hcju/isabelleps huggingface dataset:
intro: 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" assumes "bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"   and "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"   and "equivalence_pseudofunctor 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"   and "pseudofunctor 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 G \<Phi>\<^sub>G"   and "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" shows "pseudofunctor_pseudonaturally_equivalent_to_equivalence_pseudofunctor 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"
coinduct: fixes X :: "('a, 'b, 'c, 'd) converter \<Rightarrow> ('e, 'f, 'g, 'h) converter \<Rightarrow> bool"   and x1 :: "('a, 'b, 'c, 'd) converter"   and x2 :: "('e, 'f, 'g, 'h) converter"   and A :: "'a \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'f \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'d \<Rightarrow> 'h \<Rightarrow> bool" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>conv1 conv2. x1 = conv1 \<and> x2 = conv2 \<and> rel_fun A (rel_gpv'' (rel_prod B (\<lambda>uu_ uua_. X uu_ uua_ \<or> rel_converter A B C R uu_ uua_)) C R) (run_converter conv1) (run_converter conv2)" shows "rel_converter A B C R x1 x2"
if_eval\<^sub>w: fixes b :: "bexp"   and t :: "(Eg1.var, aexp, bexp) Stmt"   and e :: "(Eg1.var, aexp, bexp) Stmt"   and mds :: "Mode \<Rightarrow> Eg1.var set"   and mem :: "Eg1.var \<Rightarrow> nat" shows "eval_abv\<^sub>A \<langle>Stmt.If b t e, mds, mem\<rangle>\<^sub>A \<langle>if ev\<^sub>B mem b then t else e, mds, mem\<rangle>\<^sub>A"
well_order_on_singleton: fixes x :: "'d" shows "well_order_on {x} {(x, x)}"
for_rec2'_for_rec: fixes f :: "'b \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b nres"   and a :: "'b"   and n :: "nat" shows "for_rec2' f a n n n = for_rec (\<lambda>a i. for_rec (\<lambda>a. f a i) a n) a n"
nb\<^sub>1\<^sub>4: fixes v :: "'a"   and a :: "'b" assumes "(v, a) \<in> dom ((\<phi>\<^sub>S \<Psi> s) ++ map_of (effect_to_assignments op'))"   and "(v, a) \<notin> dom (map_of (effect_to_assignments op'))" shows "((\<phi>\<^sub>S \<Psi> s) ++ map_of (effect_to_assignments op')) (v, a) = Some (the (s v) = a)"
dvd_smult: fixes p :: "'b poly"   and q :: "'b poly"   and a :: "'b" assumes "p dvd q" shows "p dvd smult a q"
field_differentiable_within_open: fixes a :: "'a"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "a \<in> S"   and "open S" shows "(f field_differentiable at a within S) = (f field_differentiable at a)"
MemoryInvariant: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and l :: "Locs" shows "\<turnstile> MSpec ch mm rs l \<longrightarrow> \<box>MemInv mm l"
invertible_right_inverse: fixes u :: "'a" assumes "IA.additive.invertible u"   and "u \<in> A" shows "IA.subtraction u u = zeroA"
take_append: fixes n :: "nat"   and xs :: "'d list"   and ys :: "'d list" shows "take n (xs @ ys) = take n xs @ take (n - length xs) ys"
bcontfun_def: shows "bcontfun = {f. continuous_on UNIV f \<and> bounded (range f)}"
U_3: fixes i :: "nat" shows "closure (U i) \<subseteq> carrier"
left_diff_distrib: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.ring plus zero minus uminus times" shows "times (minus a b) c = minus (times a c) (times b c)"
