hcju/isabelleps huggingface dataset:
distinct_4: fixes x31 :: "nat"   and x32 :: "nat" shows "recf.Id x31 x32 \<noteq> Z"
lookup_add: fixes f :: "'d \<Rightarrow>\<^sub>0 'c"   and g :: "'d \<Rightarrow>\<^sub>0 'c"   and k :: "'d" shows "lookup (f + g) k = lookup f k + lookup g k"
uniteTrees_def: fixes t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree" shows "uniteTrees t\<^sub>1 t\<^sub>2 = Node (cachedWeight t\<^sub>1 + cachedWeight t\<^sub>2) t\<^sub>1 t\<^sub>2"
InterpEqEq: fixes I :: "('a, 'b, 'c, 'd) Interpretation"   and A :: "'a"   and e1 :: "('a, 'b) Expression"   and B :: "'a"   and g :: "'d"   and e2 :: "('a, 'b) Expression" assumes "Interpretation I"   and "L\<lbrakk>Vx : A \<turnstile> e1 : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IMor g"   and "L\<lbrakk>Vx : A \<turnstile> e2 : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IMor g" shows "L\<lbrakk>Vx : A \<turnstile> e1\<equiv>e2 : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IBool True"
merge_find_right: fixes n :: "('e \<times> 'd) list"   and k :: "'e"   and v :: "'d"   and m :: "('e \<times> 'd) list" assumes "map_of n k = Some v" shows "map_of (AList.merge m n) k = Some v"
dy: shows "CatDom (YFtor C @@ X) = Op C"
NumbI: fixes i :: "nat" shows "Numb i \<in> sexp"
finite_lattice_complete_Sup_least: fixes A :: "'a set"   and z :: "'a" assumes "\<forall>x\<in>A. x \<le> z" shows "Sup A \<le> z"
mkArr_eqI: fixes A :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and B :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and F :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr \<Rightarrow> (hf set, hf \<Rightarrow> hf) concrete_category.arr"   and A' :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and B' :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and F' :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr \<Rightarrow> (hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr (mkArr A B F)"   and "A = A'"   and "B = B'"   and "\<And>x. x \<in> A \<Longrightarrow> F x = F' x" shows "mkArr A B F = mkArr A' B' F'"
assms_3: shows "set_mset N \<subseteq> r \<down>s S"
\<R>\<^sub>1_elim: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and c\<^sub>1 :: "('b, 'a, 'c) Stmt"   and mds :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>1 :: "'b \<Rightarrow> 'd"   and c\<^sub>2 :: "('b, 'a, 'c) Stmt"   and mem\<^sub>2 :: "'b \<Rightarrow> 'd"   and dma :: "'b \<Rightarrow> Sec"   and \<Gamma> :: "'b \<Rightarrow> Sec option" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "(((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>2, mds), mem\<^sub>2) \<in> sifum_types.\<R>\<^sub>1 aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>" obtains \<Gamma>' :: "'b \<Rightarrow> Sec option" where "c\<^sub>2 = c\<^sub>1" and "sifum_types.has_type aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>' c\<^sub>1 \<Gamma>" and "sifum_types.mds_consistent dma mds \<Gamma>'" and "sifum_types.tyenv_eq dma \<Gamma>' mem\<^sub>1 mem\<^sub>2"
arrI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and a :: "'b"   and b :: "'b" assumes "partial_magma C"   and "partial_magma.in_hom C f a b" shows "partial_magma.arr C f"
smallo_powr_nonneg: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real"   and p :: "real" assumes "f \<in> o[F](g)"   and "0 < p"   and "\<forall>\<^sub>F x in F. 0 \<le> f x"   and "\<forall>\<^sub>F x in F. 0 \<le> g x" shows "(\<lambda>x. f x powr p) \<in> o[F](\<lambda>x. g x powr p)"
alw_InvGlobalInfoAtBetaUpright: fixes s :: "('c, 'd) configuration stream" assumes "spec s" shows "alw (holds InvGlobalInfoAtBetaUpright) s"
