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(* Title: CTT/arith
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1991 University of Cambridge
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Theorems for arith.thy (Arithmetic operators)
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Proofs about elementary arithmetic: addition, multiplication, etc.
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Tests definitions and simplifier.
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*)
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open Arith;
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val arith_defs = [add_def, diff_def, absdiff_def, mult_def, mod_def, div_def];
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(** Addition *)
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(*typing of add: short and long versions*)
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val add_typing = prove_goal Arith.thy
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"[| a:N; b:N |] ==> a #+ b : N"
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(fn prems=>
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[ (rewrite_goals_tac arith_defs),
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(typechk_tac prems) ]);
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val add_typingL = prove_goal Arith.thy
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"[| a=c:N; b=d:N |] ==> a #+ b = c #+ d : N"
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(fn prems=>
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[ (rewrite_goals_tac arith_defs),
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(equal_tac prems) ]);
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(*computation for add: 0 and successor cases*)
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val addC0 = prove_goal Arith.thy
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"b:N ==> 0 #+ b = b : N"
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(fn prems=>
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[ (rewrite_goals_tac arith_defs),
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(rew_tac prems) ]);
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val addC_succ = prove_goal Arith.thy
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"[| a:N; b:N |] ==> succ(a) #+ b = succ(a #+ b) : N"
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(fn prems=>
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[ (rewrite_goals_tac arith_defs),
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(rew_tac prems) ]);
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(** Multiplication *)
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(*typing of mult: short and long versions*)
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val mult_typing = prove_goal Arith.thy
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"[| a:N; b:N |] ==> a #* b : N"
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(fn prems=>
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[ (rewrite_goals_tac arith_defs),
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(typechk_tac([add_typing]@prems)) ]);
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val mult_typingL = prove_goal Arith.thy
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"[| a=c:N; b=d:N |] ==> a #* b = c #* d : N"
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(fn prems=>
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[ (rewrite_goals_tac arith_defs),
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(equal_tac (prems@[add_typingL])) ]);
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(*computation for mult: 0 and successor cases*)
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val multC0 = prove_goal Arith.thy
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"b:N ==> 0 #* b = 0 : N"
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(fn prems=>
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[ (rewrite_goals_tac arith_defs),
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(rew_tac prems) ]);
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val multC_succ = prove_goal Arith.thy
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"[| a:N; b:N |] ==> succ(a) #* b = b #+ (a #* b) : N"
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(fn prems=>
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[ (rewrite_goals_tac arith_defs),
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(rew_tac prems) ]);
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(** Difference *)
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(*typing of difference*)
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val diff_typing = prove_goal Arith.thy
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"[| a:N; b:N |] ==> a - b : N"
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(fn prems=>
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[ (rewrite_goals_tac arith_defs),
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(typechk_tac prems) ]);
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val diff_typingL = prove_goal Arith.thy
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"[| a=c:N; b=d:N |] ==> a - b = c - d : N"
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(fn prems=>
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[ (rewrite_goals_tac arith_defs),
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(equal_tac prems) ]);
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(*computation for difference: 0 and successor cases*)
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val diffC0 = prove_goal Arith.thy
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"a:N ==> a - 0 = a : N"
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(fn prems=>
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[ (rewrite_goals_tac arith_defs),
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(rew_tac prems) ]);
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(*Note: rec(a, 0, %z w.z) is pred(a). *)
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val diff_0_eq_0 = prove_goal Arith.thy
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"b:N ==> 0 - b = 0 : N"
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(fn prems=>
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[ (NE_tac "b" 1),
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(rewrite_goals_tac arith_defs),
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(hyp_rew_tac prems) ]);
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(*Essential to simplify FIRST!! (Else we get a critical pair)
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succ(a) - succ(b) rewrites to pred(succ(a) - b) *)
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val diff_succ_succ = prove_goal Arith.thy
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"[| a:N; b:N |] ==> succ(a) - succ(b) = a - b : N"
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(fn prems=>
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[ (rewrite_goals_tac arith_defs),
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(hyp_rew_tac prems),
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(NE_tac "b" 1),
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(hyp_rew_tac prems) ]);
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(*** Simplification *)
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val arith_typing_rls =
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[add_typing, mult_typing, diff_typing];
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val arith_congr_rls =
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[add_typingL, mult_typingL, diff_typingL];
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val congr_rls = arith_congr_rls@standard_congr_rls;
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val arithC_rls =
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[addC0, addC_succ,
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multC0, multC_succ,
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diffC0, diff_0_eq_0, diff_succ_succ];
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structure Arith_simp_data: TSIMP_DATA =
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struct
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val refl = refl_elem
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val sym = sym_elem
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val trans = trans_elem
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val refl_red = refl_red
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val trans_red = trans_red
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val red_if_equal = red_if_equal
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val default_rls = arithC_rls @ comp_rls
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val routine_tac = routine_tac (arith_typing_rls @ routine_rls)
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end;
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structure Arith_simp = TSimpFun (Arith_simp_data);
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fun arith_rew_tac prems = make_rew_tac
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(Arith_simp.norm_tac(congr_rls, prems));
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fun hyp_arith_rew_tac prems = make_rew_tac
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(Arith_simp.cond_norm_tac(prove_cond_tac, congr_rls, prems));
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(**********
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Addition
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**********)
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(*Associative law for addition*)
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val add_assoc = prove_goal Arith.thy
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"[| a:N; b:N; c:N |] ==> (a #+ b) #+ c = a #+ (b #+ c) : N"
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(fn prems=>
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[ (NE_tac "a" 1),
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(hyp_arith_rew_tac prems) ]);
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(*Commutative law for addition. Can be proved using three inductions.