shortest_paths_locale_step2_def: fixes G :: "('c, 'd) pre_digraph"   and s :: "'c"   and c :: "'d \<Rightarrow> real"   and num :: "'c \<Rightarrow> nat"   and parent_edge :: "'c \<Rightarrow> 'd option"   and dist :: "'c \<Rightarrow> ereal" shows "shortest_paths_locale_step2 G s c num parent_edge dist \<equiv> shortest_paths_locale_step1 G s num parent_edge dist \<and> basic_just_sp G dist c s (shortest_paths_locale_step1.enum num dist) \<and> shortest_paths_locale_step2_axioms G s num dist"
insert_absorb2: fixes x :: "'c"   and A :: "'c set" shows "insert x (insert x A) = insert x A"
weight_gt_0: fixes t :: "('f, 'b) Term.term" shows "0 < weight t"
pcr_spmf_gpv_def: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "pcr_spmf_gpv A = cr_spmf_gpv OO rel_gpv A (=)"
lunit'_naturality: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "(trg \<mu> \<star> \<mu>) \<cdot> \<l>\<^sup>-\<^sup>1[local.dom \<mu>] = \<l>\<^sup>-\<^sup>1[cod \<mu>] \<cdot> \<mu>"
rtrancl_image_unfold_right: fixes E :: "('a \<times> 'a) set"   and V :: "'a set" shows "E `` E\<^sup>* `` V \<subseteq> E\<^sup>* `` V"
simps_5: fixes var :: "nat"   and \<phi> :: "atom fm"   and \<psi> :: "atom fm"   and \<Gamma> :: "real list" shows "convert_fm var (Or \<phi> \<psi>) \<Gamma> = (case (convert_fm var \<phi> \<Gamma>, convert_fm var \<psi> \<Gamma>) of (None, ba) \<Rightarrow> None | (Some a, None) \<Rightarrow> None | (Some a, Some b) \<Rightarrow> Some (OrUni a b))"
set_3: fixes x31 :: "'d"   and x32 :: "'d ifex"   and x33 :: "'d ifex" shows "set_ifex (IF x31 x32 x33) = insert x31 (set_ifex x32 \<union> set_ifex x33)"
div_add1_eq2: fixes m :: "nat"   and a :: "nat"   and b :: "nat" assumes "0 < m"   and "m \<le> a mod m + b mod m" shows "(a + b) div m = Suc (a div m + b div m)"
c: shows "r3 = (extend p q # ps, c) \<and> r3 \<in> upRules"
intF_def: fixes f :: "'fsym"   and al :: "univ list" shows "intF f al \<equiv> eintF (Oldf f) al"
assms_3: shows "B = \<GG>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>"
DiffE: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "c \<in> A - B" obtains "c \<in> A" and "c \<notin> B"
weak_ranking_aux_unique: fixes As :: "'a set list" assumes "is_weak_ranking As"   and "of_weak_ranking As = le" shows "As = weak_ranking_aux carrier"
l_repr_independence: fixes G :: "('d, 'e) monoid_scheme"   and y :: "'d"   and x :: "'d"   and H :: "'d set" assumes "Group.group G"   and "y \<in> x <#\<^bsub>G\<^esub> H"   and "x \<in> carrier G"   and "subgroup H G" shows "x <#\<^bsub>G\<^esub> H = y <#\<^bsub>G\<^esub> H"
triv_admI: fixes P :: "'d \<Rightarrow> bool" assumes "\<forall>x. P x" shows "adm P"
Times_empty: fixes A :: "'e set"   and B :: "'f set" shows "(A \<times> B = {}) = (A = {} \<or> B = {})"
power_inject_exp: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a" shows "(power.power one times a m = power.power one times a n) = (m = n)"
halloc_no_jump': fixes G :: "prog"   and s0 :: "state"   and oi :: "obj_tag"   and a :: "loc"   and s1 :: "state"   and j :: "jump" assumes "G\<turnstile>s0 \<midarrow>halloc oi\<succ>a\<rightarrow> s1"   and "fst s1 = Some (Jump j)" shows "fst s0 = Some (Jump j)"
optimize_matches_option_append: fixes f :: "'c match_expr \<Rightarrow> 'c match_expr option"   and rs1 :: "'c rule list"   and rs2 :: "'c rule list" shows "optimize_matches_option f (rs1 @ rs2) = optimize_matches_option f rs1 @ optimize_matches_option f rs2"
