw [start_eq_stop join18.1] exact (lt_of_not_ge hstop17).le by_cases hstop18 : t ≤ stopTime R step18 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert18 hud hu18 t ⟨ht18,hstop18⟩ have h18 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert18 hud hu18 have hu19 := join_bound Q_pos join19 h18 have ht19 : startTime R step19 ≤ t := by rw [start_eq_stop join19.1] exact (lt_of_not_ge hstop18).le by_cases hstop19 : t ≤ stopTime R step19 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert19 hud hu19 t ⟨ht19,hstop19⟩ have h19 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert19 hud hu19 have hu20 := join_bound Q_pos join20 h19 have ht20 : startTime R step20 ≤ t := by rw [start_eq_stop join20.1] exact (lt_of_not_ge hstop19).le by_cases hstop20 : t ≤ stopTime R step20 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert20 hud hu20 t ⟨ht20,hstop20⟩ have h20 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert20 hud hu20 have hu21 := join_bound Q_pos join21 h20 have ht21 : startTime R step21 ≤ t := by rw [start_eq_stop join21.1] exact (lt_of_not_ge hstop20).le by_cases hstop21 : t ≤ stopTime R step21 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert21 hud hu21 t ⟨ht21,hstop21⟩ have h21 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert21 hud hu21 have hu22 := join_bound Q_pos join22 h21 have ht22 : startTime R step22 ≤ t := by rw [start_eq_stop join22.1] exact (lt_of_not_ge hstop21).le by_cases hstop22 : t ≤ stopTime R step22 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert22 hud hu22 t ⟨ht22,hstop22⟩ have h22 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert22 hud hu22 have hu23 := join_bound Q_pos join23 h22 have ht23 : startTime R step23 ≤ t := by rw [start_eq_stop join23.1] exact (lt_of_not_ge hstop22).le by_cases hstop23 : t ≤ stopTime R step23 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert23 hud hu23 t ⟨ht23,hstop23⟩ have h23 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert23 hud hu23 have hu24 := join_bound Q_pos join24 h23 have ht24 : startTime R step24 ≤ t := by rw [start_eq_stop join24.1] exact (lt_of_not_ge hstop23).le by_cases hstop24 : t ≤ stopTime R step24 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert24 hud hu24 t ⟨ht24,hstop24⟩ have h24 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert24 hud hu24 have hu25 := join_bound Q_pos join25 h24 have ht25 : startTime R step25 ≤ t := by rw [start_eq_stop join25.1] exact (lt_of_not_ge hstop24).le by_cases hstop25 : t ≤ stopTime R step25 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert25 hud hu25 t ⟨ht25,hstop25⟩ have h25 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert25 hud hu25 have hu26 := join_bound Q_pos join26 h25 have ht26 : startTime R step26 ≤ t := by rw [start_eq_stop join26.1] exact (lt_of_not_ge hstop25).le by_cases hstop26 : t ≤ stopTime R step26 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert26 hud hu26 t ⟨ht26,hstop26⟩ have h26 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert26 hud hu26 have hu27 := join_bound Q_pos join27 h26 have ht27 : startTime R step27 ≤ t := by rw [start_eq_stop join27.1] exact (lt_of_not_ge hstop26).le by_cases hstop27 : t ≤ stopTime R step27 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert27 hud hu27 t ⟨ht27,hstop27⟩ have h27 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert27 hud hu27 have hu28 := join_bound Q_pos join28 h27 have ht28 : startTime R step28 ≤ t := by rw [start_eq_stop join28.1] exact (lt_of_not_ge hstop27).le by_cases hstop28 : t ≤ stopTime R step28 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert28 hud hu28 t ⟨ht28,hstop28⟩ have h28 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert28 hud hu28 have hu29 := join_bound Q_pos join29 h28 have ht29 : startTime R step29 ≤ t := by rw [start_eq_stop join29.1] exact (lt_of_not_ge hstop28).le by_cases hstop29 : t ≤ stopTime R step29 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert29 hud hu29 t ⟨ht29,hstop29⟩ have h29 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert29 hud hu29 have hu30 := join_bound Q_pos join30 h29 have ht30 : startTime R step30 ≤ t := by rw [start_eq_stop join30.1] exact (lt_of_not_ge hstop29).le by_cases hstop30 : t ≤ stopTime R step30 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert30 hud hu30 t ⟨ht30,hstop30⟩ have h30 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert30 hud hu30 have hu31 := join_bound Q_pos join31 h30 have ht31 : startTime R step31 ≤ t := by rw [start_eq_stop join31.1] exact (lt_of_not_ge hstop30).le exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert31 hud hu31 t ⟨ht31,ht.2⟩ theorem range_32 {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u (startTime R step32) ∈ Icc ((head step32.lo:ℝ)/Q) ((head step32.hi:ℝ)/Q)) : ∀ t ∈ Icc (startTime R step32) (stopTime R step47), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have ht32 := ht.1 by_cases hstop32 : t ≤ stopTime R step32 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert32 hud hu t ⟨ht32,hstop32⟩ have h32 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert32 hud hu have hu33 := join_bound Q_pos join33 h32 have ht33 : startTime R step33 ≤ t := by rw [start_eq_stop join33.1] exact (lt_of_not_ge hstop32).le by_cases hstop33 : t ≤ stopTime R step33 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert33 hud hu33 t ⟨ht33,hstop33⟩ have h33 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert33 hud hu33 have hu34 := join_bound Q_pos join34 h33 have ht34 : startTime R step34 ≤ t := by rw [start_eq_stop join34.1] exact (lt_of_not_ge hstop33).le by_cases hstop34 : t ≤ stopTime R step34 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert34 hud hu34 t ⟨ht34,hstop34⟩ have h34 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert34 hud hu34 have hu35 := join_bound Q_pos join35 h34 have ht35 : startTime R step35 ≤ t := by rw [start_eq_stop join35.1] exact (lt_of_not_ge hstop34).le by_cases hstop35 : t ≤ stopTime R step35 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert35 hud hu35 t ⟨ht35,hstop35⟩ have h35 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert35 hud hu35 have hu36 := join_bound Q_pos join36 h35 have ht36 : startTime R step36 ≤ t := by rw [start_eq_stop join36.1] exact (lt_of_not_ge hstop35).le by_cases hstop36 : t ≤ stopTime R step36 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert36 hud hu36 t ⟨ht36,hstop36⟩ have h36 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert36 hud hu36 have hu37 := join_bound Q_pos join37 h36 have ht37 : startTime R step37 ≤ t := by rw [start_eq_stop join37.1] exact (lt_of_not_ge hstop36).le by_cases hstop37 : t ≤ stopTime R step37 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert37 hud hu37 t ⟨ht37,hstop37⟩ have h37 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert37 hud hu37 have hu38 := join_bound Q_pos join38 h37 have ht38 : startTime R step38 ≤ t := by rw [start_eq_stop join38.1] exact (lt_of_not_ge hstop37).le by_cases hstop38 : t ≤ stopTime R step38 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert38 hud hu38 t ⟨ht38,hstop38⟩ have h38 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert38 hud hu38 have hu39 := join_bound Q_pos join39 h38 have ht39 : startTime R step39 ≤ t := by rw [start_eq_stop join39.1] exact (lt_of_not_ge hstop38).le by_cases hstop39 : t ≤ stopTime R step39 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert39 hud hu39 t ⟨ht39,hstop39⟩ have h39 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert39 hud hu39 have hu40 := join_bound Q_pos join40 h39 have ht40 : startTime R step40 ≤ t := by rw [start_eq_stop join40.1] exact (lt_of_not_ge hstop39).le by_cases hstop40 : t ≤ stopTime R step40 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert40 hud hu40 t ⟨ht40,hstop40⟩ have h40 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert40 hud hu40 have hu41 := join_bound Q_pos join41 h40 have ht41 : startTime R step41 ≤ t := by rw [start_eq_stop join41.1] exact (lt_of_not_ge hstop40).le by_cases hstop41 : t ≤ stopTime R step41 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert41 hud hu41 t ⟨ht41,hstop41⟩ have h41 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert41 hud hu41 have hu42 := join_bound Q_pos join42 h41 have ht42 : startTime R step42 ≤ t := by rw [start_eq_stop join42.1] exact (lt_of_not_ge hstop41).le by_cases hstop42 : t ≤ stopTime R step42 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert42 hud hu42 t ⟨ht42,hstop42⟩ have h42 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert42 hud hu42 have hu43 := join_bound Q_pos join43 h42 have ht43 : startTime R step43 ≤ t := by rw [start_eq_stop join43.1] exact (lt_of_not_ge hstop42).le by_cases hstop43 : t ≤ stopTime R step43 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert43 hud hu43 t ⟨ht43,hstop43⟩ have h43 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert43 hud hu43 have hu44 := join_bound Q_pos join44 h43 have ht44 : startTime R step44 ≤ t := by rw [start_eq_stop join44.1] exact (lt_of_not_ge hstop43).le by_cases hstop44 : t ≤ stopTime R step44 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert44 hud hu44 t ⟨ht44,hstop44⟩ have h44 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert44 hud hu44 have hu45 := join_bound Q_pos join45 h44 have ht45 : startTime R step45 ≤ t := by rw [start_eq_stop join45.1] exact (lt_of_not_ge hstop44).le by_cases hstop45 : t ≤ stopTime R step45 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert45 hud hu45 t ⟨ht45,hstop45⟩ have h45 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert45 hud hu45 have hu46 := join_bound Q_pos join46 h45 have ht46 : startTime R step46 ≤ t := by rw [start_eq_stop join46.1] exact (lt_of_not_ge hstop45).le by_cases hstop46 : t ≤ stopTime R step46 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert46 hud hu46 t ⟨ht46,hstop46⟩ have h46 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert46 hud hu46 have hu47 := join_bound Q_pos join47 h46 have ht47 : startTime R step47 ≤ t := by rw [start_eq_stop join47.1] exact (lt_of_not_ge hstop46).le exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert47 hud hu47 t ⟨ht47,ht.2⟩ theorem range_48 {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u (startTime R step48) ∈ Icc ((head step48.lo:ℝ)/Q) ((head step48.hi:ℝ)/Q)) : ∀ t ∈ Icc (startTime R step48) (stopTime R step63), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have ht48 := ht.1 by_cases hstop48 : t ≤ stopTime R step48 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert48 hud hu t ⟨ht48,hstop48⟩ have h48 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert48 hud hu have hu49 := join_bound Q_pos join49 h48 have ht49 : startTime R step49 ≤ t := by rw [start_eq_stop join49.1] exact (lt_of_not_ge hstop48).le by_cases hstop49 : t ≤ stopTime R step49 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert49 hud hu49 t ⟨ht49,hstop49⟩ have h49 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert49 hud hu49 have hu50 := join_bound Q_pos join50 h49 have ht50 : startTime R step50 ≤ t := by rw [start_eq_stop join50.1] exact (lt_of_not_ge hstop49).le by_cases hstop50 : t ≤ stopTime R step50 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert50 hud hu50 t ⟨ht50,hstop50⟩ have h50 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert50 hud hu50 have hu51 := join_bound Q_pos join51 h50 have ht51 : startTime R step51 ≤ t := by rw [start_eq_stop join51.1] exact (lt_of_not_ge hstop50).le by_cases hstop51 : t ≤ stopTime R step51 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert51 hud hu51 t ⟨ht51,hstop51⟩ have h51 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert51 hud hu51 have hu52 := join_bound Q_pos join52 h51 have ht52 : startTime R step52 ≤ t := by rw [start_eq_stop join52.1] exact (lt_of_not_ge hstop51).le by_cases hstop52 : t ≤ stopTime R step52 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert52 hud hu52 t ⟨ht52,hstop52⟩ have h52 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert52 hud hu52 have hu53 := join_bound Q_pos join53 h52 have ht53 : startTime R step53 ≤ t := by rw [start_eq_stop join53.1] exact (lt_of_not_ge hstop52).le by_cases hstop53 : t ≤ stopTime R step53 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert53 hud hu53 t ⟨ht53,hstop53⟩ have h53 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert53 hud hu53 have hu54 := join_bound Q_pos join54 h53 have ht54 : startTime R step54 ≤ t := by rw [start_eq_stop join54.1] exact (lt_of_not_ge hstop53).le by_cases hstop54 : t ≤ stopTime R step54 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert54 hud hu54 t ⟨ht54,hstop54⟩ have h54 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert54 hud hu54 have hu55 := join_bound Q_pos join55 h54 have ht55 : startTime R step55 ≤ t := by rw [start_eq_stop join55.1] exact (lt_of_not_ge hstop54).le by_cases hstop55 : t ≤ stopTime R step55 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert55 hud hu55 t ⟨ht55,hstop55⟩ have h55 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert55 hud hu55 have hu56 := join_bound Q_pos join56 h55 have ht56 : startTime R step56 ≤ t := by rw [start_eq_stop join56.1] exact (lt_of_not_ge hstop55).le by_cases hstop56 : t ≤ stopTime R step56 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert56 hud hu56 t ⟨ht56,hstop56⟩ have h56 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert56 hud hu56 have hu57 := join_bound Q_pos join57 h56 have ht57 : startTime R step57 ≤ t := by rw [start_eq_stop join57.1] exact (lt_of_not_ge hstop56).le by_cases hstop57 : t ≤ stopTime R step57 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert57 hud hu57 t ⟨ht57,hstop57⟩ have h57 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert57 hud hu57 have hu58 := join_bound Q_pos join58 h57 have ht58 : startTime R step58 ≤ t := by rw [start_eq_stop join58.1] exact (lt_of_not_ge hstop57).le by_cases hstop58 : t ≤ stopTime R step58 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert58 hud hu58 t ⟨ht58,hstop58⟩ have h58 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert58 hud hu58 have hu59 := join_bound Q_pos join59 h58 have ht59 : startTime R step59 ≤ t := by rw [start_eq_stop join59.1] exact (lt_of_not_ge hstop58).le by_cases hstop59 : t ≤ stopTime R step59 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert59 hud hu59 t ⟨ht59,hstop59⟩ have h59 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert59 hud hu59 have hu60 := join_bound Q_pos join60 h59 have ht60 : startTime R step60 ≤ t := by rw [start_eq_stop join60.1] exact (lt_of_not_ge hstop59).le by_cases hstop60 : t ≤ stopTime R step60 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert60 hud hu60 t ⟨ht60,hstop60⟩ have h60 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert60 hud hu60 have hu61 := join_bound Q_pos join61 h60 have ht61 : startTime R step61 ≤ t := by rw [start_eq_stop join61.1] exact (lt_of_not_ge hstop60).le by_cases hstop61 : t ≤ stopTime R step61 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert61 hud hu61 t ⟨ht61,hstop61⟩ have h61 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert61 hud hu61 have hu62 := join_bound Q_pos join62 h61 have ht62 : startTime R step62 ≤ t := by rw [start_eq_stop join62.1] exact (lt_of_not_ge hstop61).le by_cases hstop62 : t ≤ stopTime R step62 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert62 hud hu62 t ⟨ht62,hstop62⟩ have h62 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert62 hud hu62 have hu63 := join_bound Q_pos join63 h62 have ht63 : startTime R step63 ≤ t := by rw [start_eq_stop join63.1] exact (lt_of_not_ge hstop62).le exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert63 hud hu63 t ⟨ht63,ht.2⟩ theorem range_64 {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u (startTime R step64) ∈ Icc ((head step64.lo:ℝ)/Q) ((head step64.hi:ℝ)/Q)) : ∀ t ∈ Icc (startTime R step64) (stopTime R step79), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have ht64 := ht.1 by_cases hstop64 : t ≤ stopTime R step64 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert64 hud hu t ⟨ht64,hstop64⟩ have h64 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert64 hud hu have hu65 := join_bound Q_pos join65 h64 have ht65 : startTime R step65 ≤ t := by rw [start_eq_stop join65.1] exact (lt_of_not_ge hstop64).le by_cases hstop65 : t ≤ stopTime R step65 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert65 hud hu65 t ⟨ht65,hstop65⟩ have h65 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert65 hud hu65 have hu66 := join_bound Q_pos join66 h65 have ht66 : startTime R step66 ≤ t := by rw [start_eq_stop join66.1] exact (lt_of_not_ge hstop65).le by_cases hstop66 : t ≤ stopTime R step66 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert66 hud hu66 t ⟨ht66,hstop66⟩ have h66 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert66 hud hu66 have hu67 := join_bound Q_pos join67 h66 have ht67 : startTime R step67 ≤ t := by rw [start_eq_stop join67.1] exact (lt_of_not_ge hstop66).le by_cases hstop67 : t ≤ stopTime R step67 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert67 hud hu67 t ⟨ht67,hstop67⟩ have h67 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert67 hud hu67 have hu68 := join_bound Q_pos join68 h67 have ht68 : startTime R step68 ≤ t := by rw [start_eq_stop join68.1] exact (lt_of_not_ge hstop67).le by_cases hstop68 : t ≤ stopTime R step68 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert68 hud hu68 t ⟨ht68,hstop68⟩ have h68 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert68 hud hu68 have hu69 := join_bound Q_pos join69 h68 have ht69 : startTime R step69 ≤ t := by rw [start_eq_stop join69.1] exact (lt_of_not_ge hstop68).le by_cases hstop69 : t ≤ stopTime R step69 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert69 hud hu69 t ⟨ht69,hstop69⟩ have h69 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert69 hud hu69 have hu70 := join_bound Q_pos join70 h69 have ht70 : startTime R step70 ≤ t := by rw [start_eq_stop join70.1] exact (lt_of_not_ge hstop69).le by_cases hstop70 : t ≤ stopTime R step70 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert70 hud hu70 t ⟨ht70,hstop70⟩ have h70 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert70 hud hu70 have hu71 := join_bound Q_pos join71 h70 have ht71 : startTime R step71 ≤ t := by rw [start_eq_stop join71.1] exact (lt_of_not_ge hstop70).le by_cases hstop71 : t ≤ stopTime R step71 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert71 hud hu71 t ⟨ht71,hstop71⟩ have h71 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert71 hud hu71 have hu72 := join_bound Q_pos join72 h71 have ht72 : startTime R step72 ≤ t := by rw [start_eq_stop join72.1] exact (lt_of_not_ge hstop71).le by_cases hstop72 : t ≤ stopTime R step72 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert72 hud hu72 t ⟨ht72,hstop72⟩ have h72 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert72 hud hu72 have hu73 := join_bound Q_pos join73 h72 have ht73 : startTime R step73 ≤ t := by rw [start_eq_stop join73.1] exact (lt_of_not_ge hstop72).le by_cases hstop73 : t ≤ stopTime R step73 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert73 hud hu73 t ⟨ht73,hstop73⟩ have h73 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert73 hud hu73 have hu74 := join_bound Q_pos join74 h73 have ht74 : startTime R step74 ≤ t := by rw [start_eq_stop join74.1] exact (lt_of_not_ge hstop73).le by_cases hstop74 : t ≤ stopTime R step74 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert74 hud hu74 t ⟨ht74,hstop74⟩ have h74 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert74 hud hu74 have hu75 := join_bound Q_pos join75 h74 have ht75 : startTime R step75 ≤ t := by rw [start_eq_stop join75.1] exact (lt_of_not_ge hstop74).le by_cases hstop75 : t ≤ stopTime R step75 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert75 hud hu75 t ⟨ht75,hstop75⟩ have h75 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert75 hud hu75 have hu76 := join_bound Q_pos join76 h75 have ht76 : startTime R step76 ≤ t := by rw [start_eq_stop join76.1] exact (lt_of_not_ge hstop75).le