hcju/isabelleps huggingface dataset:
carrier_neq: fixes R :: "'c set"   and I :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "pr_ideal R I addition multiplication zero unit" shows "I \<noteq> R"
card_degen_lotteries_equals_outcomes: fixes out :: "'b set" shows "card {x \<in> Neumann_Morgenstern_Utility_Theorem.\<P> out. card (set_pmf x) = 1} = card out"
drop_stake: fixes n :: "nat"   and m :: "nat"   and s :: "'a stream" shows "drop n (stake m s) = stake (m - n) (sdrop n s)"
a1: shows "\<not> P (i, hd xs) \<or> \<not> P (last xs, j)"
unfs: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs_fv x \<tau> s $ Fv x \<tau>) s"
inline1_left_gpv: fixes callee :: "'b \<Rightarrow> 'h \<Rightarrow> ('g \<times> 'b, 'c, 'e) gpv"   and gpv :: "('a, 'h, 'g) gpv"   and s :: "'b" shows "inline1 (\<lambda>s q. left_gpv (callee s q)) gpv s = map_spmf (map_sum id (map_prod Inl (map_prod left_rpv id))) (inline1 callee gpv s)"
IArray_sub: fixes as :: "'a iarray" shows "(!!) as = (!) (IArray.list_of as)"
Suc_2: shows "Lfilled (Suc i) lholed (vs @ [$Return]) LI"
f: shows "\<forall>n. (f n \<in> A \<and> l n \<le> f n) \<and> f n \<le> f (Suc n)"
DiffConstProof_def: shows "DiffConstProof = (([], [Equals (Differential (Const 0)) (Const 0)]), [(0, AxSubst AdConst DiffConstSubst)])"
NNF_ternary_correct: fixes t :: "ternaryformula" shows "ternary_ternary_eval (NNF_ternary t) = ternary_ternary_eval t"
compat_different_vars: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and mems :: "(('b \<Rightarrow> 'c) \<times> ('b \<Rightarrow> 'c)) list"   and i :: "nat"   and x :: "'b"   and mem\<^sub>1 :: "'b \<Rightarrow> 'c"   and mem\<^sub>2 :: "'b \<Rightarrow> 'c" assumes "sifum_security stop eval some_val some_val'"   and "fst (mems ! i) x = snd (mems ! i) x"   and "x \<notin> sifum_security.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "mem\<^sub>1 x = mem\<^sub>2 x"
strict_linear_order_on_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "strict_linear_order_on A r \<equiv> trans r \<and> irrefl r \<and> total_on A r"
injection_below: fixes f :: "'f \<rightarrow> 'e"   and g :: "'e \<rightarrow> 'f"   and x :: "'e"   and y :: "'e" assumes "\<forall>x. f\<cdot>(g\<cdot>x) = x" shows "(g\<cdot>x \<sqsubseteq> g\<cdot>y) = (x \<sqsubseteq> y)"
lang_Plus_Zero: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp"   and s :: "'a atom rexp" shows "(lang \<Sigma> n (Plus r One) = lang \<Sigma> n (Plus s One)) = (lang \<Sigma> n r - {[]} = lang \<Sigma> n s - {[]})"
bind_option_simps_2: fixes fail :: "'d"   and x :: "'e"   and f :: "'e \<Rightarrow> 'd" shows "monad.bind_option fail (Some x) f = f x"
OV_inv2_finite_obs_set: fixes s :: "'c obsv_state_scheme"   and Q :: "process set" assumes "s \<in> OV_inv2" shows "finite (vote_set (obsv_state.obs s) Q)"
def_lfp_induct: fixes A :: "'d"   and f :: "'d \<Rightarrow> 'd"   and P :: "'d" assumes "A \<equiv> lfp f"   and "mono f"   and "f (A \<sqinter> P) \<le> P" shows "A \<le> P"
less_kbo_def: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" shows "less_kbo s t = fst (kbo t s)"
powr_float_intervalI: fixes x :: "real"   and X :: "float interval"   and y :: "real"   and Y :: "float interval"   and p :: "nat" assumes "x \<in>\<^sub>r X"   and "y \<in>\<^sub>r Y" shows "x powr y \<in> set_of' (powr_float_interval p X Y)"