sum_list_zip_map_of: fixes bs :: "'a list"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" assumes "distinct bs"   and "length xs = length bs" shows "sum_list (map2 f xs bs) = (\<Sum>x\<in>set bs. f (the (map_of (zip bs xs) x)) x)"
vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_finite: fixes x :: "('d, 'c) stateful_strand_step" shows "finite (vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p x)"
spr_jviewIncr_def: shows "spr_jviewIncr \<equiv> \<lambda>a obs' tobs. tobs \<leadsto> obs'"
lit_ord_neg_lit_rhs: fixes x1 :: "'a literal"   and s :: "'a trm"   and t1 :: "'a trm"   and x2 :: "'a literal"   and t2 :: "'a trm" assumes "orient_lit x1 s t1 pos"   and "orient_lit x2 t2 s neg"   and "vars_of_lit x1 = {}"   and "vars_of_lit x2 = {}" shows "(x1, x2) \<in> lit_ord"
transpose_mat_of_cols: fixes n :: "nat"   and vs :: "'b vec list" shows "(mat_of_cols n vs)\<^sup>T = mat_of_rows n vs"
PRECOND_OPT_tag_def: fixes P :: "'a" shows "PRECOND_OPT_tag P \<equiv> P"
SLC_set: fixes s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and l :: "'l"   and v :: "('r, 'l, 'v) val"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r' :: "'r"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Assign (VE (Loc l)) (VE v)])"   and "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>(l \<mapsto> v), \<E> [VE (CV Unit)]))"   and "l \<in> dom (\<sigma>;;\<tau>)"   and "revision_step r' s\<^sub>1 s\<^sub>2'"   and "r \<noteq> r'" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
less_eq_list_drop: fixes xs :: "'b list"   and ys :: "'b list"   and y :: "'b" assumes "xs \<preceq> ys" shows "xs \<preceq> y # ys"
case_2: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> ('d \<Rightarrow> ('b, 'c, 'd) env option) \<Rightarrow> 'a"   and x21 :: "'c"   and x22 :: "'d \<Rightarrow> ('b, 'c, 'd) env option" shows "(case Env x21 x22 of Val x \<Rightarrow> f1 x | Env x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
Differentialo_undef: fixes \<theta> :: "trm option" shows "(Differentialo \<theta> = undeft) = (\<theta> = undeft)"
all_imp_eqI: fixes P :: "'d \<Rightarrow> bool"   and P' :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and Q' :: "'d \<Rightarrow> bool" assumes "P = P'"   and "\<And>x. P x \<Longrightarrow> Q x = Q' x" shows "(\<forall>x. P x \<longrightarrow> Q x) = (\<forall>x. P' x \<longrightarrow> Q' x)"
degree_reduce_fps_poly_monic: fixes F :: "'b fps poly" assumes "monic F" shows "degree (reduce_fps_poly F) = degree F"
stopping_time_less_const: fixes T :: "'a \<Rightarrow> 't"   and t :: "'t" assumes "stopping_time F T" shows "Measurable.pred (F t) (\<lambda>\<omega>. T \<omega> < t)"
qdeps_UNIV: fixes C :: "Nielson_VCG.acom" shows "qdeps (transl C) UNIV = UNIV"
f2: fixes R' :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and gpv :: "('f, 'c, 'e) gpv"   and gpv' :: "('h, 'd, 'g) gpv"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C' :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "('f, 'c, ('f, 'c, 'e) rpv) generat"   and y :: "('h, 'd, ('h, 'd, 'g) rpv) generat"   and a :: "('f, 'c, 'e) rpv"   and b :: "('h, 'd, 'g) rpv"   and a' :: "'e"   and \<alpha> :: "'g"   and \<beta> :: "'e"   and b' :: "'g" assumes "??.Generative_Probabilistic_Value.foo R' outs_gpv outs_gpv gpv gpv' C C'"   and "x \<in> set_spmf (the_gpv gpv)"   and "y \<in> set_spmf (the_gpv gpv')"   and "a \<in> generat_conts x"   and "b \<in> generat_conts y"   and "R' a' \<alpha>"   and "R' \<beta> b'" shows "??.Generative_Probabilistic_Value.foo R' outs_gpv outs_gpv (a a') (b b') C C'"
bijection_def: fixes f :: "'b \<Rightarrow> 'b" shows "bijection f \<equiv> bij f"
