hcju/isabelleps huggingface dataset:
intros_2: fixes C :: "char list"   and C' :: "char list"   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 D :: "char list"   and Cs :: "char list list" assumes "P \<turnstile> C \<preceq>\<^sup>* C'"   and "P \<turnstile> C' \<prec>\<^sub>S D"   and "Subobjs\<^sub>R P D Cs" shows "Subobjs P C Cs"
sgn_mult_dvd_iff: fixes r :: "int"   and l :: "int"   and k :: "int" shows "(sgn r * l dvd k) = (l dvd k \<and> (r = 0 \<longrightarrow> k = 0))"
range_vars_def: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "range_vars \<sigma> = \<Union> (fv ` subst_range \<sigma>)"
b': shows "\<forall>r'\<in>R. rightPrincipal r' (Compound F Fs) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')"
dB: fixes i :: "'n"   and j :: "'n" shows "degree (to_fun (\<^bold>X - \<^bold>C A) i j) = (if i = j then 1 else 0)"
ord_resolve_one_side_prem: fixes CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and E :: "'a literal multiset" assumes "ord_resolve S CAs DA AAs As \<sigma> E" shows "length CAs = 1 \<and> length AAs = 1 \<and> length As = 1"
elims: fixes x :: "fm"   and y :: "fm" assumes "not x = y" obtains p :: "fm" where "x = NOT p" and "y = p"   | "x = T" and "y = F"   | "x = F" and "y = T"   | v :: "Ferrack.num" where "x = Lt v__" and "y = NOT (Lt v__)"   | v :: "Ferrack.num" where "x = Le va__" and "y = NOT (Le va__)"   | v :: "Ferrack.num" where "x = Gt vb__" and "y = NOT (Gt vb__)"   | v :: "Ferrack.num" where "x = Ge vc__" and "y = NOT (Ge vc__)"   | v :: "Ferrack.num" where "x = Eq vd__" and "y = NOT (Eq vd__)"   | v :: "Ferrack.num" where "x = NEq ve__" and "y = NOT (NEq ve__)"   | v :: "fm" and va :: "fm" where "x = And vf__ vaa__" and "y = NOT (And vf__ vaa__)"   | v :: "fm" and va :: "fm" where "x = Or vg__ vab__" and "y = NOT (Or vg__ vab__)"   | v :: "fm" and va :: "fm" where "x = Imp vh__ vac__" and "y = NOT (Imp vh__ vac__)"   | v :: "fm" and va :: "fm" where "x = Iff vi__ va" and "y = NOT (Iff vi__ va)"   | v :: "fm" where "x = E vj__" and "y = NOT (E vj__)"   | v :: "fm" where "x = A v" and "y = NOT (A v)"
irreducibleE: fixes p :: "'a" assumes "irreducible p" obtains "p \<noteq> (0::'a)" and "\<not> p dvd (1::'a)" and "\<And>a b. p = a * b \<Longrightarrow> a dvd (1::'a) \<or> b dvd (1::'a)"
complete_lattice_ccpo': fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "class.ccpo Sup less_eq (mk_less less_eq)"
funs_term_Fun_subterm': fixes f :: "'c"   and T :: "('c, 'd) Term.term list"   and t :: "('c, 'd) Term.term" assumes "Fun f T \<sqsubseteq> t" shows "f \<in> funs_term t"
fold_graph_insertE_aux: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'a set"   and y :: "'b"   and a :: "'a" assumes "comp_fun_commute f"   and "fold_graph f z A y"   and "a \<in> A" shows "\<exists>y'. y = f a y' \<and> fold_graph f z (A - {a}) y'"
abc_list_crsp_simp4: fixes lma :: "nat list"   and lmb :: "nat list"   and m :: "nat"   and n :: "nat" assumes "abc_list_crsp lma lmb"   and "\<not> m < length lma"   and "\<not> m < length lmb" shows "abc_list_crsp (lma @ 0 \<up> (m - length lma) @ [n]) (lmb @ 0 \<up> (m - length lmb) @ [n])"
refl: fixes x :: "'a\<^sub>m" assumes "x \<in> A" shows "bo x x"
HPhase1or2ReadThen_HInv4c: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc"   and q :: "Proc" assumes "HPhase1or2ReadThen s s' p d r"   and "HInv4c s q" shows "HInv4c s' q"
returns_result_eq: fixes h :: "'l"   and f :: "('l, 'm, 'n) prog"   and y :: "'n"   and y' :: "'n" assumes "h \<turnstile> f \<rightarrow>\<^sub>r y"   and "h \<turnstile> f \<rightarrow>\<^sub>r y'" shows "y = y'"
