hcju/isabelleps huggingface dataset:
defs_2: fixes args_ivl :: "\<I>"   and args_n :: "nat"   and args_L :: "nat set"   and args_R :: "nat set"   and args_pos :: "bool" shows "args.fields args_ivl args_n args_L args_R args_pos \<equiv> \<lparr>args_ivl = args_ivl, args_n = args_n, args_L = args_L, args_R = args_R, args_pos = args_pos\<rparr>"
vector_scaleR_component: fixes r :: "real"   and x :: "('c, 'd) vec"   and i :: "'d" shows "(r *\<^sub>R x) $ i = r *\<^sub>R x $ i"
map_sq_matrix_compose: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'e"   and M :: "('f, 'd) sq_matrix" shows "map_sq_matrix f (map_sq_matrix g M) = map_sq_matrix (\<lambda>x. f (g x)) M"
d_2: shows "\<forall>x\<in>dom (discovered s). \<delta> s x = \<delta> s' x"
P'_def: fixes k :: "V" shows "P' k = P (xs \<restriction>\<^sup>l\<^sub>\<circ> k)"
h3: shows "h2 \<turnstile> set_val new_character_data_ptr text \<rightarrow>\<^sub>h h3"
narrow_PrimT2_strong: fixes G :: "prog"   and S :: "ty"   and pt :: "prim_ty" assumes "G\<turnstile>S\<succ>PrimT pt" shows "S = PrimT pt"
assn_one_left: fixes P :: "assn" shows "emp * P = P"
f3: shows "prob1 3 (Matrix.mat 8 (Suc 0) (\<lambda>(i, j). [[0, 0, 0, 0, \<phi> $$ (0, 0) / complex_of_real (sqrt 2), - (\<phi> $$ (Suc 0, 0) / complex_of_real (sqrt 2)), - (\<phi> $$ (Suc 0, 0) / complex_of_real (sqrt 2)), \<phi> $$ (0, 0) / complex_of_real (sqrt 2)]] ! j ! i)) (Suc 0) = 1 / 2"
elims_3: fixes x :: "'a extB"   and xa :: "'a extB" assumes "\<not> x \<le> xa" obtains uv_ :: "'a" where "xa__ = extB.Val uv_" and "xa = extB.Bot"   | x :: "'a" and y :: "'a" where "xa__ = extB.Val x" and "xa = extB.Val y" and "\<not> x \<le> y"
hyp2_rep_abs_examples_2: shows "hyp2_rep (hyp2_abs (vector [1 / 2, 0])) = vector [1 / 2, 0]"
prime_factors_elem: fixes n :: "nat" assumes "1 < n" shows "\<exists>p. p \<in># prime_factorization n"
disj_four_cases: fixes A :: "bool"   and B :: "bool"   and C :: "bool"   and D :: "bool" assumes "A \<or> B \<or> C \<or> D" obtains "A"   | "B"   | "C"   | "D"
Diff_not_in: fixes a :: "'e"   and A :: "'e set" shows "a \<notin> A - {a}"
induct: fixes n :: "nat"   and P :: "nat \<Rightarrow> nat \<Rightarrow> 'b mat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "'b mat" assumes "\<And>i j A. \<lbrakk>\<lbrakk>\<not> n \<le> j; i = 0\<rbrakk> \<Longrightarrow> P (j + 1) (j + 1) A; \<And>x xa xb xc xd. \<lbrakk>\<not> n \<le> j; i \<noteq> 0; x = i - 1; xa = A $$ (x, x); xb = A $$ (j, j); xc = A $$ (x, j); xd = (if xa \<noteq> xb \<and> xc \<noteq> (0::'b) then add_col_sub_row (xc / (xb - xa)) x j A else A)\<rbrakk> \<Longrightarrow> P x j xd\<rbrakk> \<Longrightarrow> P i j A" shows "P a0 a1 a2"
compl_sup: 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 "uminus (sup x y) = inf (uminus x) (uminus y)"
simps_4: fixes x41 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and x42 :: "'d"   and x43 :: "('a \<Rightarrow> 'b option) \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option"   and y41 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and y42 :: "'d"   and y43 :: "('a \<Rightarrow> 'b option) \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option" shows "(x41\<hookleftarrow>\<^bsub>x42\<^esub>x43 = y41\<hookleftarrow>\<^bsub>y42\<^esub>y43) = (x41 = y41 \<and> x42 = y42 \<and> x43 = y43)"
nonzero_mult_div_cancel_right: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and b :: "'d"   and a :: "'d" assumes "class.semidom_divide divide plus minus zero times one"   and "b \<noteq> zero" shows "divide (times a b) b = a"