eq: fixes a :: "'c" assumes "a \<in> S" shows "h (j a) = g a"
assms_6: shows "older_seniors y n \<subseteq> older_seniors x n"
assms_2: fixes i :: "nat" assumes "i < length Rs" shows "as ! i \<in> carrier (Rs ! i)"
finite_lookup: fixes f :: "'b \<Rightarrow>\<^sub>0 'c" shows "finite {k. lookup f k \<noteq> (0::'c)}"
q_Eats_def: fixes x :: "hf"   and y :: "hf" shows "q_Eats x y \<equiv> \<langle>htuple 1, x, y\<rangle>"
val2formIsEntailed: fixes F' :: "Literal list list"   and valuation :: "Literal list"   and F'' :: "Literal list list" shows "formulaEntailsValuation (F' @ val2form valuation @ F'') valuation"
order_iff: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "semilattice_order f less_eq less" shows "less_eq a b = (a = f a b)"
h_def: shows "h = monom P (g (deg R g)) (deg R g)"
ln2_ge_two_thirds: shows "2 / 3 \<le> ln 2"
prim_red_cong2: fixes e2 :: "exp"   and e2' :: "exp"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and e1 :: "exp" assumes "e2 \<longrightarrow>* e2'" shows "EPrim f e1 e2 \<longrightarrow>* EPrim f e1 e2'"
n_1: shows "isnormNum (x *\<^sub>N y *\<^sub>N z)"
not_updated: fixes r :: "nat"   and t2 :: "transition"   and s :: "vname \<Rightarrow> value option"   and s' :: "nat \<Rightarrow>f value option" assumes "not_updated r t2" shows "apply_updates (Updates t2) s s' $ r = s' $ r"
collapse_1: fixes hd :: "('a, 'b) hd" assumes "is_Var hd" shows "Var (var hd) = hd"
bij_betw_characters_dcharacters: shows "bij_betw c2dc (characters G) (dcharacters n)"
Diff_subset_conv: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" shows "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
simps_1: fixes s1 :: "'d stream"   and s2 :: "'d stream" shows "shd (sinterleave s1 s2) = shd s1"
Scan_mono_tokens: fixes P :: "(symbol \<times> character list) set"   and Q :: "(symbol \<times> character list) set"   and k :: "nat"   and I :: "item set" assumes "P \<subseteq> Q" shows "Scan P k I \<subseteq> Scan Q k I"
simps_7: fixes I :: "('b, 'c, 'a) interp"   and c :: "real" shows "dterm_sem I (Const c) = (\<lambda>v. c)"
uminus_l_inv_vec: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "- v + v = 0\<^sub>v n"
q1_def: shows "q1 \<equiv> fcompose p [:(z0 + complex_of_real r) * \<i>, z0 - complex_of_real r:] [:\<i>, 1:]"
term_p_subid_var: fixes x :: "'a" shows "(x \<cdot> (0::'a) = x) = (x \<le> 1\<^sub>\<pi>)"
matrix_to_iarray_Gram_Schmidt_upt_k_efficient: fixes k :: "nat"   and A :: "((real, 'a) vec, 'b) vec" assumes "k < ncols A" shows "matrix_to_iarray (Gram_Schmidt_upt_k A k) = foldl Gram_Schmidt_column_k_iarrays_efficient (matrix_to_iarray A) [0..<Suc k]"
finite_total_preorder_on_axioms_def: fixes carrier :: "'b set" shows "finite_total_preorder_on_axioms carrier \<equiv> finite carrier"
RD_on_mono: fixes ds :: "'d set" shows "mono (Rf (CD_on ds))"
LtKeys_parts_implSecureSet_elt: fixes K :: "ltkey"   and Ag :: "(agent \<times> agent) set" shows "LtK K \<notin> parts {implSecure A B M |A B M. (A, B) \<in> Ag \<and> M \<in> payload}"
rank_eq_suc_to_nat_greatest: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "A \<noteq> 0" shows "rank A = mod_type_class.to_nat (GREATEST a. \<not> is_zero_row a (Gauss_Jordan A)) + 1"
model': fixes h :: "nat" assumes "infinite (deriv s)"   and "init s" shows "\<forall>A. size A = h \<longrightarrow> (\<forall>m n. contains f n (m, A) \<longrightarrow> \<not> FEval mo ntou A)"
FunctorCompDef: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and f :: "'c"   and g :: "'c" assumes "Functor F"   and "f \<approx>>\<^bsub>CatDom F\<^esub> g" shows "F ## f \<approx>>\<^bsub>CatCod F\<^esub> F ## g"