invertibleRule: fixes R' :: "('b sequent list \<times> 'b sequent) set"   and R :: "('b sequent list \<times> 'b sequent) set"   and Ps :: "'b sequent list"   and C :: "'b sequent"   and n :: "nat" assumes "R' \<subseteq> upRules \<and> R = Ax \<union> R'"   and "uniqueConclusion R'"   and "(Ps, C) \<in> R*"   and "(C, n) \<in> derivable R*" shows "\<forall>p\<in>set Ps. \<exists>m\<le>n. (p, m) \<in> derivable R*"
ImpE: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<turnstile> F \<^bold>\<rightarrow> G"   and "\<Gamma> \<turnstile> F" shows "\<Gamma> \<turnstile> G"
that: fixes z :: "complex" assumes "z \<in> inside (path_image p)"   and "cmod (winding_number p z) = 1" shows "thesis"
map: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c list" shows "map_iarray f (IArray x) = IArray (map f x)"
subst_domI: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and v :: "'b" assumes "\<sigma> v \<noteq> Var v" shows "v \<in> subst_domain \<sigma>"
assms: shows "\<forall>(ps, i)\<in>set branch. \<forall>p. p on (ps, i) \<longrightarrow> M, g, w \<Turnstile> p"
inj_imp_bij_betw_inv: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d set" assumes "inj f" shows "bij_betw (inv f) (f ` M) M"
at_map_Inf_pres: fixes X :: "'a set" shows "at_map (\<Sqinter> X) = Abs_atoms (Collect atom \<inter> (\<Inter>x\<in>X. Rep_atoms (at_map x)))"
Map_comp': fixes t' :: "('b \<Rightarrow> 'a, 'b \<Rightarrow> 'a) concrete_category.arr"   and t :: "('b \<Rightarrow> 'a, 'b \<Rightarrow> 'a) concrete_category.arr" assumes "J_C.seq t' t" shows "J_C.Map (t' \<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] t) = vertical_composite.map J (\<cdot>) (J_C.Map t) (J_C.Map t')"
wa_intro: fixes \<delta> :: "('d, 'e) ta_rule set"   and addi :: "('d set \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)) set" assumes "while_algo (br_algo \<delta>)"   and "wa_invar (br'_algo \<delta>) = addi \<inter> {s. br'_\<alpha> s \<in> wa_invar (br_algo \<delta>)}"   and "\<And>s s'. \<lbrakk>s \<in> addi; s \<in> wa_cond (br'_algo \<delta>); br'_\<alpha> s \<in> wa_invar (br_algo \<delta>); (s, s') \<in> wa_step (br'_algo \<delta>)\<rbrakk> \<Longrightarrow> s' \<in> addi"   and "wa_initial (br'_algo \<delta>) \<subseteq> addi" shows "while_algo (br'_algo \<delta>)"
ParamDefs_in_Def: fixes n :: "'b"   and V :: "'c" assumes "valid_node n"   and "V \<in> set (ParamDefs n)" shows "V \<in> Def n"
lprefix_LCons_conv: fixes xs :: "'c llist"   and y :: "'c"   and ys :: "'c llist" shows "xs \<sqsubseteq> LCons y ys = (xs = LNil \<or> (\<exists>xs'. xs = LCons y xs' \<and> xs' \<sqsubseteq> ys))"
compress_normalize_primitive_def: fixes disc_sel :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and C :: "'c \<Rightarrow> 'b"   and f :: "'c negation_type list \<Rightarrow> ('c list \<times> 'c list) option"   and m :: "'b match_expr" shows "compress_normalize_primitive disc_sel C f m \<equiv> case primitive_extractor disc_sel m of (as, rst) \<Rightarrow> map_option (\<lambda>(as_pos, as_neg). MatchAnd (alist_and' (NegPos_map C (map Pos as_pos @ map Neg as_neg))) rst) (f as)"