S\<^sub>1_in_hom: shows "Maps.in_hom S\<^sub>1 (Maps.MkIde (src (tab\<^sub>0 s))) (Maps.MkIde (trg s))"
signs_smash_property_set: fixes signs1 :: "'a list list"   and signs2 :: "'a list list" shows "\<forall>elem. elem \<in> set (signs_smash signs1 signs2) \<longrightarrow> (\<exists>elem1 elem2. elem1 \<in> set signs1 \<and> elem2 \<in> set signs2 \<and> elem = elem1 @ elem2)"
charts_submanifold_def: fixes charts :: "('c, 'd) chart set"   and S :: "'c set" shows "manifold.charts_submanifold charts S = restrict_chart S ` charts"
Nmlize_Inv: fixes t :: "'a local.term" assumes "Can t" shows "\<^bold>\<lfloor>Inv t\<^bold>\<rfloor> = Inv \<^bold>\<lfloor>t\<^bold>\<rfloor>"
lunit_char_2: fixes a :: "'c" assumes "C.ide a" shows "\<I>\<^sub>C \<otimes>\<^sub>C \<l>\<^sub>C[a] = (\<iota>\<^sub>C \<otimes>\<^sub>C a) \<cdot>\<^sub>C C.assoc' \<I>\<^sub>C \<I>\<^sub>C a"
Bounded_vec_vec_set: fixes v :: "'b vec"   and b :: "'b" shows "(v \<in> Bounded_vec b) = (\<forall>x\<in>set\<^sub>v v. \<bar>x\<bar> \<le> b)"
ID_ccompare: fixes c :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "ID ccompare = Some c" shows "class.linorder (le_of_comp c) (lt_of_comp c)"
hyps: fixes x :: "real" assumes "akra_bazzi_measure x < akra_bazzi_measure xa__"   and "x\<^sub>0 \<le> x" shows "P x"
Units_def: shows "Sym = {u \<in> S \<rightarrow>\<^sub>E S. invertible u}"
Max_torder_in_set: fixes B :: "'a set" assumes "finite B"   and "B \<noteq> {}"   and "B \<subseteq> A" shows "Max_torder r B \<in> B"
wfp_on_induct: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b"   and Q :: "'b \<Rightarrow> bool" assumes "wfp_on P A"   and "x \<in> A"   and "\<And>y. \<lbrakk>y \<in> A; \<And>x. \<lbrakk>x \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x\<rbrakk> \<Longrightarrow> Q y" shows "Q x"
content_real: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "content {a..b} = b - a"
Subst_def: fixes rhs :: "'b trm set"   and X :: "'b list set"   and xrhs :: "'b trm set" shows "Subst rhs X xrhs \<equiv> rhs - {Trn X r |r. Trn X r \<in> rhs} \<union> Append_rexp_rhs xrhs (\<Uplus>{r. Trn X r \<in> rhs})"
Fpow_def: fixes A :: "'c set" shows "Fpow A \<equiv> {X. X \<subseteq> A \<and> finite X}"
preserves_hom: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and f :: "'a"   and a :: "'a"   and b :: "'a" assumes "functor A B F"   and "partial_magma.in_hom A f a b" shows "partial_magma.in_hom B (F f) (F a) (F b)"
sq_norm_pos: fixes j :: "nat" assumes "j < length (map of_int_hom.vec_hom fs')" shows "0 < \<parallel>gs2.gso j\<parallel>\<^sup>2"
remdups_remdups: fixes xs :: "'b list" shows "remdups (remdups xs) = remdups xs"
rel_advantage_def: fixes \<A> :: "'a \<times> 'a \<Rightarrow> nat spmf" shows "chaum_ped_\<Sigma>_commit.rel_advantage \<A> \<equiv> spmf (chaum_ped_\<Sigma>_commit.rel_game \<A>) True"
omega_super_id1: fixes x :: "'a" assumes "(0::'a) \<noteq> (1::'a)"   and "(1::'a) \<le> x" shows "x\<^sup>\<omega> \<noteq> (0::'a)"
add_strict_increasing: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" 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)"
is_subprob_densityD_3: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "is_subprob_density M f" shows "space M \<noteq> {}"
size_mset_mono: fixes A :: "'a multiset"   and B :: "'a multiset" assumes "A \<subseteq># B" shows "size A \<le> size B"