induct: fixes P :: "('d, 'c) stateful_strand_step list \<Rightarrow> ('c \<Rightarrow> ('d, 'c) Term.term) \<Rightarrow> (('d, 'c) Term.term \<times> ('d, 'c) Term.term) list \<Rightarrow> bool"   and a0 :: "('d, 'c) stateful_strand_step list"   and a1 :: "'c \<Rightarrow> ('d, 'c) Term.term"   and a2 :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" assumes "\<And>I D. P [] I D"   and "\<And>t s A I D. P A I (List.insert (t \<cdot> I, s \<cdot> I) D) \<Longrightarrow> P (insert\<langle>t,s\<rangle> # A) I D"   and "\<And>t s A I D. P A I (removeAll (t \<cdot> I, s \<cdot> I) D) \<Longrightarrow> P (delete\<langle>t,s\<rangle> # A) I D"   and "\<And>v A I D. P A I D \<Longrightarrow> P (send\<langle>v\<rangle> # A) I D"   and "\<And>v A I D. P A I D \<Longrightarrow> P (receive\<langle>v\<rangle> # A) I D"   and "\<And>v va vb A I D. P A I D \<Longrightarrow> P (\<langle>v: va \<doteq> vb\<rangle> # A) I D"   and "\<And>v va vb A I D. P A I D \<Longrightarrow> P (\<langle>v: va \<in> vb\<rangle> # A) I D"   and "\<And>v va vb A I D. P A I D \<Longrightarrow> P (\<forall>v\<langle>\<or>\<noteq>: va \<or>\<notin>: vb\<rangle> # A) I D" shows "P a0 a1 a2"
resolve_rename_def: fixes C :: "'a literal list"   and D :: "'a literal list" shows "resolve_rename C D = (let \<sigma>s = renamings_apart [mset D, mset C] in resolve (map (\<lambda>L. L \<cdot>l last \<sigma>s) C) (map (\<lambda>L. L \<cdot>l hd \<sigma>s) D))"
natural_isomorphism_\<phi>: shows "natural_isomorphism (subcategory.comp (\<cdot>) (\<lambda>\<mu>. \<guillemotleft>\<mu> : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright>)) (subcategory.comp (\<cdot>) (\<lambda>\<mu>. \<guillemotleft>\<mu> : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright>)) (identity_functor.map (subcategory.comp (\<cdot>) (\<lambda>\<mu>. \<guillemotleft>\<mu> : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright>))) FG.map (transformation_by_components.map (subcategory.comp (\<cdot>) (\<lambda>\<mu>. \<guillemotleft>\<mu> : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright>)) (subcategory.comp (\<cdot>) (\<lambda>\<mu>. \<guillemotleft>\<mu> : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright>)) (identity_functor.map (subcategory.comp (\<cdot>) (\<lambda>\<mu>. \<guillemotleft>\<mu> : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright>))) (\<lambda>f. (e\<^sub>1 \<star> \<a>\<^sup>-\<^sup>1[d\<^sub>1, f \<star> e\<^sub>0, d\<^sub>0]) \<cdot> \<a>[e\<^sub>1, d\<^sub>1, (f \<star> e\<^sub>0) \<star> d\<^sub>0] \<cdot> ((e\<^sub>1 \<star> d\<^sub>1) \<star> \<a>\<^sup>-\<^sup>1[f, e\<^sub>0, d\<^sub>0]) \<cdot> (local.inv \<epsilon>\<^sub>1 \<star> f \<star> local.inv \<epsilon>\<^sub>0) \<cdot> \<l>\<^sup>-\<^sup>1[f \<star> src d\<^sub>0] \<cdot> \<r>\<^sup>-\<^sup>1[f]))"
P5: shows "Bet Q Z Q' \<and> Cong Z Q' Q Z"
pos_add_strict: fixes a :: "'d multiset"   and b :: "'d multiset"   and c :: "'d multiset" assumes "{#} \<subset># a"   and "b \<subset># c" shows "b \<subset># a + c"
a_in: shows "a \<in> dom (share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<S>)"
rel_mset_def: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'b multiset"   and Y :: "'c multiset" shows "rel_mset R X Y = (\<exists>xs ys. mset xs = X \<and> mset ys = Y \<and> list_all2 R xs ys)"
insert_image: fixes x :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "x \<in> A" shows "insert (f x) (f ` A) = f ` A"
d_glb: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "d z \<le> d x \<and> d z \<le> d y" shows "d z \<le> d x \<cdot> d y"
