1274
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(* Title: HOLCF/Dnat.ML
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ID: $Id$
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Author: Franz Regensburger
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Copyright 1993 Technische Universitaet Muenchen
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Lemmas for dnat.thy
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*)
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open Dnat;
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(* ------------------------------------------------------------------------*)
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(* The isomorphisms dnat_rep_iso and dnat_abs_iso are strict *)
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(* ------------------------------------------------------------------------*)
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val dnat_iso_strict = dnat_rep_iso RS (dnat_abs_iso RS
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(allI RSN (2,allI RS iso_strict)));
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val dnat_rews = [dnat_iso_strict RS conjunct1,
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dnat_iso_strict RS conjunct2];
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(* ------------------------------------------------------------------------*)
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(* Properties of dnat_copy *)
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(* ------------------------------------------------------------------------*)
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fun prover defs thm = prove_goalw Dnat.thy defs thm
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(fn prems =>
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[
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(cut_facts_tac prems 1),
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(asm_simp_tac (!simpset addsimps
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(dnat_rews @ [dnat_abs_iso,dnat_rep_iso])) 1)
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]);
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val dnat_copy =
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[
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prover [dnat_copy_def] "dnat_copy`f`UU=UU",
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prover [dnat_copy_def,dzero_def] "dnat_copy`f`dzero= dzero",
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prover [dnat_copy_def,dsucc_def]
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"n~=UU ==> dnat_copy`f`(dsucc`n) = dsucc`(f`n)"
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];
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val dnat_rews = dnat_copy @ dnat_rews;
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(* ------------------------------------------------------------------------*)
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(* Exhaustion and elimination for dnat *)
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(* ------------------------------------------------------------------------*)
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qed_goalw "Exh_dnat" Dnat.thy [dsucc_def,dzero_def]
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"n = UU | n = dzero | (? x . x~=UU & n = dsucc`x)"
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(fn prems =>
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[
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(Simp_tac 1),
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(rtac (dnat_rep_iso RS subst) 1),
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(res_inst_tac [("p","dnat_rep`n")] ssumE 1),
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(rtac disjI1 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1),
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(rtac (disjI1 RS disjI2) 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1),
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(res_inst_tac [("p","x")] oneE 1),
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(contr_tac 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1),
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(rtac (disjI2 RS disjI2) 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1),
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(fast_tac HOL_cs 1)
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]);
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qed_goal "dnatE" Dnat.thy
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"[| n=UU ==> Q; n=dzero ==> Q; !!x.[|n=dsucc`x;x~=UU|]==>Q|]==>Q"
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(fn prems =>
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[
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(rtac (Exh_dnat RS disjE) 1),
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(eresolve_tac prems 1),
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(etac disjE 1),
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(eresolve_tac prems 1),
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(REPEAT (etac exE 1)),
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(resolve_tac prems 1),
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(fast_tac HOL_cs 1),
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(fast_tac HOL_cs 1)
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]);
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(* ------------------------------------------------------------------------*)
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(* Properties of dnat_when *)
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(* ------------------------------------------------------------------------*)
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fun prover defs thm = prove_goalw Dnat.thy defs thm
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(fn prems =>
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[
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(cut_facts_tac prems 1),
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(asm_simp_tac (!simpset addsimps
