Theory CSP_rule_rw

           (*-------------------------------------------*
            |                CSP-Prover                 |
            |              December 2004                |
            |              February 2005  (modified)    |
            |        Yoshinao Isobe (AIST JAPAN)        |
            *-------------------------------------------*)

theory CSP_rule_rw = CSP_law_etc:

(*****************************************************************

         1.
         2.
         3.
         4.

 *****************************************************************)

(*-------------------------------------------------------* 
 |                                                       |
 |       use these rules "csp_rule_..." as follows:      |
 |                                                       |
 |     lemmas new_rule = rule[THEN csp_rule_... ] or     |
 |     lemmas new_rule = csp_rule_...[OF rule]           |
 |                                                       |
 *-------------------------------------------------------*)

lemma csp_rule_rw_left_eq:
  "R1 =F R2 ==> (R1 =F R0) = (R2 =F R0)"
by (simp add: eqF_def)

lemma csp_rule_rw_left_ref:
  "R1 =F R2 ==> (R1 <=F R0) = (R2 <=F R0)"
by (simp add: eqF_def refF_def)

lemma csp_rule_rw_right_eq:
  "R1 =F R2 ==> (R0 =F R1) = (R0 =F R2)"
by (simp add: eqF_def)

lemma csp_rule_rw_right_ref:
  "R1 =F R2 ==> (R0 <=F R1) = (R0 <=F R2)"
by (simp add: eqF_def refF_def)

lemmas csp_rule_rw_left  = csp_rule_rw_left_eq csp_rule_rw_left_ref
lemmas csp_rule_rw_right = csp_rule_rw_right_eq csp_rule_rw_right_ref
lemmas csp_rule_rw       = csp_rule_rw_left csp_rule_rw_right

lemmas csp_rule_rev_rw   = csp_rule_rw[THEN sym]


(*********************************************************
                    Step laws
 *********************************************************)

lemmas csp_step = csp_STOP_step       csp_Act_prefix_step 
                  csp_Ext_choice_step csp_Parallel_step
                  csp_Hiding_step     csp_Renaming_step
                  csp_Seq_compo_step

lemmas csp_step_rw_left_eq   = csp_step[THEN csp_rule_rw_left_eq]
lemmas csp_step_rw_left_ref  = csp_step[THEN csp_rule_rw_left_ref]
lemmas csp_step_rw_right_eq  = csp_step[THEN csp_rule_rw_right_eq]
lemmas csp_step_rw_right_ref = csp_step[THEN csp_rule_rw_right_ref]

lemmas csp_step_rw_left      = csp_step_rw_left_eq csp_step_rw_left_ref
lemmas csp_step_rw_right     = csp_step_rw_right_eq csp_step_rw_right_ref

lemmas csp_step_rw           = csp_step_rw_left csp_step_rw_right

(*********************************************************
          light Step laws for Act_prefix and STOP
 *********************************************************)

lemmas csp_light_step = csp_STOP_step  csp_Act_prefix_step 

lemmas csp_light_step_rw_left_eq   = csp_light_step[THEN csp_rule_rw_left_eq]
lemmas csp_light_step_rw_left_ref  = csp_light_step[THEN csp_rule_rw_left_ref]
lemmas csp_light_step_rw_right_eq  = csp_light_step[THEN csp_rule_rw_right_eq]
lemmas csp_light_step_rw_right_ref = csp_light_step[THEN csp_rule_rw_right_ref]

lemmas csp_light_step_rw_left      = csp_light_step_rw_left_eq csp_light_step_rw_left_ref
lemmas csp_light_step_rw_right     = csp_light_step_rw_right_eq csp_light_step_rw_right_ref

lemmas csp_light_step_rw           = csp_light_step_rw_left csp_light_step_rw_right

(*********************************************************
                       SKIP laws
 *********************************************************)

lemmas csp_SKIP = csp_Seq_compo_unit_l csp_Seq_compo_unit_r
                  csp_Parallel_preterm_l csp_Parallel_preterm_r
                  csp_Parallel_term csp_SKIP_Hiding_Id
                  csp_SKIP_Renaming_Id

lemmas csp_SKIP_rw_left_eq   = csp_SKIP[THEN csp_rule_rw_left_eq]
lemmas csp_SKIP_rw_left_ref  = csp_SKIP[THEN csp_rule_rw_left_ref]
lemmas csp_SKIP_rw_right_eq  = csp_SKIP[THEN csp_rule_rw_right_eq]
lemmas csp_SKIP_rw_right_ref = csp_SKIP[THEN csp_rule_rw_right_ref]

lemmas csp_SKIP_rw_left      = csp_SKIP_rw_left_eq csp_SKIP_rw_left_ref
lemmas csp_SKIP_rw_right     = csp_SKIP_rw_right_eq csp_SKIP_rw_right_ref

lemmas csp_SKIP_rw           = csp_SKIP_rw_left csp_SKIP_rw_right

(*********************************************************
                 name unwinding function
 *********************************************************)

lemmas csp_unwind_rw_left_eq   = csp_unwind[THEN csp_rule_rw_left_eq]
lemmas csp_unwind_rw_left_ref  = csp_unwind[THEN csp_rule_rw_left_ref]
lemmas csp_unwind_rw_right_eq  = csp_unwind[THEN csp_rule_rw_right_eq]
lemmas csp_unwind_rw_right_ref = csp_unwind[THEN csp_rule_rw_right_ref]

lemmas csp_unwind_rw_left      = csp_unwind_rw_left_eq csp_unwind_rw_left_ref
lemmas csp_unwind_rw_right     = csp_unwind_rw_right_eq csp_unwind_rw_right_ref

lemmas csp_unwind_rw           = csp_unwind_rw_left csp_unwind_rw_right

(*** cpo ***)

lemmas csp_unwind_cpo_rw_left_eq   = csp_unwind_cpo[THEN csp_rule_rw_left_eq]
lemmas csp_unwind_cpo_rw_left_ref  = csp_unwind_cpo[THEN csp_rule_rw_left_ref]
lemmas csp_unwind_cpo_rw_right_eq  = csp_unwind_cpo[THEN csp_rule_rw_right_eq]
lemmas csp_unwind_cpo_rw_right_ref = csp_unwind_cpo[THEN csp_rule_rw_right_ref]

lemmas csp_unwind_cpo_rw_left      = csp_unwind_cpo_rw_left_eq csp_unwind_cpo_rw_left_ref
lemmas csp_unwind_cpo_rw_right     = csp_unwind_cpo_rw_right_eq csp_unwind_cpo_rw_right_ref

lemmas csp_unwind_cpo_rw           = csp_unwind_cpo_rw_left csp_unwind_cpo_rw_right

(*********************************************************
                     dist laws
 *********************************************************)

lemmas csp_dist = csp_Ext_choice_dist_l csp_Ext_choice_dist_r
                  csp_Parallel_dist_l   csp_Parallel_dist_r
                  csp_Hiding_dist       csp_Renaming_dist
                  csp_Seq_compo_dist

lemmas csp_dist_rw_left_eq   = csp_dist[THEN csp_rule_rw_left_eq]
lemmas csp_dist_rw_right_eq  = csp_dist[THEN csp_rule_rw_right_eq]
lemmas csp_dist_rw_left_ref  = csp_dist[THEN csp_rule_rw_left_ref]
lemmas csp_dist_rw_right_ref = csp_dist[THEN csp_rule_rw_right_ref]

lemmas csp_dist_rw_left      = csp_dist_rw_left_eq csp_dist_rw_left_ref
lemmas csp_dist_rw_right     = csp_dist_rw_right_eq csp_dist_rw_right_ref

lemmas csp_dist_rw           = csp_dist_rw_left csp_dist_rw_right

(*********************************************************
                     Rep_dist laws
 *********************************************************)

lemmas csp_Rep_dist = csp_Ext_choice_Rep_dist_l csp_Ext_choice_Rep_dist_r
                  csp_Parallel_Rep_dist_l   csp_Parallel_Rep_dist_r
                  csp_Hiding_Rep_dist       csp_Renaming_Rep_dist
                  csp_Seq_compo_Rep_dist

lemmas csp_Rep_dist_rw_left_eq   = csp_Rep_dist[THEN csp_rule_rw_left_eq]
lemmas csp_Rep_dist_rw_right_eq  = csp_Rep_dist[THEN csp_rule_rw_right_eq]
lemmas csp_Rep_dist_rw_left_ref  = csp_Rep_dist[THEN csp_rule_rw_left_ref]
lemmas csp_Rep_dist_rw_right_ref = csp_Rep_dist[THEN csp_rule_rw_right_ref]

lemmas csp_Rep_dist_rw_left      = csp_Rep_dist_rw_left_eq csp_Rep_dist_rw_left_ref
lemmas csp_Rep_dist_rw_right     = csp_Rep_dist_rw_right_eq csp_Rep_dist_rw_right_ref

lemmas csp_Rep_dist_rw           = csp_Rep_dist_rw_left csp_Rep_dist_rw_right

(*********************************************************
                    Conditional
 *********************************************************)

lemmas csp_Conditional_split_rw_left_eq = 
       csp_Conditional_split[THEN csp_rule_rw_left_eq]
lemmas csp_Conditional_split_rw_left_ref =
       csp_Conditional_split[THEN csp_rule_rw_left_ref]
lemmas csp_Conditional_split_rw_right_eq =
       csp_Conditional_split[THEN csp_rule_rw_right_eq]
lemmas csp_Conditional_split_rw_right_ref =
       csp_Conditional_split[THEN csp_rule_rw_right_ref]

lemmas csp_Conditional_split_rw_left  = csp_Conditional_split_rw_left_eq 
                                        csp_Conditional_split_rw_left_ref
lemmas csp_Conditional_split_rw_right = csp_Conditional_split_rw_right_eq
                                         csp_Conditional_split_rw_right_ref

lemmas csp_Conditional_split_rw       = csp_Conditional_split_rw_left
                                        csp_Conditional_split_rw_right

(*** True or False ***)

lemmas csp_Conditional = csp_Conditional_True csp_Conditional_False

lemmas csp_Conditional_rw_left_eq   = csp_Conditional[THEN csp_rule_rw_left_eq]
lemmas csp_Conditional_rw_left_ref  = csp_Conditional[THEN csp_rule_rw_left_ref]
lemmas csp_Conditional_rw_right_eq  = csp_Conditional[THEN csp_rule_rw_right_eq]
lemmas csp_Conditional_rw_right_ref = csp_Conditional[THEN csp_rule_rw_right_ref]

lemmas csp_Conditional_rw_left      = csp_Conditional_rw_left_eq 
                                      csp_Conditional_rw_left_ref
lemmas csp_Conditional_rw_right     = csp_Conditional_rw_right_eq 
                                      csp_Conditional_rw_right_ref

lemmas csp_Conditional_rw           = csp_Conditional_rw_left 
                                      csp_Conditional_rw_right

(*********************************************************
                      Int_simp
 *********************************************************)

lemmas csp_Int_simp = csp_DIV_unwind 
                      csp_Rep_int_choice_unique 
                      csp_Int_choice_idem_weak

lemmas csp_Int_simp_rw_left_eq   = csp_Int_simp[THEN csp_rule_rw_left_eq]
lemmas csp_Int_simp_rw_left_ref  = csp_Int_simp[THEN csp_rule_rw_left_ref]
lemmas csp_Int_simp_rw_right_eq  = csp_Int_simp[THEN csp_rule_rw_right_eq]
lemmas csp_Int_simp_rw_right_ref = csp_Int_simp[THEN csp_rule_rw_right_ref]

lemmas csp_Int_simp_rw_left      = csp_Int_simp_rw_left_eq csp_Int_simp_rw_left_ref
lemmas csp_Int_simp_rw_right     = csp_Int_simp_rw_right_eq csp_Int_simp_rw_right_ref

lemmas csp_Int_simp_rw           = csp_Int_simp_rw_left csp_Int_simp_rw_right

(*********************************************************
                      Ext_simp
 *********************************************************)

lemmas csp_Ext_simp = csp_Ext_choice_unit_l_weak
                      csp_Ext_choice_unit_r_weak

lemmas csp_Ext_simp_rw_left_eq   = csp_Ext_simp[THEN csp_rule_rw_left_eq]
lemmas csp_Ext_simp_rw_left_ref  = csp_Ext_simp[THEN csp_rule_rw_left_ref]
lemmas csp_Ext_simp_rw_right_eq  = csp_Ext_simp[THEN csp_rule_rw_right_eq]
lemmas csp_Ext_simp_rw_right_ref = csp_Ext_simp[THEN csp_rule_rw_right_ref]

lemmas csp_Ext_simp_rw_left      = csp_Ext_simp_rw_left_eq csp_Ext_simp_rw_left_ref
lemmas csp_Ext_simp_rw_right     = csp_Ext_simp_rw_right_eq csp_Ext_simp_rw_right_ref

lemmas csp_Ext_simp_rw           = csp_Ext_simp_rw_left csp_Ext_simp_rw_right

(*********************************************************
                    Timeout_simp
 *********************************************************)

lemmas csp_Mix_simp = csp_STOP_Timeout_weak

lemmas csp_Mix_simp_rw_left_eq   = csp_Mix_simp[THEN csp_rule_rw_left_eq]
lemmas csp_Mix_simp_rw_left_ref  = csp_Mix_simp[THEN csp_rule_rw_left_ref]
lemmas csp_Mix_simp_rw_right_eq  = csp_Mix_simp[THEN csp_rule_rw_right_eq]
lemmas csp_Mix_simp_rw_right_ref = csp_Mix_simp[THEN csp_rule_rw_right_ref]

lemmas csp_Mix_simp_rw_left      = csp_Mix_simp_rw_left_eq csp_Mix_simp_rw_left_ref
lemmas csp_Mix_simp_rw_right     = csp_Mix_simp_rw_right_eq csp_Mix_simp_rw_right_ref

lemmas csp_Mix_simp_rw           = csp_Mix_simp_rw_left csp_Mix_simp_rw_right

(*********************************************************
                      csp_simp
 *********************************************************)

lemmas csp_simp  = csp_Int_simp csp_Ext_simp csp_Mix_simp

lemmas csp_simp_rw_left  = csp_Int_simp_rw_left csp_Ext_simp_rw_left
                           csp_Mix_simp_rw_left
lemmas csp_simp_rw_right = csp_Int_simp_rw_right csp_Ext_simp_rw_right
                           csp_Mix_simp_rw_right
lemmas csp_simp_rw       = csp_simp_rw_left csp_simp_rw_right