continuous_limit: fixes f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "is_Zp_continuous f"   and "is_Zp_cauchy s" shows "Zp_converges_to (f \<circ> s) (f (res_lim s))"
preserves_cod: fixes f :: "'a" assumes "A.arr f" shows "inv.B.cod (\<sigma>' f) = F (A.cod f)"
magQ_qtimes: fixes x :: "'e['f, 'h]"   and y :: "'e['g, 'h]" shows "\<lbrakk>x \<^bold>\<cdot> y\<rbrakk>\<^sub>Q = \<lbrakk>x\<rbrakk>\<^sub>Q \<cdot> \<lbrakk>y\<rbrakk>\<^sub>Q"
subset_iff: fixes A :: "'b set"   and B :: "'b set" shows "(A \<subseteq> B) = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
update_by_fun_gr_pair_ko_def: fixes ko :: "'a key_order" shows "update_by_fun_gr_pair_ko ko \<equiv> comparator.update_by_fun_gr_pair (key_compare ko)"
bernoulli_unroll_all_4: fixes n :: "nat" shows "bernpoly n = (\<lambda>x. \<Sum>k\<le>n. of_nat (n choose k) * of_real (bernoulli k) * x ^ (n - k))"
supp_lift_transform: fixes a :: "'a\<^sub>\<bottom>"   and e :: "exp" shows "supp (lift_transform trans a e) \<subseteq> supp e"
i: fixes x :: "'a" assumes "x \<in> set xs"   and "\<not> P x" shows "\<not> p ((f ^^ i x) x)"
elims_3: fixes x :: "Monitor.args"   and xa :: "nat"   and xb :: "nat \<times> nat \<times> bool list \<times> bool list \<times> (nat \<times> 'a option list set) queue \<times> (nat \<times> 'a option list set) queue \<times> ('a option list, nat) mapping \<times> ('a option list, nat) mapping"   and xc :: "(nat \<times> 'a option list set) list" assumes "\<not> valid_mmsaux x xa xb xc" obtains args :: "Monitor.args" and cur :: "nat" and nt :: "nat" and gc :: "nat" and maskL :: "bool list" and maskR :: "bool list" and data_prev :: "(nat \<times> 'a option list set) queue" and data_in :: "(nat \<times> 'a option list set) queue" and tuple_in :: "('a option list, nat) mapping" and tuple_since :: "('a option list, nat) mapping" and ys :: "(nat \<times> 'a option list set) list" where "x = args" and "xa = cur" and "xb = (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since)" and "xc = ys" and "\<not> (args_L args \<subseteq> args_R args \<and> maskL = join_mask (args_n args) (args_L args) \<and> maskR = join_mask (args_n args) (args_R args) \<and> (\<forall>(t, y)\<in>set ys. table (args_n args) (args_R args) y) \<and> table (args_n args) (args_R args) (Mapping.keys tuple_in) \<and> table (args_n args) (args_R args) (Mapping.keys tuple_since) \<and> Ball (\<Union> (snd ` set (linearize data_prev))) (wf_tuple (args_n args) (args_R args)) \<and> cur = nt \<and> ts_tuple_rel (set ys) = {tas \<in> ts_tuple_rel (set (linearize data_prev) \<union> set (linearize data_in)). valid_tuple tuple_since tas} \<and> sorted (map fst (linearize data_prev)) \<and> (\<forall>t\<in>fst ` set (linearize data_prev). t \<le> nt \<and> nt - t < left (args_ivl args)) \<and> sorted (map fst (linearize data_in)) \<and> (\<forall>t\<in>fst ` set (linearize data_in). t \<le> nt \<and> left (args_ivl args) \<le> nt - t) \<and> (\<forall>as. Mapping.lookup tuple_in as = safe_max (fst ` {tas \<in> ts_tuple_rel (set (linearize data_in)). valid_tuple tuple_since tas \<and> as = snd tas})) \<and> (\<forall>as\<in>Mapping.keys tuple_since. case Mapping.lookup tuple_since as of Some t \<Rightarrow> t \<le> nt))"