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(dnat_rews @ [dnat_abs_iso,dnat_rep_iso])) 1)
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]);
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val dnat_when = [
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prover [dnat_when_def] "dnat_when`c`f`UU=UU",
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prover [dnat_when_def,dzero_def] "dnat_when`c`f`dzero=c",
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prover [dnat_when_def,dsucc_def]
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"n~=UU ==> dnat_when`c`f`(dsucc`n)=f`n"
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];
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val dnat_rews = dnat_when @ dnat_rews;
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(* ------------------------------------------------------------------------*)
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(* Rewrites for discriminators and selectors *)
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(* ------------------------------------------------------------------------*)
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fun prover defs thm = prove_goalw Dnat.thy defs thm
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(fn prems =>
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[
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(simp_tac (!simpset addsimps dnat_rews) 1)
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]);
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val dnat_discsel = [
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prover [is_dzero_def] "is_dzero`UU=UU",
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prover [is_dsucc_def] "is_dsucc`UU=UU",
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prover [dpred_def] "dpred`UU=UU"
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];
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fun prover defs thm = prove_goalw Dnat.thy defs thm
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(fn prems =>
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[
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(cut_facts_tac prems 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1)
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]);
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val dnat_discsel = [
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prover [is_dzero_def] "is_dzero`dzero=TT",
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prover [is_dzero_def] "n~=UU ==>is_dzero`(dsucc`n)=FF",
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prover [is_dsucc_def] "is_dsucc`dzero=FF",
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prover [is_dsucc_def] "n~=UU ==> is_dsucc`(dsucc`n)=TT",
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prover [dpred_def] "dpred`dzero=UU",
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prover [dpred_def] "n~=UU ==> dpred`(dsucc`n)=n"
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] @ dnat_discsel;
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val dnat_rews = dnat_discsel @ dnat_rews;
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(* ------------------------------------------------------------------------*)
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(* Definedness and strictness *)
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(* ------------------------------------------------------------------------*)
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fun prover contr thm = prove_goal Dnat.thy thm
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(fn prems =>
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[
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(res_inst_tac [("P1",contr)] classical3 1),
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(simp_tac (!simpset addsimps dnat_rews) 1),
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(dtac sym 1),
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(Asm_simp_tac 1),
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(simp_tac (!simpset addsimps (prems @ dnat_rews)) 1)
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]);
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val dnat_constrdef = [
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prover "is_dzero`UU ~= UU" "dzero~=UU",
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prover "is_dsucc`UU ~= UU" "n~=UU ==> dsucc`n~=UU"
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];
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fun prover defs thm = prove_goalw Dnat.thy defs thm
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(fn prems =>
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[
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(simp_tac (!simpset addsimps dnat_rews) 1)
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]);
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val dnat_constrdef = [
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prover [dsucc_def] "dsucc`UU=UU"
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] @ dnat_constrdef;
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val dnat_rews = dnat_constrdef @ dnat_rews;
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(* ------------------------------------------------------------------------*)
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(* Distinctness wrt. << and = *)
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(* ------------------------------------------------------------------------*)
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val temp = prove_goal Dnat.thy "~dzero << dsucc`n"
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(fn prems =>
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[
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(res_inst_tac [("P1","TT << FF")] classical3 1),
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(resolve_tac dist_less_tr 1),
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(dres_inst_tac [("fo5","is_dzero")] monofun_cfun_arg 1),
