hcju/isabelleps huggingface dataset:
subst_lam_neq: fixes y :: "nat"   and x :: "nat"   and v :: "exp"   and e :: "exp" assumes "y \<noteq> x" shows "subst x v (ELam y e) = ELam y (subst x v e)"
unfld: shows "x\<^bsub>1::'a\<^esub> + \<Sum> {x\<^bsub>i\<^esub> |i. i \<in> UNIV - {1::'a}} = (1::'b) + x\<^bsub>1::'a\<^esub> + \<Sum> {x\<^bsub>i\<^esub> |i. i \<in> UNIV - {1::'a}}"
simps: fixes G :: "nat graph_rec"   and E :: "(nat \<times> nat) set"   and a1 :: "nat list"   and a2 :: "nat" assumes "cfg_doms G" shows "path_entry E a1 a2 = (a1 = [] \<and> a2 = 0 \<or> (\<exists>u v l. a1 = u # l \<and> a2 = v \<and> (u, v) \<in> E \<and> path_entry E l u))"
n: shows "Ln (prod f A) = (\<Sum>x\<in>A. Ln (f x) + complex_of_real (real_of_int (n x) * (2 * pi)) * \<i>)"
induct: fixes x :: "'b set"   and F :: "'b program"   and B :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "x \<in> wens_set F B"   and "P B"   and "\<And>X act. \<lbrakk>X \<in> wens_set F B; P X; act \<in> Acts F\<rbrakk> \<Longrightarrow> P (wens F act X)"   and "\<And>W. \<lbrakk>W \<noteq> {}; \<forall>U\<in>W. U \<in> wens_set F B \<and> P U\<rbrakk> \<Longrightarrow> P (\<Union> W)" shows "P x"
FrobeniusReciprocity_axioms: shows "FrobeniusReciprocity H G (\<cdot>) V (\<currency>) (\<star>) W"
pred_inject_2: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and aa :: "'b list" shows "list_all P (a # aa) = (P a \<and> list_all P aa)"
isomorphic_implies_ide_2: fixes f :: "'a"   and g :: "'a" assumes "B.isomorphic f g" shows "B.ide g"
finite_list_upwards: fixes xs :: "['b]"   and ys :: "['b]" assumes "finite_list xs"   and "xs \<sqsubseteq> ys" shows "finite_list ys"
ssb: shows "S \<subseteq> inside (path_image \<gamma>)"
simps_4: fixes r :: "'a pfm"   and p :: "'a pfm"   and q :: "'a pfm" shows "reduce' r (p \<^bold>\<and>\<^sub>! q) = (reduce' r p \<^bold>\<and>\<^sub>! reduce' r q)"
fmlookup_inject: fixes x :: "('a, 'b) fmap"   and y :: "('a, 'b) fmap" shows "(($$) x = ($$) y) = (x = y)"
to0: shows "(\<lambda>n. of_nat n * (x / of_real r) ^ n) \<longlonglongrightarrow> (0::'a)"
simps_1: fixes f :: "'c \<Rightarrow> ('c, 'a, 'b) sp\<^sub>\<mu>"   and s :: "'c stream" shows "run\<^sub>\<mu> (Get f) s = run\<^sub>\<mu> (f (shd s)) (stl s)"
append_child_wf_is_l_append_child_wf: shows "l_append_child_wf ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.append_child Shadow_DOM.heap_is_wellformed"
least_mod_type: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and n :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "less_eq zero n"
Start_nsys_xcpts: shows "Start \<notin> sys_xcpts"
max_def: fixes a :: "'t"   and b :: "'t" shows "ord_term_lin.max a b = (if a \<preceq>\<^sub>t b then b else a)"
not_distinct_conv_prefix: fixes as :: "'g list" shows "(\<not> distinct as) = (\<exists>xs y ys. y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys)"
swapped_skel: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set" assumes "good X"   and "(X, Y) \<in> swapped" shows "skel Y = skel X"
intersection_numbers_def: shows "intersection_numbers \<equiv> {b1 |\<inter>| b2 |b1 b2. b1 \<in># \<B> \<and> b2 \<in># remove1_mset b1 \<B>}"
ctr: fixes s1 :: "'d stream"   and s2 :: "'e stream" shows "szip s1 s2 = (shd s1, shd s2) ## szip (stl s1) (stl s2)"