(*-------------------------------------------------------* 
 |                                                       |
 |      decompostion controlled by Not_Decompo_Flag      |
 |                                                       |
 *-------------------------------------------------------*)

consts Not_Decompo_Flag :: bool
defs   Not_Decompo_Flag_def : "Not_Decompo_Flag == True"

lemma on_Not_Decompo_Flag: "Not_Decompo_Flag & R ==> R"
by (simp)

lemma off_Not_Decompo_Flag: "R ==> Not_Decompo_Flag & R"
by (simp add: Not_Decompo_Flag_def)

lemma off_Not_Decompo_Flag_True: "Not_Decompo_Flag"
by (simp add: Not_Decompo_Flag_def)

(*----------------------------------------------*
 |   "Not_Decompo_Flag" is used for avoiding    |
 |    infinite decompositions in csp_tactics.   |
 *----------------------------------------------*)

lemma csp_rule_flag_left_eq:
  "(R1 =F R2) ==> (Not_Decompo_Flag & (R2 =F R0) ==> (R1 =F R0))"
by (simp add: eqF_def Not_Decompo_Flag_def)

lemma csp_rule_flag_left_ref:
  "R1 <=F R2 ==> (Not_Decompo_Flag & (R2 <=F R0) ==> (R1 <=F R0))"
by (simp add: eqF_def refF_def Not_Decompo_Flag_def)

lemma csp_rule_flag_right_eq:
  "R2 =F R1 ==> (Not_Decompo_Flag & (R0 =F R2) ==> (R0 =F R1))"
by (simp add: eqF_def Not_Decompo_Flag_def)

lemma csp_rule_flag_right_ref:
  "R2 <=F R1 ==> (Not_Decompo_Flag & (R0 <=F R2) ==> (R0 <=F R1))"
by (simp add: eqF_def refF_def Not_Decompo_Flag_def)

lemmas csp_rule_flag_left  = csp_rule_flag_left_eq csp_rule_flag_left_ref
lemmas csp_rule_flag_right = csp_rule_flag_right_eq csp_rule_flag_right_ref
lemmas csp_rule_flag       = csp_rule_flag_left csp_rule_flag_right

(*** no flag ***)

lemma csp_rule_tr_left_eq:
  "R1 =F R2 ==> ((R2 =F R0) ==> (R1 =F R0))"
by (simp add: eqF_def)

lemma csp_rule_tr_left_ref:
  "R1 <=F R2 ==> ((R2 <=F R0) ==> (R1 <=F R0))"
by (simp add: eqF_def refF_def)

lemma csp_rule_tr_right_eq:
  "R2 =F R1 ==> ((R0 =F R2) ==> (R0 =F R1))"
by (simp add: eqF_def)

lemma csp_rule_tr_right_ref:
  "R2 <=F R1 ==> ((R0 <=F R2) ==> (R0 <=F R1))"
by (simp add: eqF_def refF_def)

lemmas csp_rule_tr_left  = csp_rule_tr_left_eq csp_rule_tr_left_ref
lemmas csp_rule_tr_right = csp_rule_tr_right_eq csp_rule_tr_right_ref
lemmas csp_rule_tr       = csp_rule_tr_left csp_rule_tr_right

lemmas csp_rule_tr_eq   = csp_rule_tr_left_eq csp_rule_tr_right_eq
lemmas csp_rule_tr_ref  = csp_rule_tr_left_ref csp_rule_tr_right_ref

(*------------------------------------------------*
 |           laws for monotonicity (all)          |
 *------------------------------------------------*)

lemmas csp_free_mono_left  = csp_free_mono[THEN csp_rule_tr_left_ref]
lemmas csp_free_mono_right = csp_free_mono[THEN csp_rule_tr_right_ref]

lemmas csp_mono_left  = csp_mono[THEN csp_rule_tr_left_ref]
lemmas csp_mono_right = csp_mono[THEN csp_rule_tr_right_ref]

lemmas csp_free_mono_flag_left  = csp_free_mono[THEN csp_rule_flag_left_ref]
lemmas csp_free_mono_flag_right = csp_free_mono[THEN csp_rule_flag_right_ref]

lemmas csp_mono_flag_left  = csp_mono[THEN csp_rule_flag_left_ref]
lemmas csp_mono_flag_right = csp_mono[THEN csp_rule_flag_right_ref]

(*------------------------------------------------*
 |            laws for congruence (all)           |
 *------------------------------------------------*)

lemmas csp_free_cong_left  = csp_free_cong[THEN csp_rule_tr_left_eq]
lemmas csp_free_cong_right = csp_free_cong[THEN csp_rule_tr_right_eq]

lemmas csp_cong_left  = csp_cong[THEN csp_rule_tr_left_eq]
lemmas csp_cong_right = csp_cong[THEN csp_rule_tr_right_eq]

lemmas csp_free_cong_flag_left  = csp_free_cong[THEN csp_rule_flag_left_eq]
lemmas csp_free_cong_flag_right = csp_free_cong[THEN csp_rule_flag_right_eq]

lemmas csp_cong_flag_left  = csp_cong[THEN csp_rule_flag_left_eq]
lemmas csp_cong_flag_right = csp_cong[THEN csp_rule_flag_right_eq]

(*------------------------------------------------*
 |           laws for decompo (ALL)               |
 *------------------------------------------------*)

lemmas csp_free_decompo_left  = csp_free_mono_left csp_free_cong_left
lemmas csp_free_decompo_right = csp_free_mono_right csp_free_cong_right

lemmas csp_decompo_left  = csp_mono_left csp_cong_left
lemmas csp_decompo_right = csp_mono_right csp_cong_right

lemmas csp_free_decompo_flag_left  = csp_free_mono_flag_left csp_free_cong_flag_left
lemmas csp_free_decompo_flag_right = csp_free_mono_flag_right csp_free_cong_flag_right

lemmas csp_decompo_flag_left  = csp_mono_flag_left csp_cong_flag_left
lemmas csp_decompo_flag_right = csp_mono_flag_right csp_cong_flag_right

(*------------------------------------------------*
 |              trans with Flag                   |
 *------------------------------------------------*)

(*** rewrite (eq) ***)

lemma csp_rw_flag_left_eq:
  "[| R1 =F R2 ; Not_Decompo_Flag & R2 =F R3 |] ==> R1 =F R3"
by (simp add: eqF_def Not_Decompo_Flag_def)

lemma csp_rw_flag_left_ref:
  "[| R1 =F R2 ; Not_Decompo_Flag & R2 <=F R3 |] ==> R1 <=F R3"
by (simp add: refF_def eqF_def Not_Decompo_Flag_def)

lemmas csp_rw_flag_left = csp_rw_flag_left_eq csp_rw_flag_left_ref

lemma csp_rw_flag_right_eq:
  "[| R3 =F R2 ; Not_Decompo_Flag & R1 =F R2 |] ==> R1 =F R3"
by (simp add: eqF_def Not_Decompo_Flag_def)

lemma csp_rw_flag_right_ref:
  "[| R3 =F R2 ; Not_Decompo_Flag & R1 <=F R2 |] ==> R1 <=F R3"
by (simp add: refF_def eqF_def Not_Decompo_Flag_def)

lemmas csp_rw_flag_right = csp_rw_flag_right_eq csp_rw_flag_right_ref

(*** rewrite (ref) ***)

lemma csp_tr_flag_left_eq: "[| R1 =F R2 ; Not_Decompo_Flag & R2 =F R3 |] ==> R1 =F R3"
by (simp add: eqF_def Not_Decompo_Flag_def)

lemma csp_tr_flag_left_ref: "[| R1 <=F R2 ; Not_Decompo_Flag & R2 <=F R3 |] ==> R1 <=F R3"
by (simp add: refF_def eqF_def Not_Decompo_Flag_def)

lemmas csp_tr_flag_left = csp_tr_flag_left_eq csp_tr_flag_left_ref

lemma csp_tr_flag_right_eq: "[| R2 =F R3 ; Not_Decompo_Flag & R1 =F R2 |] ==> R1 =F R3"
by (simp add: eqF_def Not_Decompo_Flag_def)

lemma csp_tr_flag_right_ref: "[| R2 <=F R3 ; Not_Decompo_Flag & R1 <=F R2 |] ==> R1 <=F R3"
by (simp add: refF_def eqF_def Not_Decompo_Flag_def)

lemmas csp_tr_flag_right = csp_tr_flag_right_eq csp_tr_flag_right_ref

end

lemma csp_rule_rw_left_eq:

  R1 =F R2 ==> (R1 =F R0) = (R2 =F R0)

lemma csp_rule_rw_left_ref:

  R1 =F R2 ==> (R1 <=F R0) = (R2 <=F R0)

lemma csp_rule_rw_right_eq:

  R1 =F R2 ==> (R0 =F R1) = (R0 =F R2)

lemma csp_rule_rw_right_ref:

  R1 =F R2 ==> (R0 <=F R1) = (R0 <=F R2)

lemmas csp_rule_rw_left:

  R1 =F R2 ==> (R1 =F R0) = (R2 =F R0)

  R1 =F R2 ==> (R1 <=F R0) = (R2 <=F R0)

lemmas csp_rule_rw_right:

  R1 =F R2 ==> (R0 =F R1) = (R0 =F R2)

  R1 =F R2 ==> (R0 <=F R1) = (R0 <=F R2)

lemmas csp_rule_rw:

  R1 =F R2 ==> (R1 =F R0) = (R2 =F R0)

  R1 =F R2 ==> (R1 <=F R0) = (R2 <=F R0)

  R1 =F R2 ==> (R0 =F R1) = (R0 =F R2)

  R1 =F R2 ==> (R0 <=F R1) = (R0 <=F R2)

lemmas csp_rule_rev_rw:

  R1_1 =F R2_1 ==> (R2_1 =F R0_1) = (R1_1 =F R0_1)

  R1_1 =F R2_1 ==> (R2_1 <=F R0_1) = (R1_1 <=F R0_1)

  R1_1 =F R2_1 ==> (R0_1 =F R2_1) = (R0_1 =F R1_1)

  R1_1 =F R2_1 ==> (R0_1 <=F R2_1) = (R0_1 <=F R1_1)

lemmas csp_step:

  
  LET:fp df IN STOP =F 
  LET:fp df IN ? x:{} -> DIV

  
  LET:fp df IN a -> P =F 
  LET:fp df IN ? x:{a} -> P

  
  LET:fp df IN ? :X -> Pf [+] ? :Y -> Qf =F 
  LET:fp df IN Ext_choice_step X Y Pf Qf

  
  LET:fp df IN ? :Y -> Pf |[X]| ? :Z -> Qf =F 
  LET:fp df IN Parallel_step X Y Z Pf Qf

  
  LET:fp df IN (? :Y -> Pf) -- X =F 
  LET:fp df IN Hiding_step X Y Pf

  
  LET:fp df IN (? :X -> Pf) [[r]] =F 
  LET:fp df IN Renaming_step r X Pf

  
  LET:fp df IN ? :X -> Pf ;; Q =F 
  LET:fp df IN Seq_compo_step X Pf Q

lemmas csp_step_rw_left_eq:

  (
   LET:fp_1 df_1 IN STOP =F R0) =
  (
   LET:fp_1 df_1 IN ? x:{} -> DIV =F R0)

  (
   LET:fp_1 df_1 IN a_1 -> P_1 =F R0) =
  (
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1 =F R0)

  (
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 [+] ? :Y_1 -> Qf_1 =F R0) =
  (
   LET:fp_1 df_1 IN Ext_choice_step X_1 Y_1 Pf_1 Qf_1 =F R0)

  (
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| ? :Z_1 -> Qf_1 =F R0) =
  (
   LET:fp_1 df_1 IN Parallel_step X_1 Y_1 Z_1 Pf_1 Qf_1 =F R0)

  (
   LET:fp_1 df_1 IN (? :Y_1 -> Pf_1) -- X_1 =F R0) =
  (
   LET:fp_1 df_1 IN Hiding_step X_1 Y_1 Pf_1 =F R0)

  (
   LET:fp_1 df_1 IN (? :X_1 -> Pf_1) [[r_1]] =F R0) =
  (
   LET:fp_1 df_1 IN Renaming_step r_1 X_1 Pf_1 =F R0)

  (
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 ;; Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN Seq_compo_step X_1 Pf_1 Q_1 =F R0)

lemmas csp_step_rw_left_ref:

  (
   LET:fp_1 df_1 IN STOP <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:{} -> DIV <=F R0)

  (
   LET:fp_1 df_1 IN a_1 -> P_1 <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1 <=F R0)

  (
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 [+] ? :Y_1 -> Qf_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Ext_choice_step X_1 Y_1 Pf_1 Qf_1 <=F R0)

  (
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| ? :Z_1 -> Qf_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Parallel_step X_1 Y_1 Z_1 Pf_1 Qf_1 <=F R0)

  (
   LET:fp_1 df_1 IN (? :Y_1 -> Pf_1) -- X_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Hiding_step X_1 Y_1 Pf_1 <=F R0)

  (
   LET:fp_1 df_1 IN (? :X_1 -> Pf_1) [[r_1]] <=F R0) =
  (
   LET:fp_1 df_1 IN Renaming_step r_1 X_1 Pf_1 <=F R0)

  (
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 ;; Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Seq_compo_step X_1 Pf_1 Q_1 <=F R0)

lemmas csp_step_rw_right_eq:

  (R0 =F 
   LET:fp_1 df_1 IN STOP) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:{} -> DIV)

  (R0 =F 
   LET:fp_1 df_1 IN a_1 -> P_1) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1)

  (R0 =F 
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 [+] ? :Y_1 -> Qf_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Ext_choice_step X_1 Y_1 Pf_1 Qf_1)

  (R0 =F 
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| ? :Z_1 -> Qf_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Parallel_step X_1 Y_1 Z_1 Pf_1 Qf_1)

  (R0 =F 
   LET:fp_1 df_1 IN (? :Y_1 -> Pf_1) -- X_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Hiding_step X_1 Y_1 Pf_1)

  (R0 =F 
   LET:fp_1 df_1 IN (? :X_1 -> Pf_1) [[r_1]]) =
  (R0 =F 
   LET:fp_1 df_1 IN Renaming_step r_1 X_1 Pf_1)

  (R0 =F 
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 ;; Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Seq_compo_step X_1 Pf_1 Q_1)

lemmas csp_step_rw_right_ref:

  (R0 <=F 
   LET:fp_1 df_1 IN STOP) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:{} -> DIV)

  (R0 <=F 
   LET:fp_1 df_1 IN a_1 -> P_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 [+] ? :Y_1 -> Qf_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Ext_choice_step X_1 Y_1 Pf_1 Qf_1)

  (R0 <=F 
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| ? :Z_1 -> Qf_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Parallel_step X_1 Y_1 Z_1 Pf_1 Qf_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (? :Y_1 -> Pf_1) -- X_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Hiding_step X_1 Y_1 Pf_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (? :X_1 -> Pf_1) [[r_1]]) =
  (R0 <=F 
   LET:fp_1 df_1 IN Renaming_step r_1 X_1 Pf_1)

