hcju/isabelleps huggingface dataset:
exe_wf_sig_imp_wf_sig: fixes cto :: "(String.literal \<times> typ) list"   and tao :: "(String.literal \<times> nat) list"   and sa :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" assumes "alist_conds cto"   and "alist_conds tao"   and "exe_osig_conds sa"   and "(exe_wf_osig sa \<and> fst ` set (snd sa) = fst ` set tao \<and> (\<forall>type\<in>fst ` set (snd sa). \<forall>ars\<in>snd ` set (the (lookup (\<lambda>k. k = type) (snd sa))). the (lookup (\<lambda>k. k = type) tao) = length ars)) \<and> (\<forall>ty\<in>snd ` set cto. typ_ok_sig (map_of cto, map_of tao, translate_osig sa) ty)" shows "wf_sig (map_of cto, map_of tao, translate_osig sa)"
subset: fixes N :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "submonoid N M composition unit" shows "N \<subseteq> M"
genideal_def: fixes R :: "('a, 'b) ring_scheme"   and S :: "'a set" shows "Idl\<^bsub>R\<^esub> S = \<Inter> {I. ideal I R \<and> S \<subseteq> I}"
assoc: fixes a :: "'k list"   and b :: "'k list"   and c :: "'k list" shows "(a @ b) @ c = a @ b @ c"
maxsimp_def: fixes x :: "'a set" shows "folding_g.chamber x \<equiv> x \<in> X \<and> (\<forall>z\<in>X. order.greater_eq z x \<longrightarrow> z = x)"
lunit_char_3: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "\<exists>!\<mu>. \<guillemotleft>\<mu> : L f \<Rightarrow> f\<guillemotright> \<and> L \<mu> = (\<i>[trg f] \<star> f) \<bullet> \<a>\<^sup>-\<^sup>1[trg f, trg f, f]"
bipartite_web_of_sel_4: fixes \<Gamma> :: "('a, 'b) web_scheme"   and u :: "'a" shows "weight (bipartite_web_of \<Gamma>) (Inl u) = (if u \<in> B \<Gamma> then 0 else weight \<Gamma> u)"
direct_lim_is_ring: fixes U :: "'a set" assumes "U \<in> I" shows "ring stk1.carrier_direct_lim stk1.add_rel stk1.mult_rel (stk1.class_of U (zero_str1 U)) (stk1.class_of U (one_str1 U))"
insert_absorb: fixes a :: "'h"   and A :: "'h set" assumes "a \<in> A" shows "insert a A = A"
size_multiset_single: fixes f :: "'k \<Rightarrow> nat"   and b :: "'k" shows "size_multiset f {#b#} = Suc (f b)"
red_unicity: fixes x :: "trm"   and a :: "trm"   and b :: "trm" assumes "x \<leadsto> a"   and "x \<leadsto> b" shows "a = b"
bind_rule: fixes P :: "assn"   and f :: "'b Heap"   and Q :: "'b \<Rightarrow> assn"   and g :: "'b \<Rightarrow> 'c Heap"   and R :: "'c \<Rightarrow> assn" assumes "<P> f <Q>"   and "\<forall>x. <Q x> g x <R>" shows "<P> f \<bind> g <R>"
finite_entropy_nn: fixes S :: "'d measure"   and X :: "'a \<Rightarrow> 'd"   and Px :: "'d \<Rightarrow> real"   and x :: "'d" assumes "finite_entropy S X Px"   and "x \<in> space S" shows "0 \<le> Px x"
iwlsFSwSb_iterAll_termFSwSbImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSwSb MOD" shows "termFSwSbImorph (iter MOD) (iterAbs MOD) MOD"
sorted_nth_mono: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and i :: "nat"   and j :: "nat" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs"   and "i \<le> j"   and "j < length xs" shows "less_eq (xs ! i) (xs ! j)"
qmpt_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" shows "qmpt M T \<equiv> sigma_finite_measure M \<and> qmpt_axioms M T"