asubg_subset: fixes H :: "'a set" assumes "M +> H" shows "H \<subseteq> carrier M"
leg1_simps_4: shows "trg p\<^sub>1 = trg (k\<^sup>* \<star> h)"
ltl'_def: fixes xs :: "'c llist" shows "ltl' xs = (if lnull xs then None else Some (ltl xs))"
imonad1: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'c nres" shows "RETURN x \<bind> f = f x"
mem_eq: shows "subst (to_partial mem |` X) mem = mem"
m_less_z: fixes x :: "int" shows "- \<infinity> < ant x"
divergences_def: fixes P :: "'e process" shows "divergences P \<equiv> snd (Rep_process P)"
reset_set1: fixes cs :: "'c list"   and d :: "'d"   and u :: "'c \<Rightarrow> 'd" shows "\<forall>c\<in>set cs. ([cs\<rightarrow>d]u) c = d"
xy: shows "real_of_rat r1 * real_of_1 (p2, l2, r2) = real_of_1 (cf_pos_poly (poly_mult_rat r1 p2), l, r)"
pts_symbolic_induct: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) strand_step list"   and \<S>' :: "('fun, 'var) strand_step list set"   and \<A>' :: "('fun, 'var) strand_step list" assumes "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet> (\<S>', \<A>')" obtains "[] \<in> \<S>" and "\<S>' = update\<^sub>s\<^sub>t \<S> []" and "\<A>' = \<A>"   | t :: "('fun, 'var) Term.term" and S :: "('fun, 'var) strand_step list" where "send\<langle>t__\<rangle>\<^sub>s\<^sub>t # S__ \<in> \<S>" and "\<S>' = update\<^sub>s\<^sub>t \<S> (send\<langle>t__\<rangle>\<^sub>s\<^sub>t # S__)" and "\<A>' = \<A> @ [receive\<langle>t__\<rangle>\<^sub>s\<^sub>t]"   | t :: "('fun, 'var) Term.term" and S :: "('fun, 'var) strand_step list" where "receive\<langle>ta__\<rangle>\<^sub>s\<^sub>t # Sa__ \<in> \<S>" and "\<S>' = update\<^sub>s\<^sub>t \<S> (receive\<langle>ta__\<rangle>\<^sub>s\<^sub>t # Sa__)" and "\<A>' = \<A> @ [send\<langle>ta__\<rangle>\<^sub>s\<^sub>t]"   | a :: "poscheckvariant" and t :: "('fun, 'var) Term.term" and t' :: "('fun, 'var) Term.term" and S :: "('fun, 'var) strand_step list" where "\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # Sb__ \<in> \<S>" and "\<S>' = update\<^sub>s\<^sub>t \<S> (\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # Sb__)" and "\<A>' = \<A> @ [\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t]"   | X :: "'var list" and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" and S :: "('fun, 'var) strand_step list" where "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" and "\<S>' = update\<^sub>s\<^sub>t \<S> (\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S)" and "\<A>' = \<A> @ [\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t]"
fps_div_fps_X_nth': fixes f :: "'a fps"   and k :: "nat" assumes "inverse (1::'a) = (1::'a)" shows "(f / fps_X) $ k = f $ Suc k"
coeff_0_reflect_poly_0_iff: fixes p :: "'a poly" shows "(coeff (reflect_poly p) 0 = (0::'a)) = (p = 0)"
szip_smap_snd: fixes xs :: "'b stream"   and ys :: "'a stream" shows "smap snd (xs ||| ys) = ys"
wf\<^sub>s\<^sub>t\<^sub>s'_decomp_rm: fixes A :: "('fun, 'var) extstrand_step list"   and S :: "('fun, 'var) strand_step list set" assumes "well_analyzed A"   and "wf\<^sub>s\<^sub>t\<^sub>s' S (decomp_rm\<^sub>e\<^sub>s\<^sub>t A)" shows "wf\<^sub>s\<^sub>t\<^sub>s' S A"