  (R0 <=F 
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 ;; Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Seq_compo_step X_1 Pf_1 Q_1)

lemmas csp_step_rw_left:

  (
   LET:fp_1 df_1 IN STOP =F R0) =
  (
   LET:fp_1 df_1 IN ? x:{} -> DIV =F R0)

  (
   LET:fp_1 df_1 IN a_1 -> P_1 =F R0) =
  (
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1 =F R0)

  (
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 [+] ? :Y_1 -> Qf_1 =F R0) =
  (
   LET:fp_1 df_1 IN Ext_choice_step X_1 Y_1 Pf_1 Qf_1 =F R0)

  (
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| ? :Z_1 -> Qf_1 =F R0) =
  (
   LET:fp_1 df_1 IN Parallel_step X_1 Y_1 Z_1 Pf_1 Qf_1 =F R0)

  (
   LET:fp_1 df_1 IN (? :Y_1 -> Pf_1) -- X_1 =F R0) =
  (
   LET:fp_1 df_1 IN Hiding_step X_1 Y_1 Pf_1 =F R0)

  (
   LET:fp_1 df_1 IN (? :X_1 -> Pf_1) [[r_1]] =F R0) =
  (
   LET:fp_1 df_1 IN Renaming_step r_1 X_1 Pf_1 =F R0)

  (
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 ;; Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN Seq_compo_step X_1 Pf_1 Q_1 =F R0)

  (
   LET:fp_1 df_1 IN STOP <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:{} -> DIV <=F R0)

  (
   LET:fp_1 df_1 IN a_1 -> P_1 <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1 <=F R0)

  (
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 [+] ? :Y_1 -> Qf_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Ext_choice_step X_1 Y_1 Pf_1 Qf_1 <=F R0)

  (
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| ? :Z_1 -> Qf_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Parallel_step X_1 Y_1 Z_1 Pf_1 Qf_1 <=F R0)

  (
   LET:fp_1 df_1 IN (? :Y_1 -> Pf_1) -- X_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Hiding_step X_1 Y_1 Pf_1 <=F R0)

  (
   LET:fp_1 df_1 IN (? :X_1 -> Pf_1) [[r_1]] <=F R0) =
  (
   LET:fp_1 df_1 IN Renaming_step r_1 X_1 Pf_1 <=F R0)

  (
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 ;; Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Seq_compo_step X_1 Pf_1 Q_1 <=F R0)

lemmas csp_step_rw_right:

  (R0 =F 
   LET:fp_1 df_1 IN STOP) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:{} -> DIV)

  (R0 =F 
   LET:fp_1 df_1 IN a_1 -> P_1) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1)

  (R0 =F 
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 [+] ? :Y_1 -> Qf_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Ext_choice_step X_1 Y_1 Pf_1 Qf_1)

  (R0 =F 
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| ? :Z_1 -> Qf_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Parallel_step X_1 Y_1 Z_1 Pf_1 Qf_1)

  (R0 =F 
   LET:fp_1 df_1 IN (? :Y_1 -> Pf_1) -- X_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Hiding_step X_1 Y_1 Pf_1)

  (R0 =F 
   LET:fp_1 df_1 IN (? :X_1 -> Pf_1) [[r_1]]) =
  (R0 =F 
   LET:fp_1 df_1 IN Renaming_step r_1 X_1 Pf_1)

  (R0 =F 
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 ;; Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Seq_compo_step X_1 Pf_1 Q_1)

  (R0 <=F 
   LET:fp_1 df_1 IN STOP) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:{} -> DIV)

  (R0 <=F 
   LET:fp_1 df_1 IN a_1 -> P_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 [+] ? :Y_1 -> Qf_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Ext_choice_step X_1 Y_1 Pf_1 Qf_1)

  (R0 <=F 
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| ? :Z_1 -> Qf_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Parallel_step X_1 Y_1 Z_1 Pf_1 Qf_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (? :Y_1 -> Pf_1) -- X_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Hiding_step X_1 Y_1 Pf_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (? :X_1 -> Pf_1) [[r_1]]) =
  (R0 <=F 
   LET:fp_1 df_1 IN Renaming_step r_1 X_1 Pf_1)

  (R0 <=F 
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 ;; Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Seq_compo_step X_1 Pf_1 Q_1)

lemmas csp_step_rw:

  (
   LET:fp_1 df_1 IN STOP =F R0) =
  (
   LET:fp_1 df_1 IN ? x:{} -> DIV =F R0)

  (
   LET:fp_1 df_1 IN a_1 -> P_1 =F R0) =
  (
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1 =F R0)

  (
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 [+] ? :Y_1 -> Qf_1 =F R0) =
  (
   LET:fp_1 df_1 IN Ext_choice_step X_1 Y_1 Pf_1 Qf_1 =F R0)

  (
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| ? :Z_1 -> Qf_1 =F R0) =
  (
   LET:fp_1 df_1 IN Parallel_step X_1 Y_1 Z_1 Pf_1 Qf_1 =F R0)

  (
   LET:fp_1 df_1 IN (? :Y_1 -> Pf_1) -- X_1 =F R0) =
  (
   LET:fp_1 df_1 IN Hiding_step X_1 Y_1 Pf_1 =F R0)

  (
   LET:fp_1 df_1 IN (? :X_1 -> Pf_1) [[r_1]] =F R0) =
  (
   LET:fp_1 df_1 IN Renaming_step r_1 X_1 Pf_1 =F R0)

  (
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 ;; Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN Seq_compo_step X_1 Pf_1 Q_1 =F R0)

  (
   LET:fp_1 df_1 IN STOP <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:{} -> DIV <=F R0)

  (
   LET:fp_1 df_1 IN a_1 -> P_1 <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1 <=F R0)

  (
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 [+] ? :Y_1 -> Qf_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Ext_choice_step X_1 Y_1 Pf_1 Qf_1 <=F R0)

  (
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| ? :Z_1 -> Qf_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Parallel_step X_1 Y_1 Z_1 Pf_1 Qf_1 <=F R0)

  (
   LET:fp_1 df_1 IN (? :Y_1 -> Pf_1) -- X_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Hiding_step X_1 Y_1 Pf_1 <=F R0)

  (
   LET:fp_1 df_1 IN (? :X_1 -> Pf_1) [[r_1]] <=F R0) =
  (
   LET:fp_1 df_1 IN Renaming_step r_1 X_1 Pf_1 <=F R0)

  (
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 ;; Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Seq_compo_step X_1 Pf_1 Q_1 <=F R0)

  (R0 =F 
   LET:fp_1 df_1 IN STOP) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:{} -> DIV)

  (R0 =F 
   LET:fp_1 df_1 IN a_1 -> P_1) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1)

  (R0 =F 
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 [+] ? :Y_1 -> Qf_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Ext_choice_step X_1 Y_1 Pf_1 Qf_1)

  (R0 =F 
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| ? :Z_1 -> Qf_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Parallel_step X_1 Y_1 Z_1 Pf_1 Qf_1)

  (R0 =F 
   LET:fp_1 df_1 IN (? :Y_1 -> Pf_1) -- X_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Hiding_step X_1 Y_1 Pf_1)

  (R0 =F 
   LET:fp_1 df_1 IN (? :X_1 -> Pf_1) [[r_1]]) =
  (R0 =F 
   LET:fp_1 df_1 IN Renaming_step r_1 X_1 Pf_1)

  (R0 =F 
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 ;; Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Seq_compo_step X_1 Pf_1 Q_1)

  (R0 <=F 
   LET:fp_1 df_1 IN STOP) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:{} -> DIV)

  (R0 <=F 
   LET:fp_1 df_1 IN a_1 -> P_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 [+] ? :Y_1 -> Qf_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Ext_choice_step X_1 Y_1 Pf_1 Qf_1)

  (R0 <=F 
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| ? :Z_1 -> Qf_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Parallel_step X_1 Y_1 Z_1 Pf_1 Qf_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (? :Y_1 -> Pf_1) -- X_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Hiding_step X_1 Y_1 Pf_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (? :X_1 -> Pf_1) [[r_1]]) =
  (R0 <=F 
   LET:fp_1 df_1 IN Renaming_step r_1 X_1 Pf_1)

  (R0 <=F 
   LET:fp_1 df_1 IN ? :X_1 -> Pf_1 ;; Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Seq_compo_step X_1 Pf_1 Q_1)

lemmas csp_light_step:

  
  LET:fp df IN STOP =F 
  LET:fp df IN ? x:{} -> DIV

  
  LET:fp df IN a -> P =F 
  LET:fp df IN ? x:{a} -> P

lemmas csp_light_step_rw_left_eq:

  (
   LET:fp_1 df_1 IN STOP =F R0) =
  (
   LET:fp_1 df_1 IN ? x:{} -> DIV =F R0)

  (
   LET:fp_1 df_1 IN a_1 -> P_1 =F R0) =
  (
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1 =F R0)

lemmas csp_light_step_rw_left_ref:

  (
   LET:fp_1 df_1 IN STOP <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:{} -> DIV <=F R0)

  (
   LET:fp_1 df_1 IN a_1 -> P_1 <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1 <=F R0)

lemmas csp_light_step_rw_right_eq:

  (R0 =F 
   LET:fp_1 df_1 IN STOP) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:{} -> DIV)

  (R0 =F 
   LET:fp_1 df_1 IN a_1 -> P_1) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1)

lemmas csp_light_step_rw_right_ref:

  (R0 <=F 
   LET:fp_1 df_1 IN STOP) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:{} -> DIV)

  (R0 <=F 
   LET:fp_1 df_1 IN a_1 -> P_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1)

lemmas csp_light_step_rw_left:

  (
   LET:fp_1 df_1 IN STOP =F R0) =
  (
   LET:fp_1 df_1 IN ? x:{} -> DIV =F R0)

  (
   LET:fp_1 df_1 IN a_1 -> P_1 =F R0) =
  (
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1 =F R0)

  (
   LET:fp_1 df_1 IN STOP <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:{} -> DIV <=F R0)

  (
   LET:fp_1 df_1 IN a_1 -> P_1 <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1 <=F R0)

lemmas csp_light_step_rw_right:

  (R0 =F 
   LET:fp_1 df_1 IN STOP) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:{} -> DIV)

  (R0 =F 
   LET:fp_1 df_1 IN a_1 -> P_1) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN STOP) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:{} -> DIV)

  (R0 <=F 
   LET:fp_1 df_1 IN a_1 -> P_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1)

lemmas csp_light_step_rw:

  (
   LET:fp_1 df_1 IN STOP =F R0) =
  (
   LET:fp_1 df_1 IN ? x:{} -> DIV =F R0)

  (
   LET:fp_1 df_1 IN a_1 -> P_1 =F R0) =
  (
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1 =F R0)

  (
   LET:fp_1 df_1 IN STOP <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:{} -> DIV <=F R0)

  (
   LET:fp_1 df_1 IN a_1 -> P_1 <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1 <=F R0)

  (R0 =F 
   LET:fp_1 df_1 IN STOP) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:{} -> DIV)

  (R0 =F 
   LET:fp_1 df_1 IN a_1 -> P_1) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN STOP) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:{} -> DIV)

  (R0 <=F 
   LET:fp_1 df_1 IN a_1 -> P_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:{a_1} -> P_1)

lemmas csp_SKIP:

  
  LET:fp df IN SKIP ;; P =F 
  LET:fp df IN P

  
  LET:fp df IN P ;; SKIP =F 
  LET:fp df IN P

  
  LET:fp df IN SKIP |[X]| ? :Y -> Qf =F 
  LET:fp df IN ? x:(Y - X) -> (SKIP |[X]| Qf x)

  
  LET:fp df IN ? :Y -> Pf |[X]| SKIP =F 
  LET:fp df IN ? x:(Y - X) -> (Pf x |[X]| SKIP)

  
  LET:fp df IN SKIP |[X]| SKIP =F 
  LET:fp df IN SKIP

  
  LET:fp df IN SKIP -- X =F 
  LET:fp df IN SKIP

  
  LET:fp df IN SKIP [[r]] =F 
  LET:fp df IN SKIP

lemmas csp_SKIP_rw_left_eq:

  (
   LET:fp_1 df_1 IN SKIP ;; P_1 =F R0) =
  (
   LET:fp_1 df_1 IN P_1 =F R0)

  (
   LET:fp_1 df_1 IN P_1 ;; SKIP =F R0) =
  (
   LET:fp_1 df_1 IN P_1 =F R0)

  (
   LET:fp_1 df_1 IN SKIP |[X_1]| ? :Y_1 -> Qf_1 =F R0) =
  (
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (SKIP |[X_1]| Qf_1 x) =F R0)

  (
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| SKIP =F R0) =
  (
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (Pf_1 x |[X_1]| SKIP) =F R0)

  (
   LET:fp_1 df_1 IN SKIP |[X_1]| SKIP =F R0) =
  (
   LET:fp_1 df_1 IN SKIP =F R0)

  (
   LET:fp_1 df_1 IN SKIP -- X_1 =F R0) =
  (
   LET:fp_1 df_1 IN SKIP =F R0)

  (
   LET:fp_1 df_1 IN SKIP [[r_1]] =F R0) =
  (
   LET:fp_1 df_1 IN SKIP =F R0)

lemmas csp_SKIP_rw_left_ref:

  (
   LET:fp_1 df_1 IN SKIP ;; P_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P_1 <=F R0)

  (
   LET:fp_1 df_1 IN P_1 ;; SKIP <=F R0) =
  (
   LET:fp_1 df_1 IN P_1 <=F R0)

  (
   LET:fp_1 df_1 IN SKIP |[X_1]| ? :Y_1 -> Qf_1 <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (SKIP |[X_1]| Qf_1 x) <=F R0)

  (
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| SKIP <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (Pf_1 x |[X_1]| SKIP) <=F R0)

  (
   LET:fp_1 df_1 IN SKIP |[X_1]| SKIP <=F R0) =
  (
   LET:fp_1 df_1 IN SKIP <=F R0)

  (
   LET:fp_1 df_1 IN SKIP -- X_1 <=F R0) =
  (
   LET:fp_1 df_1 IN SKIP <=F R0)

  (
   LET:fp_1 df_1 IN SKIP [[r_1]] <=F R0) =
  (
   LET:fp_1 df_1 IN SKIP <=F R0)

lemmas csp_SKIP_rw_right_eq:

  (R0 =F 
   LET:fp_1 df_1 IN SKIP ;; P_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P_1)

  (R0 =F 
   LET:fp_1 df_1 IN P_1 ;; SKIP) =
  (R0 =F 
   LET:fp_1 df_1 IN P_1)

  (R0 =F 
   LET:fp_1 df_1 IN SKIP |[X_1]| ? :Y_1 -> Qf_1) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (SKIP |[X_1]| Qf_1 x))