comp_in_homE: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b"   and a :: "'b"   and c :: "'b" assumes "category C"   and "partial_magma.in_hom C (C g f) a c" obtains b :: "'b" where "partial_magma.in_hom C f a b" and "partial_magma.in_hom C g b c"
Lazy_llist_def: fixes xs :: "unit \<Rightarrow> ('a \<times> 'a llist) option" shows "Lazy_llist xs = (case xs () of None \<Rightarrow> LNil | Some (x, ys) \<Rightarrow> LCons x ys)"
digit_to_char_simps_8: shows "digit_to_char 7 = CHR ''7''"
one_mult_of: fixes R :: "('c, 'd) ring_scheme" shows "\<one>\<^bsub>mult_of R\<^esub> = \<one>\<^bsub>R\<^esub>"
Eq\<kappa>E: fixes v :: "i"   and x :: "\<kappa>"   and y :: "\<kappa>" assumes "[x \<^bold>=\<^sub>\<kappa> y in v]" shows "\<exists>o\<^sub>1 o\<^sub>2. Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> x \<and> Some o\<^sub>2 = Semantics.d\<^sub>\<kappa> y \<and> o\<^sub>1 = o\<^sub>2"
r_append_zeros_prim: shows "prim_recfn 2 r_append_zeros"
function_to_finmap_def: fixes J :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and f' :: "'e \<Rightarrow> 'd" shows "function_to_finmap J f f' \<equiv> finite J \<and> (\<forall>i. i \<in> J \<longrightarrow> f' (f i) = i)"
fin: shows "finite {\<bar>a $ i\<bar> |a i. a \<in> A \<and> i < n}"
tree\<^sub>i_inorder: fixes u :: "'a up\<^sub>i" shows "inorder_up\<^sub>i u = inorder (tree\<^sub>i u)"
type_definition_id_bnf_UNIV: shows "type_definition BNF_Composition.id_bnf BNF_Composition.id_bnf UNIV"
lbisimulation_def: fixes R :: "('h \<times> 'i) set"   and A :: "('h, 'j, 'k) sa_rec_scheme"   and B :: "('i, 'j, 'l) sa_rec_scheme" shows "lbisimulation R A B \<equiv> (sa A \<and> sa B) \<and> lsimulation R A B \<and> lsimulation (R\<inverse>) B A \<and> bisimulation R A B"
locally_path_connected_sphere_gen: fixes S :: "'a set" assumes "bounded S"   and "convex S" shows "locally path_connected (rel_frontier S)"
network_with_constrained_ops_axioms: shows "network_with_constrained_ops history interp initial_state valid_msg"
cblinfun_compose_id_left: fixes U :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "id_cblinfun o\<^sub>C\<^sub>L U = U"
distinct_hd_not_cons: fixes vs :: "'a list"   and x :: "'a" assumes "distinct vs"   and "\<exists>as bs. vs = as @ x # hd vs # bs" shows "False"
igSwapCls_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gSwapCls MOD" shows "igSwapCls (fromMOD MOD)"
rtc1: fixes x :: "'a" shows "(1::'a) + x\<^sup>\<star> \<cdot> x\<^sup>\<star> + x \<le> x\<^sup>\<star>"
quot_tm_def: fixes t :: "tm" shows "\<guillemotleft>t\<guillemotright> = quot_dbtm (trans_tm [] t)"
least_bounded_Suc: fixes upper :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "\<exists>x<upper. P x" shows "(LEAST x. x < upper \<and> P x) = (LEAST x. x < Suc upper \<and> P x)"
the_\<theta>_simps_1: shows "arr \<chi>.the_\<theta>"
that: fixes C :: "'a set" assumes "countable C"   and "C \<subseteq> K"   and "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"   and "negligible (S - (\<Union>i\<in>C. cball (a i) (r i)))" shows "thesis"
Sup_option_def: fixes A :: "'a option set" shows "\<Squnion> A = (if A = {} \<or> A = {None} then None else Some (\<Squnion> Option.these A))"
equivalence_in_bicategory_axioms: shows "internal_equivalence (trg f) (trg f) (local.inv \<i>[trg f]) \<i>[trg f]"