assms_4: shows "\<forall>a. \<not> disc2 (C a)"
l3_inv10I: fixes x :: "l3_state" assumes "secret x \<subseteq> payload" shows "x \<in> l3_inv10"
orbit_equality: fixes x :: "'a" assumes "x \<in> G" shows "left.transformation.orbit.Class x = {\<alpha> x |\<alpha>. \<alpha> \<in> left.translation ` G}"
less_divide_eq_numeral1_1: fixes a :: "'a"   and b :: "'a"   and w :: "num" shows "(a < b / numeral w) = (a \<cdot> numeral w < b)"
paths_ne: shows "{p. apath u p v} \<noteq> {}"
res: fixes R :: "'b \<times> 'd \<Rightarrow> 'b \<times> 'd \<Rightarrow> bool" assumes "\<And>k v. m k = Some v \<Longrightarrow> set_iterator_genord (it_b v) (S_b v) (\<lambda>_ _. True)"   and "\<And>k v u k' v' u'. \<lbrakk>m k = Some v; u \<in> S_b v; m k' = Some v'; u' \<in> S_b v'; if k = k' then True else True\<rbrakk> \<Longrightarrow> R (k, u) (k', u')" shows "set_iterator (map_iterator_product it_a it_b) {(k, e). \<exists>v. m k = Some v \<and> e \<in> S_b v}"
suc_least: fixes r :: "('a \<times> 'a) set"   and B :: "'a set"   and a :: "'a" assumes "wo_rel r"   and "B \<subseteq> Field r"   and "a \<in> Field r"   and "\<And>b. b \<in> B \<Longrightarrow> a \<noteq> b \<and> (b, a) \<in> r" shows "(wo_rel.suc r B, a) \<in> r"
mset_subset_eq_count: fixes A :: "'a multiset"   and B :: "'a multiset"   and a :: "'a" assumes "A \<subseteq># B" shows "count A a \<le> count B a"
cases: fixes a1 :: "pi"   and a2 :: "pi" assumes "bangPred a1 a2" obtains P :: "pi" where "a1 = Pa__" and "a2 = !Pa__"   | P :: "pi" where "a1 = P" and "a2 = P \<parallel> !P"
h_def: shows "h \<equiv> (\<lambda>y. a + (B / norm (y - a)) *\<^sub>R (y - a)) \<circ> g"
indexed_const_def: fixes k :: "'a" shows "indexed_const k = (\<lambda>m. if m = {#} then k else \<zero>)"
get_child_nodes_locs_def: shows "Core_DOM_Functions.get_child_nodes_locs \<equiv> l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_child_nodes_locs"
region_set'_def: fixes R :: "('a \<Rightarrow> 'b) set"   and r :: "'a list"   and c :: "'b" shows "region_set' R r c = {[r\<rightarrow>c]v |v. v \<in> R}"
has_integral_affinity: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and a :: "'a"   and b :: "'a"   and m :: "real"   and c :: "'a" assumes "(f has_integral i) (cbox a b)"   and "m \<noteq> 0" shows "((\<lambda>x. f (m *\<^sub>R x + c)) has_integral (1 / \<bar>m\<bar> ^ DIM('a)) *\<^sub>R i) ((\<lambda>x. (1 / m) *\<^sub>R x + - ((1 / m) *\<^sub>R c)) ` cbox a b)"
split_list: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs"
sorted_fst_hd_drop_less_drop: fixes ps :: "(int \<times> int) list"   and n :: "nat" assumes "sorted_fst ps" shows "\<forall>p\<in>set (drop n ps). fst (hd (drop n ps)) \<le> fst p"
strict_from_inj_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> nat" shows "strict_from_inj n f = (\<lambda>i. if i \<in> {0..<n} then sorted_list_of_set (f ` {0..<n}) ! i else i)"
dvd_add_times_triv_right_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "dvd.dvd times a (plus b (times c a)) = dvd.dvd times a b"
Cast_red: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and e\<^sub>0 :: "expr"   and e\<^sub>1 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"   and "e = \<lparr>C\<rparr>e\<^sub>0"   and "e' = \<lparr>C\<rparr>e\<^sub>1" shows "P,E \<turnstile> \<langle>e\<^sub>0,s\<rangle> \<rightarrow>* \<langle>e\<^sub>1,s'\<rangle>"