is_Arg_eqI: fixes z :: "complex"   and r :: "real"   and s :: "real" assumes "is_Arg z r"   and "is_Arg z s"   and "\<bar>r - s\<bar> < 2 * pi"   and "z \<noteq> 0" shows "r = s"
facet_of_polyhedron: fixes S :: "'b set"   and C :: "'b set" assumes "polyhedron S"   and "C facet_of S" obtains a :: "'b" and b :: "real" where "a \<noteq> (0::'b)" and "S \<subseteq> {x. a \<bullet> x \<le> b}" and "C = S \<inter> {x. a \<bullet> x = b}"
invariance_of_dimension_convex_domain: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "convex S"   and "continuous_on S f"   and "f ` S \<subseteq> affine hull T"   and "inj_on f S" shows "aff_dim S \<le> aff_dim T"
iso4: fixes x :: "'a" shows "d (x \<cdot> 1\<^sub>\<pi> \<parallel> U) \<cdot> U = x \<cdot> 1\<^sub>\<pi> \<parallel> U"
Unique_def: fixes ev :: "event"   and evs :: "event list" shows "Unique ev on evs = (ev \<notin> set (tl (dropWhile (\<lambda>z. z \<noteq> ev) evs)))"
bl_word_xor: fixes v :: "'b word"   and w :: "'b word" shows "to_bl (v XOR w) = map2 (\<noteq>) (to_bl v) (to_bl w)"
all_ptrs_in_heap_def: shows "Shadow_DOM.all_ptrs_in_heap \<equiv> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_all_ptrs_in_heap get_shadow_root"
lincomb_empty: fixes a :: "'e \<Rightarrow> 'a" shows "W.module.lincomb a {} = \<zero>\<^bsub>W\<^esub>"
find_elem_correct: fixes x :: "int"   and xs :: "int list" shows "find_elem x xs \<le> SPEC (\<lambda>i. i \<le> length xs \<and> (i < length xs \<longrightarrow> xs ! i = x))"
one_less_numeral_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less one (numeral.numeral one plus n) = (num.One < n)"
bprv_\<phi>G_eqv: shows "bprv (eqv \<phi>G (neg (wrepr.PP \<langle>\<phi>G\<rangle>)))"
injectivity: fixes x :: "real poly"   and y :: "real poly" assumes "map_poly complex_of_real x = map_poly complex_of_real y" shows "x = y"
limites_def: fixes T :: "'b set set"   and F :: "'b set set" shows "lims\<^bsub>T\<^esub> F = {x \<in> carrier\<^bsub>T\<^esub>. F \<longlongrightarrow>\<^bsub>T\<^esub> x}"
lemma4_aux: fixes rs :: "lam list"   and rs' :: "lam list"   and ss :: "lam list" assumes "listrelp (\<lambda>t u. t \<rightarrow>\<^sub>s u \<and> (\<forall>r. u \<rightarrow>\<^sub>\<beta> r \<longrightarrow> t \<rightarrow>\<^sub>s r)) rs rs'"   and "rs' [\<rightarrow>\<^sub>\<beta>]\<^sub>1 ss" shows "rs [\<rightarrow>\<^sub>s] ss"
mono_Ndet: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "(P |-| S) \<sqsubseteq> (Q |-| S)"
soundness: fixes p :: "('c, 'd) fm"   and E :: "nat \<Rightarrow> 'e"   and F :: "'c \<Rightarrow> 'e list \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e list \<Rightarrow> bool" assumes "\<turnstile> p" shows "\<lbrakk>E, F, G\<rbrakk> p"
linear_add: fixes f :: "'b \<Rightarrow> 'c"   and b1 :: "'b"   and b2 :: "'b" assumes "clinear f" shows "f (b1 + b2) = f b1 + f b2"
arityOf_lt_var_INNER: fixes delta :: "'opSym" assumes "wlsOpS delta" shows "|{i. arOf delta i \<noteq> None}| <o |UNIV|"
len_non_empty_inside: fixes v :: "view"   and ts :: "traffic"   and c :: "cars" shows "0 < \<parallel>len v ts c\<parallel> \<longrightarrow> left (space ts v c) < right (ext v) \<and> left (ext v) < right (space ts v c)"
sub: shows "(\<Union>x\<in>{x. Inl (Inr x) \<in> SIGP \<alpha>}. FVT (\<sigma> x)) \<subseteq> - U"