  (R0 =F 
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| SKIP) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (Pf_1 x |[X_1]| SKIP))

  (R0 =F 
   LET:fp_1 df_1 IN SKIP |[X_1]| SKIP) =
  (R0 =F 
   LET:fp_1 df_1 IN SKIP)

  (R0 =F 
   LET:fp_1 df_1 IN SKIP -- X_1) =
  (R0 =F 
   LET:fp_1 df_1 IN SKIP)

  (R0 =F 
   LET:fp_1 df_1 IN SKIP [[r_1]]) =
  (R0 =F 
   LET:fp_1 df_1 IN SKIP)

lemmas csp_SKIP_rw_right_ref:

  (R0 <=F 
   LET:fp_1 df_1 IN SKIP ;; P_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN P_1 ;; SKIP) =
  (R0 <=F 
   LET:fp_1 df_1 IN P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN SKIP |[X_1]| ? :Y_1 -> Qf_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (SKIP |[X_1]| Qf_1 x))

  (R0 <=F 
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| SKIP) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (Pf_1 x |[X_1]| SKIP))

  (R0 <=F 
   LET:fp_1 df_1 IN SKIP |[X_1]| SKIP) =
  (R0 <=F 
   LET:fp_1 df_1 IN SKIP)

  (R0 <=F 
   LET:fp_1 df_1 IN SKIP -- X_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN SKIP)

  (R0 <=F 
   LET:fp_1 df_1 IN SKIP [[r_1]]) =
  (R0 <=F 
   LET:fp_1 df_1 IN SKIP)

lemmas csp_SKIP_rw_left:

  (
   LET:fp_1 df_1 IN SKIP ;; P_1 =F R0) =
  (
   LET:fp_1 df_1 IN P_1 =F R0)

  (
   LET:fp_1 df_1 IN P_1 ;; SKIP =F R0) =
  (
   LET:fp_1 df_1 IN P_1 =F R0)

  (
   LET:fp_1 df_1 IN SKIP |[X_1]| ? :Y_1 -> Qf_1 =F R0) =
  (
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (SKIP |[X_1]| Qf_1 x) =F R0)

  (
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| SKIP =F R0) =
  (
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (Pf_1 x |[X_1]| SKIP) =F R0)

  (
   LET:fp_1 df_1 IN SKIP |[X_1]| SKIP =F R0) =
  (
   LET:fp_1 df_1 IN SKIP =F R0)

  (
   LET:fp_1 df_1 IN SKIP -- X_1 =F R0) =
  (
   LET:fp_1 df_1 IN SKIP =F R0)

  (
   LET:fp_1 df_1 IN SKIP [[r_1]] =F R0) =
  (
   LET:fp_1 df_1 IN SKIP =F R0)

  (
   LET:fp_1 df_1 IN SKIP ;; P_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P_1 <=F R0)

  (
   LET:fp_1 df_1 IN P_1 ;; SKIP <=F R0) =
  (
   LET:fp_1 df_1 IN P_1 <=F R0)

  (
   LET:fp_1 df_1 IN SKIP |[X_1]| ? :Y_1 -> Qf_1 <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (SKIP |[X_1]| Qf_1 x) <=F R0)

  (
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| SKIP <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (Pf_1 x |[X_1]| SKIP) <=F R0)

  (
   LET:fp_1 df_1 IN SKIP |[X_1]| SKIP <=F R0) =
  (
   LET:fp_1 df_1 IN SKIP <=F R0)

  (
   LET:fp_1 df_1 IN SKIP -- X_1 <=F R0) =
  (
   LET:fp_1 df_1 IN SKIP <=F R0)

  (
   LET:fp_1 df_1 IN SKIP [[r_1]] <=F R0) =
  (
   LET:fp_1 df_1 IN SKIP <=F R0)

lemmas csp_SKIP_rw_right:

  (R0 =F 
   LET:fp_1 df_1 IN SKIP ;; P_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P_1)

  (R0 =F 
   LET:fp_1 df_1 IN P_1 ;; SKIP) =
  (R0 =F 
   LET:fp_1 df_1 IN P_1)

  (R0 =F 
   LET:fp_1 df_1 IN SKIP |[X_1]| ? :Y_1 -> Qf_1) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (SKIP |[X_1]| Qf_1 x))

  (R0 =F 
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| SKIP) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (Pf_1 x |[X_1]| SKIP))

  (R0 =F 
   LET:fp_1 df_1 IN SKIP |[X_1]| SKIP) =
  (R0 =F 
   LET:fp_1 df_1 IN SKIP)

  (R0 =F 
   LET:fp_1 df_1 IN SKIP -- X_1) =
  (R0 =F 
   LET:fp_1 df_1 IN SKIP)

  (R0 =F 
   LET:fp_1 df_1 IN SKIP [[r_1]]) =
  (R0 =F 
   LET:fp_1 df_1 IN SKIP)

  (R0 <=F 
   LET:fp_1 df_1 IN SKIP ;; P_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN P_1 ;; SKIP) =
  (R0 <=F 
   LET:fp_1 df_1 IN P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN SKIP |[X_1]| ? :Y_1 -> Qf_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (SKIP |[X_1]| Qf_1 x))

  (R0 <=F 
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| SKIP) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (Pf_1 x |[X_1]| SKIP))

  (R0 <=F 
   LET:fp_1 df_1 IN SKIP |[X_1]| SKIP) =
  (R0 <=F 
   LET:fp_1 df_1 IN SKIP)

  (R0 <=F 
   LET:fp_1 df_1 IN SKIP -- X_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN SKIP)

  (R0 <=F 
   LET:fp_1 df_1 IN SKIP [[r_1]]) =
  (R0 <=F 
   LET:fp_1 df_1 IN SKIP)

lemmas csp_SKIP_rw:

  (
   LET:fp_1 df_1 IN SKIP ;; P_1 =F R0) =
  (
   LET:fp_1 df_1 IN P_1 =F R0)

  (
   LET:fp_1 df_1 IN P_1 ;; SKIP =F R0) =
  (
   LET:fp_1 df_1 IN P_1 =F R0)

  (
   LET:fp_1 df_1 IN SKIP |[X_1]| ? :Y_1 -> Qf_1 =F R0) =
  (
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (SKIP |[X_1]| Qf_1 x) =F R0)

  (
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| SKIP =F R0) =
  (
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (Pf_1 x |[X_1]| SKIP) =F R0)

  (
   LET:fp_1 df_1 IN SKIP |[X_1]| SKIP =F R0) =
  (
   LET:fp_1 df_1 IN SKIP =F R0)

  (
   LET:fp_1 df_1 IN SKIP -- X_1 =F R0) =
  (
   LET:fp_1 df_1 IN SKIP =F R0)

  (
   LET:fp_1 df_1 IN SKIP [[r_1]] =F R0) =
  (
   LET:fp_1 df_1 IN SKIP =F R0)

  (
   LET:fp_1 df_1 IN SKIP ;; P_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P_1 <=F R0)

  (
   LET:fp_1 df_1 IN P_1 ;; SKIP <=F R0) =
  (
   LET:fp_1 df_1 IN P_1 <=F R0)

  (
   LET:fp_1 df_1 IN SKIP |[X_1]| ? :Y_1 -> Qf_1 <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (SKIP |[X_1]| Qf_1 x) <=F R0)

  (
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| SKIP <=F R0) =
  (
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (Pf_1 x |[X_1]| SKIP) <=F R0)

  (
   LET:fp_1 df_1 IN SKIP |[X_1]| SKIP <=F R0) =
  (
   LET:fp_1 df_1 IN SKIP <=F R0)

  (
   LET:fp_1 df_1 IN SKIP -- X_1 <=F R0) =
  (
   LET:fp_1 df_1 IN SKIP <=F R0)

  (
   LET:fp_1 df_1 IN SKIP [[r_1]] <=F R0) =
  (
   LET:fp_1 df_1 IN SKIP <=F R0)

  (R0 =F 
   LET:fp_1 df_1 IN SKIP ;; P_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P_1)

  (R0 =F 
   LET:fp_1 df_1 IN P_1 ;; SKIP) =
  (R0 =F 
   LET:fp_1 df_1 IN P_1)

  (R0 =F 
   LET:fp_1 df_1 IN SKIP |[X_1]| ? :Y_1 -> Qf_1) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (SKIP |[X_1]| Qf_1 x))

  (R0 =F 
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| SKIP) =
  (R0 =F 
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (Pf_1 x |[X_1]| SKIP))

  (R0 =F 
   LET:fp_1 df_1 IN SKIP |[X_1]| SKIP) =
  (R0 =F 
   LET:fp_1 df_1 IN SKIP)

  (R0 =F 
   LET:fp_1 df_1 IN SKIP -- X_1) =
  (R0 =F 
   LET:fp_1 df_1 IN SKIP)

  (R0 =F 
   LET:fp_1 df_1 IN SKIP [[r_1]]) =
  (R0 =F 
   LET:fp_1 df_1 IN SKIP)

  (R0 <=F 
   LET:fp_1 df_1 IN SKIP ;; P_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN P_1 ;; SKIP) =
  (R0 <=F 
   LET:fp_1 df_1 IN P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN SKIP |[X_1]| ? :Y_1 -> Qf_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (SKIP |[X_1]| Qf_1 x))

  (R0 <=F 
   LET:fp_1 df_1 IN ? :Y_1 -> Pf_1 |[X_1]| SKIP) =
  (R0 <=F 
   LET:fp_1 df_1 IN ? x:(Y_1 - X_1) -> (Pf_1 x |[X_1]| SKIP))

  (R0 <=F 
   LET:fp_1 df_1 IN SKIP |[X_1]| SKIP) =
  (R0 <=F 
   LET:fp_1 df_1 IN SKIP)

  (R0 <=F 
   LET:fp_1 df_1 IN SKIP -- X_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN SKIP)

  (R0 <=F 
   LET:fp_1 df_1 IN SKIP [[r_1]]) =
  (R0 <=F 
   LET:fp_1 df_1 IN SKIP)

lemmas csp_unwind_rw_left_eq:

  [| !!C. nohide (df_1 C); !!C. guard (df_1 C) |]
  ==> (
       LET:fp_1 df_1 IN <C_1> =F R0) =
      (
       LET:fp_1 df_1 IN df_1 C_1 =F R0)

lemmas csp_unwind_rw_left_ref:

  [| !!C. nohide (df_1 C); !!C. guard (df_1 C) |]
  ==> (
       LET:fp_1 df_1 IN <C_1> <=F R0) =
      (
       LET:fp_1 df_1 IN df_1 C_1 <=F R0)

lemmas csp_unwind_rw_right_eq:

  [| !!C. nohide (df_1 C); !!C. guard (df_1 C) |]
  ==> (R0 =F 
       LET:fp_1 df_1 IN <C_1>) =
      (R0 =F 
       LET:fp_1 df_1 IN df_1 C_1)

lemmas csp_unwind_rw_right_ref:

  [| !!C. nohide (df_1 C); !!C. guard (df_1 C) |]
  ==> (R0 <=F 
       LET:fp_1 df_1 IN <C_1>) =
      (R0 <=F 
       LET:fp_1 df_1 IN df_1 C_1)

lemmas csp_unwind_rw_left:

  [| !!C. nohide (df_1 C); !!C. guard (df_1 C) |]
  ==> (
       LET:fp_1 df_1 IN <C_1> =F R0) =
      (
       LET:fp_1 df_1 IN df_1 C_1 =F R0)

  [| !!C. nohide (df_1 C); !!C. guard (df_1 C) |]
  ==> (
       LET:fp_1 df_1 IN <C_1> <=F R0) =
      (
       LET:fp_1 df_1 IN df_1 C_1 <=F R0)

lemmas csp_unwind_rw_right:

  [| !!C. nohide (df_1 C); !!C. guard (df_1 C) |]
  ==> (R0 =F 
       LET:fp_1 df_1 IN <C_1>) =
      (R0 =F 
       LET:fp_1 df_1 IN df_1 C_1)

  [| !!C. nohide (df_1 C); !!C. guard (df_1 C) |]
  ==> (R0 <=F 
       LET:fp_1 df_1 IN <C_1>) =
      (R0 <=F 
       LET:fp_1 df_1 IN df_1 C_1)

lemmas csp_unwind_rw:

  [| !!C. nohide (df_1 C); !!C. guard (df_1 C) |]
  ==> (
       LET:fp_1 df_1 IN <C_1> =F R0) =
      (
       LET:fp_1 df_1 IN df_1 C_1 =F R0)

  [| !!C. nohide (df_1 C); !!C. guard (df_1 C) |]
  ==> (
       LET:fp_1 df_1 IN <C_1> <=F R0) =
      (
       LET:fp_1 df_1 IN df_1 C_1 <=F R0)

  [| !!C. nohide (df_1 C); !!C. guard (df_1 C) |]
  ==> (R0 =F 
       LET:fp_1 df_1 IN <C_1>) =
      (R0 =F 
       LET:fp_1 df_1 IN df_1 C_1)

  [| !!C. nohide (df_1 C); !!C. guard (df_1 C) |]
  ==> (R0 <=F 
       LET:fp_1 df_1 IN <C_1>) =
      (R0 <=F 
       LET:fp_1 df_1 IN df_1 C_1)

lemmas csp_unwind_cpo_rw_left_eq:

  (
   LET:Lfp df_1 IN <C_1> =F R0) =
  (
   LET:Lfp df_1 IN df_1 C_1 =F R0)

lemmas csp_unwind_cpo_rw_left_ref:

  (
   LET:Lfp df_1 IN <C_1> <=F R0) =
  (
   LET:Lfp df_1 IN df_1 C_1 <=F R0)

lemmas csp_unwind_cpo_rw_right_eq:

  (R0 =F 
   LET:Lfp df_1 IN <C_1>) =
  (R0 =F 
   LET:Lfp df_1 IN df_1 C_1)

lemmas csp_unwind_cpo_rw_right_ref:

  (R0 <=F 
   LET:Lfp df_1 IN <C_1>) =
  (R0 <=F 
   LET:Lfp df_1 IN df_1 C_1)

lemmas csp_unwind_cpo_rw_left:

  (
   LET:Lfp df_1 IN <C_1> =F R0) =
  (
   LET:Lfp df_1 IN df_1 C_1 =F R0)

  (
   LET:Lfp df_1 IN <C_1> <=F R0) =
  (
   LET:Lfp df_1 IN df_1 C_1 <=F R0)

lemmas csp_unwind_cpo_rw_right:

  (R0 =F 
   LET:Lfp df_1 IN <C_1>) =
  (R0 =F 
   LET:Lfp df_1 IN df_1 C_1)

