--- a/src/HOL/ex/Mutil.ML Wed May 07 13:50:52 1997 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,172 +0,0 @@
-(* Title: HOL/ex/Mutil
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1996 University of Cambridge
-
-The Mutilated Chess Board Problem, formalized inductively
-*)
-
-open Mutil;
-
-Addsimps tiling.intrs;
-
-(** The union of two disjoint tilings is a tiling **)
-
-goal thy "!!t. t: tiling A ==> \
-\ u: tiling A --> t <= Compl u --> t Un u : tiling A";
-by (etac tiling.induct 1);
-by (Simp_tac 1);
-by (simp_tac (!simpset addsimps [Un_assoc]) 1);
-by (blast_tac (!claset addIs tiling.intrs) 1);
-qed_spec_mp "tiling_UnI";
-
-
-(*** Chess boards ***)
-
-val [below_0, below_Suc] = nat_recs below_def;
-Addsimps [below_0, below_Suc];
-
-goal thy "ALL i. (i: below k) = (i<k)";
-by (nat_ind_tac "k" 1);
-by (ALLGOALS (asm_simp_tac (!simpset addsimps [less_Suc_eq])));
-by (Blast_tac 1);
-qed_spec_mp "below_less_iff";
-
-Addsimps [below_less_iff];
-
-goal thy "below(Suc n) Times B = ({n} Times B) Un ((below n) Times B)";
-by (Simp_tac 1);
-by (Blast_tac 1);
-qed "Sigma_Suc1";
-
-goal thy "A Times below(Suc n) = (A Times {n}) Un (A Times (below n))";
-by (Simp_tac 1);
-by (Blast_tac 1);
-qed "Sigma_Suc2";
-
-(*Deletion is essential to allow use of Sigma_Suc1,2*)
-Delsimps [below_Suc];
-
-goal thy "{i} Times below(n + n) : tiling domino";
-by (nat_ind_tac "n" 1);
-by (ALLGOALS (asm_simp_tac (!simpset addsimps [Un_assoc RS sym, Sigma_Suc2])));
-by (resolve_tac tiling.intrs 1);
-by (assume_tac 2);
-by (subgoal_tac (*seems the easiest way of turning one to the other*)
- "({i} Times {Suc(n+n)}) Un ({i} Times {n+n}) = \
-\ {(i, n+n), (i, Suc(n+n))}" 1);
-by (Blast_tac 2);
-by (asm_simp_tac (!simpset addsimps [domino.horiz]) 1);
-by (blast_tac (!claset addEs [less_irrefl, less_asym]
- addSDs [below_less_iff RS iffD1]) 1);
-qed "dominoes_tile_row";
-
-goal thy "(below m) Times below(n + n) : tiling domino";
-by (nat_ind_tac "m" 1);
-by (ALLGOALS (asm_simp_tac (!simpset addsimps [Sigma_Suc1])));
-by (blast_tac (!claset addSIs [tiling_UnI, dominoes_tile_row]
- addSEs [below_less_iff RS iffD1 RS less_irrefl]) 1);
-qed "dominoes_tile_matrix";
-
-
-(*** Basic properties of evnodd ***)
-
-goalw thy [evnodd_def] "(i,j): evnodd A b = ((i,j): A & (i+j) mod 2 = b)";
-by (Simp_tac 1);
-qed "evnodd_iff";
-
-goalw thy [evnodd_def] "evnodd A b <= A";
-by (rtac Int_lower1 1);
-qed "evnodd_subset";
-
-(* finite X ==> finite(evnodd X b) *)
-bind_thm("finite_evnodd", evnodd_subset RS finite_subset);
-
-goalw thy [evnodd_def] "evnodd (A Un B) b = evnodd A b Un evnodd B b";
-by (Blast_tac 1);
-qed "evnodd_Un";
-
-goalw thy [evnodd_def] "evnodd (A - B) b = evnodd A b - evnodd B b";
-by (Blast_tac 1);
-qed "evnodd_Diff";
-
-goalw thy [evnodd_def] "evnodd {} b = {}";
-by (Simp_tac 1);
-qed "evnodd_empty";