by_cases hstop76 : t ≤ stopTime R step76 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert76 hud hu76 t ⟨ht76,hstop76⟩ have h76 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert76 hud hu76 have hu77 := join_bound Q_pos join77 h76 have ht77 : startTime R step77 ≤ t := by rw [start_eq_stop join77.1] exact (lt_of_not_ge hstop76).le by_cases hstop77 : t ≤ stopTime R step77 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert77 hud hu77 t ⟨ht77,hstop77⟩ have h77 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert77 hud hu77 have hu78 := join_bound Q_pos join78 h77 have ht78 : startTime R step78 ≤ t := by rw [start_eq_stop join78.1] exact (lt_of_not_ge hstop77).le by_cases hstop78 : t ≤ stopTime R step78 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert78 hud hu78 t ⟨ht78,hstop78⟩ have h78 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert78 hud hu78 have hu79 := join_bound Q_pos join79 h78 have ht79 : startTime R step79 ≤ t := by rw [start_eq_stop join79.1] exact (lt_of_not_ge hstop78).le exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert79 hud hu79 t ⟨ht79,ht.2⟩ theorem range_80 {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u (startTime R step80) ∈ Icc ((head step80.lo:ℝ)/Q) ((head step80.hi:ℝ)/Q)) : ∀ t ∈ Icc (startTime R step80) (stopTime R step95), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have ht80 := ht.1 by_cases hstop80 : t ≤ stopTime R step80 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert80 hud hu t ⟨ht80,hstop80⟩ have h80 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert80 hud hu have hu81 := join_bound Q_pos join81 h80 have ht81 : startTime R step81 ≤ t := by rw [start_eq_stop join81.1] exact (lt_of_not_ge hstop80).le by_cases hstop81 : t ≤ stopTime R step81 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert81 hud hu81 t ⟨ht81,hstop81⟩ have h81 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert81 hud hu81 have hu82 := join_bound Q_pos join82 h81 have ht82 : startTime R step82 ≤ t := by rw [start_eq_stop join82.1] exact (lt_of_not_ge hstop81).le by_cases hstop82 : t ≤ stopTime R step82 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert82 hud hu82 t ⟨ht82,hstop82⟩ have h82 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert82 hud hu82 have hu83 := join_bound Q_pos join83 h82 have ht83 : startTime R step83 ≤ t := by rw [start_eq_stop join83.1] exact (lt_of_not_ge hstop82).le by_cases hstop83 : t ≤ stopTime R step83 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert83 hud hu83 t ⟨ht83,hstop83⟩ have h83 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert83 hud hu83 have hu84 := join_bound Q_pos join84 h83 have ht84 : startTime R step84 ≤ t := by rw [start_eq_stop join84.1] exact (lt_of_not_ge hstop83).le by_cases hstop84 : t ≤ stopTime R step84 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert84 hud hu84 t ⟨ht84,hstop84⟩ have h84 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert84 hud hu84 have hu85 := join_bound Q_pos join85 h84 have ht85 : startTime R step85 ≤ t := by rw [start_eq_stop join85.1] exact (lt_of_not_ge hstop84).le by_cases hstop85 : t ≤ stopTime R step85 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert85 hud hu85 t ⟨ht85,hstop85⟩ have h85 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert85 hud hu85 have hu86 := join_bound Q_pos join86 h85 have ht86 : startTime R step86 ≤ t := by rw [start_eq_stop join86.1] exact (lt_of_not_ge hstop85).le by_cases hstop86 : t ≤ stopTime R step86 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert86 hud hu86 t ⟨ht86,hstop86⟩ have h86 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert86 hud hu86 have hu87 := join_bound Q_pos join87 h86 have ht87 : startTime R step87 ≤ t := by rw [start_eq_stop join87.1] exact (lt_of_not_ge hstop86).le by_cases hstop87 : t ≤ stopTime R step87 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert87 hud hu87 t ⟨ht87,hstop87⟩ have h87 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert87 hud hu87 have hu88 := join_bound Q_pos join88 h87 have ht88 : startTime R step88 ≤ t := by rw [start_eq_stop join88.1] exact (lt_of_not_ge hstop87).le by_cases hstop88 : t ≤ stopTime R step88 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert88 hud hu88 t ⟨ht88,hstop88⟩ have h88 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert88 hud hu88 have hu89 := join_bound Q_pos join89 h88 have ht89 : startTime R step89 ≤ t := by rw [start_eq_stop join89.1] exact (lt_of_not_ge hstop88).le by_cases hstop89 : t ≤ stopTime R step89 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert89 hud hu89 t ⟨ht89,hstop89⟩ have h89 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert89 hud hu89 have hu90 := join_bound Q_pos join90 h89 have ht90 : startTime R step90 ≤ t := by rw [start_eq_stop join90.1] exact (lt_of_not_ge hstop89).le by_cases hstop90 : t ≤ stopTime R step90 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert90 hud hu90 t ⟨ht90,hstop90⟩ have h90 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert90 hud hu90 have hu91 := join_bound Q_pos join91 h90 have ht91 : startTime R step91 ≤ t := by rw [start_eq_stop join91.1] exact (lt_of_not_ge hstop90).le by_cases hstop91 : t ≤ stopTime R step91 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert91 hud hu91 t ⟨ht91,hstop91⟩ have h91 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert91 hud hu91 have hu92 := join_bound Q_pos join92 h91 have ht92 : startTime R step92 ≤ t := by rw [start_eq_stop join92.1] exact (lt_of_not_ge hstop91).le by_cases hstop92 : t ≤ stopTime R step92 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert92 hud hu92 t ⟨ht92,hstop92⟩ have h92 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert92 hud hu92 have hu93 := join_bound Q_pos join93 h92 have ht93 : startTime R step93 ≤ t := by rw [start_eq_stop join93.1] exact (lt_of_not_ge hstop92).le by_cases hstop93 : t ≤ stopTime R step93 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert93 hud hu93 t ⟨ht93,hstop93⟩ have h93 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert93 hud hu93 have hu94 := join_bound Q_pos join94 h93 have ht94 : startTime R step94 ≤ t := by rw [start_eq_stop join94.1] exact (lt_of_not_ge hstop93).le by_cases hstop94 : t ≤ stopTime R step94 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert94 hud hu94 t ⟨ht94,hstop94⟩ have h94 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert94 hud hu94 have hu95 := join_bound Q_pos join95 h94 have ht95 : startTime R step95 ≤ t := by rw [start_eq_stop join95.1] exact (lt_of_not_ge hstop94).le exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert95 hud hu95 t ⟨ht95,ht.2⟩ theorem range_96 {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u (startTime R step96) ∈ Icc ((head step96.lo:ℝ)/Q) ((head step96.hi:ℝ)/Q)) : ∀ t ∈ Icc (startTime R step96) (stopTime R step111), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have ht96 := ht.1 by_cases hstop96 : t ≤ stopTime R step96 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert96 hud hu t ⟨ht96,hstop96⟩ have h96 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert96 hud hu have hu97 := join_bound Q_pos join97 h96 have ht97 : startTime R step97 ≤ t := by rw [start_eq_stop join97.1] exact (lt_of_not_ge hstop96).le by_cases hstop97 : t ≤ stopTime R step97 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert97 hud hu97 t ⟨ht97,hstop97⟩ have h97 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert97 hud hu97 have hu98 := join_bound Q_pos join98 h97 have ht98 : startTime R step98 ≤ t := by rw [start_eq_stop join98.1] exact (lt_of_not_ge hstop97).le by_cases hstop98 : t ≤ stopTime R step98 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert98 hud hu98 t ⟨ht98,hstop98⟩ have h98 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert98 hud hu98 have hu99 := join_bound Q_pos join99 h98 have ht99 : startTime R step99 ≤ t := by rw [start_eq_stop join99.1] exact (lt_of_not_ge hstop98).le by_cases hstop99 : t ≤ stopTime R step99 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert99 hud hu99 t ⟨ht99,hstop99⟩ have h99 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert99 hud hu99 have hu100 := join_bound Q_pos join100 h99 have ht100 : startTime R step100 ≤ t := by rw [start_eq_stop join100.1] exact (lt_of_not_ge hstop99).le by_cases hstop100 : t ≤ stopTime R step100 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert100 hud hu100 t ⟨ht100,hstop100⟩ have h100 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert100 hud hu100 have hu101 := join_bound Q_pos join101 h100 have ht101 : startTime R step101 ≤ t := by rw [start_eq_stop join101.1] exact (lt_of_not_ge hstop100).le by_cases hstop101 : t ≤ stopTime R step101 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert101 hud hu101 t ⟨ht101,hstop101⟩ have h101 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert101 hud hu101 have hu102 := join_bound Q_pos join102 h101 have ht102 : startTime R step102 ≤ t := by rw [start_eq_stop join102.1] exact (lt_of_not_ge hstop101).le by_cases hstop102 : t ≤ stopTime R step102 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert102 hud hu102 t ⟨ht102,hstop102⟩ have h102 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert102 hud hu102 have hu103 := join_bound Q_pos join103 h102 have ht103 : startTime R step103 ≤ t := by rw [start_eq_stop join103.1] exact (lt_of_not_ge hstop102).le by_cases hstop103 : t ≤ stopTime R step103 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert103 hud hu103 t ⟨ht103,hstop103⟩ have h103 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert103 hud hu103 have hu104 := join_bound Q_pos join104 h103 have ht104 : startTime R step104 ≤ t := by rw [start_eq_stop join104.1] exact (lt_of_not_ge hstop103).le by_cases hstop104 : t ≤ stopTime R step104 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert104 hud hu104 t ⟨ht104,hstop104⟩ have h104 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert104 hud hu104 have hu105 := join_bound Q_pos join105 h104 have ht105 : startTime R step105 ≤ t := by rw [start_eq_stop join105.1] exact (lt_of_not_ge hstop104).le by_cases hstop105 : t ≤ stopTime R step105 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert105 hud hu105 t ⟨ht105,hstop105⟩ have h105 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert105 hud hu105 have hu106 := join_bound Q_pos join106 h105 have ht106 : startTime R step106 ≤ t := by rw [start_eq_stop join106.1] exact (lt_of_not_ge hstop105).le by_cases hstop106 : t ≤ stopTime R step106 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert106 hud hu106 t ⟨ht106,hstop106⟩ have h106 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert106 hud hu106 have hu107 := join_bound Q_pos join107 h106 have ht107 : startTime R step107 ≤ t := by rw [start_eq_stop join107.1] exact (lt_of_not_ge hstop106).le by_cases hstop107 : t ≤ stopTime R step107 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert107 hud hu107 t ⟨ht107,hstop107⟩ have h107 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert107 hud hu107 have hu108 := join_bound Q_pos join108 h107 have ht108 : startTime R step108 ≤ t := by rw [start_eq_stop join108.1] exact (lt_of_not_ge hstop107).le by_cases hstop108 : t ≤ stopTime R step108 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert108 hud hu108 t ⟨ht108,hstop108⟩ have h108 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert108 hud hu108 have hu109 := join_bound Q_pos join109 h108 have ht109 : startTime R step109 ≤ t := by rw [start_eq_stop join109.1] exact (lt_of_not_ge hstop108).le by_cases hstop109 : t ≤ stopTime R step109 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert109 hud hu109 t ⟨ht109,hstop109⟩ have h109 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert109 hud hu109 have hu110 := join_bound Q_pos join110 h109 have ht110 : startTime R step110 ≤ t := by rw [start_eq_stop join110.1] exact (lt_of_not_ge hstop109).le by_cases hstop110 : t ≤ stopTime R step110 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert110 hud hu110 t ⟨ht110,hstop110⟩ have h110 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert110 hud hu110 have hu111 := join_bound Q_pos join111 h110 have ht111 : startTime R step111 ≤ t := by rw [start_eq_stop join111.1] exact (lt_of_not_ge hstop110).le exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert111 hud hu111 t ⟨ht111,ht.2⟩ theorem range_112 {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u (startTime R step112) ∈ Icc ((head step112.lo:ℝ)/Q) ((head step112.hi:ℝ)/Q)) : ∀ t ∈ Icc (startTime R step112) (stopTime R step127), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have ht112 := ht.1 by_cases hstop112 : t ≤ stopTime R step112 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert112 hud hu t ⟨ht112,hstop112⟩ have h112 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert112 hud hu have hu113 := join_bound Q_pos join113 h112 have ht113 : startTime R step113 ≤ t := by rw [start_eq_stop join113.1] exact (lt_of_not_ge hstop112).le by_cases hstop113 : t ≤ stopTime R step113 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert113 hud hu113 t ⟨ht113,hstop113⟩ have h113 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert113 hud hu113 have hu114 := join_bound Q_pos join114 h113 have ht114 : startTime R step114 ≤ t := by rw [start_eq_stop join114.1] exact (lt_of_not_ge hstop113).le by_cases hstop114 : t ≤ stopTime R step114 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert114 hud hu114 t ⟨ht114,hstop114⟩ have h114 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert114 hud hu114 have hu115 := join_bound Q_pos join115 h114 have ht115 : startTime R step115 ≤ t := by rw [start_eq_stop join115.1] exact (lt_of_not_ge hstop114).le by_cases hstop115 : t ≤ stopTime R step115 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert115 hud hu115 t ⟨ht115,hstop115⟩ have h115 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert115 hud hu115 have hu116 := join_bound Q_pos join116 h115 have ht116 : startTime R step116 ≤ t := by rw [start_eq_stop join116.1] exact (lt_of_not_ge hstop115).le by_cases hstop116 : t ≤ stopTime R step116 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert116 hud hu116 t ⟨ht116,hstop116⟩ have h116 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert116 hud hu116 have hu117 := join_bound Q_pos join117 h116 have ht117 : startTime R step117 ≤ t := by rw [start_eq_stop join117.1] exact (lt_of_not_ge hstop116).le by_cases hstop117 : t ≤ stopTime R step117 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert117 hud hu117 t ⟨ht117,hstop117⟩ have h117 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert117 hud hu117 have hu118 := join_bound Q_pos join118 h117 have ht118 : startTime R step118 ≤ t := by rw [start_eq_stop join118.1] exact (lt_of_not_ge hstop117).le by_cases hstop118 : t ≤ stopTime R step118 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert118 hud hu118 t ⟨ht118,hstop118⟩ have h118 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert118 hud hu118 have hu119 := join_bound Q_pos join119 h118 have ht119 : startTime R step119 ≤ t := by rw [start_eq_stop join119.1] exact (lt_of_not_ge hstop118).le by_cases hstop119 : t ≤ stopTime R step119 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert119 hud hu119 t ⟨ht119,hstop119⟩ have h119 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert119 hud hu119 have hu120 := join_bound Q_pos join120 h119 have ht120 : startTime R step120 ≤ t := by rw [start_eq_stop join120.1] exact (lt_of_not_ge hstop119).le by_cases hstop120 : t ≤ stopTime R step120 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert120 hud hu120 t ⟨ht120,hstop120⟩ have h120 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert120 hud hu120 have hu121 := join_bound Q_pos join121 h120 have ht121 : startTime R step121 ≤ t := by rw [start_eq_stop join121.1] exact (lt_of_not_ge hstop120).le by_cases hstop121 : t ≤ stopTime R step121 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert121 hud hu121 t ⟨ht121,hstop121⟩ have h121 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert121 hud hu121 have hu122 := join_bound Q_pos join122 h121 have ht122 : startTime R step122 ≤ t := by rw [start_eq_stop join122.1] exact (lt_of_not_ge hstop121).le by_cases hstop122 : t ≤ stopTime R step122 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert122 hud hu122 t ⟨ht122,hstop122⟩ have h122 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert122 hud hu122 have hu123 := join_bound Q_pos join123 h122 have ht123 : startTime R step123 ≤ t := by rw [start_eq_stop join123.1] exact (lt_of_not_ge hstop122).le by_cases hstop123 : t ≤ stopTime R step123 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert123 hud hu123 t ⟨ht123,hstop123⟩ have h123 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert123 hud hu123 have hu124 := join_bound Q_pos join124 h123 have ht124 : startTime R step124 ≤ t := by rw [start_eq_stop join124.1] exact (lt_of_not_ge hstop123).le by_cases hstop124 : t ≤ stopTime R step124 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert124 hud hu124 t ⟨ht124,hstop124⟩ have h124 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert124 hud hu124 have hu125 := join_bound Q_pos join125 h124 have ht125 : startTime R step125 ≤ t := by rw [start_eq_stop join125.1] exact (lt_of_not_ge hstop124).le by_cases hstop125 : t ≤ stopTime R step125 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert125 hud hu125 t ⟨ht125,hstop125⟩ have h125 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert125 hud hu125 have hu126 := join_bound Q_pos join126 h125 have ht126 : startTime R step126 ≤ t := by rw [start_eq_stop join126.1] exact (lt_of_not_ge hstop125).le by_cases hstop126 : t ≤ stopTime R step126 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert126 hud hu126 t ⟨ht126,hstop126⟩ have h126 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert126 hud hu126 have hu127 := join_bound Q_pos join127 h126 have ht127 : startTime R step127 ≤ t := by rw [start_eq_stop join127.1] exact (lt_of_not_ge hstop126).le exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert127 hud hu127 t ⟨ht127,ht.2⟩ theorem range_128 {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u (startTime R step128) ∈ Icc ((head step128.lo:ℝ)/Q) ((head step128.hi:ℝ)/Q)) : ∀ t ∈ Icc (startTime R step128) (stopTime R step143), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have ht128 := ht.1 by_cases hstop128 : t ≤ stopTime R step128 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert128 hud hu t ⟨ht128,hstop128⟩ have h128 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert128 hud hu have hu129 := join_bound Q_pos join129 h128 have ht129 : startTime R step129 ≤ t := by rw [start_eq_stop join129.1] exact (lt_of_not_ge hstop128).le by_cases hstop129 : t ≤ stopTime R step129 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert129 hud hu129 t ⟨ht129,hstop129⟩ have h129 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert129 hud hu129 have hu130 := join_bound Q_pos join130 h129 have ht130 : startTime R step130 ≤ t := by rw [start_eq_stop join130.1] exact (lt_of_not_ge hstop129).le by_cases hstop130 : t ≤ stopTime R step130 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert130 hud hu130 t ⟨ht130,hstop130⟩ have h130 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert130 hud hu130 have hu131 := join_bound Q_pos join131 h130 have ht131 : startTime R step131 ≤ t := by rw [start_eq_stop join131.1] exact (lt_of_not_ge hstop130).le by_cases hstop131 : t ≤ stopTime R step131 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert131 hud hu131 t ⟨ht131,hstop131⟩ have h131 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert131 hud hu131 have hu132 := join_bound Q_pos join132 h131 have ht132 : startTime R step132 ≤ t := by rw [start_eq_stop join132.1] exact (lt_of_not_ge hstop131).le by_cases hstop132 : t ≤ stopTime R step132 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert132 hud hu132 t ⟨ht132,hstop132⟩ have h132 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert132 hud hu132 have hu133 := join_bound Q_pos join133 