  (R0 <=F 
   LET:Lfp df_1 IN <C_1>) =
  (R0 <=F 
   LET:Lfp df_1 IN df_1 C_1)

lemmas csp_unwind_cpo_rw:

  (
   LET:Lfp df_1 IN <C_1> =F R0) =
  (
   LET:Lfp df_1 IN df_1 C_1 =F R0)

  (
   LET:Lfp df_1 IN <C_1> <=F R0) =
  (
   LET:Lfp df_1 IN df_1 C_1 <=F R0)

  (R0 =F 
   LET:Lfp df_1 IN <C_1>) =
  (R0 =F 
   LET:Lfp df_1 IN df_1 C_1)

  (R0 <=F 
   LET:Lfp df_1 IN <C_1>) =
  (R0 <=F 
   LET:Lfp df_1 IN df_1 C_1)

lemmas csp_dist:

  
  LET:fp df IN (P1 |~| P2) [+] Q =F 
  LET:fp df IN P1 [+] Q |~| P2 [+] Q

  
  LET:fp df IN P [+] (Q1 |~| Q2) =F 
  LET:fp df IN P [+] Q1 |~| P [+] Q2

  
  LET:fp df IN (P1 |~| P2) |[X]| Q =F 
  LET:fp df IN P1 |[X]| Q |~| P2 |[X]| Q

  
  LET:fp df IN P |[X]| (Q1 |~| Q2) =F 
  LET:fp df IN P |[X]| Q1 |~| P |[X]| Q2

  
  LET:fp df IN (P1 |~| P2) -- X =F 
  LET:fp df IN P1 -- X |~| P2 -- X

  
  LET:fp df IN (P1 |~| P2) [[r]] =F 
  LET:fp df IN P1 [[r]] |~| P2 [[r]]

  
  LET:fp df IN (P1 |~| P2) ;; Q =F 
  LET:fp df IN P1 ;; Q |~| P2 ;; Q

lemmas csp_dist_rw_left_eq:

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN P1_1 [+] Q_1 |~| P2_1 [+] Q_1 =F R0)

  (
   LET:fp_1 df_1 IN P_1 [+] (Q1_1 |~| Q2_1) =F R0) =
  (
   LET:fp_1 df_1 IN P_1 [+] Q1_1 |~| P_1 [+] Q2_1 =F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) |[X_1]| Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN P1_1 |[X_1]| Q_1 |~| P2_1 |[X_1]| Q_1 =F R0)

  (
   LET:fp_1 df_1 IN P_1 |[X_1]| (Q1_1 |~| Q2_1) =F R0) =
  (
   LET:fp_1 df_1 IN P_1 |[X_1]| Q1_1 |~| P_1 |[X_1]| Q2_1 =F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) -- X_1 =F R0) =
  (
   LET:fp_1 df_1 IN P1_1 -- X_1 |~| P2_1 -- X_1 =F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [[r_1]] =F R0) =
  (
   LET:fp_1 df_1 IN P1_1 [[r_1]] |~| P2_1 [[r_1]] =F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) ;; Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN P1_1 ;; Q_1 |~| P2_1 ;; Q_1 =F R0)

lemmas csp_dist_rw_right_eq:

  (R0 =F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P1_1 [+] Q_1 |~| P2_1 [+] Q_1)

  (R0 =F 
   LET:fp_1 df_1 IN P_1 [+] (Q1_1 |~| Q2_1)) =
  (R0 =F 
   LET:fp_1 df_1 IN P_1 [+] Q1_1 |~| P_1 [+] Q2_1)

  (R0 =F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) |[X_1]| Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P1_1 |[X_1]| Q_1 |~| P2_1 |[X_1]| Q_1)

  (R0 =F 
   LET:fp_1 df_1 IN P_1 |[X_1]| (Q1_1 |~| Q2_1)) =
  (R0 =F 
   LET:fp_1 df_1 IN P_1 |[X_1]| Q1_1 |~| P_1 |[X_1]| Q2_1)

  (R0 =F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) -- X_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P1_1 -- X_1 |~| P2_1 -- X_1)

  (R0 =F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [[r_1]]) =
  (R0 =F 
   LET:fp_1 df_1 IN P1_1 [[r_1]] |~| P2_1 [[r_1]])

  (R0 =F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) ;; Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P1_1 ;; Q_1 |~| P2_1 ;; Q_1)

lemmas csp_dist_rw_left_ref:

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P1_1 [+] Q_1 |~| P2_1 [+] Q_1 <=F R0)

  (
   LET:fp_1 df_1 IN P_1 [+] (Q1_1 |~| Q2_1) <=F R0) =
  (
   LET:fp_1 df_1 IN P_1 [+] Q1_1 |~| P_1 [+] Q2_1 <=F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) |[X_1]| Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P1_1 |[X_1]| Q_1 |~| P2_1 |[X_1]| Q_1 <=F R0)

  (
   LET:fp_1 df_1 IN P_1 |[X_1]| (Q1_1 |~| Q2_1) <=F R0) =
  (
   LET:fp_1 df_1 IN P_1 |[X_1]| Q1_1 |~| P_1 |[X_1]| Q2_1 <=F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) -- X_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P1_1 -- X_1 |~| P2_1 -- X_1 <=F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [[r_1]] <=F R0) =
  (
   LET:fp_1 df_1 IN P1_1 [[r_1]] |~| P2_1 [[r_1]] <=F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) ;; Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P1_1 ;; Q_1 |~| P2_1 ;; Q_1 <=F R0)

lemmas csp_dist_rw_right_ref:

  (R0 <=F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P1_1 [+] Q_1 |~| P2_1 [+] Q_1)

  (R0 <=F 
   LET:fp_1 df_1 IN P_1 [+] (Q1_1 |~| Q2_1)) =
  (R0 <=F 
   LET:fp_1 df_1 IN P_1 [+] Q1_1 |~| P_1 [+] Q2_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) |[X_1]| Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P1_1 |[X_1]| Q_1 |~| P2_1 |[X_1]| Q_1)

  (R0 <=F 
   LET:fp_1 df_1 IN P_1 |[X_1]| (Q1_1 |~| Q2_1)) =
  (R0 <=F 
   LET:fp_1 df_1 IN P_1 |[X_1]| Q1_1 |~| P_1 |[X_1]| Q2_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) -- X_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P1_1 -- X_1 |~| P2_1 -- X_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [[r_1]]) =
  (R0 <=F 
   LET:fp_1 df_1 IN P1_1 [[r_1]] |~| P2_1 [[r_1]])

  (R0 <=F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) ;; Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P1_1 ;; Q_1 |~| P2_1 ;; Q_1)

lemmas csp_dist_rw_left:

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN P1_1 [+] Q_1 |~| P2_1 [+] Q_1 =F R0)

  (
   LET:fp_1 df_1 IN P_1 [+] (Q1_1 |~| Q2_1) =F R0) =
  (
   LET:fp_1 df_1 IN P_1 [+] Q1_1 |~| P_1 [+] Q2_1 =F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) |[X_1]| Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN P1_1 |[X_1]| Q_1 |~| P2_1 |[X_1]| Q_1 =F R0)

  (
   LET:fp_1 df_1 IN P_1 |[X_1]| (Q1_1 |~| Q2_1) =F R0) =
  (
   LET:fp_1 df_1 IN P_1 |[X_1]| Q1_1 |~| P_1 |[X_1]| Q2_1 =F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) -- X_1 =F R0) =
  (
   LET:fp_1 df_1 IN P1_1 -- X_1 |~| P2_1 -- X_1 =F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [[r_1]] =F R0) =
  (
   LET:fp_1 df_1 IN P1_1 [[r_1]] |~| P2_1 [[r_1]] =F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) ;; Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN P1_1 ;; Q_1 |~| P2_1 ;; Q_1 =F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P1_1 [+] Q_1 |~| P2_1 [+] Q_1 <=F R0)

  (
   LET:fp_1 df_1 IN P_1 [+] (Q1_1 |~| Q2_1) <=F R0) =
  (
   LET:fp_1 df_1 IN P_1 [+] Q1_1 |~| P_1 [+] Q2_1 <=F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) |[X_1]| Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P1_1 |[X_1]| Q_1 |~| P2_1 |[X_1]| Q_1 <=F R0)

  (
   LET:fp_1 df_1 IN P_1 |[X_1]| (Q1_1 |~| Q2_1) <=F R0) =
  (
   LET:fp_1 df_1 IN P_1 |[X_1]| Q1_1 |~| P_1 |[X_1]| Q2_1 <=F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) -- X_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P1_1 -- X_1 |~| P2_1 -- X_1 <=F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [[r_1]] <=F R0) =
  (
   LET:fp_1 df_1 IN P1_1 [[r_1]] |~| P2_1 [[r_1]] <=F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) ;; Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P1_1 ;; Q_1 |~| P2_1 ;; Q_1 <=F R0)

lemmas csp_dist_rw_right:

  (R0 =F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P1_1 [+] Q_1 |~| P2_1 [+] Q_1)

  (R0 =F 
   LET:fp_1 df_1 IN P_1 [+] (Q1_1 |~| Q2_1)) =
  (R0 =F 
   LET:fp_1 df_1 IN P_1 [+] Q1_1 |~| P_1 [+] Q2_1)

  (R0 =F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) |[X_1]| Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P1_1 |[X_1]| Q_1 |~| P2_1 |[X_1]| Q_1)

  (R0 =F 
   LET:fp_1 df_1 IN P_1 |[X_1]| (Q1_1 |~| Q2_1)) =
  (R0 =F 
   LET:fp_1 df_1 IN P_1 |[X_1]| Q1_1 |~| P_1 |[X_1]| Q2_1)

  (R0 =F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) -- X_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P1_1 -- X_1 |~| P2_1 -- X_1)

  (R0 =F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [[r_1]]) =
  (R0 =F 
   LET:fp_1 df_1 IN P1_1 [[r_1]] |~| P2_1 [[r_1]])

  (R0 =F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) ;; Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P1_1 ;; Q_1 |~| P2_1 ;; Q_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P1_1 [+] Q_1 |~| P2_1 [+] Q_1)

  (R0 <=F 
   LET:fp_1 df_1 IN P_1 [+] (Q1_1 |~| Q2_1)) =
  (R0 <=F 
   LET:fp_1 df_1 IN P_1 [+] Q1_1 |~| P_1 [+] Q2_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) |[X_1]| Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P1_1 |[X_1]| Q_1 |~| P2_1 |[X_1]| Q_1)

  (R0 <=F 
   LET:fp_1 df_1 IN P_1 |[X_1]| (Q1_1 |~| Q2_1)) =
  (R0 <=F 
   LET:fp_1 df_1 IN P_1 |[X_1]| Q1_1 |~| P_1 |[X_1]| Q2_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) -- X_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P1_1 -- X_1 |~| P2_1 -- X_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [[r_1]]) =
  (R0 <=F 
   LET:fp_1 df_1 IN P1_1 [[r_1]] |~| P2_1 [[r_1]])

  (R0 <=F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) ;; Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P1_1 ;; Q_1 |~| P2_1 ;; Q_1)

lemmas csp_dist_rw:

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN P1_1 [+] Q_1 |~| P2_1 [+] Q_1 =F R0)

  (
   LET:fp_1 df_1 IN P_1 [+] (Q1_1 |~| Q2_1) =F R0) =
  (
   LET:fp_1 df_1 IN P_1 [+] Q1_1 |~| P_1 [+] Q2_1 =F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) |[X_1]| Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN P1_1 |[X_1]| Q_1 |~| P2_1 |[X_1]| Q_1 =F R0)

  (
   LET:fp_1 df_1 IN P_1 |[X_1]| (Q1_1 |~| Q2_1) =F R0) =
  (
   LET:fp_1 df_1 IN P_1 |[X_1]| Q1_1 |~| P_1 |[X_1]| Q2_1 =F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) -- X_1 =F R0) =
  (
   LET:fp_1 df_1 IN P1_1 -- X_1 |~| P2_1 -- X_1 =F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [[r_1]] =F R0) =
  (
   LET:fp_1 df_1 IN P1_1 [[r_1]] |~| P2_1 [[r_1]] =F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) ;; Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN P1_1 ;; Q_1 |~| P2_1 ;; Q_1 =F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P1_1 [+] Q_1 |~| P2_1 [+] Q_1 <=F R0)

  (
   LET:fp_1 df_1 IN P_1 [+] (Q1_1 |~| Q2_1) <=F R0) =
  (
   LET:fp_1 df_1 IN P_1 [+] Q1_1 |~| P_1 [+] Q2_1 <=F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) |[X_1]| Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P1_1 |[X_1]| Q_1 |~| P2_1 |[X_1]| Q_1 <=F R0)

  (
   LET:fp_1 df_1 IN P_1 |[X_1]| (Q1_1 |~| Q2_1) <=F R0) =
  (
   LET:fp_1 df_1 IN P_1 |[X_1]| Q1_1 |~| P_1 |[X_1]| Q2_1 <=F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) -- X_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P1_1 -- X_1 |~| P2_1 -- X_1 <=F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [[r_1]] <=F R0) =
  (
   LET:fp_1 df_1 IN P1_1 [[r_1]] |~| P2_1 [[r_1]] <=F R0)

  (
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) ;; Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P1_1 ;; Q_1 |~| P2_1 ;; Q_1 <=F R0)

  (R0 =F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P1_1 [+] Q_1 |~| P2_1 [+] Q_1)

  (R0 =F 
   LET:fp_1 df_1 IN P_1 [+] (Q1_1 |~| Q2_1)) =
  (R0 =F 
   LET:fp_1 df_1 IN P_1 [+] Q1_1 |~| P_1 [+] Q2_1)

  (R0 =F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) |[X_1]| Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P1_1 |[X_1]| Q_1 |~| P2_1 |[X_1]| Q_1)

  (R0 =F 
   LET:fp_1 df_1 IN P_1 |[X_1]| (Q1_1 |~| Q2_1)) =
  (R0 =F 
   LET:fp_1 df_1 IN P_1 |[X_1]| Q1_1 |~| P_1 |[X_1]| Q2_1)

  (R0 =F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) -- X_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P1_1 -- X_1 |~| P2_1 -- X_1)

  (R0 =F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [[r_1]]) =
  (R0 =F 
   LET:fp_1 df_1 IN P1_1 [[r_1]] |~| P2_1 [[r_1]])

  (R0 =F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) ;; Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P1_1 ;; Q_1 |~| P2_1 ;; Q_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P1_1 [+] Q_1 |~| P2_1 [+] Q_1)

  (R0 <=F 
   LET:fp_1 df_1 IN P_1 [+] (Q1_1 |~| Q2_1)) =
  (R0 <=F 
   LET:fp_1 df_1 IN P_1 [+] Q1_1 |~| P_1 [+] Q2_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) |[X_1]| Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P1_1 |[X_1]| Q_1 |~| P2_1 |[X_1]| Q_1)