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Must simplify after first induction! Orientation of rewrites is delicate*)
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val add_commute = prove_goal Arith.thy
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"[| a:N; b:N |] ==> a #+ b = b #+ a : N"
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(fn prems=>
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[ (NE_tac "a" 1),
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(hyp_arith_rew_tac prems),
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(NE_tac "b" 2),
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(resolve_tac [sym_elem] 1),
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(NE_tac "b" 1),
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(hyp_arith_rew_tac prems) ]);
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(****************
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Multiplication
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****************)
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(*Commutative law for multiplication
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val mult_commute = prove_goal Arith.thy
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"[| a:N; b:N |] ==> a #* b = b #* a : N"
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(fn prems=>
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[ (NE_tac "a" 1),
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(hyp_arith_rew_tac prems),
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(NE_tac "b" 2),
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(resolve_tac [sym_elem] 1),
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(NE_tac "b" 1),
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(hyp_arith_rew_tac prems) ]); NEEDS COMMUTATIVE MATCHING
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***************)
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(*right annihilation in product*)
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val mult_0_right = prove_goal Arith.thy
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"a:N ==> a #* 0 = 0 : N"
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(fn prems=>
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[ (NE_tac "a" 1),
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(hyp_arith_rew_tac prems) ]);
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(*right successor law for multiplication*)
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val mult_succ_right = prove_goal Arith.thy
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"[| a:N; b:N |] ==> a #* succ(b) = a #+ (a #* b) : N"
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(fn prems=>
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[ (NE_tac "a" 1),
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(*swap round the associative law of addition*)
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(hyp_arith_rew_tac (prems @ [add_assoc RS sym_elem])),
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(*leaves a goal involving a commutative law*)
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(REPEAT (assume_tac 1 ORELSE
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resolve_tac
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(prems@[add_commute,mult_typingL,add_typingL]@
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intrL_rls@[refl_elem]) 1)) ]);
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(*Commutative law for multiplication*)
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val mult_commute = prove_goal Arith.thy
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"[| a:N; b:N |] ==> a #* b = b #* a : N"
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(fn prems=>
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[ (NE_tac "a" 1),
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(hyp_arith_rew_tac (prems @ [mult_0_right, mult_succ_right])) ]);
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(*addition distributes over multiplication*)
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val add_mult_distrib = prove_goal Arith.thy
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"[| a:N; b:N; c:N |] ==> (a #+ b) #* c = (a #* c) #+ (b #* c) : N"
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(fn prems=>
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[ (NE_tac "a" 1),
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(*swap round the associative law of addition*)
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(hyp_arith_rew_tac (prems @ [add_assoc RS sym_elem])) ]);
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(*Associative law for multiplication*)
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val mult_assoc = prove_goal Arith.thy
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"[| a:N; b:N; c:N |] ==> (a #* b) #* c = a #* (b #* c) : N"
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(fn prems=>
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[ (NE_tac "a" 1),
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(hyp_arith_rew_tac (prems @ [add_mult_distrib])) ]);
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(************
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Difference
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************
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Difference on natural numbers, without negative numbers
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a - b = 0 iff a<=b a - b = succ(c) iff a>b *)
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val diff_self_eq_0 = prove_goal Arith.thy
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"a:N ==> a - a = 0 : N"
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(fn prems=>
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[ (NE_tac "a" 1),
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(hyp_arith_rew_tac prems) ]);
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(* [| c : N; 0 : N; c : N |] ==> c #+ 0 = c : N *)
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val add_0_right = addC0 RSN (3, add_commute RS trans_elem);
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(*Addition is the inverse of subtraction: if b<=x then b#+(x-b) = x.
