hcju/isabelleps huggingface dataset:
upd_list_snoc: fixes xs :: "('c \<times> 'd) list"   and x :: "'c"   and y :: "'d" assumes "sorted1 (xs @ [(x, y)])" shows "upd_list x y xs = xs @ [(x, y)]"
targets_some_trg: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "targets (some_trg \<mu>) = targets \<mu>"
inv_end_steps: fixes x :: "nat"   and cf :: "nat \<times> cell list \<times> cell list"   and stp :: "nat" assumes "0 < x"   and "inv_end x cf" shows "inv_end x (steps0 cf tcopy_end stp)"
nz: shows "\<forall>n\<ge>i. X n \<noteq> 0"
prj1_in_hom: fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and b :: "'a" assumes "C.cospan f g"   and "a = f \<down>\<down> g"   and "b = C.dom f" shows "\<guillemotleft>\<p>\<^sub>1[f, g] : a \<rightarrow>\<^sub>C b\<guillemotright>"
elims_3: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a list" assumes "\<not> sorted_by x xa" obtains cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and x1 :: "'a" and x2 :: "'a" and xs :: "'a list" where "x = cmp" and "xa = x1 # x2 # xs" and "\<not> (cmp x1 x2 \<and> sorted_by cmp (x2 # xs))"
image_hd_lclss_of_state: fixes St :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set" shows "fst ` lclss_of_state St = N_of_state St \<union> P_of_state St \<union> Q_of_state St"
distinct_fst_Cons: fixes k :: "'a"   and x :: "'b"   and kxs :: "('a \<times> 'b) list" shows "distinct_fst ((k, x) # kxs) = (distinct_fst kxs \<and> (\<forall>y. (k, y) \<notin> set kxs))"
exu_\<sigma>: shows "prv (exu out (subst \<sigma> (Var zz) inp))"
infinite: shows "\<not> (finite UNIV \<and> finite UNIV)"
deg'_pp: fixes t :: "('a, 'b) pp" shows "deg' t = deg_pp (rep_nat_pp t)"
4: fixes a :: "alt"   and b :: "alt"   and c :: "alt"   and P :: "indi \<Rightarrow> (alt \<times> alt) set"   and P' :: "indi \<Rightarrow> (alt \<times> alt) set" assumes "a \<noteq> b"   and "b \<noteq> c"   and "a \<noteq> c"   and "P \<in> Prof"   and "P' \<in> Prof"   and "\<forall>i. (a <\<^bsub>P i\<^esub> b) = (b <\<^bsub>P' i\<^esub> c)" shows "(a <\<^bsub>F P\<^esub> b) = (b <\<^bsub>F P'\<^esub> c)"
notin_nodes0: fixes k :: "int"   and t :: "int tree" shows "(k, 0) \<notin> nodes t"
i: fixes x :: "nat \<times> int \<times> nat" assumes "x \<in> E - E'" shows "subgraph \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = insert x E'\<rparr> \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>"
NSBseq_def: fixes X :: "nat \<Rightarrow> 'b" shows "NSBseq X = (\<forall>N\<in>HNatInfinite. (*f* X) N \<in> HFinite)"
test_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "test y" shows "test (x \<sqinter> y)"
span_insertD: fixes p :: "'b"   and r :: "'b"   and B :: "'b set" assumes "p \<in> span (insert r B)"   and "r \<in> span B" shows "p \<in> span B"
perm_still_cont: fixes \<pi> :: "perm"   and f :: "'c \<Rightarrow> 'd" shows "cont (\<pi> \<bullet> f) = cont f"
TupleR: fixes u :: "trm"   and u' :: "trm"   and t :: "trm" assumes "u \<longmapsto> u'" shows "\<langle>t, u\<rangle> \<longmapsto> \<langle>t, u'\<rangle>"
True: shows "emeasure lebesgue B = 2 ^ CARD('n)"