  (R0 <=F 
   LET:fp_1 df_1 IN P_1 |[X_1]| (Q1_1 |~| Q2_1)) =
  (R0 <=F 
   LET:fp_1 df_1 IN P_1 |[X_1]| Q1_1 |~| P_1 |[X_1]| Q2_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) -- X_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P1_1 -- X_1 |~| P2_1 -- X_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) [[r_1]]) =
  (R0 <=F 
   LET:fp_1 df_1 IN P1_1 [[r_1]] |~| P2_1 [[r_1]])

  (R0 <=F 
   LET:fp_1 df_1 IN (P1_1 |~| P2_1) ;; Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P1_1 ;; Q_1 |~| P2_1 ;; Q_1)

lemmas csp_Rep_dist:

  X ≠ {}
  ==> 
      LET:fp df IN (! :X .. Pf) [+] Q =F 
      LET:fp df IN ! x:X .. Pf x [+] Q

  X ≠ {}
  ==> 
      LET:fp df IN P [+] (! :X .. Qf) =F 
      LET:fp df IN ! x:X .. P [+] Qf x

  Y ≠ {}
  ==> 
      LET:fp df IN (! :Y .. Pf) |[X]| Q =F 
      LET:fp df IN ! x:Y .. Pf x |[X]| Q

  Y ≠ {}
  ==> 
      LET:fp df IN P |[X]| (! :Y .. Qf) =F 
      LET:fp df IN ! x:Y .. P |[X]| Qf x

  
  LET:fp df IN (! :Y .. Pf) -- X =F 
  LET:fp df IN ! x:Y .. Pf x -- X

  
  LET:fp df IN (! :X .. Pf) [[r]] =F 
  LET:fp df IN ! x:X .. Pf x [[r]]

  
  LET:fp df IN (! :X .. Pf) ;; Q =F 
  LET:fp df IN ! x:X .. Pf x ;; Q

lemmas csp_Rep_dist_rw_left_eq:

  X_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [+] Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [+] Q_1 =F R0)

  X_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN P_1 [+] (! :X_1 .. Qf_1) =F R0) =
      (
       LET:fp_1 df_1 IN ! x:X_1 .. P_1 [+] Qf_1 x =F R0)

  Y_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) |[X_1]| Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x |[X_1]| Q_1 =F R0)

  Y_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN P_1 |[X_1]| (! :Y_1 .. Qf_1) =F R0) =
      (
       LET:fp_1 df_1 IN ! x:Y_1 .. P_1 |[X_1]| Qf_1 x =F R0)

  (
   LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) -- X_1 =F R0) =
  (
   LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x -- X_1 =F R0)

  (
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [[r_1]] =F R0) =
  (
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [[r_1]] =F R0)

  (
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) ;; Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x ;; Q_1 =F R0)

lemmas csp_Rep_dist_rw_right_eq:

  X_1 ≠ {}
  ==> (R0 =F 
       LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [+] Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [+] Q_1)

  X_1 ≠ {}
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 [+] (! :X_1 .. Qf_1)) =
      (R0 =F 
       LET:fp_1 df_1 IN ! x:X_1 .. P_1 [+] Qf_1 x)

  Y_1 ≠ {}
  ==> (R0 =F 
       LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) |[X_1]| Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x |[X_1]| Q_1)

  Y_1 ≠ {}
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 |[X_1]| (! :Y_1 .. Qf_1)) =
      (R0 =F 
       LET:fp_1 df_1 IN ! x:Y_1 .. P_1 |[X_1]| Qf_1 x)

  (R0 =F 
   LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) -- X_1) =
  (R0 =F 
   LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x -- X_1)

  (R0 =F 
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [[r_1]]) =
  (R0 =F 
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [[r_1]])

  (R0 =F 
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) ;; Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x ;; Q_1)

lemmas csp_Rep_dist_rw_left_ref:

  X_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [+] Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [+] Q_1 <=F R0)

  X_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN P_1 [+] (! :X_1 .. Qf_1) <=F R0) =
      (
       LET:fp_1 df_1 IN ! x:X_1 .. P_1 [+] Qf_1 x <=F R0)

  Y_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) |[X_1]| Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x |[X_1]| Q_1 <=F R0)

  Y_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN P_1 |[X_1]| (! :Y_1 .. Qf_1) <=F R0) =
      (
       LET:fp_1 df_1 IN ! x:Y_1 .. P_1 |[X_1]| Qf_1 x <=F R0)

  (
   LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) -- X_1 <=F R0) =
  (
   LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x -- X_1 <=F R0)

  (
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [[r_1]] <=F R0) =
  (
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [[r_1]] <=F R0)

  (
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) ;; Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x ;; Q_1 <=F R0)

lemmas csp_Rep_dist_rw_right_ref:

  X_1 ≠ {}
  ==> (R0 <=F 
       LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [+] Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [+] Q_1)

  X_1 ≠ {}
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 [+] (! :X_1 .. Qf_1)) =
      (R0 <=F 
       LET:fp_1 df_1 IN ! x:X_1 .. P_1 [+] Qf_1 x)

  Y_1 ≠ {}
  ==> (R0 <=F 
       LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) |[X_1]| Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x |[X_1]| Q_1)

  Y_1 ≠ {}
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 |[X_1]| (! :Y_1 .. Qf_1)) =
      (R0 <=F 
       LET:fp_1 df_1 IN ! x:Y_1 .. P_1 |[X_1]| Qf_1 x)

  (R0 <=F 
   LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) -- X_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x -- X_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [[r_1]]) =
  (R0 <=F 
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [[r_1]])

  (R0 <=F 
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) ;; Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x ;; Q_1)

lemmas csp_Rep_dist_rw_left:

  X_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [+] Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [+] Q_1 =F R0)

  X_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN P_1 [+] (! :X_1 .. Qf_1) =F R0) =
      (
       LET:fp_1 df_1 IN ! x:X_1 .. P_1 [+] Qf_1 x =F R0)

  Y_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) |[X_1]| Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x |[X_1]| Q_1 =F R0)

  Y_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN P_1 |[X_1]| (! :Y_1 .. Qf_1) =F R0) =
      (
       LET:fp_1 df_1 IN ! x:Y_1 .. P_1 |[X_1]| Qf_1 x =F R0)

  (
   LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) -- X_1 =F R0) =
  (
   LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x -- X_1 =F R0)

  (
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [[r_1]] =F R0) =
  (
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [[r_1]] =F R0)

  (
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) ;; Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x ;; Q_1 =F R0)

  X_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [+] Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [+] Q_1 <=F R0)

  X_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN P_1 [+] (! :X_1 .. Qf_1) <=F R0) =
      (
       LET:fp_1 df_1 IN ! x:X_1 .. P_1 [+] Qf_1 x <=F R0)

  Y_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) |[X_1]| Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x |[X_1]| Q_1 <=F R0)

  Y_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN P_1 |[X_1]| (! :Y_1 .. Qf_1) <=F R0) =
      (
       LET:fp_1 df_1 IN ! x:Y_1 .. P_1 |[X_1]| Qf_1 x <=F R0)

  (
   LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) -- X_1 <=F R0) =
  (
   LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x -- X_1 <=F R0)

  (
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [[r_1]] <=F R0) =
  (
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [[r_1]] <=F R0)

  (
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) ;; Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x ;; Q_1 <=F R0)

lemmas csp_Rep_dist_rw_right:

  X_1 ≠ {}
  ==> (R0 =F 
       LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [+] Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [+] Q_1)

  X_1 ≠ {}
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 [+] (! :X_1 .. Qf_1)) =
      (R0 =F 
       LET:fp_1 df_1 IN ! x:X_1 .. P_1 [+] Qf_1 x)

  Y_1 ≠ {}
  ==> (R0 =F 
       LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) |[X_1]| Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x |[X_1]| Q_1)

  Y_1 ≠ {}
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 |[X_1]| (! :Y_1 .. Qf_1)) =
      (R0 =F 
       LET:fp_1 df_1 IN ! x:Y_1 .. P_1 |[X_1]| Qf_1 x)

  (R0 =F 
   LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) -- X_1) =
  (R0 =F 
   LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x -- X_1)

  (R0 =F 
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [[r_1]]) =
  (R0 =F 
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [[r_1]])

  (R0 =F 
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) ;; Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x ;; Q_1)

  X_1 ≠ {}
  ==> (R0 <=F 
       LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [+] Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [+] Q_1)

  X_1 ≠ {}
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 [+] (! :X_1 .. Qf_1)) =
      (R0 <=F 
       LET:fp_1 df_1 IN ! x:X_1 .. P_1 [+] Qf_1 x)

  Y_1 ≠ {}
  ==> (R0 <=F 
       LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) |[X_1]| Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x |[X_1]| Q_1)

  Y_1 ≠ {}
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 |[X_1]| (! :Y_1 .. Qf_1)) =
      (R0 <=F 
       LET:fp_1 df_1 IN ! x:Y_1 .. P_1 |[X_1]| Qf_1 x)

  (R0 <=F 
   LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) -- X_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x -- X_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [[r_1]]) =
  (R0 <=F 
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [[r_1]])

  (R0 <=F 
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) ;; Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x ;; Q_1)

lemmas csp_Rep_dist_rw:

  X_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [+] Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [+] Q_1 =F R0)

  X_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN P_1 [+] (! :X_1 .. Qf_1) =F R0) =
      (
       LET:fp_1 df_1 IN ! x:X_1 .. P_1 [+] Qf_1 x =F R0)

  Y_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) |[X_1]| Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x |[X_1]| Q_1 =F R0)

  Y_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN P_1 |[X_1]| (! :Y_1 .. Qf_1) =F R0) =
      (
       LET:fp_1 df_1 IN ! x:Y_1 .. P_1 |[X_1]| Qf_1 x =F R0)

  (
   LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) -- X_1 =F R0) =
  (
   LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x -- X_1 =F R0)

  (
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [[r_1]] =F R0) =
  (
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [[r_1]] =F R0)

  (
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) ;; Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x ;; Q_1 =F R0)

  X_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [+] Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [+] Q_1 <=F R0)

  X_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN P_1 [+] (! :X_1 .. Qf_1) <=F R0) =
      (
       LET:fp_1 df_1 IN ! x:X_1 .. P_1 [+] Qf_1 x <=F R0)

  Y_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) |[X_1]| Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x |[X_1]| Q_1 <=F R0)

  Y_1 ≠ {}
  ==> (
       LET:fp_1 df_1 IN P_1 |[X_1]| (! :Y_1 .. Qf_1) <=F R0) =
      (
       LET:fp_1 df_1 IN ! x:Y_1 .. P_1 |[X_1]| Qf_1 x <=F R0)

  (
   LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) -- X_1 <=F R0) =
  (
   LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x -- X_1 <=F R0)

  (
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [[r_1]] <=F R0) =
  (
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [[r_1]] <=F R0)

  (
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) ;; Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x ;; Q_1 <=F R0)

  X_1 ≠ {}
  ==> (R0 =F 
       LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [+] Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [+] Q_1)

  X_1 ≠ {}
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 [+] (! :X_1 .. Qf_1)) =
      (R0 =F 
       LET:fp_1 df_1 IN ! x:X_1 .. P_1 [+] Qf_1 x)

  Y_1 ≠ {}
  ==> (R0 =F 
       LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) |[X_1]| Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x |[X_1]| Q_1)

  Y_1 ≠ {}
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 |[X_1]| (! :Y_1 .. Qf_1)) =
      (R0 =F 
       LET:fp_1 df_1 IN ! x:Y_1 .. P_1 |[X_1]| Qf_1 x)

  (R0 =F 
   LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) -- X_1) =
  (R0 =F 
   LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x -- X_1)

  (R0 =F 
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [[r_1]]) =
  (R0 =F 
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [[r_1]])

  (R0 =F 
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) ;; Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x ;; Q_1)

  X_1 ≠ {}
  ==> (R0 <=F 
       LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [+] Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [+] Q_1)

  X_1 ≠ {}
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 [+] (! :X_1 .. Qf_1)) =
      (R0 <=F 
       LET:fp_1 df_1 IN ! x:X_1 .. P_1 [+] Qf_1 x)

  Y_1 ≠ {}
  ==> (R0 <=F 
       LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) |[X_1]| Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x |[X_1]| Q_1)

  Y_1 ≠ {}
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 |[X_1]| (! :Y_1 .. Qf_1)) =
      (R0 <=F 
       LET:fp_1 df_1 IN ! x:Y_1 .. P_1 |[X_1]| Qf_1 x)

  (R0 <=F 
   LET:fp_1 df_1 IN (! :Y_1 .. Pf_1) -- X_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN ! x:Y_1 .. Pf_1 x -- X_1)

  (R0 <=F 
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) [[r_1]]) =
  (R0 <=F 
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x [[r_1]])

  (R0 <=F 
   LET:fp_1 df_1 IN (! :X_1 .. Pf_1) ;; Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN ! x:X_1 .. Pf_1 x ;; Q_1)

lemmas csp_Conditional_split_rw_left_eq:

  (
   LET:fp_1 df_1 IN IF b_1 THEN P_1 ELSE Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN (if b_1 then P_1 else Q_1) =F R0)

lemmas csp_Conditional_split_rw_left_ref:

  (
   LET:fp_1 df_1 IN IF b_1 THEN P_1 ELSE Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN (if b_1 then P_1 else Q_1) <=F R0)

lemmas csp_Conditional_split_rw_right_eq:

  (R0 =F 
   LET:fp_1 df_1 IN IF b_1 THEN P_1 ELSE Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN (if b_1 then P_1 else Q_1))

lemmas csp_Conditional_split_rw_right_ref:

  (R0 <=F 
   LET:fp_1 df_1 IN IF b_1 THEN P_1 ELSE Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN (if b_1 then P_1 else Q_1))

lemmas csp_Conditional_split_rw_left:

  (
   LET:fp_1 df_1 IN IF b_1 THEN P_1 ELSE Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN (if b_1 then P_1 else Q_1) =F R0)

  (
   LET:fp_1 df_1 IN IF b_1 THEN P_1 ELSE Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN (if b_1 then P_1 else Q_1) <=F R0)

lemmas csp_Conditional_split_rw_right:

  (R0 =F 
   LET:fp_1 df_1 IN IF b_1 THEN P_1 ELSE Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN (if b_1 then P_1 else Q_1))

  (R0 <=F 
   LET:fp_1 df_1 IN IF b_1 THEN P_1 ELSE Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN (if b_1 then P_1 else Q_1))

lemmas csp_Conditional_split_rw:

  (
   LET:fp_1 df_1 IN IF b_1 THEN P_1 ELSE Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN (if b_1 then P_1 else Q_1) =F R0)

  (
   LET:fp_1 df_1 IN IF b_1 THEN P_1 ELSE Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN (if b_1 then P_1 else Q_1) <=F R0)