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An example of induction over a quantified formula (a product).
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Uses rewriting with a quantified, implicative inductive hypothesis.*)
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val prems =
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goal Arith.thy
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"b:N ==> ?a : PROD x:N. Eq(N, b-x, 0) --> Eq(N, b #+ (x-b), x)";
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by (NE_tac "b" 1);
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(*strip one "universal quantifier" but not the "implication"*)
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by (resolve_tac intr_rls 3);
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(*case analysis on x in
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(succ(u) <= x) --> (succ(u)#+(x-succ(u)) = x) *)
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by (NE_tac "x" 4 THEN assume_tac 4);
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(*Prepare for simplification of types -- the antecedent succ(u)<=x *)
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by (resolve_tac [replace_type] 5);
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by (resolve_tac [replace_type] 4);
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by (arith_rew_tac prems);
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(*Solves first 0 goal, simplifies others. Two sugbgoals remain.
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Both follow by rewriting, (2) using quantified induction hyp*)
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by (intr_tac[]); (*strips remaining PRODs*)
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by (hyp_arith_rew_tac (prems@[add_0_right]));
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by (assume_tac 1);
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val add_diff_inverse_lemma = result();
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(*Version of above with premise b-a=0 i.e. a >= b.
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Using ProdE does not work -- for ?B(?a) is ambiguous.
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Instead, add_diff_inverse_lemma states the desired induction scheme;
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the use of RS below instantiates Vars in ProdE automatically. *)
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val prems =
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goal Arith.thy "[| a:N; b:N; b-a = 0 : N |] ==> b #+ (a-b) = a : N";
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by (resolve_tac [EqE] 1);
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by (resolve_tac [ add_diff_inverse_lemma RS ProdE RS ProdE ] 1);
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by (REPEAT (resolve_tac (prems@[EqI]) 1));
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val add_diff_inverse = result();
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(********************
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Absolute difference
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********************)
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(*typing of absolute difference: short and long versions*)
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val absdiff_typing = prove_goal Arith.thy
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"[| a:N; b:N |] ==> a |-| b : N"
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(fn prems=>
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[ (rewrite_goals_tac arith_defs),
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(typechk_tac prems) ]);
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val absdiff_typingL = prove_goal Arith.thy
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"[| a=c:N; b=d:N |] ==> a |-| b = c |-| d : N"
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(fn prems=>
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[ (rewrite_goals_tac arith_defs),
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(equal_tac prems) ]);
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val absdiff_self_eq_0 = prove_goal Arith.thy
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"a:N ==> a |-| a = 0 : N"
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(fn prems=>
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[ (rewrite_goals_tac [absdiff_def]),
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(arith_rew_tac (prems@[diff_self_eq_0])) ]);
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val absdiffC0 = prove_goal Arith.thy
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"a:N ==> 0 |-| a = a : N"
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(fn prems=>
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[ (rewrite_goals_tac [absdiff_def]),
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(hyp_arith_rew_tac prems) ]);
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val absdiff_succ_succ = prove_goal Arith.thy
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"[| a:N; b:N |] ==> succ(a) |-| succ(b) = a |-| b : N"
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(fn prems=>
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[ (rewrite_goals_tac [absdiff_def]),
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(hyp_arith_rew_tac prems) ]);
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(*Note how easy using commutative laws can be? ...not always... *)
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val prems = goal Arith.thy "[| a:N; b:N |] ==> a |-| b = b |-| a : N";
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by (rewrite_goals_tac [absdiff_def]);
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by (resolve_tac [add_commute] 1);
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by (typechk_tac ([diff_typing]@prems));
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val absdiff_commute = result();
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(*If a+b=0 then a=0. Surprisingly tedious*)
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val prems =
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goal Arith.thy "[| a:N; b:N |] ==> ?c : PROD u: Eq(N,a#+b,0) . Eq(N,a,0)";
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by (NE_tac "a" 1);
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by (resolve_tac [replace_type] 3);
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by (arith_rew_tac prems);
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by (intr_tac[]); (*strips remaining PRODs*)
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by (resolve_tac [ zero_ne_succ RS FE ] 2);
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by (etac (EqE RS sym_elem) 3);
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by (typechk_tac ([add_typing] @prems));
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val add_eq0_lemma = result();
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(*Version of above with the premise a+b=0.