simps_9: fixes i :: "int"   and c :: "int"   and e :: "Cooper.num" shows "mirror (Dvd i (CN 0 c e)) = Dvd i (CN 0 c (Neg e))"
wlp_mono_and_close: fixes dims :: "nat list"   and S :: "com"   and P :: "complex mat"   and Q :: "complex mat" assumes "state_sig.well_com dims S"   and "state_sig.is_quantum_predicate dims P"   and "state_sig.is_quantum_predicate dims Q"   and "P \<le>\<^sub>L Q" shows "state_sig.is_quantum_predicate dims (state_sig.wlp dims S P) \<and> state_sig.wlp dims S P \<le>\<^sub>L state_sig.wlp dims S Q"
assms_2: fixes j :: "nat" assumes "d < j" shows "pdevs_apply x j = (0::'a)"
monomials_and_monoms: fixes k :: "nat \<Rightarrow>\<^sub>0 nat"   and m :: "'b mpoly" shows "(k \<in> monomials m) = (\<exists>a. a \<noteq> 0 \<and> monomials (MPoly_Type.monom k a) \<subseteq> monomials m)"
that: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> X"   and "y \<in> X"   and "z \<in> X"   and "x \<noteq> y"   and "x \<noteq> z"   and "y \<noteq> z" shows "thesis"
lm066: fixes X :: "('c \<times> 'd) set set" shows "converse ` converse ` X = X"
forall_\<Pi>\<^sub>2_def: shows "forall \<equiv> forall\<^sub>2"
fg: fixes x :: "'a" assumes "x \<in> S" shows "f x = exp (g x)"
nth'_def: fixes xs :: "'a list"   and i :: "nat" shows "nth' xs i = (if i < length xs then xs ! i else undefined)"
aless_neq: fixes x :: "ant"   and y :: "ant" assumes "x < y" shows "x \<noteq> y"
elementary_category_with_terminal_object_axioms: shows "elementary_category_with_terminal_object (\<cdot>) \<one> trm"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "hom.arr f"   and "hom.cod f = b" shows "hom.comp b f = f"
fmrel_fmdom_eq: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'g \<rightharpoonup>\<^sub>f 'e"   and y :: "'g \<rightharpoonup>\<^sub>f 'f" assumes "fmrel P x y" shows "fmdom x = fmdom y"
assms_2: shows "Marker \<in> set (msgs (S t i) cid)"
PData_select: fixes D :: "'b pdata" shows "PData (PDataValue D) (PDataSpace D)"
induct: fixes x1 :: "any_unop"   and x2 :: "'a OCL_Types.type"   and x3 :: "'a OCL_Types.type"   and P :: "any_unop \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> bool" assumes "any_unop_type x1 x2 x3"   and "\<And>\<tau>. \<tau> \<le> OclAny[?] \<Longrightarrow> P OclAsSetOp \<tau> (Set (to_required_type \<tau>))"   and "\<And>\<tau>. \<tau> \<le> OclAny[?] \<Longrightarrow> P OclIsNewOp \<tau> Boolean[1]"   and "\<And>\<tau>. \<tau> \<le> OclAny[?] \<Longrightarrow> P OclIsUndefinedOp \<tau> Boolean[1]"   and "\<And>\<tau>. \<tau> \<le> OclAny[?] \<Longrightarrow> P OclIsInvalidOp \<tau> Boolean[1]"   and "\<And>\<tau>. \<tau> \<le> OclAny[?] \<Longrightarrow> P OclLocaleOp \<tau> String[1]"   and "\<And>\<tau>. \<tau> \<le> OclAny[?] \<Longrightarrow> P ToStringOp \<tau> String[1]" shows "P x1 x2 x3"
simps_1: fixes x1 :: "'f"   and y1 :: "'f" shows "(Node x1 = Node y1) = (x1 = y1)"
