hcju/isabelleps huggingface dataset:
id: shows "add_col_sub_row a (i - 1) j A $$ (i', j') = (if i' = i - 1 \<and> j' \<noteq> j then A $$ (i', j') - a * A $$ (j, j') else if i' \<noteq> i - 1 \<and> j' = j then A $$ (i', j') + a * A $$ (i', i - 1) else A $$ (i', j'))"
strict_sorted_imp_less_sets: fixes as :: "'a list"   and bs :: "'a list" assumes "strict_sorted (as @ bs)" shows "list.set as \<lless> list.set bs"
simps_5: fixes f1 :: "'g \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> ('h, 'g) tllist \<Rightarrow> 'f"   and x1 :: "'g" shows "(case TNil x1 of TNil x \<Rightarrow> f1 x | TCons x xa \<Rightarrow> f2 x xa) = f1 x1"
simps_2: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; send\<langle>t\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>d = (\<lambda>\<I>. M \<turnstile> t \<cdot> \<I> \<and> \<lbrakk>M; S\<rbrakk>\<^sub>d \<I>)"
symmetric: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<approx> Q" shows "Q \<approx> P"
divs: fixes d :: "'a mod_ring poly" assumes "d \<in> set divs" shows "monic d \<and> 0 < degree d"
qty_single_val_process: fixes Mkt :: "('b, 'c) discrete_market"   and asset :: "'c"   and qty :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and n :: "nat"   and w :: "'b" shows "val_process Mkt (qty_single asset qty) n w = prices Mkt asset n w * qty (Suc n) w"
hinv_insrt: fixes t :: "'a tree"   and x :: "'a" assumes "hinv t" shows "hinv (insrt x t)"
shadow_root_ptr_kinds_eq_h2: shows "shadow_root_ptr_kinds h2 = shadow_root_ptr_kinds h3"
join_shunt: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "(sup x y = top) = less_eq (uminus x) y"
hseqI': fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "B.arr \<mu>"   and "B.arr \<nu>"   and "src \<nu> = trg \<mu>" shows "B.hseq \<nu> \<mu>"
msum_aform_def: fixes n :: "nat"   and f :: "'b \<times> 'b pdevs"   and g :: "'b \<times> 'b pdevs" shows "msum_aform n f g = (fst f + fst g, msum_pdevs n (snd f) (snd g))"
stl_def: fixes stream :: "'d stream" shows "stl stream = (case stream of x1 ## x2 \<Rightarrow> x2)"
sorted_list_of_fset_sort: fixes l :: "'a list" shows "sorted_list_of_fset (fset_of_list l) = sort (remdups l)"
simps: fixes v1 :: "'a"   and p1 :: "'b"   and v2 :: "'a"   and p2 :: "'c" shows "Pre_Port_Graph.edge_end ((v1, p1), v2, p2) = v2"
path2_def: fixes \<alpha>n :: "'c \<Rightarrow> 'd list"   and invar :: "'c \<Rightarrow> bool"   and inEdges' :: "'c \<Rightarrow> 'd \<Rightarrow> ('d \<times> 'e) list"   and g :: "'c"   and n :: "'d"   and ns :: "'d list"   and m :: "'d" shows "graph_path_base.path2 \<alpha>n invar inEdges' g n ns m \<equiv> graph_path_base.path \<alpha>n invar inEdges' g ns \<and> n = hd ns \<and> m = last ns"
eq_N: shows "image_mset card (mset_set P) = N"
gd_term_axioms: shows "gd_term pair_of_term term_of_pair (\<lambda>s t. le_of_nat_term_order cmp_term (term_of_pair (s, the_min)) (term_of_pair (t, the_min))) (\<lambda>s t. lt_of_nat_term_order cmp_term (term_of_pair (s, the_min)) (term_of_pair (t, the_min))) (le_of_nat_term_order cmp_term) (lt_of_nat_term_order cmp_term)"
n_absorb1: fixes x :: "'a"   and y :: "'a" shows "n x \<oplus> n x \<cdot> n y = n x"