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Again, resolution instantiates variables in ProdE *)
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val prems =
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goal Arith.thy "[| a:N; b:N; a #+ b = 0 : N |] ==> a = 0 : N";
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by (resolve_tac [EqE] 1);
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by (resolve_tac [add_eq0_lemma RS ProdE] 1);
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by (resolve_tac [EqI] 3);
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by (ALLGOALS (resolve_tac prems));
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val add_eq0 = result();
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(*Here is a lemma to infer a-b=0 and b-a=0 from a|-|b=0, below. *)
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val prems = goal Arith.thy
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"[| a:N; b:N; a |-| b = 0 : N |] ==> \
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\ ?a : SUM v: Eq(N, a-b, 0) . Eq(N, b-a, 0)";
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372 |
by (intr_tac[]);
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373 |
by eqintr_tac;
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374 |
by (resolve_tac [add_eq0] 2);
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375 |
by (resolve_tac [add_eq0] 1);
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376 |
by (resolve_tac [add_commute RS trans_elem] 6);
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377 |
by (typechk_tac (diff_typing:: map (rewrite_rule [absdiff_def]) prems));
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378 |
val absdiff_eq0_lem = result();
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379 |
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380 |
(*if a |-| b = 0 then a = b
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381 |
proof: a-b=0 and b-a=0, so b = a+(b-a) = a+0 = a*)
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382 |
val prems =
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383 |
goal Arith.thy "[| a |-| b = 0 : N; a:N; b:N |] ==> a = b : N";
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384 |
by (resolve_tac [EqE] 1);
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385 |
by (resolve_tac [absdiff_eq0_lem RS SumE] 1);
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386 |
by (TRYALL (resolve_tac prems));
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387 |
by eqintr_tac;
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388 |
by (resolve_tac [add_diff_inverse RS sym_elem RS trans_elem] 1);
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389 |
by (resolve_tac [EqE] 3 THEN assume_tac 3);
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390 |
by (hyp_arith_rew_tac (prems@[add_0_right]));
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391 |
val absdiff_eq0 = result();
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392 |
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393 |
(***********************
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394 |
Remainder and Quotient
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395 |
***********************)
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396 |
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397 |
(*typing of remainder: short and long versions*)
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398 |
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399 |
val mod_typing = prove_goal Arith.thy
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400 |
"[| a:N; b:N |] ==> a mod b : N"
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401 |
(fn prems=>
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402 |
[ (rewrite_goals_tac [mod_def]),
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403 |
(typechk_tac (absdiff_typing::prems)) ]);
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404 |
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405 |
val mod_typingL = prove_goal Arith.thy
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406 |
"[| a=c:N; b=d:N |] ==> a mod b = c mod d : N"
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407 |
(fn prems=>
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408 |
[ (rewrite_goals_tac [mod_def]),
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409 |
(equal_tac (prems@[absdiff_typingL])) ]);
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410 |
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411 |
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412 |
(*computation for mod : 0 and successor cases*)
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413 |
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414 |
val modC0 = prove_goal Arith.thy "b:N ==> 0 mod b = 0 : N"
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415 |
(fn prems=>
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416 |
[ (rewrite_goals_tac [mod_def]),
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417 |
(rew_tac(absdiff_typing::prems)) ]);
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418 |
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419 |
val modC_succ = prove_goal Arith.thy
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420 |
"[| a:N; b:N |] ==> succ(a) mod b = rec(succ(a mod b) |-| b, 0, %x y.succ(a mod b)) : N"
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|
421 |
(fn prems=>
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422 |
[ (rewrite_goals_tac [mod_def]),
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423 |