wf_cdecl_def: fixes G :: "prog" shows "wf_cdecl G = (\<lambda>(C, c). \<not> is_iface G C \<and> (\<forall>I\<in>set (superIfs c). is_acc_iface G (pid C) I \<and> (\<forall>s. \<forall>im\<in>DeclConcepts.imethds G I s. ? cm:methd G C s: G\<turnstile>resTy cm\<preceq>resTy im \<and> \<not> is_static cm \<and> accmodi im \<le> accmodi cm)) \<and> (\<forall>f\<in>set (cfields c). wf_fdecl G (pid C) f) \<and> unique (cfields c) \<and> (\<forall>m\<in>set (methods c). wf_mdecl G C m) \<and> unique (methods c) \<and> jumpNestingOkS {} (init c) \<and> (\<exists>A. \<lparr>prg = G, cls = C, lcl = Map.empty\<rparr>\<turnstile> {} \<guillemotright>\<langle>init c\<rangle>\<guillemotright> A) \<and> \<lparr>prg = G, cls = C, lcl = Map.empty\<rparr>\<turnstile>init c\<Colon>\<surd> \<and> ws_cdecl G C (super c) \<and> (C \<noteq> Object \<longrightarrow> is_acc_class G (pid C) (super c) \<and> (table_of (map (\<lambda>(s, m). (s, C, m)) (methods c)) entails \<lambda>new. \<forall>old sig. (G \<turnstile> qmdecl sig new overrides\<^sub>S qmdecl sig old \<longrightarrow> G\<turnstile>resTy new\<preceq>resTy old \<and> accmodi old \<le> accmodi new \<and> \<not> is_static old) \<and> (G,sig\<turnstile> new hides old \<longrightarrow> accmodi old \<le> accmodi new \<and> is_static old))))"
distinct_5: fixes x1 :: "'b" shows "left x1 \<noteq> ack"
less_eq_basic_type_def: shows "(\<le>) \<equiv> (\<sqsubset>\<^sub>B)\<^sup>*\<^sup>*"
v: shows "f `` V \<subseteq> vertices (S (max i j))"
D_eq: shows "{l. length l = n \<and> (\<exists>i<n. 3 \<le> i \<and> (\<forall>k<i. l ! k = R) \<and> l ! i = B \<and> valid (drop (i + 1) l))} = (\<Union>i\<in>{3..<n}. (\<lambda>l. replicate i R @ B # l) ` {l. length l = n - i - 1 \<and> valid l})"
rbt_delete_is_rbt: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "('a, 'b) RBT_Impl.rbt"   and k :: "'a" assumes "class.linorder less_eq less"   and "ord.is_rbt less t" shows "ord.is_rbt less (ord.rbt_delete less k t)"
le_fin_not_member': fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and u :: "'a list"   and v :: "'a list" assumes "traces ind"   and "traces.le_fin ind [a] (u @ v)"   and "a \<notin> set u" shows "traces.le_fin ind [a] v"
discr_ZObis: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" assumes "proper c"   and "proper d"   and "discr c"   and "discr d" shows "c \<approx>01 d"
prems_1: shows "\<Delta> \<in> set (\<Delta>3 # \<Delta>s')"
is_group_action: shows "group_action G (conjugation_action (p ^ a)) (subgroups_of_size (p ^ a))"
path_inf_priorities_LCons: fixes P :: "'a llist"   and v :: "'a" shows "path_inf_priorities P = path_inf_priorities (LCons v P)"
mono_gamma: fixes a :: "'av"   and b :: "'av" assumes "a \<le> b" shows "\<gamma> a \<subseteq> \<gamma> b"
img_is_subgroup: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h" shows "subgroup (h ` carrier G) H"
sig_irredE_dgrad_max_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading d"   and "F \<subseteq> dgrad_max_set d" obtains q :: "'t \<Rightarrow>\<^sub>0 'b" where "(sig_red sing_reg top_tail F)\<^sup>*\<^sup>* p q" and "\<not> is_sig_red sing_reg top_tail F q"
coeff_mpoly_of_poly1: fixes i :: "nat"   and p :: "'b poly"   and n :: "nat" shows "MPoly_Type.coeff (mpoly_of_poly i p) (Poly_Mapping.single i n) = poly.coeff p n"
list_updates_nth_less: fixes xs :: "nat list"   and ys :: "'a list"   and i :: "nat"   and vs :: "'a list" assumes "length xs = length ys"   and "distinct xs"   and "i < length vs" shows "list_updates xs ys vs ! i = (if i \<in> set xs then ys ! index xs i else vs ! i)"