put_env_parametric: fixes S :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and M :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool" shows "((S ===> M ===> M) ===> S ===> Monomorphic_Monad.rel_envT R M ===> Monomorphic_Monad.rel_envT R M) put_env put_env"
lfp_trans_unfold: fixes T :: "(('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real) \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and v :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>t u. \<lbrakk>le_trans t u; \<And>P. sound P \<Longrightarrow> sound (t P); \<And>P. sound P \<Longrightarrow> sound (u P)\<rbrakk> \<Longrightarrow> le_trans (T t) (T u)"   and "\<And>t P. \<lbrakk>\<And>Q. sound Q \<Longrightarrow> sound (t Q); sound P\<rbrakk> \<Longrightarrow> sound (T t P)"   and "le_trans (T v) v"   and "\<And>P. sound P \<Longrightarrow> sound (v P)" shows "equiv_trans (lfp_trans T) (T (lfp_trans T))"
increasing_conv_induction_0_pre: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "a = acc_point s" shows "\<exists>k>convergent_subseq_fun s a n. s k (Suc n) = a (Suc n)"
entropy_le: fixes MX :: "'c measure"   and X :: "'a \<Rightarrow> 'c"   and Px :: "'c \<Rightarrow> real" assumes "distributed M MX X (\<lambda>x. ennreal (Px x))"   and "\<And>x. x \<in> space MX \<Longrightarrow> 0 \<le> Px x"   and "emeasure MX {x \<in> space MX. Px x \<noteq> 0} \<noteq> top"   and "integrable MX (\<lambda>x. - Px x * log b (Px x))" shows "entropy b MX X \<le> log b \<P>(x in MX. Px x \<noteq> 0)"
healthy_equivI: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "equiv_trans t u"   and "healthy t" shows "healthy u"
ret_modifAbr: shows "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t"
pos_poly_add: fixes p :: "'b poly"   and q :: "'b poly" assumes "pos_poly p"   and "pos_poly q" shows "pos_poly (p + q)"
abs_eq: fixes x :: "integer list" shows "literal_of_asciis x = Abs_literal (map (ascii_of \<circ> char_of) x)"
c_secure_def: fixes step :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and out :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and s\<^sub>0 :: "'e"   and I :: "('h \<times> 'h) set"   and D :: "'f \<Rightarrow> 'h" shows "c_secure step out s\<^sub>0 I D \<equiv> \<forall>x xs. out (foldl step s\<^sub>0 xs) x = out (foldl step s\<^sub>0 (c_ipurge I D (D x) xs)) x"
assms_2: shows "\<D>\<^sub>\<circ> gf = 2\<^sub>\<nat>"
iprev_atLeastAtMost: fixes m :: "nat"   and t :: "nat"   and n :: "nat" assumes "m < t"   and "t \<le> n" shows "iprev t {m..n} = t - Suc 0"
lborel_real_affine: fixes c :: "real"   and t :: "real" assumes "c \<noteq> 0" shows "lborel = density (distr lborel borel (\<lambda>x. t + c * x)) (\<lambda>_. ennreal \<bar>c\<bar>)"
isLub_in_lattice: fixes S :: "'a set"   and L :: "'a" assumes "isLub S cl L" shows "L \<in> A"
mu_isotone: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "has_least_prefixpoint f"   and "has_least_prefixpoint g"   and "isotone f"   and "isotone g"   and "f \<le>\<le> g" shows "\<mu> f \<le> \<mu> g"
right_lang_Reverse: fixes q :: "hf" shows "nfa.right_lang (Reverse_nfa Canon.DFA) q = rev ` MN.left_lang q"
1: shows "\<guillemotleft>\<Phi>.map yx : Hom_FopxC.map yx \<rightarrow>\<^sub>S Hom_DopxG.map yx\<guillemotright>"
simps_2: fixes p :: "proc"   and v :: "val messageValue" shows "sends p s1 v = {# <p1, v0> }"
leg_simps_1: shows "C.arr \<mu>\<nu>\<pi>.leg0"
real_of_int_def: fixes x :: "int" shows "real_of x = real_of_int x"
insert: fixes A :: "'c set"   and x :: "'c" assumes "finite A"   and "x \<notin> A" shows "mset_set (insert x A) = add_mset x (mset_set A)"