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(etac box_less 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1),
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(res_inst_tac [("Q","n=UU")] classical2 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1)
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]);
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val dnat_dist_less = [temp];
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val temp = prove_goal Dnat.thy "n~=UU ==> ~dsucc`n << dzero"
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(fn prems =>
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[
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(cut_facts_tac prems 1),
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(res_inst_tac [("P1","TT << FF")] classical3 1),
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(resolve_tac dist_less_tr 1),
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(dres_inst_tac [("fo5","is_dsucc")] monofun_cfun_arg 1),
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(etac box_less 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1)
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]);
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val dnat_dist_less = temp::dnat_dist_less;
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val temp = prove_goal Dnat.thy "dzero ~= dsucc`n"
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(fn prems =>
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[
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(res_inst_tac [("Q","n=UU")] classical2 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1),
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(res_inst_tac [("P1","TT = FF")] classical3 1),
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(resolve_tac dist_eq_tr 1),
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(dres_inst_tac [("f","is_dzero")] cfun_arg_cong 1),
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(etac box_equals 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1)
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]);
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val dnat_dist_eq = [temp, temp RS not_sym];
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val dnat_rews = dnat_dist_less @ dnat_dist_eq @ dnat_rews;
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(* ------------------------------------------------------------------------*)
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(* Invertibility *)
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(* ------------------------------------------------------------------------*)
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val dnat_invert =
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[
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prove_goal Dnat.thy
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"[|x1~=UU; y1~=UU; dsucc`x1 << dsucc`y1 |] ==> x1<< y1"
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(fn prems =>
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[
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(cut_facts_tac prems 1),
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(dres_inst_tac [("fo5","dnat_when`c`(LAM x.x)")] monofun_cfun_arg 1),
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(etac box_less 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1)
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])
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];
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(* ------------------------------------------------------------------------*)
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(* Injectivity *)
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(* ------------------------------------------------------------------------*)
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val dnat_inject =
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[
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prove_goal Dnat.thy
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"[|x1~=UU; y1~=UU; dsucc`x1 = dsucc`y1 |] ==> x1= y1"
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(fn prems =>
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[
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(cut_facts_tac prems 1),
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(dres_inst_tac [("f","dnat_when`c`(LAM x.x)")] cfun_arg_cong 1),
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(etac box_equals 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1)
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])
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];
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(* ------------------------------------------------------------------------*)
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(* definedness for discriminators and selectors *)
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(* ------------------------------------------------------------------------*)
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fun prover thm = prove_goal Dnat.thy thm
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(fn prems =>
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[
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(cut_facts_tac prems 1),
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(rtac dnatE 1),
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(contr_tac 1),
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(REPEAT (asm_simp_tac (!simpset addsimps dnat_rews) 1))
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]);
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val dnat_discsel_def =
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[
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prover "n~=UU ==> is_dzero`n ~= UU",
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prover "n~=UU ==> is_dsucc`n ~= UU"
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];
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val dnat_rews = dnat_discsel_def @ dnat_rews;