inject_3: fixes x31 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and x32 :: "'c"   and x33 :: "'d"   and x34 :: "(('a \<Rightarrow> 'b option) \<Rightarrow> 'b option) list"   and y31 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and y32 :: "'c"   and y33 :: "'d"   and y34 :: "(('a \<Rightarrow> 'b option) \<Rightarrow> 'b option) list" shows "(x31:x32\<hookrightarrow>\<^bsub>x33\<^esub>x34 = y31:y32\<hookrightarrow>\<^bsub>y33\<^esub>y34) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33 \<and> x34 = y34)"
simps_8: fixes I :: "interp"   and x :: "char"   and \<theta> :: "trm" shows "game_sem I (ODE x \<theta>) = (\<lambda>X. {\<omega>. \<exists>F T. Vagree \<omega> (F 0) (- {DVar x}) \<and> F T \<in> X \<and> solves_ODE I F x \<theta>})"
blocking_on_allocation: fixes ds :: "'b set"   and X :: "'a set"   and h :: "'c"   and X'' :: "'a set" assumes "Singleton_for_d.blocking_on ds X h X''" shows "inj_on Xd X''"
silent_move_intra: fixes f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and a :: "'edge"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and ms :: "'node list"   and S :: "'node SDG_node set"   and ms' :: "'node list" assumes "pred (f a) s"   and "transfer (f a) s = s'"   and "valid_edge a"   and "intra_kind (kind a)"   and "(\<exists>m\<in>set (tl ms). \<exists>m'. call_of_return_node m m' \<and> m' \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>) \<or> hd ms \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "Ball (set (tl ms)) return_node"   and "length s' = length s"   and "length ms = length s"   and "hd ms = sourcenode a"   and "ms' = targetnode a # tl ms" shows "S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms',s')"
map_iterator_genord_rule_P: assumes "map_iterator_genord it m (\<lambda>_ _. True)"   and "I (dom m) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; m k = Some v; it \<subseteq> dom m; I it \<sigma>; \<forall>k' v'. k' \<in> it - {k} \<and> m k' = Some v' \<longrightarrow> True; \<forall>k' v'. k' \<notin> it \<and> m k' = Some v' \<longrightarrow> True\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>; \<forall>k v k' v'. k \<notin> it \<and> m k = Some v \<and> k' \<in> it \<and> m k' = Some v' \<longrightarrow> True\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
prems_2: shows "(\<beta> @ [c] @ \<alpha>) \<upharpoonleft> E\<^bsub>ESi\<^esub> \<in> Tr\<^bsub>ESi\<^esub>"
inverse_powr: fixes x :: "real"   and y :: "real" assumes "0 \<le> x" shows "inverse x powr y = inverse (x powr y)"
rep_eq: shows "Rep_float 1 = (0, 2 ^ (LENGTH('a) - 1) - 1, 0)"
WTrt_elim_cases_8: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and U :: "ty"   and T :: "ty" assumes "P,E,h \<turnstile> e instanceof U : T" obtains T :: "ty" where "Ta__ = Boolean" and "P,E,h \<turnstile> e : T" and "is_type P U"
minf_neq_inf: shows "- \<infinity> \<noteq> \<infinity>"
order_refl: fixes x :: "'t" shows "x \<preceq>\<^sub>t x"
assms: fixes x :: "'a" assumes "x \<in> vertices G" shows "f x = g x"
GoodImpl: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> IPImp p \<and> HistP rmhist p \<longrightarrow> Init (ImpInit p \<and> HInit rmhist p) \<and> \<box>(ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p)) \<and> \<box>(\<forall>l. $MemInv mm l) \<and> \<box>$ImpInv rmhist p \<and> ImpLive p"
map_pmf_const: fixes c :: "'a"   and M :: "'b pmf" shows "map_pmf (\<lambda>_. c) M = return_pmf c"
eval_ring_hom: fixes s :: "'c" assumes "s \<in> carrier S" shows "eval R S h s \<in> ring_hom P S"
