hcju/isabelleps huggingface dataset:
Substable_intro_impl: fixes cond :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<psi> :: "'a \<Rightarrow> \<o>"   and \<chi> :: "'a \<Rightarrow> \<o>" assumes "Substable cond \<psi>"   and "Substable cond \<chi>" shows "Substable cond (\<lambda>\<phi>. \<psi> \<phi> \<^bold>\<rightarrow> \<chi> \<phi>)"
wf_word: fixes n :: "nat"   and w :: "'a list" shows "wf_word n w = (w \<in> lists (\<Sigma> n))"
bounded_subset_ballD: fixes S :: "'c set"   and x :: "'c" assumes "bounded S" shows "\<exists>r>0. S \<subseteq> ball x r"
delete_edges_simp2: fixes G :: "'c graph"   and E :: "('c \<times> 'c) set" shows "FiniteGraph.delete_edges G E = \<lparr>nodes = nodes G, edges = edges G - E\<rparr>"
ru_6: fixes y :: "real" shows "(y = real_of_1 (p, l, r)) = root_cond (p, l, r) y"
linear: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.linorder less_eq less" shows "less_eq x y \<or> less_eq y x"
hyps_3: shows "val t\<^sub>1 \<or> Ex (eval t\<^sub>1)"
components_are_iso: fixes a :: "'a" assumes "A.ide a" shows "A.iso (A.map a)"
cblinfun_norm_approx_witness_mult: fixes \<epsilon> :: "real"   and A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "\<epsilon> < 1" shows "\<exists>\<psi>. norm A * \<epsilon> \<le> norm (A *\<^sub>V \<psi>) \<and> norm \<psi> = 1"
content_ball: fixes r :: "real"   and c :: "'b" assumes "0 \<le> r" shows "content (ball c r) = unit_ball_vol (real DIM('b)) * r ^ DIM('b)"
AlphaClosure_def: fixes X :: "'d set" shows "AlphaClosure X \<equiv> finite X"
scaleC: fixes f :: "'a \<Rightarrow> 'b"   and r :: "complex"   and x :: "'a" assumes "antilinear f" shows "f (r *\<^sub>C x) = cnj r *\<^sub>C f x"
assms_2: shows "A \<TTurnstile>\<^sub>A (I \<rightarrow>) onll \<Gamma> P"
inverse_mult_fds: fixes f :: "'b fds"   and g :: "'b fds" shows "inverse (f * g) = inverse f * inverse g"
l_set_disconnected_nodes_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<equiv> l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs \<and> l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
assms_3: shows "fresh zs z (X #[Y / y]_ys)"
non_empty_cycle_root_subset_equals_change_root: fixes r :: "'a"   and x :: "'a"   and q :: "'a"   and y :: "'a" assumes "non_empty_cycle_root r x"   and "non_empty_cycle_root q y"   and "x \<le> y" shows "x = y"
sets_distr: fixes M :: "'b measure"   and N :: "'a measure"   and f :: "'b \<Rightarrow> 'a" shows "sets (distr M N f) = sets N"
vars1'_def: shows "b.vars1' = ind_in_set b.vars0 ` (vars1 \<union> vars2)"
consistent_prfOf_iff_PPf: fixes "prf" :: "'proof"   and \<phi> :: "'fmla" assumes "consistent"   and "prf \<in> proof"   and "\<phi> \<in> fmla" shows "Fvars \<phi> = {} \<longrightarrow> prfOf prf \<phi> = bprv (PPf (encPf prf) \<langle>\<phi>\<rangle>)"
mcont_const: fixes leq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lub :: "'b set \<Rightarrow> 'b"   and luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'b" assumes "partial_function_definitions leq lub" shows "mcont luba orda lub leq (\<lambda>x. c)"
bin_lcp_pref_snd_fst: shows "\<alpha> \<le>p u\<^sub>1 \<cdot> u\<^sub>0"
resource_of_oracle_run_resource: fixes res :: "('a, 'b) resource" shows "RES run_resource res = res"