NR\<^sub>N_def: fixes n :: "nat"   and u :: "nat"   and l :: "nat" shows "NR\<^sub>N n u l = \<integral>\<^sup>+ x. ennreal (real x) \<partial>measure_pmf (R\<^sub>N n u l)"
Tree\<^sub>\<alpha>_abs_rep: fixes t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" shows "abs_Tree\<^sub>\<alpha> (rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha>) = t\<^sub>\<alpha>"
MStep_returnElimAux: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and k :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, s, k, ll, t) \<in> MStep"   and "get_ins M l = Some vreturn" shows "t = s \<and> ll = l"
out_in_arcs_reflexive_uniq: fixes v :: "'a"   and e :: "'b" assumes "v \<in> verts G"   and "e \<in> out_arcs G v" shows "\<exists>!e'. e' \<in> in_arcs G v \<and> tail G e' = head G e"
simps_1: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and xs :: "'a list" shows "proper_intrvl.proper_interval_set_aux less proper_interval xs [] = False"
bn_abs_residual_fresh: fixes \<alpha> :: "'b"   and P :: "'c" shows "bn \<alpha> \<sharp>* \<langle>\<alpha>,P\<rangle>"
IH: fixes Ts :: "ty list" assumes "P,E,hp s'' \<turnstile> es'' [:] Ts"   and "P,E \<turnstile> s'' \<surd>" shows "P,E \<turnstile> s' \<surd>"
S_irrefl: fixes t :: "('f, 'a) Term.term" shows "\<not> S t t"
comp_assoc: fixes h :: "'a1"   and g :: "'a1"   and f :: "'a1" shows "(h \<cdot>\<^sub>1 g) \<cdot>\<^sub>1 f = h \<cdot>\<^sub>1 g \<cdot>\<^sub>1 f"
thetaParTT_def: shows "thetaParTT \<equiv> {(c1 | c2, d1 | d2) |c1 c2 d1 d2. c1 \<approx>T d1 \<and> c2 \<approx>T d2}"
assms_2: shows "\<alpha> = \<phi> $$ (0, 0)"
elims_3: fixes x :: "nat"   and xa :: "cell list \<times> cell list" assumes "\<not> inv_loop5 x xa" obtains n :: "nat" and l :: "cell list" and r :: "cell list" where "x = n" and "xa = (l, r)" and "\<not> (inv_loop5_loop n (l, r) \<or> inv_loop5_exit n (l, r))"
I_and: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm"   and xs :: "real list" shows "R.I (and \<phi>\<^sub>1 \<phi>\<^sub>2) xs = R.I (And \<phi>\<^sub>1 \<phi>\<^sub>2) xs"
Pair_equalI: fixes x :: "'r"   and p :: "'r \<times> 's"   and y :: "'s" assumes "x \<equiv> prod.fst p"   and "y \<equiv> prod.snd p" shows "(x, y) \<equiv> p"
plus_enat_simps_1: fixes m :: "nat"   and n :: "nat" shows "enat m + enat n = enat (m + n)"
outgoing_transitions_def: fixes e :: "(('a \<times> 'b) \<times> 'c) fset"   and s :: "'a" shows "EFSM.outgoing_transitions e s = ffilter (\<lambda>((o, uu_), uu_). o = s) e"
simps_3: fixes \<phi> :: "'c" shows "S_transform_Tree (tPred \<phi>) = Act\<^sub>\<alpha> (S_action.Pred \<phi>) (Conj\<^sub>\<alpha> bempty)"
r_amalgamation_recfn: shows "recfn 3 r_amalgamation"
arity_sym\<^sub>h_if_\<delta>\<^sub>h_gt_0_E: fixes f :: "'s" assumes "0 < \<delta>\<^sub>h" obtains n :: "nat" where "arity_sym\<^sub>h f = of_nat n"
grdOf: fixes c :: "('fsym, 'psym) lit list"   and l :: "('fsym, 'psym) lit"   and x :: "Preliminaries.var" assumes "c \<in> \<Phi>"   and "l \<in> set c"   and "x \<in> nv2L l"   and "\<not> infTp (tpOfV x)" shows "grdOf c l x \<in> set c"