  (R0 =F 
   LET:fp_1 df_1 IN IF b_1 THEN P_1 ELSE Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN (if b_1 then P_1 else Q_1))

  (R0 <=F 
   LET:fp_1 df_1 IN IF b_1 THEN P_1 ELSE Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN (if b_1 then P_1 else Q_1))

lemmas csp_Conditional:

  
  LET:fp df IN IF True THEN P ELSE Q =F 
  LET:fp df IN P

  
  LET:fp df IN IF False THEN P ELSE Q =F 
  LET:fp df IN Q

lemmas csp_Conditional_rw_left_eq:

  (
   LET:fp_1 df_1 IN IF True THEN P_1 ELSE Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN P_1 =F R0)

  (
   LET:fp_1 df_1 IN IF False THEN P_1 ELSE Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN Q_1 =F R0)

lemmas csp_Conditional_rw_left_ref:

  (
   LET:fp_1 df_1 IN IF True THEN P_1 ELSE Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P_1 <=F R0)

  (
   LET:fp_1 df_1 IN IF False THEN P_1 ELSE Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Q_1 <=F R0)

lemmas csp_Conditional_rw_right_eq:

  (R0 =F 
   LET:fp_1 df_1 IN IF True THEN P_1 ELSE Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P_1)

  (R0 =F 
   LET:fp_1 df_1 IN IF False THEN P_1 ELSE Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Q_1)

lemmas csp_Conditional_rw_right_ref:

  (R0 <=F 
   LET:fp_1 df_1 IN IF True THEN P_1 ELSE Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN IF False THEN P_1 ELSE Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Q_1)

lemmas csp_Conditional_rw_left:

  (
   LET:fp_1 df_1 IN IF True THEN P_1 ELSE Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN P_1 =F R0)

  (
   LET:fp_1 df_1 IN IF False THEN P_1 ELSE Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN Q_1 =F R0)

  (
   LET:fp_1 df_1 IN IF True THEN P_1 ELSE Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P_1 <=F R0)

  (
   LET:fp_1 df_1 IN IF False THEN P_1 ELSE Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Q_1 <=F R0)

lemmas csp_Conditional_rw_right:

  (R0 =F 
   LET:fp_1 df_1 IN IF True THEN P_1 ELSE Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P_1)

  (R0 =F 
   LET:fp_1 df_1 IN IF False THEN P_1 ELSE Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Q_1)

  (R0 <=F 
   LET:fp_1 df_1 IN IF True THEN P_1 ELSE Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN IF False THEN P_1 ELSE Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Q_1)

lemmas csp_Conditional_rw:

  (
   LET:fp_1 df_1 IN IF True THEN P_1 ELSE Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN P_1 =F R0)

  (
   LET:fp_1 df_1 IN IF False THEN P_1 ELSE Q_1 =F R0) =
  (
   LET:fp_1 df_1 IN Q_1 =F R0)

  (
   LET:fp_1 df_1 IN IF True THEN P_1 ELSE Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN P_1 <=F R0)

  (
   LET:fp_1 df_1 IN IF False THEN P_1 ELSE Q_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Q_1 <=F R0)

  (R0 =F 
   LET:fp_1 df_1 IN IF True THEN P_1 ELSE Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN P_1)

  (R0 =F 
   LET:fp_1 df_1 IN IF False THEN P_1 ELSE Q_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Q_1)

  (R0 <=F 
   LET:fp_1 df_1 IN IF True THEN P_1 ELSE Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN IF False THEN P_1 ELSE Q_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Q_1)

lemmas csp_Int_simp:

  
  LET:fp df IN DIV =F 
  LET:fp df IN ! x:{} .. DIV

  
  LET:fp df IN ! :{a} .. Pf =F 
  LET:fp df IN Pf a

  
  LET:fp df IN P =F 
  LET:fp df IN Q
  ==> 
      LET:fp df IN P |~| Q =F 
      LET:fp df IN P

lemmas csp_Int_simp_rw_left_eq:

  (
   LET:fp_1 df_1 IN DIV =F R0) =
  (
   LET:fp_1 df_1 IN ! x:{} .. DIV =F R0)

  (
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1 =F R0) =
  (
   LET:fp_1 df_1 IN Pf_1 a_1 =F R0)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (
       LET:fp_1 df_1 IN P_1 |~| Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN P_1 =F R0)

lemmas csp_Int_simp_rw_left_ref:

  (
   LET:fp_1 df_1 IN DIV <=F R0) =
  (
   LET:fp_1 df_1 IN ! x:{} .. DIV <=F R0)

  (
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Pf_1 a_1 <=F R0)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (
       LET:fp_1 df_1 IN P_1 |~| Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN P_1 <=F R0)

lemmas csp_Int_simp_rw_right_eq:

  (R0 =F 
   LET:fp_1 df_1 IN DIV) =
  (R0 =F 
   LET:fp_1 df_1 IN ! x:{} .. DIV)

  (R0 =F 
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Pf_1 a_1)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 |~| Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN P_1)

lemmas csp_Int_simp_rw_right_ref:

  (R0 <=F 
   LET:fp_1 df_1 IN DIV) =
  (R0 <=F 
   LET:fp_1 df_1 IN ! x:{} .. DIV)

  (R0 <=F 
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Pf_1 a_1)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 |~| Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN P_1)

lemmas csp_Int_simp_rw_left:

  (
   LET:fp_1 df_1 IN DIV =F R0) =
  (
   LET:fp_1 df_1 IN ! x:{} .. DIV =F R0)

  (
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1 =F R0) =
  (
   LET:fp_1 df_1 IN Pf_1 a_1 =F R0)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (
       LET:fp_1 df_1 IN P_1 |~| Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN P_1 =F R0)

  (
   LET:fp_1 df_1 IN DIV <=F R0) =
  (
   LET:fp_1 df_1 IN ! x:{} .. DIV <=F R0)

  (
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Pf_1 a_1 <=F R0)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (
       LET:fp_1 df_1 IN P_1 |~| Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN P_1 <=F R0)

lemmas csp_Int_simp_rw_right:

  (R0 =F 
   LET:fp_1 df_1 IN DIV) =
  (R0 =F 
   LET:fp_1 df_1 IN ! x:{} .. DIV)

  (R0 =F 
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Pf_1 a_1)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 |~| Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN DIV) =
  (R0 <=F 
   LET:fp_1 df_1 IN ! x:{} .. DIV)

  (R0 <=F 
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Pf_1 a_1)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 |~| Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN P_1)

lemmas csp_Int_simp_rw:

  (
   LET:fp_1 df_1 IN DIV =F R0) =
  (
   LET:fp_1 df_1 IN ! x:{} .. DIV =F R0)

  (
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1 =F R0) =
  (
   LET:fp_1 df_1 IN Pf_1 a_1 =F R0)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (
       LET:fp_1 df_1 IN P_1 |~| Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN P_1 =F R0)

  (
   LET:fp_1 df_1 IN DIV <=F R0) =
  (
   LET:fp_1 df_1 IN ! x:{} .. DIV <=F R0)

  (
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Pf_1 a_1 <=F R0)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (
       LET:fp_1 df_1 IN P_1 |~| Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN P_1 <=F R0)

  (R0 =F 
   LET:fp_1 df_1 IN DIV) =
  (R0 =F 
   LET:fp_1 df_1 IN ! x:{} .. DIV)

  (R0 =F 
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Pf_1 a_1)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 |~| Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN DIV) =
  (R0 <=F 
   LET:fp_1 df_1 IN ! x:{} .. DIV)

  (R0 <=F 
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Pf_1 a_1)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 |~| Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN P_1)

lemmas csp_Ext_simp:

  
  LET:fp df IN Q =F 
  LET:fp df IN STOP
  ==> 
      LET:fp df IN Q [+] P =F 
      LET:fp df IN P

  
  LET:fp df IN Q =F 
  LET:fp df IN STOP
  ==> 
      LET:fp df IN P [+] Q =F 
      LET:fp df IN P

lemmas csp_Ext_simp_rw_left_eq:

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN Q_1 [+] P_1 =F R0) =
      (
       LET:fp_1 df_1 IN P_1 =F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN P_1 [+] Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN P_1 =F R0)

lemmas csp_Ext_simp_rw_left_ref:

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN Q_1 [+] P_1 <=F R0) =
      (
       LET:fp_1 df_1 IN P_1 <=F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN P_1 [+] Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN P_1 <=F R0)

lemmas csp_Ext_simp_rw_right_eq:

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 =F 
       LET:fp_1 df_1 IN Q_1 [+] P_1) =
      (R0 =F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 [+] Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN P_1)

lemmas csp_Ext_simp_rw_right_ref:

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 <=F 
       LET:fp_1 df_1 IN Q_1 [+] P_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 [+] Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN P_1)

lemmas csp_Ext_simp_rw_left:

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN Q_1 [+] P_1 =F R0) =
      (
       LET:fp_1 df_1 IN P_1 =F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN P_1 [+] Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN P_1 =F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN Q_1 [+] P_1 <=F R0) =
      (
       LET:fp_1 df_1 IN P_1 <=F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN P_1 [+] Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN P_1 <=F R0)

lemmas csp_Ext_simp_rw_right:

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 =F 
       LET:fp_1 df_1 IN Q_1 [+] P_1) =
      (R0 =F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 [+] Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 <=F 
       LET:fp_1 df_1 IN Q_1 [+] P_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 [+] Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN P_1)

lemmas csp_Ext_simp_rw:

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN Q_1 [+] P_1 =F R0) =
      (
       LET:fp_1 df_1 IN P_1 =F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN P_1 [+] Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN P_1 =F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN Q_1 [+] P_1 <=F R0) =
      (
       LET:fp_1 df_1 IN P_1 <=F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN P_1 [+] Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN P_1 <=F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 =F 
       LET:fp_1 df_1 IN Q_1 [+] P_1) =
      (R0 =F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 [+] Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 <=F 
       LET:fp_1 df_1 IN Q_1 [+] P_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 [+] Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN P_1)

lemmas csp_Mix_simp:

  [| 
     LET:fp df IN P1 =F 
     LET:fp df IN STOP;
     
     LET:fp df IN P2 =F 
     LET:fp df IN STOP |]
  ==> 
      LET:fp df IN (P1 |~| P2) [+] Q =F 
      LET:fp df IN Q

lemmas csp_Mix_simp_rw_left_eq:

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN Q_1 =F R0)

lemmas csp_Mix_simp_rw_left_ref:

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN Q_1 <=F R0)

lemmas csp_Mix_simp_rw_right_eq:

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (R0 =F 
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN Q_1)

lemmas csp_Mix_simp_rw_right_ref:

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (R0 <=F 
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN Q_1)

lemmas csp_Mix_simp_rw_left:

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN Q_1 =F R0)

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN Q_1 <=F R0)

lemmas csp_Mix_simp_rw_right:

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (R0 =F 
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN Q_1)

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (R0 <=F 
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN Q_1)

lemmas csp_Mix_simp_rw:

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN Q_1 =F R0)

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN Q_1 <=F R0)

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (R0 =F 
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN Q_1)

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (R0 <=F 
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN Q_1)

lemmas csp_simp:

  
  LET:fp df IN DIV =F 
  LET:fp df IN ! x:{} .. DIV

  
  LET:fp df IN ! :{a} .. Pf =F 
  LET:fp df IN Pf a

  
  LET:fp df IN P =F 
  LET:fp df IN Q
  ==> 
      LET:fp df IN P |~| Q =F 
      LET:fp df IN P

  
  LET:fp df IN Q =F 
  LET:fp df IN STOP
  ==> 
      LET:fp df IN Q [+] P =F 
      LET:fp df IN P

  
  LET:fp df IN Q =F 
  LET:fp df IN STOP
  ==> 
      LET:fp df IN P [+] Q =F 
      LET:fp df IN P

  [| 
     LET:fp df IN P1 =F 
     LET:fp df IN STOP;
     
     LET:fp df IN P2 =F 
     LET:fp df IN STOP |]
  ==> 
      LET:fp df IN (P1 |~| P2) [+] Q =F 
      LET:fp df IN Q

lemmas csp_simp_rw_left:

  (
   LET:fp_1 df_1 IN DIV =F R0) =
  (
   LET:fp_1 df_1 IN ! x:{} .. DIV =F R0)

  (
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1 =F R0) =
  (
   LET:fp_1 df_1 IN Pf_1 a_1 =F R0)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (
       LET:fp_1 df_1 IN P_1 |~| Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN P_1 =F R0)

  (
   LET:fp_1 df_1 IN DIV <=F R0) =
  (
   LET:fp_1 df_1 IN ! x:{} .. DIV <=F R0)

  (
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Pf_1 a_1 <=F R0)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (
       LET:fp_1 df_1 IN P_1 |~| Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN P_1 <=F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN Q_1 [+] P_1 =F R0) =
      (
       LET:fp_1 df_1 IN P_1 =F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN P_1 [+] Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN P_1 =F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN Q_1 [+] P_1 <=F R0) =
      (
       LET:fp_1 df_1 IN P_1 <=F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN P_1 [+] Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN P_1 <=F R0)

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN Q_1 =F R0)

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN Q_1 <=F R0)

lemmas csp_simp_rw_right:

  (R0 =F 
   LET:fp_1 df_1 IN DIV) =
  (R0 =F 
   LET:fp_1 df_1 IN ! x:{} .. DIV)

  (R0 =F 
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Pf_1 a_1)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 |~| Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN DIV) =
  (R0 <=F 
   LET:fp_1 df_1 IN ! x:{} .. DIV)

  (R0 <=F 
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Pf_1 a_1)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 |~| Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 =F 
       LET:fp_1 df_1 IN Q_1 [+] P_1) =
      (R0 =F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 [+] Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 <=F 
       LET:fp_1 df_1 IN Q_1 [+] P_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 [+] Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN P_1)

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (R0 =F 
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN Q_1)

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (R0 <=F 
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN Q_1)

lemmas csp_simp_rw:

  (
   LET:fp_1 df_1 IN DIV =F R0) =
  (
   LET:fp_1 df_1 IN ! x:{} .. DIV =F R0)

  (
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1 =F R0) =
  (
   LET:fp_1 df_1 IN Pf_1 a_1 =F R0)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (
       LET:fp_1 df_1 IN P_1 |~| Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN P_1 =F R0)

  (
   LET:fp_1 df_1 IN DIV <=F R0) =
  (
   LET:fp_1 df_1 IN ! x:{} .. DIV <=F R0)

  (
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1 <=F R0) =
  (
   LET:fp_1 df_1 IN Pf_1 a_1 <=F R0)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (
       LET:fp_1 df_1 IN P_1 |~| Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN P_1 <=F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN Q_1 [+] P_1 =F R0) =
      (
       LET:fp_1 df_1 IN P_1 =F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN P_1 [+] Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN P_1 =F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN Q_1 [+] P_1 <=F R0) =
      (
       LET:fp_1 df_1 IN P_1 <=F R0)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (
       LET:fp_1 df_1 IN P_1 [+] Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN P_1 <=F R0)