holomorphic_EM_remainder: fixes a :: "int"   and G :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and U :: "complex set"   and f :: "complex \<Rightarrow> real \<Rightarrow> complex"   and f' :: "complex \<Rightarrow> real \<Rightarrow> complex"   and n :: "nat" assumes "\<And>y. real_of_int a \<le> y \<Longrightarrow> (G has_real_derivative g y) (at y within {real_of_int a..})"   and "\<And>z t x. \<lbrakk>z \<in> U; a \<le> x; t \<in> {real_of_int a..real_of_int x}\<rbrakk> \<Longrightarrow> ((\<lambda>z. f z t) has_field_derivative f' z t) (at z within U)"   and "continuous_on (U \<times> {real_of_int a..}) (\<lambda>(z, t). f' z t)"   and "\<And>b c z e. \<lbrakk>real_of_int a \<le> b; z \<in> U\<rbrakk> \<Longrightarrow> (\<lambda>t. complex_of_real (bernpoly n (t - e)) * f z t) integrable_on {b..c}"   and "\<And>a' b y. \<lbrakk>y \<in> U; real_of_int a \<le> a'; a' \<le> b\<rbrakk> \<Longrightarrow> (\<lambda>t. pbernpoly n t *\<^sub>R f y t) integrable_on {a'..b}"   and "convergent (\<lambda>y. G (real y))"   and "\<forall>\<^sub>F x in at_top. \<forall>y\<in>U. cmod (f y x) \<le> g x"   and "open U" shows "(\<lambda>s. EM_remainder n (f s) a) holomorphic_on U"
IH: fixes as' :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) list"   and vs :: "'a set" assumes "subseq as' (as_proj as vs)" shows "as_proj as' vs = as'"
stl_sset: fixes x :: "'d"   and a :: "'d stream" assumes "x \<in> sset (stl a)" shows "x \<in> sset a"
pw_ASSUME_2: fixes \<Phi> :: "bool"   and x :: "unit" shows "inres (ASSUME \<Phi>) x = \<Phi>"
filterlim_at_within_not_equal: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and s :: "'c set"   and F :: "'b filter"   and b :: "'c" assumes "filterlim f (at a within s) F" shows "\<forall>\<^sub>F w in F. f w \<in> s \<and> f w \<noteq> b"
make\<Pi>\<^sub>1_cases: fixes x :: "\<Pi>\<^sub>1" obtains y :: "\<upsilon> \<Rightarrow> j \<Rightarrow> i \<Rightarrow> bool" where "x = make\<Pi>\<^sub>1 y" and "y \<in> UNIV"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "sup (sup a b) c = sup a (sup b c)"
cardinalityOneTheElemIdentity: fixes X :: "'c set" shows "(card X = 1) = (X = {the_elem X})"
hyps_1: shows "P \<Longrightarrow>\<^sub>\<tau> P''"
SDG_Use_parent_Use: fixes V :: "'var"   and n :: "'node SDG_node" assumes "V \<in> Use\<^bsub>SDG\<^esub> n" shows "V \<in> Use (parent_node n)"
fset_of_list_def: shows "fset_of_list \<equiv> map_fun id Abs_fset set"
oracle\<^sub>n_def: fixes b :: "bool"   and \<eta> :: "nat" shows "oracle\<^sub>n b \<eta> = family_oracle (\<lambda>_. oracle\<^sub>1 b \<eta>)"
Assign_le: fixes x :: "char list"   and e :: "aexp"   and S :: "'b"   and c :: "'b acom" shows "(x ::= e {S} \<le> c) = (\<exists>S'. c = x ::= e {S'} \<and> S \<le> S')"
simps_3: fixes Mkt :: "('c, 'd) discrete_market"   and v :: "real"   and pf :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and asset :: "'d"   and n :: "nat" shows "remaining_qty Mkt v pf asset (Suc (Suc n)) = (\<lambda>w. remaining_qty Mkt v pf asset (Suc n) w + (cls_val_process Mkt pf (Suc n) w - val_process Mkt pf (Suc n) w) / prices Mkt asset (Suc n) w)"
RedCallNull: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and M :: "String.literal"   and vs :: "'addr val list"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" shows "extTA,P,t \<turnstile> \<langle>null\<bullet>M(map Val vs),s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer),s\<rangle>"