suffix_ConsD: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "suffix (x # xs) ys" shows "suffix xs ys"
int_numeral_bitXOR_num: fixes m :: "num"   and n :: "num" shows "numeral m XOR numeral n = (case bitXOR_num m n of None \<Rightarrow> 0 | Some n' \<Rightarrow> numeral n')"
act_rews_2: fixes v :: "state \<Rightarrow> 'c"   and s :: "state"   and t :: "state" shows "(v$) (s, t) \<equiv> v t"
exists_next_marker_free_state: fixes cid :: "nat"   and p :: "'a"   and q :: "'a"   and t :: "('a, 'b, 'c) event list"   and i :: "nat" assumes "channel cid = Some (p, q)"   and "trace init t final" shows "\<exists>!j. next_marker_free_state t i cid = j \<and> i \<le> j \<and> Marker \<notin> set (msgs (S t j) cid)"
measurable_from_subalg: fixes M :: "'c measure"   and F :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and N :: "'d measure" assumes "subalgebra M F"   and "f \<in> F \<rightarrow>\<^sub>M N" shows "f \<in> M \<rightarrow>\<^sub>M N"
is_Groebner_basis_def: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "is_Groebner_basis F \<equiv> relation.is_ChurchRosser (red F)"
echelon_form_of_upt_k_det_def: fixes bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c"   and A' :: "'f \<times> (('c, 'd) vec, 'e) vec"   and k :: "nat" shows "echelon_form_of_upt_k_det bezout A' k = (let A = snd A'; f = foldl (echelon_form_of_column_k_det bezout) (1::'c, A, 0) [0..<Suc k] in (fst f, fst (snd f)))"
good_def: fixes v :: "'a" shows "good v = {\<sigma>. strategy p \<sigma> \<and> strategy_attracts_via p \<sigma> v S W}"
less_ref_def: fixes P :: "'a process"   and Q :: "'a process" shows "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
distrib_right_numeral: fixes a :: "'b"   and b :: "'b"   and v :: "num" shows "(a + b) * numeral v = a * numeral v + b * numeral v"
lfp_eq_fixp: fixes f :: "'a \<Rightarrow> 'a" assumes "mono f" shows "lfp f = ccpo_class.fixp f"
wf_digraph: shows "wf_digraph (G \<restriction> {v. u \<rightarrow>\<^sup>* v})"
pow_mono_one: fixes a :: "'d"   and n :: "nat" assumes "(1::'d) \<le> a" shows "(1::'d) \<le> a ^ n"
continuous_on_Un_local: fixes s :: "'a set"   and t :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "closedin (top_of_set (s \<union> t)) s"   and "closedin (top_of_set (s \<union> t)) t"   and "continuous_on s f"   and "continuous_on t f" shows "continuous_on (s \<union> t) f"
intro: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a P. \<lbrakk>a \<in> A; \<And>x. \<lbrakk>x \<in> A; \<And>y. \<lbrakk>y \<in> A; less y x\<rbrakk> \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x\<rbrakk> \<Longrightarrow> P a" shows "well_founded A less"
is_Sup_binary: fixes x :: "'c"   and y :: "'c"   and sup :: "'c" shows "is_Sup {x, y} sup = is_sup x y sup"
chordal_circle_radius_positive: fixes A :: "complex"   and B :: "complex"   and C :: "complex"   and D :: "complex"   and dsc :: "real"   and a1 :: "complex"   and a2 :: "complex" assumes "hermitean (A, B, C, D)"   and "Re (mat_det (A, B, C, D)) \<le> 0"   and "B \<noteq> 0"   and "dsc = sqrt (Re ((D - A)\<^sup>2 + 4 * (B * cnj B)))"   and "a1 = (A - D + cor dsc) / (2 * C)"   and "a2 = (A - D - cor dsc) / (2 * C)" shows "- 1 \<le> Re (A * a1 / B) \<and> - 1 \<le> Re (A * a2 / B)"
