-- Generated by the EML-lang Lean backend
-- Source module: rc_filter
-- Source file:   /home/monogate/monogate/forge/examples/rc_filter.eml
-- Verified fns:  tau, vout_steady, vout_initial, vout_charging, vout_charging_at_zero

import MachLib.EML
import MachLib.Trig
import MachLib.Forge
import MachLib.Linarith
import MachLib.FixedPoint
import MachLib.SignTactic

open MachLib
open MachLib.Real

axiom circuit  : Real  -- helper (axiomatised in MachLib/Discovered)

-- ── tau ──

noncomputable def tau (r : Real) (c : Real) : Real :=
  (r * c)

theorem rc_time_constant_def (r : Real) (c : Real)
    (h1 : (r > (0 : Real)))
    (h2 : (c > (0 : Real))) :
    ((tau r c) = (r * c)) := by
  unfold tau
  first
  | (apply lo_le_clamp <;> (first | assumption | mach_positivity))
  | apply clamp_le_hi
  | mach_positivity
  | mach_sign
  | rfl
  | sorry  -- out of reach; left for the prover

-- ── vout_steady ──

noncomputable def vout_steady (vin : Real) : Real :=
  vin

theorem rc_steady_state_equals_input (vin : Real) :
    ((vout_steady vin) = vin) := by
  unfold vout_steady
  first
  | (apply lo_le_clamp <;> (first | assumption | mach_positivity))
  | apply clamp_le_hi
  | mach_positivity
  | mach_sign
  | rfl
  | sorry  -- out of reach; left for the prover

-- ── vout_initial ──

noncomputable def vout_initial  : Real :=
  (0 : Real)

theorem rc_initial_output_zero  :
    ((vout_initial ) = (0 : Real)) := by
  unfold vout_initial
  first
  | (apply lo_le_clamp <;> (first | assumption | mach_positivity))
  | apply clamp_le_hi
  | mach_positivity
  | mach_sign
  | rfl
  | sorry  -- out of reach; left for the prover

-- ── vout_charging ──

noncomputable def vout_charging (vin : Real) (t : Real) (tau_val : Real) : Real :=
  (vin * ((1 : Real) - (Real.exp ((-t) / tau_val))))

theorem rc_step_response_form (vin : Real) (t : Real) (tau_val : Real)
    (h1 : (tau_val > (0 : Real))) :
    ((vout_charging vin t tau_val) = (vin * ((1 : Real) - (Real.exp ((-t) / tau_val))))) := by
  unfold vout_charging
  first
  | (apply lo_le_clamp <;> (first | assumption | mach_positivity))
  | apply clamp_le_hi
  | mach_positivity
  | mach_sign
  | rfl
  | sorry  -- out of reach; left for the prover

-- ── vout_charging_at_zero ──

noncomputable def vout_charging_at_zero (vin : Real) (tau_val : Real) : Real :=
  (vin * ((1 : Real) - (Real.exp ((0 : Real) / tau_val))))

theorem rc_step_response_at_zero (vin : Real) (tau_val : Real)
    (h1 : (tau_val > (0 : Real))) :
    ((vout_charging_at_zero vin tau_val) = (0 : Real)) := by
  unfold vout_charging_at_zero
  first
  | (apply lo_le_clamp <;> (first | assumption | mach_positivity))
  | apply clamp_le_hi
  | mach_positivity
  | mach_sign
  | rfl
  | sorry  -- out of reach; left for the prover