msg_id_unique: fixes m1 :: "'a \<times> ('a, 'b) operation"   and i :: "nat"   and m2 :: "'a \<times> ('a, 'b) operation"   and j :: "nat" assumes "Broadcast m1 \<in> set (history i)"   and "Broadcast m2 \<in> set (history j)"   and "fst m1 = fst m2" shows "i = j \<and> m1 = m2"
m2r_pred_def: fixes p :: "'b \<Rightarrow> 'b option \<Rightarrow> bool" shows "m2r_pred p \<equiv> \<lambda>x. case x of None \<Rightarrow> \<lambda>_. False | Some a \<Rightarrow> p a"
opnet_disconnect: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d net_state"   and i :: "nat"   and i' :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d net_state"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set"   and t :: "'d net_state"   and t' :: "'d net_state"   and T :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set" assumes "((\<sigma>, s), disconnect(i, i'), \<sigma>', s') \<in> S"   and "((\<sigma>, t), disconnect(i, i'), \<sigma>', t') \<in> T" shows "((\<sigma>, SubnetS s t), disconnect(i, i'), \<sigma>', SubnetS s' t') \<in> opnet_sos S T"
lcp_ext_right: fixes r :: "'b list"   and r' :: "'b list"   and u :: "'b list"   and v :: "'b list" obtains "r \<bowtie> r'"   | "r \<cdot> u \<and>\<^sub>p r' \<cdot> v = r \<and>\<^sub>p r'"
eq3: shows "Sigma_Algebra.measure lebesgue (cbox a (\<chi>i. if i = m then b $ m + b $ n else b $ i)) = Sigma_Algebra.measure lebesgue (cbox a (\<chi>i. if i = m then a $ m + b $ n else b $ i)) + Sigma_Algebra.measure lebesgue (cbox a b)"
starlike_negligible_compact: fixes S :: "'c set" assumes "compact S"   and "\<And>c x. \<lbrakk>c *\<^sub>R x \<in> S; 0 \<le> c; x \<in> S\<rbrakk> \<Longrightarrow> c = 1" shows "S \<in> null_sets lebesgue"
error_free_supd1: fixes a :: "abopt"   and s :: "st"   and f :: "st \<Rightarrow> st" assumes "error_free (a, s)" shows "error_free (a, f s)"
filterlim_floor_at_bot: shows "filterlim floor at_bot at_bot"
sin_total: fixes y :: "real" assumes "- 1 \<le> y"   and "y \<le> 1" shows "\<exists>!x. - (pi / 2) \<le> x \<and> x \<le> pi / 2 \<and> sin x = y"
A4: shows "\<forall>TP TQ. TP \<in>T P \<and> TQ \<in>T Q \<longrightarrow> (TP, TQ) \<in> TRel\<^sup>+"
elims_2: fixes x :: "char list"   and xa :: "char list" assumes "internal_iface_name_match x xa" obtains "x = []" and "xa = []"   | i :: "char" and "is" :: "char list" where "x = i__ # is__" and "xa = []" and "i__ = CHR ''+'' \<and> is__ = []"   | i :: "char" and "is" :: "char list" and p_i :: "char" and p_is :: "char list" where "x = i # is" and "xa = p_i # p_is" and "if i = CHR ''+'' \<and> is = [] then True else p_i = i \<and> internal_iface_name_match is p_is"
simps_2: fixes n :: "nat"   and c :: "int"   and t :: "Ferrack.num" shows "maxcoeff (CN n c t) = max \<bar>c\<bar> (maxcoeff t)"
sup_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion>2 y \<squnion>2 z = x \<squnion>2 (y \<squnion>2 z)"
lbound_def: fixes Us :: "'a set set" shows "lbound Us = Inf (principal_subs I ` {S \<in> I. \<forall>u\<in>Us. S \<subseteq> u})"
OF_match_fields_unsafe_def: fixes m :: "of_match_field set"   and p :: "(32, 'b) simple_packet_ext_scheme" shows "OF_match_fields_unsafe m p = (\<forall>f\<in>m. match_no_prereq f p)"
set_2: fixes x21 :: "'a"   and x22 :: "'a llist" shows "lset (LCons x21 x22) = insert x21 (lset x22)"
phl_n_while: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "\<lparr>!x \<cdot> !y\<rparr>z\<lparr>z \<cdot> !x\<rparr>" shows "\<lparr>!x\<rparr>(!y \<cdot> z)\<^sup>\<infinity> \<cdot> t y\<lparr>(!y \<cdot> z)\<^sup>\<infinity> \<cdot> t y \<cdot> (!x \<cdot> t y)\<rparr>"