rep_eq: fixes x :: "'d"   and xa :: "'b" shows "Rep_formula (Pred x xa) = Pred\<^sub>\<alpha> x xa"
tr_par_tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p: fixes A' :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "A' \<in> set (tr\<^sub>p\<^sub>c A D)"   and "list_all tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p (unlabel A)"   and "fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"   and "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (unlabel D) \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"   and "\<forall>t\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t (unlabel A) \<union> pair ` snd ` set D. \<forall>t'\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t (unlabel A) \<union> pair ` snd ` set D. (\<exists>\<delta>. Unifier \<delta> t t') \<longrightarrow> \<Gamma> t = \<Gamma> t'" shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p (unlabel A')"
order_less_imp_le: fixes x :: "'c"   and y :: "'c" assumes "x < y" shows "x \<le> y"
complex_inner_1_left: fixes x :: "complex" shows "\<langle>1, x\<rangle> = x"
stake_nil: fixes i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "(stake i \<sigma> = []) = (i = 0)"
sorted_merge: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "transp f"   and "\<And>x y. f x y \<or> f y x" shows "sorted_wrt f (PAC_Checker_Init.merge f xs ys) = (sorted_wrt f xs \<and> sorted_wrt f ys)"
subset_Diff_insert: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b"   and C :: "'b set" shows "order.greater_eq (B - insert x C) A = (order.greater_eq (B - C) A \<and> x \<notin> A)"
monotonicity_L: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "Cn A \<subseteq> Cn B"
measures_less: fixes f :: "'b \<Rightarrow> nat"   and x :: "'b"   and y :: "'b"   and fs :: "('b \<Rightarrow> nat) list" assumes "f x < f y" shows "(x, y) \<in> measures (f # fs)"
m_ac_2: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "Group.comm_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"
elims: fixes x :: "nat \<times> nat list"   and xa :: "abc_inst list"   and xb :: "nat"   and y :: "nat \<times> nat list" assumes "abc_steps_l x xa xb = y" obtains s :: "nat" and lm :: "nat list" and p :: "abc_inst list" where "x = (s__, lma__)" and "xa = p__" and "xb = 0" and "y = (s__, lma__)"   | s :: "nat" and lm :: "nat list" and p :: "abc_inst list" and n :: "nat" where "x = (s, lm)" and "xa = p" and "xb = Suc n" and "y = abc_steps_l (abc_step_l (s, lm) (abc_fetch s p)) p n"
im_sheaf_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and S' :: "'e set"   and is_open' :: "'e set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e" shows "Comm_Ring.im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S' is_open' f \<equiv> sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<and> continuous_map S is_open S' is_open' f"
mod_partition_iT_Div_Int_one_segment: fixes k :: "nat"   and A :: "nat set"   and n :: "nat" assumes "0 < k" shows "A \<inter> [n * k\<dots>,k - Suc 0] \<oslash> k = (A \<oslash> k) \<inter> ([n * k\<dots>,k - Suc 0] \<oslash> k)"