imp_imp_compat_eqv: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi>1 :: "'fmla"   and \<chi>2 :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla" shows "prv (imp (eqv \<phi>1 \<phi>2) (imp (eqv \<chi>1 \<chi>2) (eqv (imp \<phi>1 \<chi>1) (imp \<phi>2 \<chi>2))))"
wt_e1: shows "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e1\<Colon>-e1T"
prism_lens_def: fixes X :: "'c \<Longrightarrow>\<^sub>\<triangle> 'd" shows "prism_lens X = \<lparr>lens_get = \<lambda>s. the (match\<^bsub>X\<^esub> s), lens_put = \<lambda>s. build\<^bsub>X\<^esub>\<rparr>"
is_empty_stack_impl_def: fixes s :: "('e, 'f) simple_state_scheme" shows "is_empty_stack_impl s \<equiv> ss_stack s = []"
root_cancel: fixes d :: "nat"   and n :: "nat"   and k :: "nat" assumes "0 < d" shows "FFT.root (d * n) ^ (d * k) = FFT.root n ^ k"
maxcoeff_nz: fixes t :: "Ferrack.num" assumes "nozerocoeff t"   and "maxcoeff t = 0" shows "t = C 0"
fls_X_inv_def: shows "fls_X_inv \<equiv> Abs_fls (\<lambda>n. if n = - 1 then 1::'a else (0::'a))"
igConsIPresIGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b, 'c, 'd) model_scheme" shows "igConsIPresIGWls MOD \<equiv> igVarIPresIGWls MOD \<and> igAbsIPresIGWls MOD \<and> igOpIPresIGWls MOD"
cut_ge_Max_eq: fixes I :: "'b set"   and t :: "'b" assumes "finite I"   and "I \<down>\<ge> t \<noteq> {}" shows "Max (I \<down>\<ge> t) = Max I"
total_lexord: fixes r :: "('b \<times> 'b) set" assumes "total r" shows "total (lexord r)"
CLS_chine: shows "\<lbrakk>\<lbrakk>TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = tuple_ABC"
neutr_eq_iff: fixes a :: "'d multiset"   and b :: "'d multiset" shows "({#} = a \<union># b) = (a = {#} \<and> b = {#})"
infdist_triangle: fixes x :: "'c"   and A :: "'c set"   and y :: "'c" shows "infdist x A \<le> infdist y A + dist x y"
ind_hyp_l1: fixes l2 :: "'a list" assumes "Sorted_Less.sorted (rev l2)" shows "Sorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \<and> set (inter_sorted_rev l1 l2) = set l1 \<inter> set l2"
set1_F_bound: fixes x :: "('n, 'v, 'w, 'o, 'p, 'q, 'r, 's, 't, 'u) F" shows "|set1_F x| \<le>o bd_F"
minus_power_mult_self: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "times (power.power one times (uminus a) n) (power.power one times (uminus a) n) = power.power one times a (2 * n)"
size_Un_disjoint: fixes A :: "'a multiset"   and B :: "'a multiset" assumes "A \<inter># B = {#}" shows "size (A \<union># B) = size A + size B"
in_mset_sum_list: fixes L :: "'b"   and LL :: "'b multiset"   and Ci :: "'b multiset list" assumes "L \<in># LL"   and "LL \<in> set Ci" shows "L \<in># sum_list Ci"
other: fixes \<sigma> :: "'a"   and p :: "('b, 'c, 'd, 'e) seqp"   and \<sigma>' :: "'a" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "\<forall>l\<in>labels \<Gamma> p. P (\<sigma>, l)"   and "U \<sigma> \<sigma>'" shows "\<forall>l\<in>labels \<Gamma> p. P (\<sigma>', l)"
Suc: shows "higher_differentiable_on UNIV f k \<and> (\<forall>t\<le>0. nth_derivative k f t 1 = 0) \<and> (\<exists>p. real_polynomial_function p \<and> (\<forall>t>0. nth_derivative k f t 1 = p t / t ^ (2 * k) * exp (- inverse t)))"
Confidentiality_A_temporal: fixes A :: "agent"   and T :: "nat"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Number T, Agent B, Key K, X\<rbrace> \<in> parts (knows Spy evs)"   and "\<not> expiredK T evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerb_gets" shows "Key K \<notin> analz (knows Spy evs)"