heap_sort_fst_part_bigger_snd_part_step: fixes l\<^sub>0 :: "int"   and m :: "int"   and r\<^sub>0 :: "int"   and arr' :: "int \<Rightarrow> int"   and arr :: "int \<Rightarrow> int"   and mod_arr :: "int \<Rightarrow> int"   and x :: "int"   and y :: "int" assumes "l\<^sub>0 < m"   and "m \<le> r\<^sub>0"   and "arr_mset arr' l\<^sub>0 (m - 1) = arr_mset arr l\<^sub>0 m - {#Min_mset (arr_mset arr l\<^sub>0 m)#}"   and "arr = arr' on - {l\<^sub>0..<m}"   and "\<forall>x\<in>#arr_mset arr l\<^sub>0 m. \<forall>y\<in>#arr_mset arr m r\<^sub>0. \<not> x < y"   and "mod_arr = arr'(m - 1 := Min_mset (arr_mset arr l\<^sub>0 m))"   and "x \<in># arr_mset mod_arr l\<^sub>0 (m - 1)"   and "y \<in># arr_mset mod_arr (m - 1) r\<^sub>0" shows "\<not> x < y"
ordinal_times_left_cancel_less: fixes w :: "ordinal"   and x :: "ordinal"   and y :: "ordinal" assumes "0 < w" shows "(w * x < w * y) = (x < y)"
image_mset_is_empty_iff: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b multiset" shows "(image_mset f M = {#}) = (M = {#})"
Derivs_pderivs: fixes s :: "'a list"   and r :: "'a rexp" shows "Derivs s (lang r) = \<Union> (lang ` pderivs s r)"
mkM_def: fixes T :: "'s Trace set" shows "SPR.mkM T \<equiv> \<lparr>worlds = T, relations = \<lambda>a. rel_ext (\<lambda>(t, t'). {t, t'} \<subseteq> T \<and> spr_jview a t = spr_jview a t'), valuation = envVal \<circ> tLast\<rparr>"
degrees_append_leD1: fixes xs :: "'a pdevs list"   and ys :: "'a pdevs list" shows "degrees xs \<le> degrees (xs @ ys)"
cons_lim_def: fixes s :: "nat \<Rightarrow> nat option" shows "cons_lim s \<equiv> s \<in> \<R>"
SeqThrow: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>0 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and e :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and e\<^sub>1 :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>0;; e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>"
assms: shows "FL_Formula.hereditarily_fs t\<^sub>\<alpha>"
B: fixes x :: "'a"   and y :: "'b" shows "norm (h x y) \<le> B * norm x * norm y"
dom_insert: shows "subst_domain (\<theta>(w := t)) \<subseteq> insert w (subst_domain \<theta>)"
lim_any: fixes Xa :: "'a set"   and Ya :: "'a set"   and xa :: "'a"   and ya :: "'a"   and Z :: "'a set" assumes "derivation Xa"   and "derivation Ya"   and "xa \<in> Xa"   and "ya \<in> Ya"   and "Z = {z \<in> Xa. z \<sqsubset> xa}"   and "f ` Z \<subseteq> Z"   and "extreme_bound A (\<sqsubseteq>) Z xa"   and "\<forall>z\<in>Xa. z \<sqsubset> xa \<longrightarrow> z \<sqsubset> ya \<and> z \<in> Ya \<or> z = ya \<or> ya \<sqsubset> z \<and> ya \<in> Xa" shows "ya \<sqsubset> xa \<and> ya \<in> Xa \<or> xa \<sqsubseteq> ya"
card_UNIV_option: shows "CARD('a option) = (if CARD('a) = 0 then 0 else CARD('a) + 1)"
transitive_closure_trans_3: fixes x :: "'c"   and y :: "'c"   and r :: "('c \<times> 'c) set"   and z :: "'c" assumes "(x, y) \<in> r\<^sup>*"   and "(y, z) \<in> r\<^sup>*" shows "(x, z) \<in> r\<^sup>*"
qfree_list_conj: fixes fs :: "'b fm list" assumes "\<forall>\<phi>\<in>set fs. qfree \<phi>" shows "qfree (list_conj fs)"
invertible_left_inverse2: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c"   and v :: "'c" assumes "monoid M composition unit"   and "monoid.invertible M composition unit u"   and "u \<in> M"   and "v \<in> M" shows "composition (monoid.inverse M composition unit u) (composition u v) = v"