h132 have ht133 : startTime R step133 ≤ t := by rw [start_eq_stop join133.1] exact (lt_of_not_ge hstop132).le by_cases hstop133 : t ≤ stopTime R step133 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert133 hud hu133 t ⟨ht133,hstop133⟩ have h133 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert133 hud hu133 have hu134 := join_bound Q_pos join134 h133 have ht134 : startTime R step134 ≤ t := by rw [start_eq_stop join134.1] exact (lt_of_not_ge hstop133).le by_cases hstop134 : t ≤ stopTime R step134 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert134 hud hu134 t ⟨ht134,hstop134⟩ have h134 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert134 hud hu134 have hu135 := join_bound Q_pos join135 h134 have ht135 : startTime R step135 ≤ t := by rw [start_eq_stop join135.1] exact (lt_of_not_ge hstop134).le by_cases hstop135 : t ≤ stopTime R step135 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert135 hud hu135 t ⟨ht135,hstop135⟩ have h135 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert135 hud hu135 have hu136 := join_bound Q_pos join136 h135 have ht136 : startTime R step136 ≤ t := by rw [start_eq_stop join136.1] exact (lt_of_not_ge hstop135).le by_cases hstop136 : t ≤ stopTime R step136 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert136 hud hu136 t ⟨ht136,hstop136⟩ have h136 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert136 hud hu136 have hu137 := join_bound Q_pos join137 h136 have ht137 : startTime R step137 ≤ t := by rw [start_eq_stop join137.1] exact (lt_of_not_ge hstop136).le by_cases hstop137 : t ≤ stopTime R step137 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert137 hud hu137 t ⟨ht137,hstop137⟩ have h137 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert137 hud hu137 have hu138 := join_bound Q_pos join138 h137 have ht138 : startTime R step138 ≤ t := by rw [start_eq_stop join138.1] exact (lt_of_not_ge hstop137).le by_cases hstop138 : t ≤ stopTime R step138 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert138 hud hu138 t ⟨ht138,hstop138⟩ have h138 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert138 hud hu138 have hu139 := join_bound Q_pos join139 h138 have ht139 : startTime R step139 ≤ t := by rw [start_eq_stop join139.1] exact (lt_of_not_ge hstop138).le by_cases hstop139 : t ≤ stopTime R step139 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert139 hud hu139 t ⟨ht139,hstop139⟩ have h139 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert139 hud hu139 have hu140 := join_bound Q_pos join140 h139 have ht140 : startTime R step140 ≤ t := by rw [start_eq_stop join140.1] exact (lt_of_not_ge hstop139).le by_cases hstop140 : t ≤ stopTime R step140 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert140 hud hu140 t ⟨ht140,hstop140⟩ have h140 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert140 hud hu140 have hu141 := join_bound Q_pos join141 h140 have ht141 : startTime R step141 ≤ t := by rw [start_eq_stop join141.1] exact (lt_of_not_ge hstop140).le by_cases hstop141 : t ≤ stopTime R step141 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert141 hud hu141 t ⟨ht141,hstop141⟩ have h141 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert141 hud hu141 have hu142 := join_bound Q_pos join142 h141 have ht142 : startTime R step142 ≤ t := by rw [start_eq_stop join142.1] exact (lt_of_not_ge hstop141).le by_cases hstop142 : t ≤ stopTime R step142 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert142 hud hu142 t ⟨ht142,hstop142⟩ have h142 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert142 hud hu142 have hu143 := join_bound Q_pos join143 h142 have ht143 : startTime R step143 ≤ t := by rw [start_eq_stop join143.1] exact (lt_of_not_ge hstop142).le exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert143 hud hu143 t ⟨ht143,ht.2⟩ theorem range_144 {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u (startTime R step144) ∈ Icc ((head step144.lo:ℝ)/Q) ((head step144.hi:ℝ)/Q)) : ∀ t ∈ Icc (startTime R step144) (stopTime R step159), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have ht144 := ht.1 by_cases hstop144 : t ≤ stopTime R step144 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert144 hud hu t ⟨ht144,hstop144⟩ have h144 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert144 hud hu have hu145 := join_bound Q_pos join145 h144 have ht145 : startTime R step145 ≤ t := by rw [start_eq_stop join145.1] exact (lt_of_not_ge hstop144).le by_cases hstop145 : t ≤ stopTime R step145 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert145 hud hu145 t ⟨ht145,hstop145⟩ have h145 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert145 hud hu145 have hu146 := join_bound Q_pos join146 h145 have ht146 : startTime R step146 ≤ t := by rw [start_eq_stop join146.1] exact (lt_of_not_ge hstop145).le by_cases hstop146 : t ≤ stopTime R step146 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert146 hud hu146 t ⟨ht146,hstop146⟩ have h146 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert146 hud hu146 have hu147 := join_bound Q_pos join147 h146 have ht147 : startTime R step147 ≤ t := by rw [start_eq_stop join147.1] exact (lt_of_not_ge hstop146).le by_cases hstop147 : t ≤ stopTime R step147 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert147 hud hu147 t ⟨ht147,hstop147⟩ have h147 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert147 hud hu147 have hu148 := join_bound Q_pos join148 h147 have ht148 : startTime R step148 ≤ t := by rw [start_eq_stop join148.1] exact (lt_of_not_ge hstop147).le by_cases hstop148 : t ≤ stopTime R step148 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert148 hud hu148 t ⟨ht148,hstop148⟩ have h148 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert148 hud hu148 have hu149 := join_bound Q_pos join149 h148 have ht149 : startTime R step149 ≤ t := by rw [start_eq_stop join149.1] exact (lt_of_not_ge hstop148).le by_cases hstop149 : t ≤ stopTime R step149 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert149 hud hu149 t ⟨ht149,hstop149⟩ have h149 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert149 hud hu149 have hu150 := join_bound Q_pos join150 h149 have ht150 : startTime R step150 ≤ t := by rw [start_eq_stop join150.1] exact (lt_of_not_ge hstop149).le by_cases hstop150 : t ≤ stopTime R step150 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert150 hud hu150 t ⟨ht150,hstop150⟩ have h150 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert150 hud hu150 have hu151 := join_bound Q_pos join151 h150 have ht151 : startTime R step151 ≤ t := by rw [start_eq_stop join151.1] exact (lt_of_not_ge hstop150).le by_cases hstop151 : t ≤ stopTime R step151 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert151 hud hu151 t ⟨ht151,hstop151⟩ have h151 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert151 hud hu151 have hu152 := join_bound Q_pos join152 h151 have ht152 : startTime R step152 ≤ t := by rw [start_eq_stop join152.1] exact (lt_of_not_ge hstop151).le by_cases hstop152 : t ≤ stopTime R step152 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert152 hud hu152 t ⟨ht152,hstop152⟩ have h152 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert152 hud hu152 have hu153 := join_bound Q_pos join153 h152 have ht153 : startTime R step153 ≤ t := by rw [start_eq_stop join153.1] exact (lt_of_not_ge hstop152).le by_cases hstop153 : t ≤ stopTime R step153 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert153 hud hu153 t ⟨ht153,hstop153⟩ have h153 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert153 hud hu153 have hu154 := join_bound Q_pos join154 h153 have ht154 : startTime R step154 ≤ t := by rw [start_eq_stop join154.1] exact (lt_of_not_ge hstop153).le by_cases hstop154 : t ≤ stopTime R step154 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert154 hud hu154 t ⟨ht154,hstop154⟩ have h154 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert154 hud hu154 have hu155 := join_bound Q_pos join155 h154 have ht155 : startTime R step155 ≤ t := by rw [start_eq_stop join155.1] exact (lt_of_not_ge hstop154).le by_cases hstop155 : t ≤ stopTime R step155 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert155 hud hu155 t ⟨ht155,hstop155⟩ have h155 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert155 hud hu155 have hu156 := join_bound Q_pos join156 h155 have ht156 : startTime R step156 ≤ t := by rw [start_eq_stop join156.1] exact (lt_of_not_ge hstop155).le by_cases hstop156 : t ≤ stopTime R step156 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert156 hud hu156 t ⟨ht156,hstop156⟩ have h156 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert156 hud hu156 have hu157 := join_bound Q_pos join157 h156 have ht157 : startTime R step157 ≤ t := by rw [start_eq_stop join157.1] exact (lt_of_not_ge hstop156).le by_cases hstop157 : t ≤ stopTime R step157 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert157 hud hu157 t ⟨ht157,hstop157⟩ have h157 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert157 hud hu157 have hu158 := join_bound Q_pos join158 h157 have ht158 : startTime R step158 ≤ t := by rw [start_eq_stop join158.1] exact (lt_of_not_ge hstop157).le by_cases hstop158 : t ≤ stopTime R step158 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert158 hud hu158 t ⟨ht158,hstop158⟩ have h158 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert158 hud hu158 have hu159 := join_bound Q_pos join159 h158 have ht159 : startTime R step159 ≤ t := by rw [start_eq_stop join159.1] exact (lt_of_not_ge hstop158).le exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert159 hud hu159 t ⟨ht159,ht.2⟩ theorem range_160 {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u (startTime R step160) ∈ Icc ((head step160.lo:ℝ)/Q) ((head step160.hi:ℝ)/Q)) : ∀ t ∈ Icc (startTime R step160) (stopTime R step175), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have ht160 := ht.1 by_cases hstop160 : t ≤ stopTime R step160 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert160 hud hu t ⟨ht160,hstop160⟩ have h160 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert160 