induct: fixes P1 :: "'b exp \<Rightarrow> bool"   and P2 :: "'b exp list \<Rightarrow> bool"   and exp_list :: "'b exp list" assumes "\<And>x. P1 (new x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (Cast x1 x2)"   and "\<And>x. P1 (Val x)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x3\<rbrakk> \<Longrightarrow> P1 (x1 \<guillemotleft>x2\<guillemotright> x3)"   and "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (x1:=x2)"   and "\<And>x1 x2 x3. P1 x1 \<Longrightarrow> P1 (x1\<bullet>x2{x3})"   and "\<And>x1 x2 x3. P1 (x1\<bullet>\<^sub>sx2{x3})"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2{x3} := x4)"   and "\<And>x1 x2 x3 x4. P1 x4 \<Longrightarrow> P1 (x1\<bullet>\<^sub>sx2{x3} := x4)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P2 x3\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2(x3))"   and "\<And>x1 x2 x3. P2 x3 \<Longrightarrow> P1 (x1\<bullet>\<^sub>sx2(x3))"   and "\<And>x1 x2 x3. P1 x3 \<Longrightarrow> P1 {x1:x2; x3}"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (if (x1) x2 else x3)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (while (x1) x2)"   and "\<And>x. P1 x \<Longrightarrow> P1 (throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (try x1 catch(x2 x3) x4)"   and "\<And>x1 x2 x3 x4. P1 x4 \<Longrightarrow> P1 (INIT x1 (x2,x3) \<leftarrow> x4)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x2; P1 x4\<rbrakk> \<Longrightarrow> P1 (RI (x1,x2) ; x3 \<leftarrow> x4)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P2 exp_list"
wn_nz: shows "winding_number \<gamma> z \<noteq> 0"
rep_eq: fixes x :: "'b \<Longrightarrow> 'c"   and xa :: "('d, 'c) uexpr" shows "(x \<sharp> xa) = (\<forall>b v. \<lbrakk>xa\<rbrakk>\<^sub>e (put\<^bsub>x\<^esub> b v) = \<lbrakk>xa\<rbrakk>\<^sub>e b)"
perm_swap_def: fixes x :: "'a"   and y :: "'a"   and f :: "'a \<Rightarrow> 'a" shows "perm_swap x y f \<equiv> x \<rightleftharpoons>\<^sub>F y \<circ> f \<circ> x \<rightleftharpoons>\<^sub>F y"
prod_cases: fixes P :: "'c \<times> 'd \<Rightarrow> bool"   and p :: "'c \<times> 'd" assumes "\<And>a b. P (a, b)" shows "P p"
length_xs_gt_1: fixes xs :: "'b list" assumes "1 < length xs" shows "\<exists>x y ys. xs = x # y # ys"
bind_has_subprob_density: fixes M :: "'a measure"   and N :: "'b measure"   and f :: "'a \<Rightarrow> 'b measure"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal" assumes "subprob_space M"   and "sigma_finite_measure N"   and "space M \<noteq> {}"   and "\<And>x. x \<in> space M \<Longrightarrow> has_density (f x) N (g x)"   and "(\<lambda>(x, y). g x y) \<in> borel_measurable (M \<Otimes>\<^sub>M N)"   and "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N" shows "has_subprob_density (M \<bind> f) N (\<lambda>y. \<integral>\<^sup>+ x. g x y \<partial>M)"
basic_cong_3: fixes p :: "int" assumes "2 < p" shows "[0 \<noteq> 1] (mod p)"
simps_3: fixes m :: "nat"   and n :: "nat"   and xs :: "nat list" shows "rec_exec (recf.id m n) xs = xs ! n"
functor_HoHV: shows "functor VVV.comp (\<cdot>) HoHV"
Pareto_strictI': fixes dom :: "'a set"   and carrier :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and i :: "'a" assumes "preorder_family dom carrier R"   and "\<And>i. i \<in> dom \<Longrightarrow> R i x y"   and "i \<in> dom"   and "\<not> R i y x" shows "x \<prec>[Pareto R] y"
continuous_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "continuous_map X Y f = (f ` topspace X \<subseteq> topspace Y \<and> (\<forall>U. openin Y U \<longrightarrow> openin X {x \<in> topspace X. f x \<in> U}))"
Close2: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<Longrightarrow>a<\<nu>x> \<prec> P'"   and "Q \<Longrightarrow>a<x> \<prec> Q'"   and "x \<sharp> Q" shows "P \<parallel> Q \<Longrightarrow>\<tau> \<prec> <\<nu>x>(P' \<parallel> Q')"