seqI: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr f"   and "Span.arr g"   and "Span.dom g = Span.cod f" shows "Span.seq g f"
gcd_diff1: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and m :: "'a"   and n :: "'a" assumes "class.ring_gcd one zero times gcd lcm divide plus minus unit_factor normalize uminus" shows "gcd (minus m n) n = gcd m n"
nn_integral_fst_count_space: fixes f :: "'c \<times> 'd \<Rightarrow> ennreal" shows "\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV = integral\<^sup>N (count_space UNIV) f"
strips_op_to_sasp_def: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op :: "('c \<times> 'd) strips_operator" shows "\<phi>\<^sub>O\<inverse> \<Psi> op \<equiv> let precondition = strips_operator.precondition_of op; effect = add_effects_of op in \<lparr>sas_plus_operator.precondition_of = precondition, effect_of = effect\<rparr>"
intro: fixes r :: "('a \<times> 'a) set" assumes "Linear_order r" shows "MaxR r"
filter_mset_cong: fixes M :: "'a multiset"   and M' :: "'a multiset"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "M = M'"   and "\<And>a. a \<in># M \<Longrightarrow> P a = Q a" shows "filter_mset P M = filter_mset Q M"
cancel_le: fixes F :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal"   and y :: "ordinal" assumes "normal F" shows "(F x \<le> F y) = (x \<le> y)"
sprod_one: fixes M :: "('h, 'i, 'j) Module_scheme"   and R :: "('i, 'k) Ring_scheme"   and m :: "'h" assumes "R module M"   and "m \<in> carrier M" shows "1\<^sub>r\<^bsub>R\<^esub> \<cdot>\<^sub>s\<^bsub>M\<^esub> m = m"
bisimWeakCoinductAux: fixes P :: "ccs"   and Q :: "ccs"   and X :: "(ccs \<times> ccs) set" assumes "(P, Q) \<in> X"   and "\<And>R S. (R, S) \<in> X \<Longrightarrow> R \<leadsto>[X] S \<and> (S, R) \<in> X" shows "P \<sim> Q"
newton_seq_fact5: fixes p :: "int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and l :: "nat" assumes "hensel p f a" shows "padic_integers.divide p (UP_cring.to_fun (padic_int p) f (hensel.newton_seq p f a l)) (UP_cring.to_fun (padic_int p) (UP_cring.pderiv (padic_int p) f) (hensel.newton_seq p f a l)) \<in> carrier (padic_int p)"
axioms_1: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and altc :: "'c cset \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_altc return bind altc" shows "monad return bind"
permute_list_def: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'c list" shows "permute_list f xs = map (\<lambda>i. xs ! f i) [0..<length xs]"
strict_implies_order: fixes b :: "'a option \<Rightarrow>\<^sub>0 nat"   and a :: "'a option \<Rightarrow>\<^sub>0 nat" assumes "extended_ord_strict b a" shows "extended_ord b a"
simps_1: fixes c :: "'b"   and c' :: "'b" shows "(c)\<^sub>p +\<^sub>p (c')\<^sub>p = (c + c')\<^sub>p"
Image_Un: fixes R :: "('d \<times> 'c) set"   and A :: "'d set"   and B :: "'d set" shows "R `` (A \<union> B) = R `` A \<union> R `` B"
mat_mult_plus_distrib_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and nr :: "nat"   and nc :: "nat"   and m1 :: "'c list list"   and m2 :: "'c list list"   and ncc :: "nat"   and m3 :: "'c list list" assumes "class.semiring_0 plus zero times"   and "mat nr nc m1"   and "mat nr nc m2"   and "mat nc ncc m3" shows "mat_multI zero plus times nr (mat_plusI plus m1 m2) m3 = mat_plusI plus (mat_multI zero plus times nr m1 m3) (mat_multI zero plus times nr m2 m3)"