hud hu have hu161 := join_bound Q_pos join161 h160 have ht161 : startTime R step161 ≤ t := by rw [start_eq_stop join161.1] exact (lt_of_not_ge hstop160).le by_cases hstop161 : t ≤ stopTime R step161 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert161 hud hu161 t ⟨ht161,hstop161⟩ have h161 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert161 hud hu161 have hu162 := join_bound Q_pos join162 h161 have ht162 : startTime R step162 ≤ t := by rw [start_eq_stop join162.1] exact (lt_of_not_ge hstop161).le by_cases hstop162 : t ≤ stopTime R step162 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert162 hud hu162 t ⟨ht162,hstop162⟩ have h162 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert162 hud hu162 have hu163 := join_bound Q_pos join163 h162 have ht163 : startTime R step163 ≤ t := by rw [start_eq_stop join163.1] exact (lt_of_not_ge hstop162).le by_cases hstop163 : t ≤ stopTime R step163 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert163 hud hu163 t ⟨ht163,hstop163⟩ have h163 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert163 hud hu163 have hu164 := join_bound Q_pos join164 h163 have ht164 : startTime R step164 ≤ t := by rw [start_eq_stop join164.1] exact (lt_of_not_ge hstop163).le by_cases hstop164 : t ≤ stopTime R step164 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert164 hud hu164 t ⟨ht164,hstop164⟩ have h164 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert164 hud hu164 have hu165 := join_bound Q_pos join165 h164 have ht165 : startTime R step165 ≤ t := by rw [start_eq_stop join165.1] exact (lt_of_not_ge hstop164).le by_cases hstop165 : t ≤ stopTime R step165 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert165 hud hu165 t ⟨ht165,hstop165⟩ have h165 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert165 hud hu165 have hu166 := join_bound Q_pos join166 h165 have ht166 : startTime R step166 ≤ t := by rw [start_eq_stop join166.1] exact (lt_of_not_ge hstop165).le by_cases hstop166 : t ≤ stopTime R step166 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert166 hud hu166 t ⟨ht166,hstop166⟩ have h166 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert166 hud hu166 have hu167 := join_bound Q_pos join167 h166 have ht167 : startTime R step167 ≤ t := by rw [start_eq_stop join167.1] exact (lt_of_not_ge hstop166).le by_cases hstop167 : t ≤ stopTime R step167 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert167 hud hu167 t ⟨ht167,hstop167⟩ have h167 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert167 hud hu167 have hu168 := join_bound Q_pos join168 h167 have ht168 : startTime R step168 ≤ t := by rw [start_eq_stop join168.1] exact (lt_of_not_ge hstop167).le by_cases hstop168 : t ≤ stopTime R step168 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert168 hud hu168 t ⟨ht168,hstop168⟩ have h168 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert168 hud hu168 have hu169 := join_bound Q_pos join169 h168 have ht169 : startTime R step169 ≤ t := by rw [start_eq_stop join169.1] exact (lt_of_not_ge hstop168).le by_cases hstop169 : t ≤ stopTime R step169 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert169 hud hu169 t ⟨ht169,hstop169⟩ have h169 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert169 hud hu169 have hu170 := join_bound Q_pos join170 h169 have ht170 : startTime R step170 ≤ t := by rw [start_eq_stop join170.1] exact (lt_of_not_ge hstop169).le by_cases hstop170 : t ≤ stopTime R step170 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert170 hud hu170 t ⟨ht170,hstop170⟩ have h170 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert170 hud hu170 have hu171 := join_bound Q_pos join171 h170 have ht171 : startTime R step171 ≤ t := by rw [start_eq_stop join171.1] exact (lt_of_not_ge hstop170).le by_cases hstop171 : t ≤ stopTime R step171 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert171 hud hu171 t ⟨ht171,hstop171⟩ have h171 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert171 hud hu171 have hu172 := join_bound Q_pos join172 h171 have ht172 : startTime R step172 ≤ t := by rw [start_eq_stop join172.1] exact (lt_of_not_ge hstop171).le by_cases hstop172 : t ≤ stopTime R step172 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert172 hud hu172 t ⟨ht172,hstop172⟩ have h172 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert172 hud hu172 have hu173 := join_bound Q_pos join173 h172 have ht173 : startTime R step173 ≤ t := by rw [start_eq_stop join173.1] exact (lt_of_not_ge hstop172).le by_cases hstop173 : t ≤ stopTime R step173 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert173 hud hu173 t ⟨ht173,hstop173⟩ have h173 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert173 hud hu173 have hu174 := join_bound Q_pos join174 h173 have ht174 : startTime R step174 ≤ t := by rw [start_eq_stop join174.1] exact (lt_of_not_ge hstop173).le by_cases hstop174 : t ≤ stopTime R step174 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert174 hud hu174 t ⟨ht174,hstop174⟩ have h174 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert174 hud hu174 have hu175 := join_bound Q_pos join175 h174 have ht175 : startTime R step175 ≤ t := by rw [start_eq_stop join175.1] exact (lt_of_not_ge hstop174).le exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert175 hud hu175 t ⟨ht175,ht.2⟩ theorem range_176 {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u (startTime R step176) ∈ Icc ((head step176.lo:ℝ)/Q) ((head step176.hi:ℝ)/Q)) : ∀ t ∈ Icc (startTime R step176) (stopTime R step191), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have ht176 := ht.1 by_cases hstop176 : t ≤ stopTime R step176 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert176 hud hu t ⟨ht176,hstop176⟩ have h176 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert176 hud hu have hu177 := join_bound Q_pos join177 h176 have ht177 : startTime R step177 ≤ t := by rw [start_eq_stop join177.1] exact (lt_of_not_ge hstop176).le by_cases hstop177 : t ≤ stopTime R step177 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert177 hud hu177 t ⟨ht177,hstop177⟩ have h177 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert177 hud hu177 have hu178 := join_bound Q_pos join178 h177 have ht178 : startTime R step178 ≤ t := by rw [start_eq_stop join178.1] exact (lt_of_not_ge hstop177).le by_cases hstop178 : t ≤ stopTime R step178 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert178 hud hu178 t ⟨ht178,hstop178⟩ have h178 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert178 hud hu178 have hu179 := join_bound Q_pos join179 h178 have ht179 : startTime R step179 ≤ t := by rw [start_eq_stop join179.1] exact (lt_of_not_ge hstop178).le by_cases hstop179 : t ≤ stopTime R step179 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert179 hud hu179 t ⟨ht179,hstop179⟩ have h179 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert179 hud hu179 have hu180 := join_bound Q_pos join180 h179 have ht180 : startTime R step180 ≤ t := by rw [start_eq_stop join180.1] exact (lt_of_not_ge hstop179).le by_cases hstop180 : t ≤ stopTime R step180 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert180 hud hu180 t ⟨ht180,hstop180⟩ have h180 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert180 hud hu180 have hu181 := join_bound Q_pos join181 h180 have ht181 : startTime R step181 ≤ t := by rw [start_eq_stop join181.1] exact (lt_of_not_ge hstop180).le by_cases hstop181 : t ≤ stopTime R step181 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert181 hud hu181 t ⟨ht181,hstop181⟩ have h181 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert181 hud hu181 have hu182 := join_bound Q_pos join182 h181 have ht182 : startTime R step182 ≤ t := by rw [start_eq_stop join182.1] exact (lt_of_not_ge hstop181).le by_cases hstop182 : t ≤ stopTime R step182 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert182 hud hu182 t ⟨ht182,hstop182⟩ have h182 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert182 hud hu182 have hu183 := join_bound Q_pos join183 h182 have ht183 : startTime R step183 ≤ t := by rw [start_eq_stop join183.1] exact (lt_of_not_ge hstop182).le by_cases hstop183 : t ≤ stopTime R step183 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert183 hud hu183 t ⟨ht183,hstop183⟩ have h183 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert183 hud hu183 have hu184 := join_bound Q_pos join184 h183 have ht184 : startTime R step184 ≤ t := by rw [start_eq_stop join184.1] exact (lt_of_not_ge hstop183).le by_cases hstop184 : t ≤ stopTime R step184 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert184 hud hu184 t ⟨ht184,hstop184⟩ have h184 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert184 hud hu184 have hu185 := join_bound Q_pos join185 h184 have ht185 : startTime R step185 ≤ t := by rw [start_eq_stop join185.1] exact (lt_of_not_ge hstop184).le by_cases hstop185 : t ≤ stopTime R step185 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert185 hud hu185 t ⟨ht185,hstop185⟩ have h185 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert185 hud hu185 have hu186 := join_bound Q_pos join186 h185 have ht186 : startTime R step186 ≤ t := by rw [start_eq_stop join186.1] exact (lt_of_not_ge hstop185).le by_cases hstop186 : t ≤ stopTime R step186 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert186 hud hu186 t ⟨ht186,hstop186⟩ have h186 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert186 hud hu186 have hu187 := join_bound Q_pos join187 h186 have ht187 : startTime R step187 ≤ t := by rw [start_eq_stop join187.1] exact (lt_of_not_ge hstop186).le by_cases hstop187 : t ≤ stopTime R step187 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert187 hud hu187 t ⟨ht187,hstop187⟩ have h187 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert187 hud hu187 have hu188 := join_bound Q_pos join188 h187 have ht188 : startTime R step188 ≤ t := by rw [start_eq_stop join188.1] exact (lt_of_not_ge hstop187).le by_cases hstop188 : t ≤ stopTime R step188 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert188 hud hu188 t ⟨ht188,hstop188⟩ have h188 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert188 hud hu188 have hu189 := join_bound Q_pos join189 h188 have ht189 : startTime R step189 ≤ t := by rw [start_eq_stop join189.1] exact (lt_of_not_ge hstop188).le by_cases hstop189 : t ≤ stopTime R step189 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert189 hud hu189 t ⟨ht189,hstop189⟩ have h189 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert189 hud hu189 have hu190 := join_bound Q_pos join190 h189 have ht190 : startTime R step190 ≤ t := by rw [start_eq_stop join190.1] exact (lt_of_not_ge hstop189).le by_cases hstop190 : t ≤ stopTime R step190 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert190 hud hu190 t ⟨ht190,hstop190⟩ have h190 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert190 hud hu190 have hu191 := join_bound Q_pos join191 h190 have ht191 : startTime R step191 ≤ t := by rw [start_eq_stop join191.1] exact (lt_of_not_ge hstop190).le exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert191 hud hu191 t ⟨ht191,ht.2⟩ theorem range_192 {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u (startTime R step192) ∈ Icc ((head step192.lo:ℝ)/Q) ((head step192.hi:ℝ)/Q)) : ∀ t ∈ Icc (startTime R step192) (stopTime R step207), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have ht192 := ht.1 by_cases hstop192 : t ≤ stopTime R step192 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert192 hud hu t ⟨ht192,hstop192⟩ have h192 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert192 hud hu have hu193 := join_bound Q_pos join193 h192 have ht193 : startTime R step193 ≤ t := by rw [start_eq_stop join193.1] exact (lt_of_not_ge hstop192).le by_cases hstop193 : t ≤ stopTime R step193 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert193 hud hu193 t ⟨ht193,hstop193⟩ have h193 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert193 hud hu193 have hu194 := join_bound Q_pos join194 h193 have ht194 : startTime R step194 ≤ t := by rw [start_eq_stop join194.1] exact (lt_of_not_ge hstop193).le by_cases hstop194 : t ≤ stopTime R step194 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert194 hud hu194 t ⟨ht194,hstop194⟩ have h194 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert194 hud hu194 have hu195 := join_bound Q_pos join195 h194 have ht195 : startTime R step195 ≤ t := by rw [start_eq_stop join195.1] exact (lt_of_not_ge hstop194).le by_cases hstop195 : t ≤ stopTime R step195 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert195 hud hu195 t ⟨ht195,hstop195⟩ have h195 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert195 hud hu195 have hu196 := join_bound Q_pos join196 h195 have ht196 : startTime R step196 ≤ t := by rw [start_eq_stop join196.1] exact (lt_of_not_ge hstop195).le by_cases hstop196 : t ≤ stopTime R step196 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert196 hud hu196 t ⟨ht196,hstop196⟩ have h196 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert196 hud hu196 have hu197 := join_bound Q_pos join197 h196 have ht197 : startTime R step197 ≤ t := by rw [start_eq_stop join197.1] exact (lt_of_not_ge hstop196).le by_cases hstop197 : t ≤ stopTime R step197 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert197 hud hu197 t ⟨ht197,hstop197⟩ have h197 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert197 hud hu197 have hu198 := join_bound Q_pos join198 h197 have ht198 : startTime R step198 ≤ t := by rw [start_eq_stop join198.1] exact (lt_of_not_ge hstop197).le by_cases hstop198 : t ≤ stopTime R step198 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert198 hud hu198 t ⟨ht198,hstop198⟩ have h198 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert198 hud hu198 have hu199 := join_bound Q_pos join199 h198 have ht199 : startTime R step199 ≤ t := by rw [start_eq_stop join199.1] exact (lt_of_not_ge hstop198).le by_cases hstop199 : t ≤ stopTime R step199 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert199 hud hu199 t ⟨ht199,hstop199⟩ have h199 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert199 hud hu199 have hu200 := join_bound Q_pos join200 h199 have ht200 : startTime R step200 ≤ t := by rw [start_eq_stop join200.1] exact (lt_of_not_ge hstop199).le by_cases hstop200 : t ≤ stopTime R step200 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert200 hud hu200 t ⟨ht200,hstop200⟩ have h200 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert200 hud hu200 have hu201 := join_bound Q_pos join201 h200 have ht201 : startTime R step201 ≤ t := by rw [start_eq_stop join201.1] exact (lt_of_not_ge hstop200).le by_cases hstop201 : t ≤ stopTime R step201 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert201 hud hu201 t ⟨ht201,hstop201⟩ have h201 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert201 hud hu201 have hu202 := join_bound Q_pos join202 h201 have ht202 : startTime R step202 ≤ t := by rw [start_eq_stop join202.1] exact (lt_of_not_ge hstop201).le by_cases hstop202 : t ≤ stopTime R step202 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert202 hud hu202 t ⟨ht202,hstop202⟩ have h202 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert202 hud hu202 have hu203 := join_bound Q_pos join203 h202 have ht203 : startTime R step203 ≤ t := by rw [start_eq_stop join203.1] exact (lt_of_not_ge hstop202).le by_cases hstop203 : t ≤ stopTime R step203 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert203 hud hu203 t ⟨ht203,hstop203⟩ have h203 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert203 hud hu203 have hu204 := join_bound Q_pos join204 h203 have ht204 : startTime R step204 ≤ t := by rw [start_eq_stop join204.1] exact (lt_of_not_ge hstop203).le by_cases hstop204 : t ≤ stopTime R step204 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert204 hud hu204 t ⟨ht204,hstop204⟩ have h204 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert204 hud hu204 have hu205 := join_bound Q_pos join205 h204 have ht205 : startTime R step205 ≤ t := by rw [start_eq_stop join205.1] exact (lt_of_not_ge hstop204).le by_cases hstop205 : t ≤ stopTime R step205 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert205 hud hu205 t ⟨ht205,hstop205⟩ have h205 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert205 hud hu205 have hu206 := join_bound Q_pos join206 h205 have ht206 : startTime R step206 ≤ t := by rw [start_eq_stop join206.1] exact (lt_of_not_ge hstop205).le by_cases hstop206 : t ≤ stopTime R step206 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert206 hud hu206 t ⟨ht206,hstop206⟩ have h206 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert206 hud hu206 have hu207 := join_bound Q_pos join207 h206 have ht207 : startTime R step207 ≤ t := by rw [start_eq_stop join207.1] exact (lt_of_not_ge hstop206).le exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert207 hud hu207 t ⟨ht207,ht.2⟩ theorem range_208 {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u (startTime R step208) ∈ Icc ((head step208.lo:ℝ)/Q) ((head step208.hi:ℝ)/Q)) : ∀ t ∈ Icc (startTime R step208) (stopTime R step223), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have ht208 := ht.1 by_cases hstop208 : t ≤ stopTime R step208 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert208 hud hu t ⟨ht208,hstop208⟩ have h208 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert208 hud hu have hu209 := join_bound Q_pos join209 h208 have ht209 : startTime R step209 ≤ t := by rw [start_eq_stop join209.1] exact (lt_of_not_ge hstop208).le by_cases hstop209 : t ≤ stopTime R step209 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert209 hud hu209 t ⟨ht209,hstop209⟩ have h209 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert209 hud hu209 have hu210 := join_bound Q_pos join210 h209 have ht210 : startTime R step210 ≤ t := by rw [start_eq_stop join210.1] exact (lt_of_not_ge hstop209).le by_cases hstop210 : t ≤ stopTime R step210 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert210 hud hu210 t ⟨ht210,hstop210⟩ have h210 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert210 hud hu210 have hu211 := join_bound Q_pos join211 h210 have ht211 : startTime R step211 ≤ t := by rw [start_eq_stop join211.1] exact (lt_of_not_ge hstop210).le by_cases hstop211 : t ≤ stopTime R step211 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert211 hud hu211 t ⟨ht211,hstop211⟩ have h211 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert211 hud hu211 have hu212 := join_bound Q_pos join212 h211 have ht212 : startTime R step212 ≤ t := by rw [start_eq_stop join212.1] exact (lt_of_not_ge hstop211).le by_cases hstop212 : t ≤ stopTime R step212 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert212 hud hu212 t ⟨ht212,hstop212⟩ have h212 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert212 hud hu212 have hu213 := join_bound Q_pos join213 h212 have ht213 : startTime R step213 ≤ t := by rw [start_eq_stop join213.1] exact (lt_of_not_ge hstop212).le by_cases hstop213 : t ≤ stopTime R step213 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert213 hud hu213 t ⟨ht213,hstop213⟩ have h213 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert213 hud hu213 have hu214 := join_bound Q_pos join214 h213 have ht214 : startTime R step214 ≤ t := by rw [start_eq_stop join214.1] exact (lt_of_not_ge hstop213).le by_cases hstop214 : t ≤ stopTime R step214 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert214 hud hu214 t ⟨ht214,hstop214⟩ have h214 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert214 hud hu214 have hu215 := join_bound Q_pos join215 h214 have ht215 : startTime R step215 ≤ t := by rw [start_eq_stop join215.1] exact (lt_of_not_ge hstop214).le by_cases hstop215 : t ≤ stopTime R step215 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert215 hud hu215 t ⟨ht215,hstop215⟩ have h215 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert215 hud hu215 have hu216 := join_bound Q_pos join216 h215 have ht216 : startTime R step216 ≤ t := by rw [start_eq_stop join216.1] exact (lt_of_not_ge hstop215).le by_cases hstop216 : t ≤ stopTime R step216 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert216 