reduce_argument_rel_def: shows "reduce_argument_rel TYPE('a) = UNIV"
ith_in_extension: fixes f :: "nat \<Rightarrow> 'a list"   and i :: "nat"   and n\<^sub>1 :: "nat"   and n\<^sub>2 :: "nat" assumes "wf_infpath f"   and "i < length (f n\<^sub>1)"   and "n\<^sub>1 \<le> n\<^sub>2" shows "f n\<^sub>1 ! i = f n\<^sub>2 ! i"
compose_Cons: fixes x :: "brick list"   and xs :: "wall"   and ys :: "wall" shows "(x * xs) \<circ> ys = x * xs \<circ> ys"
doesnt_have_mode\<^sub>C: fixes x :: "Eg1.var"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and m :: "Mode" assumes "x \<notin> mds\<^sub>A_of mds\<^sub>C m" shows "Eg2_var\<^sub>C_of_Eg1 x \<notin> mds\<^sub>C m"
rel_mono_strong: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R3 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "('d, 'f, 'h) generat"   and y :: "('e, 'g, 'i) generat"   and R1a :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2a :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R3a :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "rel_generat R1 R2 R3 x y"   and "\<And>z1 y1. \<lbrakk>z1 \<in> generat_pures x; y1 \<in> generat_pures y; R1 z1 y1\<rbrakk> \<Longrightarrow> R1a z1 y1"   and "\<And>z2 y2. \<lbrakk>z2 \<in> generat_outs x; y2 \<in> generat_outs y; R2 z2 y2\<rbrakk> \<Longrightarrow> R2a z2 y2"   and "\<And>z3 y3. \<lbrakk>z3 \<in> generat_conts x; y3 \<in> generat_conts y; R3 z3 y3\<rbrakk> \<Longrightarrow> R3a z3 y3" shows "rel_generat R1a R2a R3a x y"
shows_prec_list_def: fixes p :: "nat"   and xs :: "'a list" shows "shows_prec p xs = shows_list xs"
is_pascal_def: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points"   and E :: "Points"   and F :: "Points" shows "is_pascal A B C D E F \<equiv> distinct6 A B C D E F \<longrightarrow> line B C \<noteq> line E F \<longrightarrow> line C D \<noteq> line A F \<longrightarrow> line A B \<noteq> line D E \<longrightarrow> (let P = Pascal_Property.inter (line B C) (line E F); Q = Pascal_Property.inter (line C D) (line A F) in Let (Pascal_Property.inter (line A B) (line D E)) (col P Q))"
OAlist_tc_lex_ord_EqI: fixes xs :: "('b, 'c) oalist_tc"   and ys :: "('b, 'd) oalist_tc"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> order option" assumes "\<And>k. k \<in> fst ` set (list_of_oalist_tc xs) \<union> fst ` set (list_of_oalist_tc ys) \<Longrightarrow> f k (OAlist_tc_lookup xs k) (OAlist_tc_lookup ys k) = Some Eq" shows "OAlist_tc_lex_ord f xs ys = Some Eq"
times_nd_fun_def: shows "(\<cdot>) \<equiv> id (\<lambda>f g. (f\<^sub>\<bullet> \<circ>\<^sub>K g\<^sub>\<bullet>)\<^sup>\<bullet>)"
l: assumes "set (map snd txs) \<subseteq> var"   and "set (map fst txs) \<subseteq> trm"   and "set [\<phi>] \<subseteq> fmla" shows "length (getFrN (map snd txs) (map fst txs) [\<phi>] (length txs)) = length txs"
whisker_left: fixes f :: "'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "D.ide f"   and "D.seq \<nu> \<mu>" shows "f \<star>\<^sub>D \<nu> \<cdot>\<^sub>D \<mu> = (f \<star>\<^sub>D \<nu>) \<cdot>\<^sub>D (f \<star>\<^sub>D \<mu>)"
leg_simps_2: shows "Maps.dom SPN_fgh.\<mu>.leg0 = SPN_fgh.\<mu>.apex"