simps_3: fixes p :: "int \<times> 'a"   and q :: "int \<times> 'a" shows "del None (Branch3 Empty p Empty q Empty) = Some (p, False, Branch2 Empty q Empty)"
slo_defs_1: shows "Lin \<equiv> {L. strict_linear_order L}"
pnop_n: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "pnop A n f" shows "n \<in>\<^sub>\<circ> \<omega>"
max_number_of_1: fixes u :: "num"   and v :: "num" shows "max (numeral u) (numeral v) = (if numeral u \<le> numeral v then numeral v else numeral u)"
distinct_199: fixes x81 :: "Syntax.term"   and x82 :: "Syntax.var"   and x83 :: "Syntax.term"   and x101 :: "Syntax.term"   and x102 :: "Syntax.term"   and x103 :: "Syntax.term" shows "Syntax.Let x81 x82 x83 \<noteq> Case x101 x102 x103"
Beta_plus1_left: fixes x :: "'a"   and y :: "'a" assumes "x \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "(x + y) * Beta (x + (1::'a)) y = x * Beta x y"
add: fixes b1 :: "'a"   and b2 :: "'a" shows "h (b1 + b2) = h b1 + h b2"
alpha_A': fixes A :: "type_scheme list"   and n :: "nat" shows "$ (\<lambda>x. TVar (if x \<in> free_tv A then x else n + x)) A = $ id_subst A"
scaleC_le_cancel_left: fixes c :: "complex"   and a :: "'b"   and b :: "'b" assumes "c \<in> \<real>" shows "(c *\<^sub>C a \<le> c *\<^sub>C b) = ((0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a))"
Dag_dag: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag" assumes "Dag p l r t" shows "dag p l r = t"
fst_vimage_eq_Times: fixes S :: "'b set" shows "fst -` S = S \<times> UNIV"
map_G_mapl_G: fixes l1 :: "'u \<Rightarrow> 'n"   and l2 :: "'v \<Rightarrow> 'o"   and co1 :: "'w \<Rightarrow> 'p"   and co2 :: "'x \<Rightarrow> 'q"   and contra1 :: "'r \<Rightarrow> 'y"   and contra2 :: "'s \<Rightarrow> 'z"   and x :: "('u, 'v, 'w, 'x, 'y, 'z, 't) G" shows "map_G l1 l2 co1 co2 contra1 contra2 x = map_G id id co1 co2 contra1 contra2 (mapl_G l1 l2 x)"
finalD: fixes c :: "com"   and s :: "char list \<Rightarrow> int" assumes "final (c, s)" shows "c = SKIP"
indiff_trans: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "preference carrier relation"   and "x \<approx>[relation] y"   and "y \<approx>[relation] z" shows "x \<approx>[relation] z"
lim_Ln_over_power: fixes s :: "complex" assumes "0 < Re s" shows "(\<lambda>n. Ln (of_nat n) / of_nat n powr s) \<longlonglongrightarrow> 0"
no0_2: shows "coeff lp' x \<noteq> 0"
mgp_dist_facesAt: fixes g :: "graph"   and v :: "nat" assumes "minGraphProps g"   and "v \<in> \<V> g" shows "distinct (facesAt g v)"
igAbsCongS_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igAbsCongS MOD \<equiv> \<forall>xs x x' y s X X'. isInBar (xs, s) \<and> igWls MOD s X \<and> igWls MOD s X' \<longrightarrow> igFresh MOD xs y X \<and> igFresh MOD xs y X' \<and> igSwap MOD xs y x X = igSwap MOD xs y x' X' \<longrightarrow> igAbs MOD xs x X = igAbs MOD xs x' X'"
NS_least_least: fixes f :: "'f"   and t :: "('f, 'b) Term.term" assumes "least f"   and "NS (Fun f []) t" shows "\<exists>g. t = Fun g [] \<and> least g"
bounded: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_antilinear f" shows "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