trcl_pair_induct: fixes xc1 :: "'c"   and xc2 :: "'d"   and xb :: "'e list"   and xa1 :: "'c"   and xa2 :: "'d"   and t :: "(('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd) set"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> 'e list \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "((xc1, xc2), xb, xa1, xa2) \<in> trcl t"   and "\<And>a b. P a b [] a b"   and "\<And>a b aa ab ba w ac bb. \<lbrakk>((a, b), aa, ab, ba) \<in> t; ((ab, ba), w, ac, bb) \<in> trcl t; P ab ba w ac bb\<rbrakk> \<Longrightarrow> P a b (aa # w) ac bb" shows "P xc1 xc2 xb xa1 xa2"
tau_def: fixes x :: "'a" shows "\<tau> x = x \<cdot> (0::'a)"
elts_vcard_VSigma_eqpoll: fixes i :: "V"   and j :: "V" shows "elts (vcard (vtimes i j)) \<approx> elts i \<times> elts j"
gConsInj_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gConsInj MOD \<equiv> gVarInj MOD \<and> gAbsInj MOD \<and> gOpInj MOD \<and> gVarGOpInj MOD"
insert_Var_Zero: fixes f :: "nat \<Rightarrow> 'a"   and v :: "nat" shows "insertion f (Var v) = f v"
minus_pi_half_less_zero: shows "- (pi / 2) < 0"
ax_do_not_a_star_reduce: fixes x :: "name"   and a :: "coname"   and M :: "trm" assumes "Ax x a \<longrightarrow>\<^sub>a* M" shows "M = Ax x a"
poly_add_zero'_1: fixes p :: "'a list" assumes "set p \<subseteq> carrier R" shows "poly_add p [] = local.normalize p"
not_inf_Stuck: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com" shows "\<not> \<Gamma>\<turnstile> (c, Stuck) \<rightarrow> \<dots>(\<infinity>)"
disconnected_nodes_eq_h: fixes doc_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" shows "h \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes = h2 \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes"
uminus_poly_i_def: fixes ops :: "'b arith_ops_record" shows "uminus_poly_i ops = map (arith_ops_record.uminus ops)"
domo_def: fixes S :: "'b Abs_State.st option" shows "domo S = (case S of None \<Rightarrow> {} | Some S' \<Rightarrow> set (Abs_State.dom S'))"
eventually_ae_filter: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b measure" shows "almost_everywhere M P = (\<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N)"
rel_distinct_3: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x11 :: "'a"   and x12 :: "bool"   and y31 :: "'b"   and y32 :: "'d" shows "\<not> rel_message R1 R2 <x11, inM x12> <y31, y32>"
neutr_eq_iff: fixes a :: "'b"   and b :: "'b" shows "(\<top> = a \<sqinter> b) = (a = \<top> \<and> b = \<top>)"
ivl_disj_int_two_7: fixes l :: "'b"   and m :: "'b"   and u :: "'b" shows "{l..<m} \<inter> {m..u} = {}"
ESPEC_cons: fixes m :: "('c + 'd) nres"   and E :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and E' :: "'c \<Rightarrow> bool"   and Q' :: "'d \<Rightarrow> bool" assumes "m \<le> ESPEC E Q"   and "\<And>e. E e \<Longrightarrow> E' e"   and "\<And>x. Q x \<Longrightarrow> Q' x" shows "m \<le> ESPEC E' Q'"
connected_clopen: fixes S :: "'a set" shows "connected S = (\<forall>T. openin (top_of_set S) T \<and> closedin (top_of_set S) T \<longrightarrow> T = {} \<or> T = S)"