liminf_SUP_INF: fixes f :: "nat \<Rightarrow> 'b" shows "liminf f = (SUP n. Inf (f ` {n..}))"
bo_hk: shows "bounded {x. \<not> (h x = x \<and> k x = x)}"
inv_M_def: fixes m :: "int" shows "poly_mod.inv_M m = (\<lambda>x. if x + x \<le> m then x else x - m)"
span_G1_G: shows "gs.span (gs2.gso ` {0..<i}) = gs.span (gs1.gso ` {0..<i})"
valof_topfloat: assumes "1 < LENGTH('a)" shows "valof topfloat = largest TYPE(('a, 'b) IEEE.float)"
bang_is_le: fixes x :: "'b word"   and m :: "nat" assumes "x !! m" shows "2 ^ m \<le> x"
real_transfer_operator_add: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "integrable M f"   and "integrable M g" shows "AE x in M. real_transfer_operator (\<lambda>x. f x + g x) x = real_transfer_operator f x + real_transfer_operator g x"
fold_closed_eq: fixes A :: "'b set"   and B :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> B\<rbrakk> \<Longrightarrow> f a b = g a b"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> B\<rbrakk> \<Longrightarrow> g a b \<in> B"   and "z \<in> B" shows "Finite_Set.fold f z A = Finite_Set.fold g z A"
prems: shows "\<forall>xa\<in>set (x # xs). \<forall>y\<in>set (x # xs). g xa = g y \<longrightarrow> f xa = f y"
limf: fixes x :: "'n" assumes "x \<in> S" shows "(\<lambda>k. min (max (- h x) (f k x)) (h x)) \<longlonglongrightarrow> g x"
I_qe_pres\<^sub>1_o: fixes as :: "atom list"   and e :: "int list" assumes "\<forall>a\<in>set as. divisor a \<noteq> 0"   and "\<forall>a\<in>set as. hd_coeff a \<noteq> 0" shows "Z.I ((qe_pres\<^sub>1 \<circ> hd_coeffs1) as) e = (\<exists>x. \<forall>a\<in>set as. I\<^sub>Z a (x # e))"
lemma_2_5_8_b: fixes b :: "'a"   and a :: "'a" shows "b * a \<le> b"
Cons_5: shows "(lookup_pair xs k = (0::'b)) = (k \<notin> fst ` set xs)"
set_eq_is_eq: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set"   and B :: "'a set" assumes "partial_order L" shows "(A {.=}\<^bsub>L\<^esub> B) = (A = B)"
csqrt_2_sq: shows "complex_of_real (sqrt 2) * complex_of_real (sqrt 2) = 2"
paint_cl_inv': fixes t :: "('a, 'b) rbt"   and c :: "color" assumes "cl_inv' t" shows "cl_inv' (paint c t)"
distinct_1: fixes x1 :: "\<omega>"   and x2 :: "\<sigma>" shows "\<omega>\<upsilon> x1 \<noteq> \<sigma>\<upsilon> x2"
same_suffix_nil: fixes ys :: "'b list"   and xs :: "'b list" shows "ys \<cdot> xs \<le>s xs = (ys = \<epsilon>)"
expand_name_propag__name_ident: fixes n_ns :: "'b node \<times> 'b node set" assumes "((\<forall>nm\<in>incoming (fst n_ns). nm < name (fst n_ns)) \<and> 0 < name (fst n_ns) \<and> (\<forall>q\<in>snd n_ns. name q < name (fst n_ns) \<and> (\<forall>nm\<in>incoming q. nm < name (fst n_ns)))) \<and> expand_name_ident (snd n_ns)" shows "expand n_ns \<le> SPEC (\<lambda>r. expand_name_ident (snd r))"
pe_cond1: fixes q :: "nat"   and i :: "nat"   and p :: "nat"   and l :: "nat" assumes "te q (i + 1) \<le> te p (i + 1)"   and "correct p (te p (i + 1))"   and "correct q (te p (i + 1))"   and "correct l (te p (i + 1))" shows "\<bar>\<theta> q (i + 1) l + (PC q (te p (i + 1)) - PC q (te q (i + 1))) - \<theta> p (i + 1) l\<bar> \<le> 2 * \<rho> * \<beta> + 2 * \<Lambda>"
lift_self_disjoint: fixes x :: "V"   and u :: "V" shows "vdisjnt x (lift x u)"