in_dom: fixes x :: "'b" assumes "x \<in> S" shows "Domainp A (f x)"
a: shows "\<forall>x. \<not> t x < s x \<or> (\<exists>y<x. t y \<noteq> s y)"
DDH1_def: fixes \<G> :: "'b cyclic_group"   and \<A> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool spmf" shows "ddh.DDH1 \<G> \<A> = sample_uniform (order \<G>) \<bind> (\<lambda>s. sample_uniform (order \<G>) \<bind> (\<lambda>r. let h = \<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> s in \<A> h (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> r) (h [^]\<^bsub>\<G>\<^esub> r \<otimes>\<^bsub>\<G>\<^esub> \<^bold>g\<^bsub>\<G>\<^esub>)))"
matchers_insert: fixes e :: "('e, 'd) Term.term \<times> ('e, 'f) Term.term"   and P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) set" shows "matchers (insert e P) = {\<sigma>. fst e \<cdot> \<sigma> = snd e} \<inter> matchers P"
leftmost_def: fixes i :: "nat"   and s :: "symbol list" shows "leftmost i s = (i < length s \<and> is_word (take i s) \<and> is_nonterminal (s ! i))"
vec_lambda_inject: fixes x :: "'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'c" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(vec_lambda x = vec_lambda y) = (x = y)"
assms_1: shows "(\<Psi>, P, Q) \<in> Rel"
lemma_2_6_21_a: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a r\<rightarrow> b \<sqinter> c = (a r\<rightarrow> b) \<sqinter> (a r\<rightarrow> c)"
norm_prime_divisor: fixes q :: "gauss_int" assumes "prime_elem q"   and "q dvd of_nat p" shows "gauss_int_norm q = p"
assms: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', \<tau>.P, P \<oplus>\<^sub>\<top> \<tau>.P) \<in> Rel"
rt_fresh_asD2: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "rt1 \<approx>\<^bsub>dip\<^esub> rt2" shows "rt2 \<sqsubseteq>\<^bsub>dip\<^esub> rt1"
norm_1_bound_mignotte: fixes f :: "int poly" shows "real_of_int (norm1 f) \<le> 2 ^ degree f * mahler_measure f"
perm_restrict_simps_1: fixes x :: "'b"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'b" assumes "x \<in> S" shows "perm_restrict f S x = f x"
Gateway_L0: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list" assumes "Gateway req dt a stop lose d ack i vc" shows "GatewayReq req dt a stop lose d ack i vc"
ennreal_mono_minus: fixes c :: "ennreal"   and b :: "ennreal"   and a :: "ennreal" assumes "c \<le> b" shows "a - b \<le> a - c"
AsCatchExpr1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and Q :: "'e set"   and a2 :: "('e, 'd, 'f) ann"   and a1 :: "('e, 'd, 'f) ann"   and c1 :: "('e, 'd, 'f) com"   and a :: "'e set"   and A :: "'e set"   and c2 :: "('e, 'd, 'f) com" assumes "assertionsR \<Gamma> \<Theta> Q (pre a2) a1 c1 a" shows "assertionsR \<Gamma> \<Theta> Q A (AnnComp a1 a2) (Catch c1 c2) a"
polychain_def: fixes xs :: "('a \<times> 'a) list" shows "polychain xs = (\<forall>i. Suc i < length xs \<longrightarrow> snd (xs ! i) = fst (xs ! Suc i))"
Suc: shows "d_OUT (h_plus i) x \<le> d_IN (h_plus i) x"
inv_correspondence_conv_apply: fixes G :: "'a set"   and binop :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and e :: "'a"   and x :: "'a permutation" assumes "BinOpSetGroup G binop e"   and "x \<in> BinOpSetGroup.Abs_G_perm G binop ` G" shows "the_inv_into G (BinOpSetGroup.Abs_G_perm G binop) x = x \<rightarrow> e"
size_neq: fixes x :: "'b ptrm" shows "size x \<noteq> 0"