y2: shows "y\<^sub>2\<^sup>2 = x\<^sub>2 ^ 3 + a * x\<^sub>2 + b"
continuous_within_closed_nontrivial: fixes s :: "'b set"   and a :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "closed s"   and "a \<notin> s" shows "continuous (at a within s) f"
mult2_altE: fixes X :: "'b multiset"   and Y :: "'b multiset"   and b :: "bool"   and ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "(X, Y) \<in> mult2_alt b ns s" obtains X1 :: "'b multiset" and X2 :: "'b multiset" and Y1 :: "'b multiset" and Y2 :: "'b multiset" where "X = X1 + X2" and "Y = Y1 + Y2" and "(X1, Y1) \<in> multpw ns" and "b \<or> Y2 \<noteq> {#}" and "\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s)"
sym_preproc_aux_Nil: fixes gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ks :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "sym_preproc_aux gs ks ([], fs) = (ks, fs)"
lock_thread_ok_upd: fixes ls :: "('d, 'e) locks"   and ts :: "'e \<Rightarrow> ('f \<times> 'd \<Rightarrow>f nat) option"   and t :: "'e"   and xw :: "'f \<times> 'd \<Rightarrow>f nat" assumes "lock_thread_ok ls ts" shows "lock_thread_ok ls (ts(t \<mapsto> xw))"
desargues_config_2D_def: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points" shows "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma> \<equiv> rk {A, B, C} = 3 \<and> rk {A', B', C'} = 3 \<and> rk {A, A', P} = 2 \<and> rk {B, B', P} = 2 \<and> rk {C, C', P} = 2 \<and> rk {A, B, \<gamma>} = 2 \<and> rk {A', B', \<gamma>} = 2 \<and> rk {A, C, \<beta>} = 2 \<and> rk {A', C', \<beta>} = 2 \<and> rk {B, C, \<alpha>} = 2 \<and> rk {B', C', \<alpha>} = 2 \<and> rk {A, B, C, A', B', C'} = 3 \<and> rk {A, B, P} = 3 \<and> rk {A, C, P} = 3 \<and> rk {B, C, P} = 3 \<and> rk {A, A'} = 2 \<and> rk {B, B'} = 2 \<and> rk {C, C'} = 2"
countable_acc: fixes X :: "'a set" assumes "countable X" shows "countable (K1.acc `` X)"
gfp_prod: fixes F :: "'d \<times> 'e \<Rightarrow> 'd \<times> 'e" assumes "mono F" shows "gfp F = (gfp (\<lambda>x. fst (F (x, gfp (\<lambda>y. snd (F (x, y)))))), gfp (\<lambda>y. snd (F (gfp (\<lambda>x. fst (F (x, gfp (\<lambda>y. snd (F (x, y)))))), y))))"
sWlsAbsDisj_def: fixes SEM :: "('b, 'c, 'd, 'sort, 'e, 'f, 'g) semDom_scheme" shows "sWlsAbsDisj SEM \<equiv> \<forall>us s us' s' sA. isInBar (us, s) \<and> isInBar (us', s') \<and> sWlsAbs SEM (us, s) sA \<and> sWlsAbs SEM (us', s') sA \<longrightarrow> us = us' \<and> s = s'"
l3_obs_iagreement_Resp: shows "oreach l3 \<subseteq> l2_iagreement_Resp"
mzeromap_mHom: fixes M :: "('i, 'a, 'j) Module_scheme"   and N :: "('k, 'a, 'l) Module_scheme" assumes "R module M"   and "R module N" shows "mzeromap M N \<in> mHom R M N"
c_eq: fixes i :: "nat" shows "fpxs_of_fls (c i) = fpxs_compose_power (coeff p i) (rat_of_nat b)"
document_ptrs_def: fixes heap :: "(_) heap" shows "document_ptrs heap = ffilter is_document_ptr (document_ptr_kinds heap)"
inf_not_le_Le: fixes x :: "'c" shows "(\<infinity> \<le> Le x) = False"
is_mgu_insert_swap: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" shows "is_mgu \<sigma> (insert (s, t) E) = is_mgu \<sigma> (insert (t, s) E)"
intro: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lub :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" assumes "\<And>f n. le (f n) (lub f)"   and "\<And>f x. \<forall>n. le (f n) x \<Longrightarrow> le (lub f) x" shows "omega_complete_axioms le lub"