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(* ------------------------------------------------------------------------*)
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(* Properties dnat_take *)
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(* ------------------------------------------------------------------------*)
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val temp = prove_goalw Dnat.thy [dnat_take_def] "dnat_take n`UU = UU"
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(fn prems =>
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[
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(res_inst_tac [("n","n")] natE 1),
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(Asm_simp_tac 1),
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(Asm_simp_tac 1),
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(simp_tac (!simpset addsimps dnat_rews) 1)
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]);
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val dnat_take = [temp];
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val temp = prove_goalw Dnat.thy [dnat_take_def] "dnat_take 0`xs = UU"
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(fn prems =>
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[
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(Asm_simp_tac 1)
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]);
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val dnat_take = temp::dnat_take;
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val temp = prove_goalw Dnat.thy [dnat_take_def]
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"dnat_take (Suc n)`dzero=dzero"
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(fn prems =>
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[
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(Asm_simp_tac 1),
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(simp_tac (!simpset addsimps dnat_rews) 1)
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]);
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val dnat_take = temp::dnat_take;
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val temp = prove_goalw Dnat.thy [dnat_take_def]
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"dnat_take (Suc n)`(dsucc`xs)=dsucc`(dnat_take n ` xs)"
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(fn prems =>
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[
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(res_inst_tac [("Q","xs=UU")] classical2 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1),
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(Asm_simp_tac 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1),
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(res_inst_tac [("n","n")] natE 1),
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(Asm_simp_tac 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1),
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(Asm_simp_tac 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1),
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(Asm_simp_tac 1),
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(asm_simp_tac (!simpset addsimps dnat_rews) 1)
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]);
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val dnat_take = temp::dnat_take;
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val dnat_rews = dnat_take @ dnat_rews;
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(* ------------------------------------------------------------------------*)
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(* take lemma for dnats *)
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(* ------------------------------------------------------------------------*)
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fun prover reach defs thm = prove_goalw Dnat.thy defs thm
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(fn prems =>
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[
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(res_inst_tac [("t","s1")] (reach RS subst) 1),
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(res_inst_tac [("t","s2")] (reach RS subst) 1),
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(rtac (fix_def2 RS ssubst) 1),
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(rtac (contlub_cfun_fun RS ssubst) 1),
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(rtac is_chain_iterate 1),
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(rtac (contlub_cfun_fun RS ssubst) 1),
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(rtac is_chain_iterate 1),
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(rtac lub_equal 1),
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(rtac (is_chain_iterate RS ch2ch_fappL) 1),
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(rtac (is_chain_iterate RS ch2ch_fappL) 1),
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(rtac allI 1),
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(resolve_tac prems 1)
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]);
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val dnat_take_lemma = prover dnat_reach [dnat_take_def]
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"(!!n.dnat_take n`s1 = dnat_take n`s2) ==> s1=s2";
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(* ------------------------------------------------------------------------*)
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(* Co -induction for dnats *)
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(* ------------------------------------------------------------------------*)
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qed_goalw "dnat_coind_lemma" Dnat.thy [dnat_bisim_def]
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"dnat_bisim R ==> ! p q. R p q --> dnat_take n`p = dnat_take n`q"
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(fn prems =>
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[
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(cut_facts_tac prems 1),
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(nat_ind_tac "n" 1),