prems_2: shows "a \<in> U \<rightarrow> carrier K"
simps: fixes x :: "nat"   and n :: "nat" shows "token_run x n = run \<delta> q\<^sub>0 (suffix x w) (n - x)"
while_option_finite_increasing_Some: fixes f :: "'d \<Rightarrow> 'd"   and s :: "'d" assumes "mono f"   and "finite UNIV"   and "s \<le> f s" shows "\<exists>P. while_option (\<lambda>A. f A \<noteq> A) f s = Some P"
add_strict_increasing: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.ordered_cancel_comm_monoid_add plus minus zero less_eq less"   and "less zero a"   and "less_eq b c" shows "less b (plus a c)"
ih_3: shows "map (\<lambda>(s, l, t). l) xs' \<preceq> map (\<lambda>(s, l, t). l) xs"
DomainpE: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b" assumes "Domainp r a" obtains b :: "'c" where "r a b"
convol_def: shows "convol = (\<Lambda> f g x. (f\<cdot>x, g\<cdot>x))"
qfree_lift_dnf_qe: fixes qe :: "atom list \<Rightarrow> atom fm"   and \<phi> :: "atom fm" assumes "\<And>as. \<forall>a\<in>set as. hd_coeff a \<noteq> 0 \<Longrightarrow> qfree (qe as)" shows "qfree (Z.lift_dnf_qe qe \<phi>)"
Cong_cases: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Cong A B C D \<or> Cong A B D C \<or> Cong B A C D \<or> Cong B A D C \<or> Cong C D A B \<or> Cong C D B A \<or> Cong D C A B \<or> Cong D C B A" shows "Cong A B C D"
compet_rand_def: fixes A :: "('a list \<Rightarrow> 'b pmf) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> ((nat \<times> nat list) \<times> 'b) pmf)"   and c :: "real"   and S0 :: "'a list set" shows "Partial_Cost_Model.compet_rand A c S0 = (\<forall>s\<in>S0. \<exists>b\<ge>0. \<forall>rs. static s rs \<longrightarrow> T\<^sub>p_on_rand' A (fst A s \<bind> (\<lambda>is. return_pmf (s, is))) rs \<le> c * real (T\<^sub>p_opt s rs) + b)"
hom_one: shows "to_fract (1::'b) = 1"
reduction_step1': fixes c0 :: "bool"   and A1 :: "'v_OT121 \<Rightarrow> bool \<times> bool \<Rightarrow> bool spmf" shows "\<bar>spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). R1_OT12 (m0, m1) c0 \<bind> (\<lambda>view. A1 view (m0, m1)))) True - spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). S1_OT12 (m0, m1) () \<bind> (\<lambda>view. A1 view (m0, m1)))) True\<bar> \<le> adv_OT12"
lnull_llexord: fixes xs :: "'c llist"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ys :: "'c llist" assumes "lnull xs" shows "llexord r xs ys"
fst_diag_id: fixes z :: "'d" shows "(fst \<circ> (\<lambda>x. (x, x))) z = id z"
left_le_right2: fixes i :: "int"   and j :: "int" assumes "opt_bst_wpl2_dom (i, j)" shows "(i = j \<longrightarrow> root (fst (opt_bst_wpl2 i j)) = i) \<and> (i < j \<longrightarrow> root (fst (opt_bst_wpl2 i j)) \<in> {root (fst (opt_bst_wpl2 i (j - 1)))..root (fst (opt_bst_wpl2 (i + 1) j))})"
is_homography_def: fixes f :: "complex_homo \<Rightarrow> complex_homo" shows "is_homography f = is_moebius f"
rquot_def: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c" assumes "class.cancellative_partial_semigroup times D" shows "cancellative_partial_semigroup.rquot times D y x = (THE z. D x z \<and> times x z = y)"
Nmlize_Cod: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Cod t\<^bold>\<rfloor> = Cod \<^bold>\<lfloor>t\<^bold>\<rfloor>"