Gamma_series_def: fixes z :: "'a"   and n :: "nat" shows "Gamma_series z n = fact n * exp (z * of_real (ln (real n))) / pochhammer z (n + 1)"
i_less_a: shows "mod_type_class.to_nat i \<le> a"
distinct_rev: fixes xs :: "'a list" shows "distinct (rev xs) = distinct xs"
bind_splits_1: fixes P :: "'c option \<Rightarrow> bool"   and m :: "'d option"   and f :: "'d \<Rightarrow> 'c option" shows "P (m \<bind> f) = ((m = None \<longrightarrow> P None) \<and> (\<forall>v. m = Some v \<longrightarrow> P (f v)))"
lower_circle_01_def: shows "lower_circle_01 = {x. cmod (x - (1 / 2 - complex_of_real (sqrt 3 / 6) * \<i>)) < sqrt 3 / 3}"
sec: shows "constructive_security_aux_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common adv"
vars_llist_alt_def: fixes xs :: "(char list list \<times> int) list"   and \<V> :: "char list set" shows "(vars_llist xs \<subseteq> \<V>) = vars_of_poly_in xs \<V>"
Torder_axioms_def: fixes D :: "('c, 'd) Order_scheme" shows "Torder_axioms D \<equiv> \<forall>a b. a \<in> carrier D \<longrightarrow> b \<in> carrier D \<longrightarrow> a \<preceq>\<^bsub>D\<^esub> b \<or> b \<preceq>\<^bsub>D\<^esub> a"
cone_elem_sum: fixes Vs :: "'a vec set"   and x :: "'a vec"   and y :: "'a vec" assumes "Vs \<subseteq> carrier_vec (n + 1)"   and "x \<in> next_dim.cone Vs"   and "y \<in> next_dim.cone Vs" shows "x + y \<in> next_dim.cone Vs"
det_mk_poly: fixes A :: "'b mat" shows "det (mk_poly A) = [:det A:]"
prism_1: fixes q :: "nat" shows "prism q 0 = 0"
beta_reducible_imp_beta_step: fixes t :: "Core.term" assumes "beta_reducible t" shows "\<exists>t'. t \<rightarrow>\<^sub>\<beta> t'"
c: shows "AE x in M. subcocycle_lim w x = 0"
elt_set_times_def: fixes a :: "'c"   and B :: "'c set" shows "a *o B = {c. \<exists>b\<in>B. c = a * b}"
comp_mkArr: fixes A :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr set"   and B :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr set"   and F :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr \<Rightarrow> (('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr"   and C :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr set"   and G :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr \<Rightarrow> (('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr" assumes "S.arr (S.mkArr A B F)"   and "S.arr (S.mkArr B C G)" shows "S.mkArr B C G \<cdot>\<^sub>S S.mkArr A B F = S.mkArr A C (G \<circ> F)"
assms: shows "s \<in> init (pnet np \<langle>i; R\<rangle>)"
weakBisimE_2: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(\<Psi>, P, Q) \<in> env.weakBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" shows "env.weakSimulation substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P (env.weakBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq') Q"
L7_Controller_Zero: fixes y :: "nat \<Rightarrow> nat list"   and l :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and t :: "nat" assumes "Controller_L y (fin_inf_append [Zero] l) l z"   and "l t = Zero" shows "last (fin_make_untimed (inf_truncate z t)) = Zero"
seq_stop_elim: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>Stop ;; c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "c' = c \<and> mds' = mds \<and> mem' = mem"
H_def: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" assumes "bidel_arc G M a" shows "bidel_arc.H G M a \<equiv> pre_digraph.del_arc (pre_digraph.del_arc G (edge_rev M a)) a"