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(simp_tac (!simpset addsimps dnat_take) 1),
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(strip_tac 1),
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((etac allE 1) THEN (etac allE 1) THEN (etac (mp RS disjE) 1)),
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(atac 1),
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(asm_simp_tac (!simpset addsimps dnat_take) 1),
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(etac disjE 1),
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(asm_simp_tac (!simpset addsimps dnat_take) 1),
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(etac exE 1),
|
|
376 |
(etac exE 1),
|
|
377 |
(asm_simp_tac (!simpset addsimps dnat_take) 1),
|
|
378 |
(REPEAT (etac conjE 1)),
|
|
379 |
(rtac cfun_arg_cong 1),
|
|
380 |
(fast_tac HOL_cs 1)
|
|
381 |
]);
|
|
382 |
|
|
383 |
qed_goal "dnat_coind" Dnat.thy "[|dnat_bisim R;R p q|] ==> p = q"
|
|
384 |
(fn prems =>
|
|
385 |
[
|
|
386 |
(rtac dnat_take_lemma 1),
|
|
387 |
(rtac (dnat_coind_lemma RS spec RS spec RS mp) 1),
|
|
388 |
(resolve_tac prems 1),
|
|
389 |
(resolve_tac prems 1)
|
|
390 |
]);
|
|
391 |
|
|
392 |
|
|
393 |
(* ------------------------------------------------------------------------*)
|
|
394 |
(* structural induction for admissible predicates *)
|
|
395 |
(* ------------------------------------------------------------------------*)
|
|
396 |
|
|
397 |
(* not needed any longer
|
|
398 |
qed_goal "dnat_ind" Dnat.thy
|
|
399 |
"[| adm(P);\
|
|
400 |
\ P(UU);\
|
|
401 |
\ P(dzero);\
|
|
402 |
\ !! s1.[|s1~=UU ; P(s1)|] ==> P(dsucc`s1)|] ==> P(s)"
|
|
403 |
(fn prems =>
|
|
404 |
[
|
|
405 |
(rtac (dnat_reach RS subst) 1),
|
|
406 |
(res_inst_tac [("x","s")] spec 1),
|
|
407 |
(rtac fix_ind 1),
|
|
408 |
(rtac adm_all2 1),
|
|
409 |
(rtac adm_subst 1),
|
|
410 |
(cont_tacR 1),
|
|
411 |
(resolve_tac prems 1),
|
|
412 |
(Simp_tac 1),
|
|
413 |
(resolve_tac prems 1),
|
|
414 |
(strip_tac 1),
|
|
415 |
(res_inst_tac [("n","xa")] dnatE 1),
|
|
416 |
(asm_simp_tac (!simpset addsimps dnat_copy) 1),
|
|
417 |
(resolve_tac prems 1),
|
|
418 |
(asm_simp_tac (!simpset addsimps dnat_copy) 1),
|
|
419 |
(resolve_tac prems 1),
|
|
420 |
(asm_simp_tac (!simpset addsimps dnat_copy) 1),
|
|
421 |
(res_inst_tac [("Q","x`xb=UU")] classical2 1),
|
|
422 |
(asm_simp_tac (!simpset addsimps dnat_rews) 1),
|
|
423 |
(resolve_tac prems 1),
|
|
424 |
(eresolve_tac prems 1),
|
|
425 |
(etac spec 1)
|
|
426 |
]);
|
|
427 |
*)
|
|
428 |
|
|
429 |
qed_goal "dnat_finite_ind" Dnat.thy
|
|
430 |
"[|P(UU);P(dzero);\
|
|
431 |
\ !! s1.[|s1~=UU;P(s1)|] ==> P(dsucc`s1)\
|
|
432 |
\ |] ==> !s.P(dnat_take n`s)"
|
|
433 |
(fn prems =>
|
|
434 |
[
|
|
435 |
(nat_ind_tac "n" 1),
|
|
436 |
(simp_tac (!simpset addsimps dnat_rews) 1),
|
|
437 |
(resolve_tac prems 1),
|
|
438 |
(rtac allI 1),
|
|
439 |
(res_inst_tac [("n","s")] dnatE 1),
|
|
440 |
(asm_simp_tac (!simpset addsimps dnat_rews) 1),
|
|
441 |
(resolve_tac prems 1),
|
|
442 |
(asm_simp_tac (!simpset addsimps dnat_rews) 1),
|
|
443 |
(resolve_tac prems 1),
|
|
444 |
(asm_simp_tac (!simpset addsimps dnat_rews) 1),
|
|
445 |
(res_inst_tac [("Q","dnat_take n1`x=UU")] classical2 1),
|
|
446 |
(asm_simp_tac (!simpset addsimps dnat_rews) 1),
|
|
447 |
(resolve_tac prems 1),
|
|
448 |
(resolve_tac prems 1),
|
|
449 |
(atac 1),
|
|
450 |
(etac spec 1)
|
|
451 |
]);
|
|
452 |
|
|
453 |
qed_goal "dnat_all_finite_lemma1" Dnat.thy
|
|
454 |
"!s.dnat_take n`s=UU |dnat_take n`s=s"
|
|
455 |
(fn prems =>
|
|
456 |
[
|
|
457 |
(nat_ind_tac "n" 1),
|
|
458 |
(simp_tac (!simpset addsimps dnat_rews) 1),
|
|
459 |
(rtac allI 1),
|
|
460 |
(res_inst_tac [("n","s")] dnatE 1),
|
|
461 |
(asm_simp_tac (!simpset addsimps dnat_rews) 1),
|
|
462 |
(asm_simp_tac (!simpset addsimps dnat_rews) 1),
|
|
463 |
(asm_simp_tac (!simpset addsimps dnat_rews) 1),
|
|
464 |
(eres_inst_tac [("x","x")] allE 1),
|
|
465 |
(etac disjE 1),
|
|
466 |
(asm_simp_tac (!simpset addsimps dnat_rews) 1),
|
|
467 |
(asm_simp_tac (!simpset addsimps dnat_rews) 1)
|
|
468 |
]);
|
|
469 |
|
|
470 |
qed_goal "dnat_all_finite_lemma2" Dnat.thy "? n.dnat_take n`s=s"
|
|
471 |
(fn prems =>
|
|
472 |
[
|
|
473 |
(res_inst_tac [("Q","s=UU")] classical2 1),
|
|
474 |
(asm_simp_tac (!simpset addsimps dnat_rews) 1),
|
|
475 |
(subgoal_tac "(!n.dnat_take(n)`s=UU) |(? n.dnat_take(n)`s=s)" 1),
|
|
476 |
(etac disjE 1),
|
|
477 |
(eres_inst_tac [("P","s=UU")] notE 1),
|
|
478 |
(rtac dnat_take_lemma 1),
|
|
479 |
(asm_simp_tac (!simpset addsimps dnat_rews) 1),
|
|
480 |
(atac 1),
|
|
481 |
(subgoal_tac "!n.!s.dnat_take(n)`s=UU |dnat_take(n)`s=s" 1),
|
|
482 |
(fast_tac HOL_cs 1),
|
|
483 |
(rtac allI 1),
|
|
484 |
(rtac dnat_all_finite_lemma1 1)
|
|
485 |
]);
|
|
486 |
|
|
487 |
|
|
488 |
qed_goal "dnat_ind" Dnat.thy
|
|
489 |
"[|P(UU);P(dzero);\
|
|
490 |
\ !! s1.[|s1~=UU;P(s1)|] ==> P(dsucc`s1)\
|
|
491 |
\ |] ==> P(s)"
|
|
492 |
(fn prems =>
|
|
493 |
[
|
|
494 |
(rtac (dnat_all_finite_lemma2 RS exE) 1),
|
|
495 |
(etac subst 1),
|
|
496 |
(rtac (dnat_finite_ind RS spec) 1),
|
|
497 |
(REPEAT (resolve_tac prems 1)),
|
|
498 |
(REPEAT (atac 1))
|
|
499 |
]);
|
|
500 |
|
|
501 |
|
|
502 |
qed_goalw "dnat_flat" Dnat.thy [flat_def] "flat(dzero)"
|
|
503 |
(fn prems =>
|
|
504 |
[
|
|
505 |
(rtac allI 1),
|
|
506 |
(res_inst_tac [("s","x")] dnat_ind 1),
|
|
507 |
(fast_tac HOL_cs 1),
|
|
508 |
(rtac allI 1),
|
|
509 |
(res_inst_tac [("n","y")] dnatE 1),
|
|
510 |
(fast_tac (HOL_cs addSIs [UU_I]) 1),
|
|
511 |
(Asm_simp_tac 1),
|
|
512 |
(asm_simp_tac (!simpset addsimps dnat_dist_less) 1),
|
|
513 |
(rtac allI 1),
|
|
514 |
(res_inst_tac [("n","y")] dnatE 1),
|
|
515 |
(fast_tac (HOL_cs addSIs [UU_I]) 1),
|
|
516 |
(asm_simp_tac (!simpset addsimps dnat_dist_less) 1),
|
|
517 |
(asm_simp_tac (!simpset addsimps dnat_rews) 1),
|
|
518 |
(strip_tac 1),
|
|
519 |
(subgoal_tac "s1<<xa" 1),
|
|
520 |
(etac allE 1),
|
|
521 |
(dtac mp 1),
|
|
522 |
(atac 1),
|
|
523 |
(etac disjE 1),
|
|
524 |
(contr_tac 1),
|
|
525 |
(Asm_simp_tac 1),
|
|
526 |
(resolve_tac dnat_invert 1),
|
|
527 |
(REPEAT (atac 1))
|
|
528 |
]);
|
|
529 |
|
|
530 |
|
|
531 |
|
|
532 |
|
|
533 |
|
|
534 |
|