seqSubs'_def: fixes I :: "('a, 'b, 'c) input"   and \<sigma> :: "(name list \<times> 'a list) list" shows "seqSubs' I \<sigma> \<equiv> foldl (\<lambda>Q (xvec, Tvec). subs' Q xvec Tvec) I \<sigma>"
entails_def: fixes atom_ordering :: "('a \<times> 'a) set"   and S :: "'a Literal set set"   and C :: "'a Literal set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.entails S C \<equiv> \<forall>I. propositional_atoms.validate_formula I S \<longrightarrow> propositional_atoms.validate_clause I C"
sees_field_i_i_i_o_oE: fixes x :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and xa :: "char list"   and xb :: "char list"   and xc :: "ty"   and xd :: "char list" assumes "pred.eval (sees_field_i_i_i_o_o x xa xb) (xc, xd)" obtains "x \<turnstile> xa sees xb:xc in xd"
kat_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "d x \<cdot> y \<cdot> ad z = (zero_class.zero::'a)" shows "d x \<cdot> y = d x \<cdot> y \<cdot> d z"
R_lin_independent_ConsD2: fixes m :: "'m"   and ms :: "'m list"   and r :: "'r"   and rs :: "'r list" assumes "R_lin_independent (m # ms)"   and "set (r # rs) \<subseteq> R"   and "(r # rs) \<bullet>\<cdot> (m # ms) = (0::'m)" shows "r = (0::'r)"
derivation_ge_shift_plus: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and u :: "nat"   and v :: "nat" assumes "derivation_ge D u"   and "derivation_ge (derivation_shift D u 0) v" shows "derivation_ge D (u + v)"
ipurge_tr_rev_t_input_2: fixes P :: "'c process"   and I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and u :: "'d" shows "ipurge_tr_rev_t_inv_2 P I D xs \<lparr>Pol = I, Map = D, Dom = u, In = xs, Out = []\<rparr>"
ivl_disj_un_one_7: fixes l :: "'c"   and u :: "'c" assumes "l \<le> u" shows "{l..u} \<union> {u<..} = {l..}"
search_empty: fixes b :: "nat"   and i :: "nat" assumes "s b \<down>= i"   and "\<And>x. \<phi> i x \<up>" shows "search b \<up>"
mat_of_cols_def: fixes n :: "nat"   and cs :: "'a Matrix.vec list" shows "mat_of_cols n cs = Matrix.mat n (length cs) (\<lambda>(i, j). cs ! j $v i)"
isPORdb_def: fixes e :: "db" shows "isPORdb e \<equiv> closed e \<and> DBApp (DBApp e DBtt) DBDiverge \<Down> DBtt \<and> DBApp (DBApp e DBDiverge) DBtt \<Down> DBtt \<and> DBApp (DBApp e DBff) DBff \<Down> DBff"
case_eq_if: fixes f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f"   and option :: "'g option" shows "(case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (if option = None then f1 else f2 (the option))"
assms_1: shows "inj_on f ({..<q} - {0})"
star_denest_var: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<star> \<cdot> (y \<cdot> x\<^sup>\<star>)\<^sup>\<star> = (x + y)\<^sup>\<star>"
pathSuc: fixes subs :: "'b \<Rightarrow> 'b set"   and gamma :: "'b"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and n :: "nat" shows "path subs gamma P (Suc n) = (if terminal subs (path subs gamma P n) then path subs gamma P n else failingSub subs P (path subs gamma P n))"
nth_Cons: fixes x :: "'f"   and xs :: "'f list"   and n :: "nat" shows "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
P2: shows "Bet A DD E \<and> (DD = B \<or> B Out DD D)"
Rep_Store_inject: fixes x :: "Store"   and y :: "Store" shows "(Rep_Store x = Rep_Store y) = (x = y)"