quasi_ordered_set_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "quasi_ordered_set A less_eq \<equiv> reflexive A less_eq \<and> transitive A less_eq"
eq_o_iE: fixes xa :: "'a"   and x :: "'a" assumes "pred.eval (eq_o_i xa) x" obtains "Predicate.eq x xa"
safe_free_flowing_implies_safe_delayed: fixes c\<^sub>0 :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and n :: "nat" assumes "initial c\<^sub>0"   and "simple_ownership_distinct (fst c\<^sub>0)"   and "read_only_unowned (snd (snd c\<^sub>0)) (fst c\<^sub>0)"   and "unowned_shared (snd (snd c\<^sub>0)) (fst c\<^sub>0)"   and "safe_reach_upto n safe_free_flowing c\<^sub>0" shows "safe_reach_upto n safe_delayed c\<^sub>0"
mmulti_join'_opt_False: fixes n :: "nat"   and A_zs :: "nat set list"   and A_x :: "nat set"   and A_xs :: "nat set list"   and A_ws :: "nat set list"   and A_y :: "nat set"   and A_ys :: "nat set list"   and zs :: "'b option list set list"   and x :: "'b option list set"   and xs :: "'b option list set list"   and ws :: "'b option list set list"   and y :: "'b option list set"   and ys :: "'b option list set list" assumes "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) ((A_zs @ A_x # A_xs) @ A_ws @ A_y # A_ys) ((zs @ x # xs) @ ws @ y # ys)"   and "length A_ws = length ws"   and "length A_xs = length xs"   and "length A_ys = length ys"   and "length A_zs = length zs"   and "A_y \<subseteq> A_x" shows "mmulti_join' (A_zs @ A_x # A_xs) (A_ws @ A_y # A_ys) ((zs @ x # xs) @ ws @ y # ys) = mmulti_join' (A_zs @ A_x # A_xs) (A_ws @ A_ys) ((zs @ join x False y # xs) @ ws @ ys)"
prefix_match_semantics_def: fixes m :: "'b prefix_match"   and a :: "'b word" shows "prefix_match_semantics m a \<equiv> pfxm_prefix m = ~~ pfxm_mask m && a"
chosen_adjoint_equivalence: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd"   and a :: "'c" assumes "converse_pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"   and "horizontal_homs.obj V\<^sub>C src\<^sub>C a" shows "adjoint_equivalence_in_bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<tau>\<^sub>0 a) (converse_pseudonatural_equivalence.map\<^sub>0 V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<tau>\<^sub>0 a) (converse_pseudonatural_equivalence.unit V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<tau>\<^sub>0 a) (converse_pseudonatural_equivalence.counit V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<tau>\<^sub>0 a)"
Inv_in_Hom: fixes t :: "'c local.term" assumes "Can t" shows "Inv t \<in> Hom (Cod t) (Dom t)"
select_object_any\<^sub>S\<^sub>e\<^sub>q_def: fixes f :: "'f \<Rightarrow> 'd state \<times> 'd state \<Rightarrow> 'e"   and s_set :: "'f list" shows "select_object_any\<^sub>S\<^sub>e\<^sub>q f s_set = select_object\<^sub>S\<^sub>e\<^sub>q f s_set->any\<^sub>S\<^sub>e\<^sub>q()"
tarjan_def: fixes G :: "('c, 'd) graph_rec_scheme" shows "tarjan G \<equiv> ASSERT (fb_graph G) \<bind> (\<lambda>_. param_DFS_defs.it_dfs G (Tarjan_def.tarjan_params G) \<bind> (\<lambda>s. RETURN (sccs s)))"