null_space_eq_ker: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "linear (*s) (*s) f" shows "null_space (matrix f) = {x. f x = 0}"
diff_le_0_iff_le: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (minus a b) zero = less_eq a b"
F_unfold: fixes i :: "name" shows "F i = (if i \<in> Vs then Var (p \<bullet> i) else Var i)"
proves_eta_step: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and t :: "Core.term"   and u :: "Core.term" assumes "wf_theory \<Theta>"   and "finite \<Gamma>"   and "term_ok \<Theta> t"   and "t \<rightarrow>\<^sub>\<eta> u"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq t u"
S_transform_Tree_eqvt: fixes p :: "perm"   and t :: "('a, 'c, 'b) Tree" shows "p \<bullet> S_transform_Tree t = S_transform_Tree (p \<bullet> t)"
env_restr_fun_upd: fixes x :: "'c"   and S :: "'c set"   and m1 :: "'c \<Rightarrow> 'd"   and v :: "'d" assumes "x \<in> S" shows "m1(x := v) f|` S = (m1 f|` S)(x := v)"
confs_hext: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and vs :: "val list"   and Ts :: "ty list"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "P,h \<turnstile> vs [:\<le>] Ts"   and "h \<unlhd> h'" shows "P,h' \<turnstile> vs [:\<le>] Ts"
HInv4a1_def: fixes s :: "state"   and p :: "Proc" shows "HInv4a1 s p = (\<forall>bk\<in>blocksOf s p. bal bk \<le> mbal (dblock s p))"
hom_closed: fixes x :: "'a" assumes "x \<in> carrier R" shows "h x \<in> carrier S"
1: shows "\<Union> (Range (a - (X \<union> {i}) \<times> Range a \<union> ({(i, \<Union> (a `` (X \<union> {i})))} - {(i, {})}))) = \<Union> (Range (a - (X \<union> {i}) \<times> Range a)) \<union> \<Union> (Range ({(i, \<Union> (a `` (X \<union> {i})))} - {(i, {})}))"
list_of_oalist_OAlist_tc_map_val: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "('a, 'c) oalist_tc" shows "list_of_oalist_tc (OAlist_tc_map_val f xs) = map_val_pair f (list_of_oalist_tc xs)"
map_dual_func1: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "\<partial>\<^sub>F (f \<circ> g) = \<partial>\<^sub>F f \<circ> \<partial>\<^sub>F g"
keys_higherPowers: fixes b :: "nat"   and i :: "nat"   and m :: "nat \<Rightarrow>\<^sub>0 'b" shows "keys (higherPowers b i m) = dec_above b i ` (keys m \<inter> {x. x \<notin> {b..<b + i}})"
sorted_merge_all: fixes xs :: "'a list list" assumes "\<forall>x\<in>set xs. sorted (map key x)" shows "sorted (map key (local.merge_all xs))"
valid_append_atomics_reusables: fixes w :: "nat \<times> nat"   and t :: "'a tree" assumes "0 < fst w"   and "local.valid t"   and "l t \<le> fst w"   and "r t \<le> snd w"   and "snd w \<le> length as" shows "Ball (set (local.atomics (max (fst w) (Suc (r t))) (snd w) @ reusables t w)) local.valid"
nstut_after: fixes F :: "'a \<Rightarrow> 'b" shows "NSTUTINV F$"
pdivmod_pdivmodrel: fixes p :: "'b poly"   and q :: "'b poly"   and r :: "'b poly"   and s :: "'b poly" shows "eucl_rel_poly p q (r, s) = ((p div q, p mod q) = (r, s))"
outputPermSubject: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list"   and yvec :: "name list"   and zvec :: "name list" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "set p \<subseteq> set yvec \<times> set zvec"   and "yvec \<sharp>* P"   and "zvec \<sharp>* P"   and "yvec \<sharp>* \<Psi>"   and "zvec \<sharp>* \<Psi>" shows "\<Psi> \<rhd> P \<longmapsto> (p \<bullet> M)\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"
