(*  Title:      HOL/Auth/n_german_lemma_on_inv__58.thy
    Author:     Yongjian Li and Kaiqiang Duan, State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
    Copyright    2016 State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences
*)

(*header{*The n_german Protocol Case Study*}*) 

theory n_german_lemma_on_inv__58 imports n_german_base
begin
section{*All lemmas on causal relation between inv__58 and some rule r*}
lemma n_RecvInvAckVsinv__58:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_RecvInvAck  i)" and
a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_RecvInvAck  i" apply fastforce done
from a2 obtain p__Inv0 p__Inv2 where a2:"p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i=p__Inv0)\<or>(i~=p__Inv0\<and>i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))\<or>((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))" by auto
  moreover {
    assume c1: "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))"
    have "?P1 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  moreover {
    assume c1: "((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))"
    have "?P1 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  ultimately have "invHoldForRule s f r (invariants N)" by satx
}
moreover {
  assume b1: "(i=p__Inv0)"
  have "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))\<or>((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))" by auto
  moreover {
    assume c1: "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))"
    have "?P1 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  moreover {
    assume c1: "((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))"
    have "?P1 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  ultimately have "invHoldForRule s f r (invariants N)" by satx
}
moreover {
  assume b1: "(i~=p__Inv0\<and>i~=p__Inv2)"
  have "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))\<or>((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))" by auto
  moreover {
    assume c1: "((formEval (eqn (IVar (Ident ''ExGntd'')) (Const true)) s))"
    have "?P1 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  moreover {
    assume c1: "((formEval (neg (eqn (IVar (Ident ''ExGntd'')) (Const true))) s))"
    have "?P2 s"
    proof(cut_tac a1 a2 b1 c1, auto) qed
    then have "invHoldForRule s f r (invariants N)" by auto
  }
  ultimately have "invHoldForRule s f r (invariants N)" by satx
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendGntSVsinv__58:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendGntS  i)" and
a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendGntS  i" apply fastforce done
from a2 obtain p__Inv0 p__Inv2 where a2:"p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i=p__Inv0)\<or>(i~=p__Inv0\<and>i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i=p__Inv0)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv0\<and>i~=p__Inv2)"
  have "?P2 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendGntEVsinv__58:
assumes a1: "(\<exists> i. i\<le>N\<and>r=n_SendGntE N i)" and
a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2)"
shows "invHoldForRule s f r (invariants N)" (is "?P1 s \<or> ?P2 s \<or> ?P3 s")
proof -
from a1 obtain i where a1:"i\<le>N\<and>r=n_SendGntE N i" apply fastforce done
from a2 obtain p__Inv0 p__Inv2 where a2:"p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2" apply fastforce done
have "(i=p__Inv2)\<or>(i=p__Inv0)\<or>(i~=p__Inv0\<and>i~=p__Inv2)" apply (cut_tac a1 a2, auto) done
moreover {
  assume b1: "(i=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i=p__Inv0)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
moreover {
  assume b1: "(i~=p__Inv0\<and>i~=p__Inv2)"
  have "?P1 s"
  proof(cut_tac a1 a2 b1, auto) qed
  then have "invHoldForRule s f r (invariants N)" by auto
}
ultimately show "invHoldForRule s f r (invariants N)" by satx
qed

lemma n_SendReqE__part__1Vsinv__58:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendReqE__part__1  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
  

lemma n_StoreVsinv__58:
  assumes a1: "\<exists> i d. i\<le>N\<and>d\<le>N\<and>r=n_Store  i d" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
  

lemma n_RecvGntSVsinv__58:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_RecvGntS  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
  

lemma n_RecvReqEVsinv__58:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_RecvReqE N i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
  

lemma n_RecvGntEVsinv__58:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_RecvGntE  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
  

lemma n_SendInvAckVsinv__58:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendInvAck  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
  

lemma n_SendInv__part__0Vsinv__58:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendInv__part__0  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
  

lemma n_SendReqE__part__0Vsinv__58:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendReqE__part__0  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
  

lemma n_SendInv__part__1Vsinv__58:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendInv__part__1  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
  

lemma n_SendReqSVsinv__58:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_SendReqS  i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
  

lemma n_RecvReqSVsinv__58:
  assumes a1: "\<exists> i. i\<le>N\<and>r=n_RecvReqS N i" and
  a2: "(\<exists> p__Inv0 p__Inv2. p__Inv0\<le>N\<and>p__Inv2\<le>N\<and>p__Inv0~=p__Inv2\<and>f=inv__58  p__Inv0 p__Inv2)"
  shows "invHoldForRule s f r (invariants N)"
  apply (rule noEffectOnRule, cut_tac a1 a2, auto) done
  
end