H2_def: fixes P :: "'b alpha_d_scheme \<times> 'b alpha_d_scheme \<Rightarrow> bool" shows "H2 P \<equiv> P ;; J"
args_Nil_iff_is_Hd: fixes s :: "('c, 'd) tm" shows "(args s = []) = is_Hd s"
Last_in_L: fixes u :: "'a"   and c :: "'b"   and out :: "'d \<Rightarrow> 'a \<Rightarrow> 'c"   and s :: "('d, 'a, 'b) state" shows "(Last u c \<in> GM_sec_model.L out s) = (\<exists>st. s = State st u c)"
f: fixes i :: "nat" shows "f i = lead_coeff (F i)"
pd_path: fixes y :: "'n"   and x :: "'n" assumes "y pd\<rightarrow> x" obtains \<pi> :: "nat \<Rightarrow> 'n" and n :: "nat" and k :: "nat" where "is_path \<pi>" and "\<pi> 0 = x" and "\<pi> n = local.return" and "\<pi> k = y" and "k \<le> n"
wf_typeE2: fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmark.type" assumes "\<Gamma> \<turnstile> t : T" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T"
HStartBallot_valueChosen: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and v :: "InputsOrNi" assumes "HStartBallot s s' q"   and "valueChosen s v"   and "v \<in> Inputs" shows "valueChosen s' v"
bisimCasePres: fixes CsP :: "('c \<times> ('a, 'b, 'c) psi) list"   and CsQ :: "('c \<times> ('a, 'b, 'c) psi) list"   and \<Psi> :: "'b" assumes "\<And>\<phi> P. (\<phi>, P) mem CsP \<Longrightarrow> \<exists>Q. (\<phi>, Q) mem CsQ \<and> guarded Q \<and> \<Psi> \<rhd> P \<sim> Q"   and "\<And>\<phi> Q. (\<phi>, Q) mem CsQ \<Longrightarrow> \<exists>P. (\<phi>, P) mem CsP \<and> guarded P \<and> \<Psi> \<rhd> P \<sim> Q" shows "\<Psi> \<rhd> Cases CsP \<sim> Cases CsQ"
intros_2: fixes i :: "nat"   and l :: "color list" assumes "1 \<le> i"   and "i + 1 < length l"   and "l ! (i - 1) = R"   and "l ! i = R"   and "l ! (i + 1) = R" shows "valid_point l i"
E_App1: fixes t :: "trm"   and t' :: "trm"   and u :: "trm" assumes "t \<longmapsto> t'" shows "t \<bullet> u \<longmapsto> t' \<bullet> u"
ipresWls_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and MOD :: "('c, 'd, 'e, 'sort, 'f, 'g, 'b, 'h, 'i) model_scheme" shows "ipresWls h MOD \<equiv> \<forall>s X. wls s X \<longrightarrow> igWls MOD s (h X)"
holomorphic_Dirichlet_L: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and A :: "complex set" assumes "dcharacter n \<chi>"   and "\<chi> \<noteq> principal_dchar n" shows "Dirichlet_L n \<chi> holomorphic_on A"
type_definition_zet: shows "type_definition Rep_zet Abs_zet zet"
leftderives_def: fixes u :: "symbol list"   and v :: "symbol list" shows "leftderives u v = ((u, v) \<in> leftderivations)"
matchCases: fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual"   and F :: "name \<Rightarrow> name \<Rightarrow> bool" assumes "[a\<frown>b]P \<longmapsto> Rs"   and "\<lbrakk>P \<longmapsto> Rs; a = b\<rbrakk> \<Longrightarrow> F a a" shows "F a b"
simplices_map: shows "order.greater_eq Y (f \<turnstile> X)"
bind_fv2_subst_bv1_eq_subst_term: fixes t :: "Core.term"   and n :: "nat"   and b :: "Core.term"   and idn :: "variable"   and T :: "typ" assumes "\<not> loose_bvar t n"   and "is_closed b" shows "subst_term [((idn, T), b)] t = subst_bv1 (bind_fv2 (idn, T) n t) n b"