diff -r 722bf1319be5 -r fd1be45b64bf IOA/example/Lemmas.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/IOA/example/Lemmas.ML Wed Nov 02 11:50:09 1994 +0100 @@ -0,0 +1,245 @@ +(* Logic *) +val prems = goal HOL.thy "(P ==> Q-->R) ==> P&Q --> R"; + by(fast_tac (HOL_cs addDs prems) 1); +val imp_conj_lemma = result(); + +goal HOL.thy "(P --> (? x. Q(x))) = (? x. P --> Q(x))"; + by(fast_tac HOL_cs 1); +val imp_ex_equiv = result(); + +goal HOL.thy "(A --> B & C) = ((A --> B) & (A --> C))"; + by (fast_tac HOL_cs 1); +val fork_lemma = result(); + +goal HOL.thy "((A --> B) & (C --> B)) = ((A | C) --> B)"; + by (fast_tac HOL_cs 1); +val imp_or_lem = result(); + +goal HOL.thy "(X = (~ Y)) = ((~X) = Y)"; + by (fast_tac HOL_cs 1); +val neg_flip = result(); + +goal HOL.thy "P --> Q(M) --> Q(if(P,M,N))"; + by (rtac impI 1); + by (rtac impI 1); + by (rtac (expand_if RS iffD2) 1); + by (fast_tac HOL_cs 1); +val imp_true_decompose = result(); + +goal HOL.thy "(~P) --> Q(N) --> Q(if(P,M,N))"; + by (rtac impI 1); + by (rtac impI 1); + by (rtac (expand_if RS iffD2) 1); + by (fast_tac HOL_cs 1); +val imp_false_decompose = result(); + + +(* Sets *) +val set_lemmas = + map (fn s => prove_goal Set.thy s (fn _ => [fast_tac set_cs 1])) + ["f(x) : (UN x. {f(x)})", + "f(x,y) : (UN x y. {f(x,y)})", + "!!a. (!x. a ~= f(x)) ==> a ~: (UN x. {f(x)})", + "!!a. (!x y. a ~= f(x,y)) ==> a ~: (UN x y. {f(x,y)})"]; + + +(* Arithmetic *) +goal Arith.thy "n ~= 0 --> Suc(m+pred(n)) = m+n"; + by (nat_ind_tac "n" 1); + by (REPEAT(simp_tac arith_ss 1)); +val Suc_pred_lemma = result() RS mp; + +goal Arith.thy "x <= y --> x <= Suc(y)"; + by (rtac impI 1); + by (rtac (le_eq_less_or_eq RS iffD2) 1); + by (rtac disjI1 1); + by (dtac (le_eq_less_or_eq RS iffD1) 1); + by (etac disjE 1); + by (etac less_SucI 1); + by (asm_simp_tac nat_ss 1); +val leq_imp_leq_suc = result() RS mp; + +(* Same as previous! *) +goal Arith.thy "(x::nat)<=y --> x<=Suc(y)"; + by (simp_tac (arith_ss addsimps [le_eq_less_or_eq]) 1); +val leq_suc = result(); + +goal Arith.thy "((m::nat) + n = m + p) = (n = p)"; + by (nat_ind_tac "m" 1); + by (simp_tac arith_ss 1); + by (asm_simp_tac arith_ss 1); +val left_plus_cancel = result(); + +goal Arith.thy "((x::nat) + y = Suc(x + z)) = (y = Suc(z))"; + by (nat_ind_tac "x" 1); + by (simp_tac arith_ss 1); + by (asm_simp_tac arith_ss 1); +val left_plus_cancel_inside_succ = result(); + +goal Arith.thy "(x ~= 0) = (? y. x = Suc(y))"; + by (nat_ind_tac "x" 1); + by (simp_tac arith_ss 1); + by (asm_simp_tac arith_ss 1); + by (fast_tac HOL_cs 1); +val nonzero_is_succ = result(); + +goal Arith.thy "(m::nat) < n --> m + p < n + p"; + by (nat_ind_tac "p" 1); + by (simp_tac arith_ss 1); + by (asm_simp_tac arith_ss 1); +val less_add_same_less = result(); + +goal Arith.thy "(x::nat)<= y --> x<=y+k"; + by (nat_ind_tac "k" 1); + by (simp_tac arith_ss 1); + by (asm_full_simp_tac (arith_ss addsimps [leq_suc]) 1); +val leq_add_leq = result(); + +goal Arith.thy "(x::nat) + y <= z --> x <= z"; + by (nat_ind_tac "y" 1); + by (simp_tac arith_ss 1); + by (asm_simp_tac arith_ss 1); + by (rtac impI 1); + by (dtac Suc_leD 1); + by (fast_tac HOL_cs 1); +val left_add_leq = result(); + +goal Arith.thy "(A::nat) < B --> C < D --> A + C < B + D"; + by (rtac impI 1); + by (rtac impI 1); + by (rtac less_trans 1); + by (rtac (less_add_same_less RS mp) 1); + by (assume_tac 1); + by (rtac (add_commute RS ssubst)1);; + by (res_inst_tac [("m1","B")] (add_commute RS ssubst) 1); + by (rtac (less_add_same_less RS mp) 1); + by (assume_tac 1); +val less_add_cong = result(); + +goal Nat.thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)"; + by (dtac le_imp_less_or_eq 1); + by (fast_tac (HOL_cs addIs [less_trans]) 1); +val less_leq_less = result(); + +goal Arith.thy "(A::nat) <= B --> C <= D --> A + C <= B + D"; + by (rtac impI 1); + by (rtac impI 1); + by (asm_full_simp_tac (arith_ss addsimps [le_eq_less_or_eq]) 1); + by (safe_tac HOL_cs); + by (rtac (less_add_cong RS mp RS mp) 1); + by (assume_tac 1); + by (assume_tac 1); + by (rtac (less_add_same_less RS mp) 1); + by (assume_tac 1); + by (rtac (add_commute RS ssubst)1);; + by (res_inst_tac [("m1","B")] (add_commute RS ssubst) 1); + by (rtac (less_add_same_less RS mp) 1); + by (assume_tac 1); +val less_eq_add_cong = result(); + +goal Arith.thy "(w <= y) --> ((x::nat) + y <= z) --> (x + w <= z)"; + by (rtac impI 1); + by (dtac (less_eq_add_cong RS mp) 1); + by (cut_facts_tac [le_refl] 1); + by (dres_inst_tac [("P","x<=x")] mp 1);by (assume_tac 1); + by (asm_full_simp_tac (HOL_ss addsimps [add_commute]) 1); + by (rtac impI 1); + by (etac le_trans 1); + by (assume_tac 1); +val leq_add_left_cong = result(); + +goal Arith.thy "(? x. y = Suc(x)) = (~(y = 0))"; + by (nat_ind_tac "y" 1); + by (simp_tac arith_ss 1); + by (rtac iffI 1); + by (asm_full_simp_tac arith_ss 1); + by (fast_tac HOL_cs 1); +val suc_not_zero = result(); + +goal Arith.thy "Suc(x) <= y --> (? z. y = Suc(z))"; + by (rtac impI 1); + by (asm_full_simp_tac (arith_ss addsimps [le_eq_less_or_eq]) 1); + by (safe_tac HOL_cs); + by (fast_tac HOL_cs 2); + by (asm_simp_tac (arith_ss addsimps [suc_not_zero]) 1); + by (rtac ccontr 1); + by (asm_full_simp_tac (arith_ss addsimps [suc_not_zero]) 1); + by (hyp_subst_tac 1); + by (asm_full_simp_tac arith_ss 1); +val suc_leq_suc = result(); + +goal Arith.thy "~0 n = 0"; + by (nat_ind_tac "n" 1); + by (asm_simp_tac arith_ss 1); + by (safe_tac HOL_cs); + by (asm_full_simp_tac arith_ss 1); + by (asm_full_simp_tac arith_ss 1); +val zero_eq = result(); + +goal Arith.thy "x < Suc(y) --> x<=y"; + by (nat_ind_tac "n" 1); + by (asm_simp_tac arith_ss 1); + by (safe_tac HOL_cs); + by (etac less_imp_le 1); +val less_suc_imp_leq = result(); + +goal Arith.thy "0 Suc(pred(x)) = x"; + by (nat_ind_tac "x" 1); + by (simp_tac arith_ss 1); + by (asm_simp_tac arith_ss 1); +val suc_pred_id = result(); + +goal Arith.thy "0 (pred(x) = y) = (x = Suc(y))"; + by (nat_ind_tac "x" 1); + by (simp_tac arith_ss 1); + by (asm_simp_tac arith_ss 1); +val pred_suc = result(); + +goal Arith.thy "(x ~= 0) = (0 y<=z --> x<(z::nat)"; + by (rtac impI 1); by (rtac impI 1); + by (dtac le_imp_less_or_eq 1); + by (fast_tac (HOL_cs addIs [less_trans]) 1); +val less_leq_less = result(); + +goal Arith.thy "(Suc(n) <= Suc(m)) = (n <= m)"; + by (simp_tac (arith_ss addsimps [le_eq_less_or_eq]) 1); +val succ_leq_mono = result(); + +(* Odd proof. Why do induction? *) +goal Arith.thy "((x::nat) = y + z) --> (y <= x)"; + by (nat_ind_tac "y" 1); + by (simp_tac arith_ss 1); + by (simp_tac (arith_ss addsimps + [succ_leq_mono, le_refl RS (leq_add_leq RS mp)]) 1); +val plus_leq_lem = result(); + +(* Lists *) + +goal List.thy "(L @ (x#M)) ~= []"; + by (list_ind_tac "L" 1); + by (simp_tac list_ss 1); + by (asm_simp_tac list_ss 1); +val append_cons = result(); + +goal List.thy "(X ~= hd(L@M)) = (X ~= if(L = [], hd(M), hd(L)))"; + by (list_ind_tac "L" 1); + by (simp_tac list_ss 1); + by (asm_full_simp_tac list_ss 1); +val not_hd_append = result(); + +goal List.thy "(L = (x#rst)) --> (L = []) --> P"; + by (simp_tac list_ss 1); +val list_cases = result(); + +goal List.thy "(? L2. L1 = x#L2) --> (L1 ~= [])"; + by (strip_tac 1); + by (etac exE 1); + by (asm_simp_tac list_ss 1); +val cons_imp_not_null = result();