body_lc_FV: fixes t :: "sterm"   and s :: "char list"   and p :: "char list" assumes "body t" shows "lc (t\<^bsup>[Fvar s,Fvar p]\<^esup>)"
trg_src: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "trg\<^sub>D (src\<^sub>D \<mu>) = src\<^sub>D \<mu>"
simple_fw_iff_generalized_fw_accept: fixes fw :: "'b simple_rule list"   and p :: "('b, 'c) simple_packet_scheme" shows "(simple_fw fw p = Decision FinalAllow) = (\<exists>r. generalized_sfw (map simple_rule_dtor fw) p = Some (r, Accept))"
convex_hyperplane: fixes a :: "'b"   and b :: "real" shows "convex {x. a \<bullet> x = b}"
eq_up_to_def: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and \<pi>' :: "nat \<Rightarrow> 'n" shows "(\<pi> =\<^bsub>k\<^esub> \<pi>') = (\<forall>i\<le>k. \<pi> i = \<pi>' i)"
is_extensional: fixes f :: "'a" assumes "\<not> arr f" shows "f \<otimes> cod \<iota> = null"
fresh_star_empty_elim: fixes c :: "'b"   and C :: "prop" shows "({} \<sharp>* c \<Longrightarrow> PROP C) \<equiv> PROP C"
divmod_via_sdivmod: fixes y :: "'a word"   and x :: "'a word" assumes "y \<noteq> 0" shows "(x div y, x mod y) = (if 1 << LENGTH('a) - 1 \<le> y then if x < y then (0, x) else (1, x - y) else let q = (x >> 1) sdiv y << 1; r = x - q * y in if y \<le> r then (q + 1, r - y) else (q, r))"
check_nsc_code_def: shows "check_nsc_code = check_nsc check_s"
simps_2: fixes x :: "nat"   and xs :: "nat list" shows "Partial_Recursive.list_encode (x # xs) = Suc (prod_encode (length xs, tuple_encode (length xs) (x # xs)))"
lemma3_6_v: fixes r :: "('a \<times> 'a) set"   and \<sigma> :: "'a list"   and \<tau> :: "'a list"   and \<sigma>' :: "'a list"   and \<tau>' :: "'a list"   and \<upsilon> :: "'a list" assumes "trans r"   and "irrefl r"   and "\<sigma> \<noteq> []"   and "Decreasing_Diagrams.D r \<tau> \<sigma> \<sigma>' \<tau>'" shows "(r|\<tau>'| + r|\<upsilon>|, r|\<tau>| + r|\<sigma> @ \<upsilon>|) \<in> mul r"
mkArr_Fun: fixes f :: "'s" assumes "arr f" shows "mkArr (Dom f) (Cod f) (Fun f) = f"
separable_imp_square_free: fixes f :: "'a poly" assumes "separable f" shows "square_free f"
bdd_below_def: fixes A :: "'b multiset set" shows "subset_mset.bdd_below A = (\<exists>m. \<forall>x\<in>A. m \<subseteq># x)"
I_def: shows "I \<equiv> {f \<in> Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. cbox (a' f) (b' f) \<subseteq> M}"
total_pseudo_ordered_set_axioms: shows "total_pseudo_ordered_set X (\<sqsubseteq>)"
I_in_na_def: fixes s :: "'c list"   and t :: "'c list"   and i :: "nat" shows "I_in_na s t i \<equiv> \<lambda>(j, pos). case pos of None \<Rightarrow> j < length s \<and> (\<forall>jj<j. t ! (i + jj) = s ! jj) | Some p \<Rightarrow> sublist_at s t i"
simps: fixes a1 :: "bool list \<times> nat \<times> thread"   and a2 :: "bool list \<times> nat \<times> thread" shows "cstep_sng a1 a2 = ((\<exists>p n. a1 = (p, n, INIT) \<and> a2 = (p, (n + 1) mod (k * N), WAIT (n mod N))) \<or> (\<exists>p tk n. a1 = (p, n, WAIT tk) \<and> a2 = (p, n, WAIT tk) \<and> \<not> p ! tk) \<or> (\<exists>p tk n. a1 = (p, n, WAIT tk) \<and> a2 = (p, n, HOLD tk) \<and> p ! tk) \<or> (\<exists>p n tk. a1 = (p, n, HOLD tk) \<and> a2 = (p[tk := False], n, REL tk)) \<or> (\<exists>p n tk. a1 = (p, n, REL tk) \<and> a2 = (p[(tk + 1) mod N := True], n, INIT)))"