IH: assumes "crdt_ops (xs @ [(oid, oper)] @ ys) deps" shows "oid \<notin> set (map fst xs) \<and> oid \<notin> set (map fst ys)"
BPd'_simps_3: fixes d :: "'b" shows "(d \<notin> {Xd d1h1, Xd d3h1}) = (BPd' d = {})"
HPhase0Read_HInv4a: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase0Read s s' p d"   and "HInv4a s q" shows "HInv4a s' q"
pseudo_proj_preimage': fixes g :: "bool stream \<Rightarrow> 'a"   and n :: "nat"   and N :: "'a measure"   and z :: "bool stream" assumes "g \<in> F n \<rightarrow>\<^sub>M N"   and "set_discriminating n g N" shows "pseudo_proj_True n -` g -` {g z} = g -` {g z}"
loxodromic_isometryD_3: fixes f :: "'b \<Rightarrow> 'b" assumes "loxodromic_isometry f" shows "0 < stable_translation_length f"
lang_eq_ext_Nil_fold_Deriv: fixes K :: "'b language"   and A :: "'b set"   and L :: "'c language"   and B :: "'c set"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>w. in_language K w \<Longrightarrow> w \<in> lists A"   and "\<And>w. in_language L w \<Longrightarrow> w \<in> lists B"   and "\<And>a b. R a b \<Longrightarrow> (a \<in> A) = (b \<in> B)" shows "rel_language R K L = (\<forall>(K, L)\<in>{(\<dd>s w1 K, \<dd>s w2 L) |w1 w2. w1 \<in> lists A \<and> w2 \<in> lists B \<and> list_all2 R w1 w2}. \<oo> K = \<oo> L)"
pred_intros_conj1': fixes K :: "bool"   and P :: "'c \<Rightarrow> bool"   and M :: "'c measure" assumes "K \<Longrightarrow> pred M P" shows "pred M (\<lambda>x. K \<and> P x)"
\<alpha>ah_ileq: fixes w1 :: "('b set \<times> 'b set) list"   and w2 :: "('b set \<times> 'b set) list" assumes "w1 \<preceq> w2" shows "\<alpha>ah w1 \<le> \<alpha>ah w2"
rotate_id: fixes n :: "nat"   and xs :: "'g list" assumes "n mod length xs = 0" shows "rotate n xs = xs"
lprefix_conv_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" shows "lprefix xs ys = (\<exists>zs. ys = xs $ zs)"
dem_def: fixes mac_alg :: "'b \<Rightarrow> 'a \<Rightarrow> 'c spmf" shows "macode.dem mac_alg \<equiv> CNV (stateless_callee (\<lambda>_. Pause (Inr ()) (\<lambda>am'. case am' of Inl aa \<Rightarrow> Done None | Inr None \<Rightarrow> Done None | Inr (Some (a, m)) \<Rightarrow> Pause (Inl m) Done \<bind> (\<lambda>r. lift_spmf (mac_alg (projl r) m) \<bind> (\<lambda>a'. Done (if a' = a then Some m else None)))))) ()"
o_apply: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "(f \<circ> g) x = f (g x)"
wf_digraph_wp_iff: fixes G :: "'c pair_pre_digraph" shows "wf_digraph (with_proj G) = pair_wf_digraph G"
seq_char: fixes g :: "'a"   and f :: "'a" shows "Right_a.seq g f = (Right_a.arr f \<and> Right_a.arr g \<and> seq g f)"
S_def: shows "S \<equiv> \<Sum>k = a..b. f (real_of_int k)"
new_document_ptr_not_in_heap: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_document_ptr :: "(_) document_ptr" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr" shows "new_document_ptr |\<notin>| document_ptr_kinds h"
unfold_nat_to_bv_helper: fixes b :: "nat"   and l :: "bit list" shows "nat_to_bv_helper b l = nat_to_bv_helper b [] @ l"
card_left_less_pair: fixes A :: "'a set" assumes "finite A" shows "card {(a, b). a \<in> A \<and> b \<in> A \<and> a < b} = card A * (card A - 1) div 2"