refl_on_def': fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "refl_on A r = ((\<forall>(x, y)\<in>r. x \<in> A \<and> y \<in> A) \<and> (\<forall>x\<in>A. (x, x) \<in> r))"
assms_1: shows "CT;Map.empty \<turnstile> e : C"
a1: shows "sqrt (a\<^sup>2 + b\<^sup>2) * 2 < 1"
reflp_on_subset: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "A \<subseteq> B"   and "reflp_on P B" shows "reflp_on P A"
AbGroup_length_inner_dirsum_el_decomp: fixes Gs :: "'a set list"   and x :: "'a" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "x \<in> (\<Oplus>G\<leftarrow>Gs. G)" shows "length (\<Oplus>Gs\<leftarrow> x) = length Gs"
exists_map_of: fixes m :: "'a \<Rightarrow> 'b option" assumes "finite (dom m)" shows "\<exists>xs. map_of xs = m"
well_constrained_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and C :: "'b eclause" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.well_constrained trm_ord C = (\<forall>y. y \<in> trms_ecl C \<longrightarrow> basic_superposition.dom_trm trm_ord y (cl_ecl C))"
FullStruct_axioms: shows "FFullStruct intT intF intP"
phl_cons1: fixes x :: "'a"   and w :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> w"   and "\<lparr>w\<rparr>y\<lparr>y \<cdot> z\<rparr>" shows "\<lparr>x\<rparr>y\<lparr>y \<cdot> z\<rparr>"
base_simps_2: shows "hseq g f\<^sup>*"
typing_alt_simps_14: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type"   and op :: "metaop"   and \<sigma> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E MetaOperationCall \<tau> op : \<sigma>) = mataop_type \<tau> op \<sigma>"
five_segment_with_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D' :: "'p" assumes "A B C D OFSC A' B' C' D'"   and "A \<noteq> B" shows "Cong C D C' D'"
set: fixes x1 :: "'a"   and x2 :: "'a tree list" shows "set_tree (NODE x1 x2) = insert x1 (\<Union> (set_tree ` set x2))"
vec_times_mat_eqD_1: fixes y :: "'a vec"   and A :: "'a mat"   and c :: "'a vec" assumes "[y\<^sub>v*A]=c" shows "\<forall>i<dim_vec c. (A\<^sup>T *\<^sub>v y) $ i = c $ i"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "prefixes (x # xs) = [] # map ((#) x) (prefixes xs)"
sigma_finite_interval_measure: fixes F :: "real \<Rightarrow> real" assumes "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"   and "\<And>a. continuous (at_right a) F" shows "sigma_finite_measure (interval_measure F)"
path_not_Nil: fixes g :: "'g"   and ns :: "'node list" assumes "old.path g ns" shows "ns \<noteq> []"
closed_interval_as_convex_hull: fixes a :: "'c"   and b :: "'c" obtains S :: "'c set" where "finite S" and "cbox a b = convex hull S"
pdg_digraph_dg_prod: shows "digraph \<alpha> (\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>ZFC_in_HOL.set {j}. \<CC>)"
cs_sym: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "relation.cs r x y" shows "relation.cs r y x"
closed_subset_Ints: fixes A :: "'a set" assumes "A \<subseteq> \<int>" shows "closed A"
sum_nonpos: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># {#}" shows "comm_monoid_add.sum (+) {#} f A \<subseteq># {#}"
WT_restrict_gpv: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv" shows "\<I> \<turnstile>g restrict_gpv \<I> gpv \<surd>"
sorted_take: fixes xs :: "'a list"   and n :: "nat" assumes "sorted xs" shows "sorted (take n xs)"
aval_afold: fixes t :: "char list \<Rightarrow> int option"   and s :: "char list \<Rightarrow> int"   and a :: "aexp" assumes "approx t s" shows "aval (afold a t) s = aval a s"