diff_strict_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and d :: "'a"   and c :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less a b"   and "less d c" shows "less (minus a c) (minus b d)"
conversep_inject: fixes r :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "(r\<inverse>\<inverse> = s\<inverse>\<inverse>) = (r = s)"
alw_valid_iff_valid: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "(\<turnstile> \<box>F) = (\<turnstile> F)"
nAct_mono_lNact: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat" assumes "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'" shows "\<langle>c #\<^bsub>enat n\<^esub>inf_llist t\<rangle> \<le> \<langle>c #\<^bsub>enat n'\<^esub>inf_llist t\<rangle>"
ListLenDrop: fixes i :: "nat"   and lb :: "'a list"   and la :: "nat" assumes "i < length lb"   and "la \<le> i" shows "lb ! i \<in> set (drop la lb)"
b: shows "\<Gamma>1 \<subseteq> \<Gamma>2"
aux_lemma: fixes na :: "nat"   and ma :: "nat" assumes "dimension na K E"   and "dimension ma K E" shows "na \<le> ma"
R21_R23: shows "pmf (sds R21) a \<le> pmf (sds R23) a"
axioms_1: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure" assumes "filtrated_prob_space M F" shows "prob_space M"
cet_fst: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and t :: "pdf_type"   and t' :: "pdf_type" assumes "\<Gamma> \<turnstile>\<^sub>c e : PRODUCT t t'" shows "\<Gamma> \<turnstile>\<^sub>c fst\<^sub>c e : t"
complement_wf: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "incidence_system point_set block_collection" shows "incidence_system point_set (incidence_system.complement_blocks point_set block_collection)"
ltl_const_equiv_const_2: fixes \<phi> :: "'a ltln" shows "\<phi> \<sim>\<^sub>C false\<^sub>n = (eval \<phi> = No)"
subject_reduction: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and T :: "ty" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>"   and "P,E \<turnstile> s \<surd>"   and "P,E,hp s \<turnstile> e : T" shows "P,E,hp s' \<turnstile> e' :\<^bsub>NT\<^esub> T"
alphaAll_sym: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and A :: "('f, 'g, 'e, 'd, 'h) qAbs" assumes "FixVars TYPE('d) TYPE('e)" shows "(\<forall>X'. X #= X' \<longrightarrow> X' #= X) \<and> (\<forall>A'. A $= A' \<longrightarrow> A' $= A)"
gd_powerprod_axioms: fixes cmp_term :: "'a nat_term_order" shows "gd_powerprod (\<lambda>s t. le_of_nat_term_order cmp_term (fst (s, the_min)) (fst (t, the_min))) (\<lambda>s t. lt_of_nat_term_order cmp_term (fst (s, the_min)) (fst (t, the_min)))"
bezout_lemma_nat: fixes d :: "nat"   and a :: "nat"   and b :: "nat"   and x :: "nat"   and y :: "nat" assumes "d dvd a"   and "d dvd b"   and "a * x = b * y + d \<or> b * x = a * y + d" shows "\<exists>x y. d dvd a \<and> d dvd a + b \<and> (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
continuous_minus: fixes y :: "ordinal" shows "continuous (\<lambda>x. x - y)"
iso_tuple_update_accessor_cong_assist_id: fixes upd :: "('b \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'c"   and ac :: "'c \<Rightarrow> 'b" assumes "Record.iso_tuple_update_accessor_cong_assist upd ac" shows "upd id = id"