(rew_tac(absdiff_typing::prems)) ]);
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424 |
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425 |
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426 |
(*typing of quotient: short and long versions*)
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|
427 |
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|
428 |
val div_typing = prove_goal Arith.thy "[| a:N; b:N |] ==> a div b : N"
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|
429 |
(fn prems=>
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430 |
[ (rewrite_goals_tac [div_def]),
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431 |
(typechk_tac ([absdiff_typing,mod_typing]@prems)) ]);
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432 |
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|
433 |
val div_typingL = prove_goal Arith.thy
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|
434 |
"[| a=c:N; b=d:N |] ==> a div b = c div d : N"
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|
435 |
(fn prems=>
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436 |
[ (rewrite_goals_tac [div_def]),
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437 |
(equal_tac (prems @ [absdiff_typingL, mod_typingL])) ]);
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438 |
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|
439 |
val div_typing_rls = [mod_typing, div_typing, absdiff_typing];
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|
440 |
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441 |
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|
442 |
(*computation for quotient: 0 and successor cases*)
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|
443 |
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|
444 |
val divC0 = prove_goal Arith.thy "b:N ==> 0 div b = 0 : N"
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|
445 |
(fn prems=>
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|
446 |
[ (rewrite_goals_tac [div_def]),
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|
447 |
(rew_tac([mod_typing, absdiff_typing] @ prems)) ]);
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448 |
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|
449 |
val divC_succ =
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|
450 |
prove_goal Arith.thy "[| a:N; b:N |] ==> succ(a) div b = \
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|
451 |
\ rec(succ(a) mod b, succ(a div b), %x y. a div b) : N"
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|
452 |
(fn prems=>
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|
453 |
[ (rewrite_goals_tac [div_def]),
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|
454 |
(rew_tac([mod_typing]@prems)) ]);
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455 |
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|
456 |
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|
457 |
(*Version of above with same condition as the mod one*)
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|
458 |
val divC_succ2 = prove_goal Arith.thy
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|
459 |
"[| a:N; b:N |] ==> \
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|
460 |
\ succ(a) div b =rec(succ(a mod b) |-| b, succ(a div b), %x y. a div b) : N"
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|
461 |
(fn prems=>
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|
462 |
[ (resolve_tac [ divC_succ RS trans_elem ] 1),
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|
463 |
(rew_tac(div_typing_rls @ prems @ [modC_succ])),
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|
464 |
(NE_tac "succ(a mod b)|-|b" 1),
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|
465 |
(rew_tac ([mod_typing, div_typing, absdiff_typing] @prems)) ]);
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|
466 |
|
|
467 |
(*for case analysis on whether a number is 0 or a successor*)
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|
468 |
val iszero_decidable = prove_goal Arith.thy
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|
469 |
"a:N ==> rec(a, inl(eq), %ka kb.inr(<ka, eq>)) : \
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|
470 |
\ Eq(N,a,0) + (SUM x:N. Eq(N,a, succ(x)))"
|
|
471 |
(fn prems=>
|
|
472 |
[ (NE_tac "a" 1),
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|
473 |
(resolve_tac [PlusI_inr] 3),
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|
474 |
(resolve_tac [PlusI_inl] 2),
|
|
475 |
eqintr_tac,
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|
476 |
(equal_tac prems) ]);
|
|
477 |
|
|
478 |
(*Main Result. Holds when b is 0 since a mod 0 = a and a div 0 = 0 *)
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|
479 |
val prems =
|
|
480 |
goal Arith.thy "[| a:N; b:N |] ==> a mod b #+ (a div b) #* b = a : N";
|
|
481 |
by (NE_tac "a" 1);
|
|
482 |
by (arith_rew_tac (div_typing_rls@prems@[modC0,modC_succ,divC0,divC_succ2]));
|
|
483 |
by (resolve_tac [EqE] 1);
|
|
484 |
(*case analysis on succ(u mod b)|-|b *)
|
|
485 |
by (res_inst_tac [("a1", "succ(u mod b) |-| b")]
|
|
486 |
(iszero_decidable RS PlusE) 1);
|
|
487 |
by (etac SumE 3);
|
|
488 |
by (hyp_arith_rew_tac (prems @ div_typing_rls @
|
|
489 |
[modC0,modC_succ, divC0, divC_succ2]));
|
|
490 |
(*Replace one occurence of b by succ(u mod b). Clumsy!*)
|
|
491 |
by (resolve_tac [ add_typingL RS trans_elem ] 1);
|
|
492 |
by (eresolve_tac [EqE RS absdiff_eq0 RS sym_elem] 1);
|
|
493 |
by (resolve_tac [refl_elem] 3);
|
|
494 |
by (hyp_arith_rew_tac (prems @ div_typing_rls));
|
|
495 |
val mod_div_equality = result();
|
|
496 |
|
|
497 |
writeln"Reached end of file.";
|