permute_set_eq: fixes p :: "perm"   and X :: "'a set" shows "p \<bullet> X = {x. - p \<bullet> x \<in> X}"
B1: shows "\<forall>S. (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel"
f'_def: shows "f' \<equiv> \<lambda>k. k # replicate (a k) 0 @ [1] \<odot> 0\<^sup>\<infinity>"
path_image_part_circlepath_subset': fixes r :: "real"   and z :: "complex"   and s :: "real"   and t :: "real" assumes "0 \<le> r" shows "path_image (part_circlepath z r s t) \<subseteq> sphere z r"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" shows "a1 \<R>\<lbrakk>\<cdot>\<rbrakk>L a2 = (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T a1 \<and> S \<in>S a2)"
geodesic_segment_unique: fixes G :: "'a set"   and x :: "'a"   and y :: "'a" shows "geodesic_segment_between G x y = (G = {x--y})"
b: shows "Inum bs (Add (fst (split_int b)) (snd (split_int b))) = Inum bs b"
H_loop: fixes p :: "'a"   and r :: "'a"   and x :: "'a" assumes "H (tp \<cdot> tr) x p" shows "H p (while r do x od) (tp \<cdot> at r)"
class_add_app: fixes i :: "instr"   and P :: "jvm_prog"   and mxs :: "nat"   and T :: "ty"   and pc :: "nat"   and mpc :: "nat"   and xt :: "ex_table"   and t :: "(ty list \<times> ty err list) option"   and wf_md :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and C :: "char list"   and cdec :: "jvm_method class" assumes "app i P mxs T pc mpc xt t"   and "wf_prog wf_md P"   and "\<not> is_class P C" shows "app i (class_add P (C, cdec)) mxs T pc mpc xt t"
simps_18: fixes dwrite' :: "(int \<times> word32 \<times> CPU_register) list \<Rightarrow> (int \<times> word32 \<times> CPU_register) list"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "dwrite_update dwrite' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite' dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr>"
power_0_Suc: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "power.power one times zero (Suc n) = zero"
act_quant_uniq: fixes \<phi> :: "'a \<Rightarrow> \<o>"   and x :: "'a" shows "[(\<^bold>\<forall>z. \<^bold>\<A>\<phi> z \<^bold>\<equiv> z \<^bold>= x) \<^bold>\<equiv> (\<^bold>\<forall>z. \<phi> z \<^bold>\<equiv> z \<^bold>= x) in dw]"
contractible_space_top_of_set: fixes S :: "'b set" shows "contractible_space (top_of_set S) = contractible S"
poly_y_x_const: fixes p :: "'c poly" shows "poly_y_x [:p:] = poly_lift p"
isodefl_fmap: fixes d :: "'a \<rightarrow> 'a"   and t :: "udom defl" assumes "isodefl d t" shows "isodefl (fmap\<cdot>d) (TC('b)\<cdot>t)"
cring_sub_deg: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "f (deg R f) \<otimes> g (deg R g) [^] deg R f \<noteq> \<zero>" shows "deg R (Cring_Poly.compose R f g) = deg R f * deg R g"
normal_iff_subgroup: fixes N :: "'a set" shows "N \<lhd> G = subgroup N G"
fold_A_min_gallery_betw_map: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set"   and F :: "'b set"   and G :: "'b set"   and Fs :: "'b set list" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z"   and "SimplicialComplex.maxsimp X F"   and "SimplicialComplex.maxsimp X G"   and "order.greater_eq F z"   and "ChamberComplex.face_distance X z G = ChamberComplex.chamber_distance X F G"   and "SimplicialComplex.min_maxsimpchain X (F # Fs @ [G])" shows "SimplicialComplex.min_maxsimpchain X ((ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C) \<Turnstile> (F # Fs @ [G]))"