index_tensor_mat: fixes A :: "complex Matrix.mat"   and rA :: "nat"   and cA :: "nat"   and B :: "complex Matrix.mat"   and rB :: "nat"   and cB :: "nat"   and i :: "nat"   and j :: "nat" assumes "dim_row A = rA"   and "dim_col A = cA"   and "dim_row B = rB"   and "dim_col B = cB"   and "i < rA * rB"   and "j < cA * cB"   and "0 < cA"   and "0 < cB" shows "(A \<Otimes> B) $$ (i, j) = A $$ (i div rB, j div cB) * B $$ (i mod rB, j mod cB)"
lazy_def: fixes return :: "int \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> (int \<Rightarrow> 'c) \<Rightarrow> 'c"   and fail :: "'c"   and get :: "(('b \<Rightarrow> int option) \<Rightarrow> 'c) \<Rightarrow> 'c"   and put :: "('b \<Rightarrow> int option) \<Rightarrow> 'c \<Rightarrow> 'c"   and sample :: "int pmf \<Rightarrow> (int \<Rightarrow> 'c) \<Rightarrow> 'c"   and X :: "'b \<Rightarrow> int pmf" shows "prob_exp_base.lazy return bind fail get put sample X \<equiv> exp_base.eval return bind fail (memoization_base.memo return bind get put (prob_exp_base.sample_var return sample X))"
hyps_2: shows "v\<^sub>1 @ l =\<^sub>F w\<^sub>1"
f: shows "\<forall>e>0. finite (f e) \<and> f e \<subseteq> S \<and> S \<subseteq> (\<Union>x\<in>f e. ball x e)"
iface_conjunct_commute: fixes i1 :: "iface"   and i2 :: "iface" shows "iface_conjunct i1 i2 = iface_conjunct i2 i1"
wtC_append: fixes c1 :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list"   and c2 :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list" shows "GE.wtC (c1 @ c2) = (GE.wtC c1 \<and> GE.wtC c2)"
discrete_topology_frontier_of: fixes U :: "'c set"   and S :: "'c set" shows "discrete_topology U frontier_of S = {}"
intros_2: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and xs :: "'c llist"   and ys :: "'c llist" assumes "r x y" shows "llexord r (LCons x xs) (LCons y ys)"
list_update_id: fixes xs :: "'f list"   and i :: "nat" shows "xs[i := xs ! i] = xs"
Cons_5: shows "map getGMUserCom (tl (ss # sl)) = ucl"
ipresIGOp_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'e"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'f"   and MOD :: "('index, 'bindex, 'g, 'h, 'opSym, 'i, 'e, 'f, 'j) model_scheme" shows "ipresIGOp h hA termMOD MOD = ipresOp h hA MOD"
poly_lift_hom_comm_taylor_expansion: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and p :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "p \<in> carrier P"   and "a \<in> carrier R" shows "poly_lift_hom R S \<phi> (taylor_expansion R a p) = taylor_expansion S (\<phi> a) (poly_lift_hom R S \<phi> p)"
Join_stable: fixes F :: "'d program"   and G :: "'d program"   and A :: "'d set" shows "(F \<squnion> G \<in> stable A) = (F \<in> stable A \<and> G \<in> stable A)"
simps_3: fixes v :: "'b"   and va :: "'b"   and vb :: "'c"   and l :: "'b set list" shows "in_list (AllowPortFromTo v va vb) l = (bothNet (AllowPortFromTo v va vb) \<in> set l)"
\<sigma>_5: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma>"
matrix_vector_mul_assoc: fixes A :: "(('a, 'c) vec, 'b) vec"   and B :: "(('a, 'd) vec, 'c) vec"   and x :: "('a, 'd) vec" shows "A *v (B *v x) = A ** B *v x"
lookup_def: shows "lookup \<equiv> map_fun impl_of id (rbt_comp_lookup ccomp)"
elims_1: fixes x :: "('a, 'b) rbt"   and y :: "bool" assumes "bd_inv x = y" obtains "x = Leaf" and "y"   | l :: "('a, 'b) rbt" and c :: "color" and k :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "x = Node l c k v r" and "y = (bd_inv l \<and> bd_inv r \<and> black_depth l = black_depth r)"
vrs_of_eqvt_2: fixes pi' :: "(vrs \<times> vrs) list"   and x :: "binding" shows "pi' \<bullet> tyvrs_of x = tyvrs_of (pi' \<bullet> x)"
simps_15: fixes x21 :: "'b"   and x22 :: "'b list" shows "set (x21 # x22) = x21 \<triangleright> set x22"