B: shows "(Owner m, Crypt (SesK SK) (Pwd m)) \<in> s"
sqr_ci: fixes ls :: "'a list"   and P :: "pol" assumes "in_carrier ls" shows "Ipol ls (sqr P) = Ipol ls P \<otimes> Ipol ls P"
entries_rbtreeify_f_append: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" assumes "n \<le> length kvs" shows "entries (fst (rbtreeify_f n kvs)) @ snd (rbtreeify_f n kvs) = kvs"
pcompose_add: fixes p :: "'a poly"   and q :: "'a poly"   and r :: "'a poly" shows "(p + q) \<circ>\<^sub>p r = p \<circ>\<^sub>p r + q \<circ>\<^sub>p r"
wf_SNa_restricted: shows "wf (A_Redu_set \<inter> UNIV \<times> SNa_set)"
simps_17: fixes \<C> :: "t_context"   and tn :: "t list"   and tm :: "t list"   and es1 :: "b_e list"   and es2 :: "b_e list"   and ts :: "checker_type" shows "check_single \<C> (b_e.If (tn _> tm) es1 es2) ts = (if b_e_type_checker (\<C>\<lparr>label := [tm] @ label \<C>\<rparr>) es1 (tn _> tm) \<and> b_e_type_checker (\<C>\<lparr>label := [tm] @ label \<C>\<rparr>) es2 (tn _> tm) then type_update ts (to_ct_list (tn @ [T_i32])) (Type tm) else Bot)"
distrib_right: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "(a \<squnion> b) * c = a * c \<squnion> b * c"
pth_c_2: fixes c :: "real"   and x :: "'a"   and z :: "'a"   and d :: "real"   and y :: "'a" shows "c *\<^sub>R x + z + d *\<^sub>R y \<equiv> d *\<^sub>R y + (c *\<^sub>R x + z)"
tpD_eq: shows "tp D = \<omega>_sum \<alpha>s"
assms: fixes x :: "real" assumes "x \<in> einterval (min a b) (max a b)" shows "f x = g x"
isnpolyh_polybound0: fixes p :: "'b poly"   and n :: "nat" assumes "isnpolyh p (Suc n)" shows "polybound0 p"
disjI1: fixes \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and \<gamma>' :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<gamma> t n" shows "(\<gamma> \<or>\<^sup>c \<gamma>') t n"
prj_chine_assoc'_1: shows "f_src_src.Prj\<^sub>1\<^sub>1 \<cdot> f_src_src.chine_assoc' = f_src_src.Prj\<^sub>1"
fun_eq_on_cong: fixes f :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "fun_eq_on f h A"   and "fun_eq_on g h A" shows "fun_eq_on f g A"
subsetI: fixes A :: "'c set"   and B :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B" shows "A \<subseteq> B"
perm_pi_simp_1: fixes pi :: "(name \<times> name) list"   and x :: "'a" shows "pi \<bullet> rev pi \<bullet> x = x"
length_Cons: fixes x :: "'a"   and xs :: "'a list" shows "length (x # xs) = Suc (length xs)"
mult_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq c d"   and "less_eq zero b"   and "less_eq zero c" shows "less_eq (times a c) (times b d)"
FAccThrow: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D},s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
rtrancl_path_last: fixes R :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and xs :: "'k list"   and y :: "'k" assumes "rtrancl_path R x xs y"   and "xs \<noteq> []" shows "last xs = y"
ltl_prop_unfold_equiv_def: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<phi> \<sim>\<^sub>Q \<psi> \<equiv> Unf \<phi> \<sim>\<^sub>P Unf \<psi>"
zero_assoc3: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> y \<cdot> (0::'a) = x \<cdot> (y \<cdot> (0::'a))"
listset_mono: fixes xs :: "'b set list"   and ys :: "'b set list" assumes "length xs = length ys"   and "\<And>i. i < length ys \<Longrightarrow> xs ! i \<subseteq> ys ! i" shows "listset xs \<subseteq> listset ys"
variants_imp_exists_substitution: fixes D :: "'a literal multiset"   and D' :: "'a literal multiset" assumes "variants D D'" shows "\<exists>\<sigma>. D \<cdot> \<sigma> = D'"