sumlist_Cons: fixes x :: "'a"   and xs :: "'a list" shows "sumlist (x # xs) = x \<oplus> sumlist xs"
squanderFace_distr2: fixes g :: "graph"   and V :: "nat list" assumes "Invariants.inv g"   and "final g"   and "noExceptionals g (set V)"   and "separated g (set V)"   and "distinct V"   and "set V \<subseteq> \<V> g" shows "(\<Sum>\<^bsub>f\<in>filter (\<lambda>f. \<exists>v\<in>set V. f \<in> set (facesAt g v)) (faces g)\<^esub> \<d> |vertices f|) = (\<Sum>\<^bsub>v\<in>V\<^esub> tri g v * \<d> 3 + quad g v * \<d> 4)"
analz_Decrypt': fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> analz H"   and "Key K \<in> analz H" shows "X \<in> analz H"
hg: fixes x :: "(real, 'n) vec" assumes "x \<in> S" shows "g x \<in> T \<and> h (g x) = x"
cos_def: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event llist"   and i :: "nat" shows "local.cos c t i = s c (Co_Snapshot.ltake i t) i"
DFS_invar_step': fixes s :: "('v, 'es) state_scheme" assumes "nofail (step s)"   and "DFS_invar G param s"   and "cond s" shows "step s \<le> SPEC (DFS_invar G param)"
atLeastAtMost_rev: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" shows "sum g {n..m} = (\<Sum>i = n..m. g (m + n - i))"
reds_ApplicationI: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and L :: "Vars.var list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and y :: "Vars.var"   and e' :: "exp"   and x :: "Vars.var"   and \<Theta> :: "(Vars.var \<times> exp) list"   and z :: "exp" assumes "\<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : Lam [y]. e'"   and "\<Delta> : e'[y::=x] \<Down>\<^bsub>L\<^esub> \<Theta> : z" shows "\<Gamma> : App e x \<Down>\<^bsub>L\<^esub> \<Theta> : z"
leI: fixes x :: "'a \<Rightarrow>\<^sub>0 nat"   and y :: "'a \<Rightarrow>\<^sub>0 nat" assumes "\<not> x \<prec> y" shows "y \<preceq> x"
fresh_2: fixes a :: "name" shows "a \<sharp> tBOOL = True"
pq': shows "map_spmf snd pq = pair_spmf p' q'"
normalizer_def: fixes m :: "nat"   and x :: "nat \<Rightarrow> int" shows "normalizer m x = (\<lambda>k. if k = 0 then 0 else THE u. u \<in> carrier (residue_ring (p ^ k)) \<and> u * p ^ m = x (m + k))"
infinite: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "infinite A" shows "prod g A = (1::'b)"
F2: shows "\<forall>x\<^sub>1. (1::'a) * x\<^sub>1 = x\<^sub>1"
sfis_present: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and a :: "real"   and f :: "'b \<Rightarrow> real"   and b :: "real"   and g :: "'b \<Rightarrow> real" assumes "measure_space M"   and "a \<in> sfis f M"   and "b \<in> sfis g M" shows "\<exists>z1 z2 C K. f = (\<lambda>t. \<Sum>i\<in>K. z1 i * \<chi> (C i) t) \<and> g = (\<lambda>t. \<Sum>i\<in>K. z2 i * \<chi> (C i) t) \<and> a = (\<Sum>i\<in>K. z1 i * Measure.measure M (C i)) \<and> b = (\<Sum>i\<in>K. z2 i * Measure.measure M (C i)) \<and> finite K \<and> (\<forall>i\<in>K. \<forall>j\<in>K. i \<noteq> j \<longrightarrow> C i \<inter> C j = {}) \<and> (\<forall>i\<in>K. C i \<in> measurable_sets M) \<and> \<Union> (C ` K) = UNIV \<and> nonnegative z1 \<and> nonnegative z2"
assms: shows "\<forall>x\<in>insert y A. P x"