set_sorted_domain: fixes ko :: "'b nat_term_order"   and xs :: "('b, 'c) oalist_ntm" shows "set (OAlist_sorted_domain_ntm ko xs) = fst ` set (fst (list_of_oalist_ntm xs))"
s_3: shows "\<forall>n. 0 \<le> s n"
ID_Some: fixes x :: "'c" shows "ID (Some x) = Some x"
phiDefNode_is_join_node: fixes n :: "'node"   and g :: "'g"   and v :: "'var" assumes "n \<in> phiDefNodes g v" shows "length (predecessors g n) \<noteq> 1"
sublist_trans: fixes l1 :: "'c list"   and l2 :: "'c list"   and l3 :: "'c list" assumes "subseq l1 l2"   and "subseq l2 l3" shows "subseq l1 l3"
satisfiable_def: fixes S :: "'at Literal set set" shows "satisfiable S \<equiv> \<exists>I. I \<Turnstile> S"
scaleC_mono: fixes a :: "complex"   and b :: "complex"   and x :: "'a"   and y :: "'a" assumes "a \<le> b"   and "x \<le> y"   and "0 \<le> b"   and "(0::'a) \<le> x" shows "a *\<^sub>C x \<le> b *\<^sub>C y"
unique_liftrT: fixes n :: "nat"   and k :: "nat"   and fs :: "(char list \<times> POPLmarkRecord.type) list" shows "unique (\<up>\<^sub>r\<^sub>\<tau> n k fs) = unique fs"
eq_zero_iff: fixes x :: "nat" shows "(\<chi> x = 0) = (\<not> coprime x n)"
integrable1_aux: fixes x :: "real"   and z :: "complex"   and y :: "real" assumes "- 2 < x"   and "Re z \<in> {x..y}" shows "complex_interval_lebesgue_integrable lborel 0 (ereal (exp (- 1))) (f z)"
equal_mod_timpls_iff_ex_in_timpl_closure: fixes TI' :: "('c set \<times> 'c set) list"   and TI :: "('c set \<times> 'c set) set"   and s :: "(('d, 'e, 'c) prot_fun, 'f) Term.term"   and t :: "(('d, 'e, 'c) prot_fun, 'f) Term.term" assumes "set TI' = {(a, b). (a, b) \<in> TI\<^sup>+ \<and> a \<noteq> b}" shows "equal_mod_timpls TI' s t = (\<exists>u. u \<in> timpl_closure s TI \<and> u \<in> timpl_closure t TI)"
numeral_eq_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "num"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "(numeral.numeral one plus m = numeral.numeral one plus n) = (m = n)"
adopt_node_node_in_disconnected_nodes: fixes h :: "(_) heap"   and owner_document :: "(_) document_ptr"   and node_ptr :: "(_) node_ptr"   and h' :: "(_) heap"   and disc_nodes :: "(_) node_ptr list" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.adopt_node owner_document node_ptr \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disc_nodes"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "node_ptr \<in> set disc_nodes"
finite_group_def: fixes G :: "('c, 'd) monoid_scheme" shows "finite_group G \<equiv> Group.group G \<and> finite_group_axioms G"
forall_7: fixes P :: "7 \<Rightarrow> bool" shows "(\<forall>i. P i) = (P 1 \<and> P 2 \<and> P 3 \<and> P 4 \<and> P 5 \<and> P 6 \<and> P 7)"
that: fixes ns' :: "'node SDG_node list" assumes "realizable (CFG_node (_Entry_)) ns' n'"   and "n \<in> set ns'" shows "thesis"
Ball_Collect: fixes A :: "'g set"   and P :: "'g \<Rightarrow> bool" shows "Ball A P = (A \<subseteq> Collect P)"
ennreal2_cases: fixes x :: "ennreal"   and xa :: "ennreal" obtains r :: "real" and ra :: "real" where "0 \<le> r__" and "x = ennreal r__" and "0 \<le> ra" and "xa = ennreal ra"   | r :: "real" where "0 \<le> rb__" and "x = ennreal rb__" and "xa = \<top>"   | r :: "real" where "x = \<top>" and "0 \<le> r" and "xa = ennreal r"   | "x = \<top>" and "xa = \<top>"
dvd_dvd_imp_unit_mult: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.idom times one plus zero minus uminus"   and "dvd.dvd times x y"   and "dvd.dvd times y x" shows "\<exists>z. dvd.dvd times z one \<and> y = times x z"
face_of_imp_convex: fixes T :: "'b set"   and S :: "'b set" assumes "T face_of S" shows "convex T"