ideal_field_cases: fixes B :: "'b set" obtains "ideal B = {0::'b}"   | "ideal B = UNIV"
Conj_commute: fixes H :: "fm set"   and B :: "fm"   and A :: "fm" assumes "H \<turnstile> B AND A" shows "H \<turnstile> A AND B"
p1_5: fixes x1 :: "'a"   and K1 :: "'a set"   and x2 :: "'a"   and K2 :: "'a set" assumes "(x1, K1) \<in> p1"   and "(x2, K2) \<in> p1"   and "(x1, K1) \<noteq> (x2, K2)" shows "interior K1 \<inter> interior K2 = {}"
proj_def: fixes n :: "'e"   and S :: "('e strand_label \<times> 'f) list" shows "proj n S \<equiv> filter (\<lambda>s. is_LabelN n s \<or> is_LabelS s) S"
I_mono: fixes ab :: "'d"   and bb :: "'d set"   and xb :: "'e" assumes "ab \<in> bb"   and "bb \<subseteq> S"   and "I (insert ab (S - bb)) xb" shows "I (S - bb) xb"
fspec_invar2: fixes aa :: "'b"   and b :: "'a set"   and a :: "'a"   and z :: "'b nres" assumes "I aa b"   and "a \<in> b"   and "SPEC (\<lambda>xb. c aa xb \<le> SPEC (\<lambda>r. I r b)) \<le> z" shows "f a \<le> z"
block_sizes: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "int (card bl) \<in> \<K>"
rel_gpv''_map_gpv1: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'a"   and g :: "'h \<Rightarrow> 'c"   and gpv :: "('g, 'h, 'e) gpv"   and gpv' :: "('b, 'd, 'f) gpv" shows "rel_gpv'' A C R (map_gpv f g gpv) gpv' = rel_gpv'' (\<lambda>a. A (f a)) (\<lambda>c. C (g c)) R gpv gpv'"
A: shows "[reset' (FW M n) n cs v d]\<^bsub>v,n\<^esub> = {}"
meval_sched_ConsI: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val"   and k :: "nat"   and cms' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem' :: "'Var \<Rightarrow> 'Val"   and sched :: "nat list"   and cms'' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem'' :: "'Var \<Rightarrow> 'Val" assumes "(cms, mem) \<leadsto>\<^bsub>k\<^esub> (cms', mem')"   and "(cms', mem') \<rightarrow>\<^bsub>sched\<^esub> (cms'', mem'')" shows "(cms, mem) \<rightarrow>\<^bsub>k # sched\<^esub> (cms'', mem'')"
simps: fixes xs :: "('a \<times> real) msllist"   and uu :: "real \<Rightarrow> real" shows "dominant_term (MS xs uu) = dominant_term_ms_aux xs"
eq2: fixes x :: "'a" assumes "f x \<notin> f ` A" shows "f -` {f x} \<inter> space M \<inter> A = {}"
safe: shows "fst ` CY \<union> fst ` Y1 \<subseteq> Csafe"
lfp_lowerbound: fixes a :: "'a acom"   and c :: "com"   and f :: "'a acom \<Rightarrow> 'a acom" assumes "a \<in> {C. strip C = c}"   and "f a \<le> a" shows "Collecting.lfp c f \<le> a"
colossless_gpv_continuationD: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and out :: "'d"   and c :: "('f, 'd, 'e) rpv"   and input :: "'e" assumes "colossless_gpv \<I> gpv"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out" shows "colossless_gpv \<I> (c input)"
refine: fixes xsi :: "'b list list"   and xs :: "'c set" assumes "(xsi, xs) \<in> clw_rel appr_rel" shows "(nres_of (ivls_of_sets_impl xsi), ivls_of_sets $ xs) \<in> \<langle>clw_rel lvivl_rel\<rangle>nres_rel"