a_rcos_module_rev: fixes x :: "'a"   and x' :: "'a" assumes "x \<in> carrier R"   and "x' \<in> carrier R"   and "x' \<oplus> \<ominus> x \<in> a_kernel R S h" shows "x' \<in> a_kernel R S h +> x"
subset_filter_fset: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b fset"   and Q :: "'b \<Rightarrow> bool" shows "(filter_fset P xs |\<subseteq>| filter_fset Q xs) = (\<forall>x. x |\<in>| xs \<longrightarrow> P x \<longrightarrow> Q x)"
pdevs_domain_def: fixes x :: "'a pdevs" shows "pdevs_domain x = {i. pdevs_apply x i \<noteq> (0::'a)}"
Cons: shows "list_constr I (length qs) \<longrightarrow> (0 < poly (prod_list (retrieve_polys qs I)) x) = (z I sign = 1)"
qtable_empty: fixes n :: "nat"   and A :: "nat set"   and P :: "'a option list \<Rightarrow> bool"   and Q :: "'a option list \<Rightarrow> bool" assumes "\<And>x. \<lbrakk>wf_tuple n A x; P x; Q x\<rbrakk> \<Longrightarrow> False" shows "qtable n A P Q empty_table"
det_nonzero_congruence: fixes A :: "'a mat"   and M :: "'a mat"   and B :: "'a mat"   and n :: "nat" assumes "A * M = B * M"   and "det M \<noteq> (0::'a)"   and "M \<in> carrier_mat n n"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "A = B"
mbisim_thrNone_eq: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and bisim_wait :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and s1 :: "('f, 'a) locks \<times> (('a \<Rightarrow> ('b \<times> 'f \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'g wait_set_status option) \<times> 'a set"   and s2 :: "('f, 'a) locks \<times> (('a \<Rightarrow> ('d \<times> 'f \<Rightarrow>f nat) option) \<times> 'e) \<times> ('a \<Rightarrow> 'g wait_set_status option) \<times> 'a set"   and t :: "'a" assumes "FWbisimulation_base.mbisim bisim bisim_wait s1 s2" shows "(thr s1 t = None) = (thr s2 t = None)"
ccField_ccBind: fixes v :: "Vars.var"   and e :: "exp"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and G :: "CoCalls" shows "ccField (ccBind v e\<cdot>(ae, G)) \<subseteq> fv e"
inj_on_finite: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" assumes "inj_on f A"   and "f ` A \<subseteq> B"   and "finite B" shows "finite A"
assms_2: shows "\<guillemotleft>x : unity \<rightarrow> local.dom f\<guillemotright>"
\<Gamma>'_sel_3: shows "B \<Gamma>' = B \<Gamma>"
finite_imageD: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "finite (f ` A)"   and "inj_on f A" shows "finite A"
moni_def: shows "moni = Poly_Mapping.single i 1"
w_simps_3: shows "src \<pi>.chine = src s\<^sub>0"
resp_analz_image_freshK: fixes RB :: "msg"   and evs :: "event list"   and evsa :: "event list" assumes "RB \<in> responses evs"   and "evsa \<in> recur" shows "\<forall>K KK. KK \<subseteq> - range shrK \<longrightarrow> (Key K \<in> analz (insert RB (Key ` KK \<union> knows Spy evsa))) = (K \<in> KK \<or> Key K \<in> analz (insert RB (knows Spy evsa)))"
is_extensional: fixes f :: "'c local.term set" assumes "\<not> arr f" shows "E\<^sub>S.map f = D.null"
assms: fixes C :: "'a set set" assumes "C \<noteq> {}"   and "subset.chain {I. ideal I R} C" shows "\<Union> C \<in> C"
proj_setI': fixes y :: "'b"   and A :: "'b set"   and x :: "'b" assumes "y \<in> A"   and "\<And>z. z \<in> A \<Longrightarrow> dist x y \<le> dist x z" shows "y \<in> proj_set x A"
Bonk_Schramm_extension_unfolded_wo_props_5: shows "\<forall>u n. (u n, would_be_Cauchy u) \<in> Bonk_Schramm_extension_unfolded_wo"
entailsD: fixes P :: "assn"   and Q :: "assn"   and h :: "pheap" assumes "P \<Longrightarrow>\<^sub>A Q"   and "h \<Turnstile> P" shows "h \<Turnstile> Q"