hud hu216 t ⟨ht216,hstop216⟩ have h216 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert216 hud hu216 have hu217 := join_bound Q_pos join217 h216 have ht217 : startTime R step217 ≤ t := by rw [start_eq_stop join217.1] exact (lt_of_not_ge hstop216).le by_cases hstop217 : t ≤ stopTime R step217 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert217 hud hu217 t ⟨ht217,hstop217⟩ have h217 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert217 hud hu217 have hu218 := join_bound Q_pos join218 h217 have ht218 : startTime R step218 ≤ t := by rw [start_eq_stop join218.1] exact (lt_of_not_ge hstop217).le by_cases hstop218 : t ≤ stopTime R step218 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert218 hud hu218 t ⟨ht218,hstop218⟩ have h218 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert218 hud hu218 have hu219 := join_bound Q_pos join219 h218 have ht219 : startTime R step219 ≤ t := by rw [start_eq_stop join219.1] exact (lt_of_not_ge hstop218).le by_cases hstop219 : t ≤ stopTime R step219 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert219 hud hu219 t ⟨ht219,hstop219⟩ have h219 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert219 hud hu219 have hu220 := join_bound Q_pos join220 h219 have ht220 : startTime R step220 ≤ t := by rw [start_eq_stop join220.1] exact (lt_of_not_ge hstop219).le by_cases hstop220 : t ≤ stopTime R step220 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert220 hud hu220 t ⟨ht220,hstop220⟩ have h220 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert220 hud hu220 have hu221 := join_bound Q_pos join221 h220 have ht221 : startTime R step221 ≤ t := by rw [start_eq_stop join221.1] exact (lt_of_not_ge hstop220).le by_cases hstop221 : t ≤ stopTime R step221 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert221 hud hu221 t ⟨ht221,hstop221⟩ have h221 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert221 hud hu221 have hu222 := join_bound Q_pos join222 h221 have ht222 : startTime R step222 ≤ t := by rw [start_eq_stop join222.1] exact (lt_of_not_ge hstop221).le by_cases hstop222 : t ≤ stopTime R step222 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert222 hud hu222 t ⟨ht222,hstop222⟩ have h222 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert222 hud hu222 have hu223 := join_bound Q_pos join223 h222 have ht223 : startTime R step223 ≤ t := by rw [start_eq_stop join223.1] exact (lt_of_not_ge hstop222).le exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert223 hud hu223 t ⟨ht223,ht.2⟩ theorem range_224 {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u (startTime R step224) ∈ Icc ((head step224.lo:ℝ)/Q) ((head step224.hi:ℝ)/Q)) : ∀ t ∈ Icc (startTime R step224) (stopTime R step231), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have ht224 := ht.1 by_cases hstop224 : t ≤ stopTime R step224 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert224 hud hu t ⟨ht224,hstop224⟩ have h224 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert224 hud hu have hu225 := join_bound Q_pos join225 h224 have ht225 : startTime R step225 ≤ t := by rw [start_eq_stop join225.1] exact (lt_of_not_ge hstop224).le by_cases hstop225 : t ≤ stopTime R step225 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert225 hud hu225 t ⟨ht225,hstop225⟩ have h225 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert225 hud hu225 have hu226 := join_bound Q_pos join226 h225 have ht226 : startTime R step226 ≤ t := by rw [start_eq_stop join226.1] exact (lt_of_not_ge hstop225).le by_cases hstop226 : t ≤ stopTime R step226 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert226 hud hu226 t ⟨ht226,hstop226⟩ have h226 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert226 hud hu226 have hu227 := join_bound Q_pos join227 h226 have ht227 : startTime R step227 ≤ t := by rw [start_eq_stop join227.1] exact (lt_of_not_ge hstop226).le by_cases hstop227 : t ≤ stopTime R step227 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert227 hud hu227 t ⟨ht227,hstop227⟩ have h227 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert227 hud hu227 have hu228 := join_bound Q_pos join228 h227 have ht228 : startTime R step228 ≤ t := by rw [start_eq_stop join228.1] exact (lt_of_not_ge hstop227).le by_cases hstop228 : t ≤ stopTime R step228 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert228 hud hu228 t ⟨ht228,hstop228⟩ have h228 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert228 hud hu228 have hu229 := join_bound Q_pos join229 h228 have ht229 : startTime R step229 ≤ t := by rw [start_eq_stop join229.1] exact (lt_of_not_ge hstop228).le by_cases hstop229 : t ≤ stopTime R step229 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert229 hud hu229 t ⟨ht229,hstop229⟩ have h229 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert229 hud hu229 have hu230 := join_bound Q_pos join230 h229 have ht230 : startTime R step230 ≤ t := by rw [start_eq_stop join230.1] exact (lt_of_not_ge hstop229).le by_cases hstop230 : t ≤ stopTime R step230 · exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert230 hud hu230 t ⟨ht230,hstop230⟩ have h230 := global_step_endpoint S_pos K_pos R_pos Q_pos sigma_mem cert230 hud hu230 have hu231 := join_bound Q_pos join231 h230 have ht231 : startTime R step231 ≤ t := by rw [start_eq_stop join231.1] exact (lt_of_not_ge hstop230).le exact global_step_range S_pos K_pos R_pos Q_pos sigma_mem cert231 hud hu231 t ⟨ht231,ht.2⟩ theorem full_range {u : ℝ → ℝ} (hud : ∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (globalField (SN/S) (KN/K) t (u t)) (Icc 0 (9/100)) t) (hu : u 0 ∈ Icc ((head step0.lo:ℝ)/Q) ((head step0.hi:ℝ)/Q)) : ∀ t ∈ Icc (0:ℝ) (9/100), u t ∈ Icc (0:ℝ) (999/1000) := by intro t ht have hu0 : u (startTime R step0) ∈ Icc ((head step0.lo:ℝ)/Q) ((head step0.hi:ℝ)/Q) := first_time.symm ▸ hu have ht0 : startTime R step0 ≤ t := first_time.symm ▸ ht.1 by_cases hstop0 : t ≤ stopTime R step15 · exact range_0 hud hu0 t ⟨ht0,hstop0⟩ have h0 := propagate_0 hud hu0 have hu16 := join_bound Q_pos join16 h0 have ht16 : startTime R step16 ≤ t := by rw [start_eq_stop join16.1] exact (lt_of_not_ge hstop0).le by_cases hstop16 : t ≤ stopTime R step31 · exact range_16 hud hu16 t ⟨ht16,hstop16⟩ have h16 := propagate_16 hud hu16 have hu32 := join_bound Q_pos join32 h16 have ht32 : startTime R step32 ≤ t := by rw [start_eq_stop join32.1] exact (lt_of_not_ge hstop16).le by_cases hstop32 : t ≤ stopTime R step47 · exact range_32 hud hu32 t ⟨ht32,hstop32⟩ have h32 := propagate_32 hud hu32 have hu48 := join_bound Q_pos join48 h32 have ht48 : startTime R step48 ≤ t := by rw [start_eq_stop join48.1] exact (lt_of_not_ge hstop32).le by_cases hstop48 : t ≤ stopTime R step63 · exact range_48 hud hu48 t ⟨ht48,hstop48⟩ have h48 := propagate_48 hud hu48 have hu64 := join_bound Q_pos join64 h48 have ht64 : startTime R step64 ≤ t := by rw [start_eq_stop join64.1] exact (lt_of_not_ge hstop48).le by_cases hstop64 : t ≤ stopTime R step79 · exact range_64 hud hu64 t ⟨ht64,hstop64⟩ have h64 := propagate_64 hud hu64 have hu80 := join_bound Q_pos join80 h64 have ht80 : startTime R step80 ≤ t := by rw [start_eq_stop join80.1] exact (lt_of_not_ge hstop64).le by_cases hstop80 : t ≤ stopTime R step95 · exact range_80 hud hu80 t ⟨ht80,hstop80⟩ have h80 := propagate_80 hud hu80 have hu96 := join_bound Q_pos join96 h80 have ht96 : startTime R step96 ≤ t := by rw [start_eq_stop join96.1] exact (lt_of_not_ge hstop80).le by_cases hstop96 : t ≤ stopTime R step111 · exact range_96 hud hu96 t ⟨ht96,hstop96⟩ have h96 := propagate_96 hud hu96 have hu112 := join_bound Q_pos join112 h96 have ht112 : startTime R step112 ≤ t := by rw [start_eq_stop join112.1] exact (lt_of_not_ge hstop96).le by_cases hstop112 : t ≤ stopTime R step127 · exact range_112 hud hu112 t ⟨ht112,hstop112⟩ have h112 := propagate_112 hud hu112 have hu128 := join_bound Q_pos join128 h112 have ht128 : startTime R step128 ≤ t := by rw [start_eq_stop join128.1] exact (lt_of_not_ge hstop112).le by_cases hstop128 : t ≤ stopTime R step143 · exact range_128 hud hu128 t ⟨ht128,hstop128⟩ have h128 := propagate_128 hud hu128 have hu144 := join_bound Q_pos join144 h128 have ht144 : startTime R step144 ≤ t := by rw [start_eq_stop join144.1] exact (lt_of_not_ge hstop128).le by_cases hstop144 : t ≤ stopTime R step159 · exact range_144 hud hu144 t ⟨ht144,hstop144⟩ have h144 := propagate_144 hud hu144 have hu160 := join_bound Q_pos join160 h144 have ht160 : startTime R step160 ≤ t := by rw [start_eq_stop join160.1] exact (lt_of_not_ge hstop144).le by_cases hstop160 : t ≤ stopTime R step175 · exact range_160 hud hu160 t ⟨ht160,hstop160⟩ have h160 := propagate_160 hud hu160 have hu176 := join_bound Q_pos join176 h160 have ht176 : startTime R step176 ≤ t := by rw [start_eq_stop join176.1] exact (lt_of_not_ge hstop160).le by_cases hstop176 : t ≤ stopTime R step191 · exact range_176 hud hu176 t ⟨ht176,hstop176⟩ have h176 := propagate_176 hud hu176 have hu192 := join_bound Q_pos join192 h176 have ht192 : startTime R step192 ≤ t := by rw [start_eq_stop join192.1] exact (lt_of_not_ge hstop176).le by_cases hstop192 : t ≤ stopTime R step207 · exact range_192 hud hu192 t ⟨ht192,hstop192⟩ have h192 := propagate_192 hud hu192 have hu208 := join_bound Q_pos join208 h192 have ht208 : startTime R step208 ≤ t := by rw [start_eq_stop join208.1] exact (lt_of_not_ge hstop192).le by_cases hstop208 : t ≤ stopTime R step223 · exact range_208 hud hu208 t ⟨ht208,hstop208⟩ have h208 := propagate_208 hud hu208 have hu224 := join_bound Q_pos join224 h208 have ht224 : startTime R step224 ≤ t := by rw [start_eq_stop join224.1] exact (lt_of_not_ge hstop208).le exact range_224 hud hu224 t ⟨ht224,last_time.symm ▸ ht.2⟩ theorem exists_actual_regular (u0 : ℝ) (hu0 : u0 ∈ Icc ((head step0.lo:ℝ)/Q) ((head step0.hi:ℝ)/Q)) : ∃ u : ℝ → ℝ, u 0=u0 ∧ ContinuousOn u (Icc (0:ℝ) (9/100)) ∧ (∀ t ∈ Icc (0:ℝ) (9/100), u t ∈ Icc (0:ℝ) (999/1000)) ∧ (∀ t ∈ Icc (0:ℝ) (9/100), HasDerivWithinAt u (-numer (KN/K) (99/100-t) (u t)/denom (SN/S) (99/100-t) (u t)) (Icc 0 (9/100)) t) ∧ 858/1000