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 =F R0) =
      (
       LET:fp_1 df_1 IN Q_1 =F R0)

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1 <=F R0) =
      (
       LET:fp_1 df_1 IN Q_1 <=F R0)

  (R0 =F 
   LET:fp_1 df_1 IN DIV) =
  (R0 =F 
   LET:fp_1 df_1 IN ! x:{} .. DIV)

  (R0 =F 
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1) =
  (R0 =F 
   LET:fp_1 df_1 IN Pf_1 a_1)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 |~| Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN P_1)

  (R0 <=F 
   LET:fp_1 df_1 IN DIV) =
  (R0 <=F 
   LET:fp_1 df_1 IN ! x:{} .. DIV)

  (R0 <=F 
   LET:fp_1 df_1 IN ! :{a_1} .. Pf_1) =
  (R0 <=F 
   LET:fp_1 df_1 IN Pf_1 a_1)

  
  LET:fp_1 df_1 IN P_1 =F 
  LET:fp_1 df_1 IN Q_1
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 |~| Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 =F 
       LET:fp_1 df_1 IN Q_1 [+] P_1) =
      (R0 =F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 =F 
       LET:fp_1 df_1 IN P_1 [+] Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 <=F 
       LET:fp_1 df_1 IN Q_1 [+] P_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN P_1)

  
  LET:fp_1 df_1 IN Q_1 =F 
  LET:fp_1 df_1 IN STOP
  ==> (R0 <=F 
       LET:fp_1 df_1 IN P_1 [+] Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN P_1)

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (R0 =F 
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
      (R0 =F 
       LET:fp_1 df_1 IN Q_1)

  [| 
     LET:fp_1 df_1 IN P1_1 =F 
     LET:fp_1 df_1 IN STOP;
     
     LET:fp_1 df_1 IN P2_1 =F 
     LET:fp_1 df_1 IN STOP |]
  ==> (R0 <=F 
       LET:fp_1 df_1 IN (P1_1 |~| P2_1) [+] Q_1) =
      (R0 <=F 
       LET:fp_1 df_1 IN Q_1)

lemma on_Not_Decompo_Flag:

  Not_Decompo_Flag ∧ R ==> R

lemma off_Not_Decompo_Flag:

  R ==> Not_Decompo_Flag ∧ R

lemma off_Not_Decompo_Flag_True:

  Not_Decompo_Flag

lemma csp_rule_flag_left_eq:

  [| R1 =F R2; Not_Decompo_Flag ∧ R2 =F R0 |] ==> R1 =F R0

lemma csp_rule_flag_left_ref:

  [| R1 <=F R2; Not_Decompo_Flag ∧ R2 <=F R0 |] ==> R1 <=F R0

lemma csp_rule_flag_right_eq:

  [| R2 =F R1; Not_Decompo_Flag ∧ R0 =F R2 |] ==> R0 =F R1

lemma csp_rule_flag_right_ref:

  [| R2 <=F R1; Not_Decompo_Flag ∧ R0 <=F R2 |] ==> R0 <=F R1

lemmas csp_rule_flag_left:

  [| R1 =F R2; Not_Decompo_Flag ∧ R2 =F R0 |] ==> R1 =F R0

  [| R1 <=F R2; Not_Decompo_Flag ∧ R2 <=F R0 |] ==> R1 <=F R0

lemmas csp_rule_flag_right:

  [| R2 =F R1; Not_Decompo_Flag ∧ R0 =F R2 |] ==> R0 =F R1

  [| R2 <=F R1; Not_Decompo_Flag ∧ R0 <=F R2 |] ==> R0 <=F R1

lemmas csp_rule_flag:

  [| R1 =F R2; Not_Decompo_Flag ∧ R2 =F R0 |] ==> R1 =F R0

  [| R1 <=F R2; Not_Decompo_Flag ∧ R2 <=F R0 |] ==> R1 <=F R0

  [| R2 =F R1; Not_Decompo_Flag ∧ R0 =F R2 |] ==> R0 =F R1

  [| R2 <=F R1; Not_Decompo_Flag ∧ R0 <=F R2 |] ==> R0 <=F R1

lemma csp_rule_tr_left_eq:

  [| R1 =F R2; R2 =F R0 |] ==> R1 =F R0

lemma csp_rule_tr_left_ref:

  [| R1 <=F R2; R2 <=F R0 |] ==> R1 <=F R0

lemma csp_rule_tr_right_eq:

  [| R2 =F R1; R0 =F R2 |] ==> R0 =F R1

lemma csp_rule_tr_right_ref:

  [| R2 <=F R1; R0 <=F R2 |] ==> R0 <=F R1

lemmas csp_rule_tr_left:

  [| R1 =F R2; R2 =F R0 |] ==> R1 =F R0

  [| R1 <=F R2; R2 <=F R0 |] ==> R1 <=F R0

lemmas csp_rule_tr_right:

  [| R2 =F R1; R0 =F R2 |] ==> R0 =F R1

  [| R2 <=F R1; R0 <=F R2 |] ==> R0 <=F R1

lemmas csp_rule_tr:

  [| R1 =F R2; R2 =F R0 |] ==> R1 =F R0

  [| R1 <=F R2; R2 <=F R0 |] ==> R1 <=F R0

  [| R2 =F R1; R0 =F R2 |] ==> R0 =F R1

  [| R2 <=F R1; R0 <=F R2 |] ==> R0 <=F R1

lemmas csp_rule_tr_eq:

  [| R1 =F R2; R2 =F R0 |] ==> R1 =F R0

  [| R2 =F R1; R0 =F R2 |] ==> R0 =F R1

lemmas csp_rule_tr_ref:

  [| R1 <=F R2; R2 <=F R0 |] ==> R1 <=F R0

  [| R2 <=F R1; R0 <=F R2 |] ==> R0 <=F R1

lemmas csp_free_mono_left:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 -- Y_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 <=F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 [[r2_1]] <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 <=F R0

lemmas csp_free_mono_right:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

lemmas csp_mono_left:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 -- Y_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 <=F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 [[r2_1]] <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 <=F R0

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN b_1 -> Q_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN a_1 -> P_1 <=F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     
     LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 <=F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     
     LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 <=F R0

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 <=F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 <=F R0

lemmas csp_mono_right:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     R0 <=F 
     LET:fp1_1 df_1 IN a_1 -> P_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN b_1 -> Q_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     R0 <=F 
     LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     R0 <=F 
     LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 <=F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1

lemmas csp_free_mono_flag_left:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 -- Y_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 <=F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 [[r2_1]] <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 <=F R0

lemmas csp_free_mono_flag_right:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

lemmas csp_mono_flag_left:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 -- Y_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 <=F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 [[r2_1]] <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 <=F R0

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN b_1 -> Q_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN a_1 -> P_1 <=F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 <=F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 <=F R0

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 <=F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 <=F R0

lemmas csp_mono_flag_right:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN a_1 -> P_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN b_1 -> Q_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 <=F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1

lemmas csp_free_cong_left:

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 -- Y_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 =F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 [[r2_1]] =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 =F R0

lemmas csp_free_cong_right:

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     R0 =F 
     LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     R0 =F 
     LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

lemmas csp_cong_left:

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 -- Y_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 =F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 [[r2_1]] =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 =F R0

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN b_1 -> Q_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN a_1 -> P_1 =F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     
     LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 =F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     
     LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 =F R0

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 =F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 =F R0

lemmas csp_cong_right:

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     R0 =F 
     LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     R0 =F 
     LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     R0 =F 
     LET:fp1_1 df_1 IN a_1 -> P_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN b_1 -> Q_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     R0 =F 
     LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     R0 =F 
     LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 =F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1

lemmas csp_free_cong_flag_left:

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 -- Y_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 =F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 [[r2_1]] =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 =F R0

lemmas csp_free_cong_flag_right:

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

lemmas csp_cong_flag_left:

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 -- Y_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 =F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 [[r2_1]] =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 =F R0

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN b_1 -> Q_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN a_1 -> P_1 =F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 =F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 =F R0

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 =F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 =F R0

lemmas csp_cong_flag_right:

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN a_1 -> P_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN b_1 -> Q_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 =F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1

lemmas csp_free_decompo_left:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 -- Y_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 <=F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 [[r2_1]] <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 -- Y_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 =F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 [[r2_1]] =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 =F R0

lemmas csp_free_decompo_right:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     R0 =F 
     LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     R0 =F 
     LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

lemmas csp_decompo_left:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 -- Y_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 <=F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 [[r2_1]] <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 <=F R0

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN b_1 -> Q_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN a_1 -> P_1 <=F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     
     LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 <=F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     
     LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 <=F R0

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 <=F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 -- Y_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 =F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN Q_1 [[r2_1]] =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 =F R0

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     
     LET:fp2_1 df_1 IN b_1 -> Q_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN a_1 -> P_1 =F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     
     LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 =F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     
     LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 =F R0

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 =F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     
     LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 =F R0

lemmas csp_decompo_right:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     R0 <=F 
     LET:fp1_1 df_1 IN a_1 -> P_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN b_1 -> Q_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     R0 <=F 
     LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     R0 <=F 
     LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 <=F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 <=F 
     LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     R0 =F 
     LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     R0 =F 
     LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     R0 =F 
     LET:fp1_1 df_1 IN a_1 -> P_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN b_1 -> Q_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     R0 =F 
     LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     R0 =F 
     LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 =F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     R0 =F 
     LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1

lemmas csp_free_decompo_flag_left:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 -- Y_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 <=F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 [[r2_1]] <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 -- Y_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 =F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 [[r2_1]] =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 =F R0

lemmas csp_free_decompo_flag_right:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

lemmas csp_decompo_flag_left:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 <=F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 -- Y_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 <=F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 [[r2_1]] <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 <=F R0

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN b_1 -> Q_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN a_1 -> P_1 <=F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 <=F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 <=F R0

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 <=F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1 <=F R0 |]
  ==> 
      LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 <=F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 [+] Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 [+] P2_1 =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |~| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |~| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 =F R0

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 -- Y_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 -- X_1 =F R0

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q_1 [[r2_1]] =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P_1 [[r1_1]] =F R0

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN Q1_1 ;; Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN P1_1 ;; P2_1 =F R0

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN b_1 -> Q_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN a_1 -> P_1 =F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 =F R0

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 =F R0

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 =F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ 
                        LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1 =F R0 |]
  ==> 
      LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 =F R0

lemmas csp_decompo_flag_right:

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 <=F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 <=F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 <=F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 <=F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN a_1 -> P_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN b_1 -> Q_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a <=F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 <=F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 <=F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 <=F 
                        LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 |]
  ==> R0 <=F 
      LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 [+] P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 [+] Q2_1

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 |~| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |~| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P1_1 =F 
                LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 |[X_1]| P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 |[Y_1]| Q2_1

  [| X_1 = Y_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P_1 -- X_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 -- Y_1

  [| r1_1 = r2_1; 
                  LET:fp1_1 df_1 IN P_1 =F 
                  LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P_1 [[r1_1]] |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q_1 [[r2_1]]

  [| 
     LET:fp1_1 df_1 IN P1_1 =F 
     LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN P1_1 ;; P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN Q1_1 ;; Q2_1

  [| a_1 = b_1; 
                LET:fp1_1 df_1 IN P_1 =F 
                LET:fp2_1 df_1 IN Q_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN a_1 -> P_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN b_1 -> Q_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN ? :X_1 -> Pf_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN ? :Y_1 -> Qf_1

  [| X_1 = Y_1;
     !!a. a ∈ Y_1 ==> 
                      LET:fp1_1 df_1 IN Pf_1 a =F 
                      LET:fp2_1 df_1 IN Qf_1 a;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN ! :X_1 .. Pf_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN ! :Y_1 .. Qf_1

  [| b1_1 = b2_1; 
                  LET:fp1_1 df_1 IN P1_1 =F 
                  LET:fp2_1 df_1 IN Q1_1;
     
     LET:fp1_1 df_1 IN P2_1 =F 
     LET:fp2_1 df_1 IN Q2_1;
     Not_Decompo_Flag ∧ R0 =F 
                        LET:fp1_1 df_1 IN IF b1_1 THEN P1_1 ELSE P2_1 |]
  ==> R0 =F 
      LET:fp2_1 df_1 IN IF b2_1 THEN Q1_1 ELSE Q2_1

lemma csp_rw_flag_left_eq:

  [| R1 =F R2; Not_Decompo_Flag ∧ R2 =F R3 |] ==> R1 =F R3

lemma csp_rw_flag_left_ref:

  [| R1 =F R2; Not_Decompo_Flag ∧ R2 <=F R3 |] ==> R1 <=F R3

lemmas csp_rw_flag_left:

  [| R1 =F R2; Not_Decompo_Flag ∧ R2 =F R3 |] ==> R1 =F R3

  [| R1 =F R2; Not_Decompo_Flag ∧ R2 <=F R3 |] ==> R1 <=F R3

lemma csp_rw_flag_right_eq:

  [| R3 =F R2; Not_Decompo_Flag ∧ R1 =F R2 |] ==> R1 =F R3

lemma csp_rw_flag_right_ref:

  [| R3 =F R2; Not_Decompo_Flag ∧ R1 <=F R2 |] ==> R1 <=F R3

lemmas csp_rw_flag_right:

  [| R3 =F R2; Not_Decompo_Flag ∧ R1 =F R2 |] ==> R1 =F R3

  [| R3 =F R2; Not_Decompo_Flag ∧ R1 <=F R2 |] ==> R1 <=F R3

lemma csp_tr_flag_left_eq:

  [| R1 =F R2; Not_Decompo_Flag ∧ R2 =F R3 |] ==> R1 =F R3

lemma csp_tr_flag_left_ref:

  [| R1 <=F R2; Not_Decompo_Flag ∧ R2 <=F R3 |] ==> R1 <=F R3

lemmas csp_tr_flag_left:

  [| R1 =F R2; Not_Decompo_Flag ∧ R2 =F R3 |] ==> R1 =F R3

  [| R1 <=F R2; Not_Decompo_Flag ∧ R2 <=F R3 |] ==> R1 <=F R3

lemma csp_tr_flag_right_eq:

  [| R2 =F R3; Not_Decompo_Flag ∧ R1 =F R2 |] ==> R1 =F R3

lemma csp_tr_flag_right_ref:

  [| R2 <=F R3; Not_Decompo_Flag ∧ R1 <=F R2 |] ==> R1 <=F R3

lemmas csp_tr_flag_right:

  [| R2 =F R3; Not_Decompo_Flag ∧ R1 =F R2 |] ==> R1 =F R3

  [| R2 <=F R3; Not_Decompo_Flag ∧ R1 <=F R2 |] ==> R1 <=F R3