Conseq: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and F :: "'f set"   and P :: "('d, 'e, 'f) ann"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set" assumes "\<exists>P' Q' A'. \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (weaken_pre P P') c Q', A' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P c Q, A"
\<Phi>_def: shows "\<Phi> \<equiv> \<lambda>NL N. N \<subseteq> hd NL \<and> Inf (hd NL) < Inf N \<and> infinite N \<and> insert_closed NL N"
subset_subseqs: fixes X :: "'a set"   and xs :: "'a list" assumes "X \<subseteq> set xs" shows "X \<in> set ` set (subseqs xs)"
DirProds_is_finite_group: fixes I :: "'d set"   and G :: "'d \<Rightarrow> ('e, 'f) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> finite_group (G i)"   and "finite I" shows "finite_group (DirProds G I)"
cmp_interpretation: shows "identity_arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg ((f \<star> g) \<star> h) THfgh.\<rho>\<sigma>.tab (tab\<^sub>0 h \<star> THfgh.p\<^sub>0) (tab\<^sub>1 (f \<star> g) \<star> THfgh.p\<^sub>1) ((f \<star> g) \<star> h) HHfgh.tab (tab\<^sub>0 ((f \<star> g) \<star> h)) (tab\<^sub>1 ((f \<star> g) \<star> h)) ((f \<star> g) \<star> h)"
gmd_correct: fixes d :: "('c, 'd) Digit" assumes "is_measured_digit d" shows "gmd d = sum_list (map snd (digitToList d))"
T_def: shows "T \<equiv> \<lambda>f. f \<circ> Suc"
perm_nat_def: fixes pi :: "('a \<times> 'a) list"   and i :: "nat" shows "pi \<bullet> i = i"
cop_per2__col: fixes A :: "'p"   and X :: "'p"   and Y :: "'p"   and Z :: "'p" assumes "Coplanar A X Y Z"   and "A \<noteq> Z"   and "Per X Z A"   and "Per Y Z A" shows "Col X Y Z"
complement_sum_in_product_complement: fixes x :: "i"   and y :: "i" assumes "O x y"   and "x \<noteq> u"   and "y \<noteq> u" shows "P (\<midarrow> x \<oplus> \<midarrow> y) (\<midarrow> (x \<otimes> y))"
index_zero_vec_2: fixes n :: "nat" shows "dim_vec (0\<^sub>v n) = n"
reads_restrict: fixes \<sigma> :: "'var \<Rightarrow> 'a"   and n :: "'n"   and \<sigma>' :: "'var \<Rightarrow> 'a" shows "(\<sigma> \<restriction> local.reads n = \<sigma>' \<restriction> local.reads n) = (\<forall>x\<in>local.reads n. \<sigma> x = \<sigma>' x)"
lemma_2_8_i: fixes a :: "'a"   and A :: "'a set"   and b :: "'a" assumes "a \<in> supremum A" shows "a r\<rightarrow> b \<in> infimum ((\<lambda>x. x r\<rightarrow> b) ` A)"
splus_sminus: fixes t :: "'a"   and v :: "'t" shows "t \<oplus> v \<ominus> t = v"
pos_in_model: fixes M :: "nat \<Rightarrow> bool"   and n :: "nat"   and n_vars :: "nat" assumes "M n"   and "0 < n"   and "n < n_vars" shows "int n \<in> set (model_to_dimacs_model M [1..<n_vars])"
lex_not_sym: fixes xs :: "nat list"   and ys :: "nat list" assumes "xs <\<^sub>l\<^sub>e\<^sub>x ys" shows "(ys, xs) \<notin> lex less_than"
Meet_insert: fixes S :: "'a set"   and l :: "'a"   and x :: "'a"   and l2 :: "'a" assumes "S >>| l"   and "{x, l} >>| l2" shows "insert x S >>| l2"
intro: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "\<And>x y. invert_order (comp x y) = comp y x"   and "\<And>x y. comp x y = Eq \<Longrightarrow> x = y"   and "\<And>x y z. \<lbrakk>comp x y = Lt; comp y z = Lt\<rbrakk> \<Longrightarrow> comp x z = Lt" shows "comparator comp"
ProblemIkTpart_TEModel_axioms: shows "ProblemIkTpart_TEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw eintT eintF eintP"
finite_params: fixes p :: "('a, 'b) form" shows "finite (params p)"