top_le_conv: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'c"   and B :: "'c"   and x :: "'c" assumes "lbv A r f T B" shows "(T \<sqsubseteq>\<^bsub>r\<^esub> x) = (x = T)"
return_ok: fixes h :: "'l"   and x :: "'n" shows "h \<turnstile> ok return x"
that: fixes xs :: "'a list"   and ys :: "'a list" assumes "length xs = length ys"   and "X = mset xs"   and "Y = mset ys"   and "\<forall>i<length ys. (xs ! i, ys ! i) \<in> ns" shows "thesis"
presAtm_def: fixes indis :: "('a \<times> 'a) set"   and aval :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and atm :: "'b" assumes "PL_Indis indis" shows "PL_Indis.presAtm aval indis atm \<equiv> \<forall>s. (s, aval atm s) \<in> indis"
subspace_def: fixes Sa :: "'a set" assumes "Sa \<subseteq> S" shows "subspace.subspace Sa = ((0::'a) \<in> Sa \<and> (\<forall>x\<in>Sa. \<forall>y\<in>Sa. x + y \<in> Sa) \<and> (\<forall>c. \<forall>x\<in>Sa. c *\<^sub>R x \<in> Sa))"
face_of_convex_hull_insert2: fixes S :: "'b set"   and a :: "'b"   and F :: "'b set" assumes "finite S"   and "a \<notin> affine hull S"   and "F face_of convex hull S" shows "convex hull insert a F face_of convex hull insert a S"
intro: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" assumes "compat_pair S NS"   and "SN_ars S" shows "SN_pair S NS"
ptrm_nprv_exi_all: fixes \<sigma> :: "'fmla"   and n :: "nat"   and \<phi> :: "'fmla" assumes "\<sigma> \<in> ptrm n"   and "\<phi> \<in> fmla" shows "nprv {exi out (cnj \<sigma> \<phi>)} (all out (imp \<sigma> \<phi>))"
intT_int_transE: fixes T :: "'fsym trm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "wt T"   and "I.wtE \<xi>"   and "\<And>x. \<lbrakk>\<not> infTp (tpOfV x); x \<in> nv2T T\<rbrakk> \<Longrightarrow> \<exists>l. wtL l \<and> \<not> I.satL \<xi> l \<and> isGuard x l" shows "Ik_intT (tpOf T) (I.int \<xi> T)"
assms: shows "w\<^sub>1 =\<^sub>S w\<^sub>2"
Abs_asTerm_asAbs_qAbs: fixes qX :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var" assumes "qGood qX" shows "Abs xs x (asTerm qX) = asAbs (qAbs xs x qX)"
Sup_least: fixes A :: "V set"   and z :: "V" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<subseteq>\<^sub>\<circ> z" shows "\<Squnion> A \<subseteq>\<^sub>\<circ> z"
Union_eq: fixes A :: "'b set set" shows "\<Union> A = {x. \<exists>B\<in>A. x \<in> B}"
SigmaI: fixes a :: "'c"   and A :: "'c set"   and b :: "'d"   and B :: "'c \<Rightarrow> 'd set" assumes "a \<in> A"   and "b \<in> B a" shows "(a, b) \<in> Sigma A B"
linear_scale: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f"   and r :: "'d"   and b :: "'e" assumes "vector_space_pair s1 s2"   and "linear s1 s2 f" shows "f (s1 r b) = s2 r (f b)"
jumpF_const: fixes F :: "real filter"   and c :: "real" assumes "F \<noteq> bot" shows "jumpF (\<lambda>_. c) F = 0"
InterI: fixes C :: "'d set set"   and A :: "'d" assumes "\<And>X. X \<in> C \<Longrightarrow> A \<in> X" shows "A \<in> \<Inter> C"
cert_plus_def: fixes dict :: "'b dict_plus" shows "cert_plus dict = (param_plus dict = Introduction.plus_class.plus)"