-
-goalw thy [evnodd_def]
- "evnodd (insert (i,j) C) b = \
-\ (if (i+j) mod 2 = b then insert (i,j) (evnodd C b) else evnodd C b)";
-by (asm_full_simp_tac (!simpset addsimps [evnodd_def]
- setloop (split_tac [expand_if] THEN' Step_tac)) 1);
-qed "evnodd_insert";
-
-
-(*** Dominoes ***)
-
-goal thy "!!d. [| d:domino; b<2 |] ==> EX i j. evnodd d b = {(i,j)}";
-by (eresolve_tac [domino.elim] 1);
-by (res_inst_tac [("k1", "i+j")] (mod2_cases RS disjE) 2);
-by (res_inst_tac [("k1", "i+j")] (mod2_cases RS disjE) 1);
-by (REPEAT_FIRST assume_tac);
-(*Four similar cases: case (i+j) mod 2 = b, 2#-b, ...*)
-by (REPEAT (asm_full_simp_tac (!simpset addsimps
- [less_Suc_eq, evnodd_insert, evnodd_empty, mod_Suc]
- setloop split_tac [expand_if]) 1
- THEN Blast_tac 1));
-qed "domino_singleton";
-
-goal thy "!!d. d:domino ==> finite d";
-by (blast_tac (!claset addSIs [finite_insertI, finite_emptyI]
- addSEs [domino.elim]) 1);
-qed "domino_finite";
-
-
-(*** Tilings of dominoes ***)
-
-goal thy "!!t. t:tiling domino ==> finite t";
-by (eresolve_tac [tiling.induct] 1);
-by (rtac finite_emptyI 1);
-by (blast_tac (!claset addSIs [finite_UnI] addIs [domino_finite]) 1);
-qed "tiling_domino_finite";
-
-goal thy "!!t. t: tiling domino ==> card(evnodd t 0) = card(evnodd t 1)";
-by (eresolve_tac [tiling.induct] 1);
-by (simp_tac (!simpset addsimps [evnodd_def]) 1);
-by (res_inst_tac [("b1","0")] (domino_singleton RS exE) 1);
-by (Simp_tac 2 THEN assume_tac 1);
-by (res_inst_tac [("b1","1")] (domino_singleton RS exE) 1);
-by (Simp_tac 2 THEN assume_tac 1);
-by (Step_tac 1);
-by (subgoal_tac "ALL p b. p : evnodd a b --> p ~: evnodd ta b" 1);
-by (asm_simp_tac (!simpset addsimps [evnodd_Un, Un_insert_left,
- tiling_domino_finite,
- evnodd_subset RS finite_subset,
- card_insert_disjoint]) 1);
-by (blast_tac (!claset addSDs [evnodd_subset RS subsetD] addEs [equalityE]) 1);
-qed "tiling_domino_0_1";
-
-goal thy "!!m n. [| t = below(Suc m + Suc m) Times below(Suc n + Suc n); \
-\ t' = t - {(0,0)} - {(Suc(m+m), Suc(n+n))} \
-\ |] ==> t' ~: tiling domino";
-by (rtac notI 1);
-by (dtac tiling_domino_0_1 1);
-by (subgoal_tac "card(evnodd t' 0) < card(evnodd t' 1)" 1);
-by (Asm_full_simp_tac 1);
-by (subgoal_tac "t : tiling domino" 1);
-(*Requires a small simpset that won't move the Suc applications*)
-by (asm_simp_tac (HOL_ss addsimps [dominoes_tile_matrix]) 2);
-by (subgoal_tac "(m+m)+(n+n) = (m+n)+(m+n)" 1);
-by (asm_simp_tac (!simpset addsimps add_ac) 2);
-by (asm_full_simp_tac
- (!simpset addsimps [evnodd_Diff, evnodd_insert, evnodd_empty,
- mod_less, tiling_domino_0_1 RS sym]) 1);
-by (rtac less_trans 1);
-by (REPEAT
- (rtac card_Diff 1
- THEN
- asm_simp_tac (!simpset addsimps [tiling_domino_finite, finite_evnodd]) 1
- THEN
- asm_simp_tac (!simpset addsimps [mod_less, evnodd_iff]) 1));
-qed "mutil_not_tiling";
-