Mul_Blacken_Roots_def: fixes n :: "nat" shows "Mul_Blacken_Roots n \<equiv> \<lbrace>\<acute>Mul_Proper n\<rbrace> \<acute>ind := 0;; \<lbrace>\<acute>Mul_Proper n \<and> \<acute>ind = 0\<rbrace>  WHILE \<acute>ind < length \<acute>M INV \<lbrace>\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind. i \<in> Roots \<longrightarrow> \<acute>M ! i = Black) \<and> \<acute>ind \<le> length \<acute>M\<rbrace>  DO \<lbrace>\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind. i \<in> Roots \<longrightarrow> \<acute>M ! i = Black) \<and> \<acute>ind < length \<acute>M\<rbrace>  IF \<acute>ind \<in> Roots THEN \<lbrace>\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind. i \<in> Roots \<longrightarrow> \<acute>M ! i = Black) \<and> \<acute>ind < length \<acute>M \<and> \<acute>ind \<in> Roots\<rbrace> \<acute>M := \<acute>M[\<acute>ind := Black] FI;; \<lbrace>\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind + 1. i \<in> Roots \<longrightarrow> \<acute>M ! i = Black) \<and> \<acute>ind < length \<acute>M\<rbrace> \<acute>ind := \<acute>ind + 1 OD"
coincidence_dterm: fixes \<theta> :: "('a, 'b) trm"   and \<nu> :: "(real, 'b) vec \<times> (real, 'b) vec"   and \<nu>' :: "(real, 'b) vec \<times> (real, 'b) vec"   and I :: "('a, 'c, 'b) interp" assumes "dsafe \<theta>"   and "Vagree \<nu> \<nu>' (FVT \<theta>)" shows "dterm_sem I \<theta> \<nu> = dterm_sem I \<theta> \<nu>'"
iterate_axiom_def: shows "iterate_axiom \<equiv> \<langle> Game hgid1** \<rangle> pusall \<leftrightarrow> (pusall || \<langle> Game hgid1 \<rangle> \<langle> Game hgid1** \<rangle> pusall)"
fact_nonzero: fixes n :: "nat" shows "fact n \<noteq> (0::'c)"
Unifier_def: fixes \<sigma> :: "('b \<times> 'b trm) list"   and t :: "'b trm"   and u :: "'b trm" shows "Unifier \<sigma> t u = (t \<lhd> \<sigma> = u \<lhd> \<sigma>)"
fps_left_inverse_idempotent_ring1: fixes x :: "'a"   and f :: "'a fps"   and y :: "'a" assumes "x * fps_nth f 0 = (1::'a)"   and "y * x = (1::'a)" shows "fps_left_inverse (fps_left_inverse f x) y = f"
bij_lens_equiv_id: fixes X :: "'a \<Longrightarrow> 'b" shows "bij_lens X = X \<approx>\<^sub>L 1\<^sub>L"
preserves_modes_mem_mem\<^sub>A_simp: fixes mem\<^sub>A :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>C :: "'Var \<Rightarrow> 'Val" assumes "\<forall>x\<^sub>A. mem\<^sub>A x\<^sub>A = mem\<^sub>C (id x\<^sub>A)" shows "mem\<^sub>A = gen_refine.mem\<^sub>A_of mem\<^sub>C"
simps_10: fixes Vs :: "char list list"   and C :: "char list"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "char list exp" shows "compE\<^sub>1 Vs (C\<bullet>\<^sub>sF{D} := e\<^sub>2) = C\<bullet>\<^sub>sF{D} := compE\<^sub>1 Vs e\<^sub>2"
cl_Int_subset: fixes L :: "'b set set"   and r :: "'b set"   and s :: "'b set" shows "cl L (r \<inter> s) \<subseteq> cl L r \<inter> cl L s"
proj_set_geodesic_same_basepoint: fixes p :: "'b"   and x :: "'b"   and A :: "'b set"   and G :: "'b set"   and y :: "'b" assumes "p \<in> proj_set x A"   and "geodesic_segment_between G p x"   and "y \<in> G" shows "p \<in> proj_set y A"
c3_3: shows "fst (conv_mirror \<gamma>3) = lst_conv \<delta>3"