up_ucast_inj: fixes x :: "'c word"   and y :: "'c word" assumes "UCAST('c \<rightarrow> 'b) x = UCAST('c \<rightarrow> 'b) y"   and "LENGTH('c) \<le> LENGTH('b)" shows "x = y"
healthy_inf: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel"   and A :: "'a hrel set" assumes "utp_theory_mono hcond"   and "A \<subseteq> \<lbrakk>hcond\<rbrakk>\<^sub>H" shows "Lattice.inf (utp_order hcond) A = hcond (\<Or>A)"
lsol_4: shows "cball (flow t0 x0 t) ex \<subseteq> X"
wf_findnth_LE: shows "wf findnth_LE"
simps: fixes e :: "'c DBMEntry"   and t :: "'c" shows "norm_upper e t = (if Le t \<prec> e then \<infinity> else e)"
simps_3: fixes T1 :: "ty"   and T2 :: "ty" shows "size_ty (T1 \<rightarrow> T2) = size_ty T1 + size_ty T2 + 1"
hom: fixes S :: "'a set" assumes "S \<in> \<F>" shows "\<exists>a. homotopic_with_canon (\<lambda>x. True) S T f (\<lambda>x. a)"
only_call_get_return_edges: fixes a :: "'edge"   and a' :: "'edge" assumes "valid_edge a"   and "a' \<in> get_return_edges a" shows "\<exists>Q r p fs. kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"
orthogonal_current: fixes S :: "'a set"   and \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "\<And>x. \<lbrakk>x \<in> S; x \<notin> A \<Gamma>\<rbrakk> \<Longrightarrow> weight \<Gamma> x \<le> d_IN f x"   and "\<And>x. \<lbrakk>x \<in> S; x \<in> A \<Gamma>; x \<notin> B \<Gamma>\<rbrakk> \<Longrightarrow> d_OUT f x = weight \<Gamma> x"   and "\<And>u v. \<lbrakk>v \<in> RF\<^bsub>\<Gamma>\<^esub> S; u \<notin> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> S\<rbrakk> \<Longrightarrow> f (u, v) = 0" shows "orthogonal_current \<Gamma> f S"
fmmerge_def: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and xm :: "'d \<rightharpoonup>\<^sub>f 'f"   and ym :: "'d \<rightharpoonup>\<^sub>f 'g" shows "fmmerge f xm ym \<equiv> fmap_of_list (map (\<lambda>k. (k, f (the (fmlookup xm k)) (the (fmlookup ym k)))) (sorted_list_of_fset (fmdom xm |\<inter>| fmdom ym)))"
bind_returns_result_I: fixes h :: "'ap"   and f :: "('ap, 'aq, 'ar) prog"   and x :: "'ar"   and h' :: "'ap"   and g :: "'ar \<Rightarrow> ('ap, 'aq, 'as) prog"   and y :: "'as" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> g x \<rightarrow>\<^sub>r y" shows "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"
mult_Limit: fixes \<gamma> :: "V"   and x :: "V" assumes "Limit \<gamma>" shows "x * \<gamma> = \<Union>\<^sub>\<circ> (VLambda \<gamma> ((*) x) `\<^sub>\<circ> \<gamma>)"
abs_summable_finiteI0_1: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "real" assumes "\<And>F. \<lbrakk>finite F; F \<subseteq> S\<rbrakk> \<Longrightarrow> (\<Sum>x\<in>F. norm (f x)) \<le> B"   and "0 \<le> B" shows "f abs_summable_on S"
sel: fixes x :: "'b \<Rightarrow> 'c \<times> ('b, 'c) dds" shows "run (DDS x) = x"
QTrans: shows "Q \<Longrightarrow>\<^sub>l\<^sup>^a[b] \<prec> Q'"
vertex_weight_update: fixes f :: "('a \<Rightarrow> ennreal) \<Rightarrow> 'a \<Rightarrow> ennreal"   and \<Gamma> :: "('a, 'b) web_scheme" shows "vertex (weight_update f \<Gamma>) = vertex \<Gamma>"
proper_designII: fixes V :: "'a set"   and B :: "'a set multiset" assumes "design V B"   and "B \<noteq> {#}" shows "proper_design V B"