abs_eq: fixes x :: "'b" shows "interval_of x = Interval (x, x)"
c2: assumes "sqn rt ip < \<pi>\<^sub>2 r" shows "P (rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 (the (rt ip)))))"
hoare_apply: fixes P :: "'a set"   and R :: "('a \<times> 'a) set"   and Q :: "'a set"   and x :: "'a"   and y :: "'a" assumes "{P} R {> Q}"   and "x \<in> P"   and "(x, y) \<in> R" shows "y \<in> Q"
triv_iso: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme" assumes "Group.group G"   and "Group.group H"   and "carrier G = {\<one>\<^bsub>G\<^esub>}"   and "carrier H = {\<one>\<^bsub>H\<^esub>}" shows "G \<cong> H"
induct: fixes P :: "'d tree \<Rightarrow> bool"   and a0 :: "'d tree" assumes "P \<langle>\<rangle>"   and "\<And>l a r. \<lbrakk>P l; P r\<rbrakk> \<Longrightarrow> P \<langle>l, a, r\<rangle>" shows "P a0"
Konig: fixes t :: "'a tree" assumes "\<not> tfinite t" shows "ipath t (konig t)"
col_mat: fixes j :: "nat"   and nc :: "nat"   and nr :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'd" assumes "j < nc" shows "col (Matrix.mat nr nc f) j = Matrix.vec nr (\<lambda>i. f (i, j))"
do_store_action_def: fixes wact :: "('e, 'g, 'h) mem_store_action" shows "do_store_action wact = (\<lambda>s. case wact of mw_Mark r gc_mark \<Rightarrow> s\<lparr>heap := (heap s)(r := map_option (obj_mark_update (\<lambda>_. gc_mark)) (heap s r))\<rparr> | mw_Mutate r f new_r \<Rightarrow> s\<lparr>heap := (heap s)(r := map_option (\<lambda>obj. obj\<lparr>obj_fields := (obj_fields obj)(f := new_r)\<rparr>) (heap s r))\<rparr> | mw_Mutate_Payload r f pl \<Rightarrow> s\<lparr>heap := (heap s)(r := map_option (\<lambda>obj. obj\<lparr>obj_payload := (obj_payload obj)(f := pl)\<rparr>) (heap s r))\<rparr> | mw_fA gc_mark \<Rightarrow> s\<lparr>fA := gc_mark\<rparr> | mw_fM gc_mark \<Rightarrow> s\<lparr>fM := gc_mark\<rparr> | mw_Phase gc_phase \<Rightarrow> s\<lparr>phase := gc_phase\<rparr>)"
dim_mult_mat_vec: fixes A :: "'b mat"   and v :: "'b vec" shows "dim_vec (A *\<^sub>v v) = dim_row A"
a: shows "length (MGF (sha1 (generate_M' (sha1 m) salt)) (length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))))) = length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt)))))"
hyps_4: shows "fv t \<inter> insert w (subst_domain \<theta>) = {}"
strict_implies_order: fixes a :: "'a list"   and b :: "'a list" assumes "a <lex b" shows "a \<le>lex b"
STA_add: fixes A :: "'c set"   and n :: "nat"   and branch :: "(('d, 'c) fm list \<times> 'c) list"   and m :: "nat" assumes "A, n \<turnstile> branch" shows "A, m + n \<turnstile> branch"
abs_if_raw: shows "abs \<equiv> \<lambda>a. if a < (0::'b) then - a else a"
ts_process_d: fixes T :: "'a list set" assumes "trace_set T" shows "deterministic (ts_process T)"
intro: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>x y. 0 \<le> f x y"   and "\<And>x y. f x y = f y x"   and "\<And>x. f x x = 0"   and "\<And>x z y. f x z \<le> sqrt 2 * max (f x y) (f y z)" shows "Turn_into_distance f"
the_elem_wf: fixes X :: "'a set" assumes "X \<in> carrier (R Quot a_kernel R S h)" shows "\<exists>y\<in>carrier S. h ` X = {y}"
fBNsep1: fixes z :: "('b, 'c) Combinators list"   and x :: "('b, 'd) Combinators"   and a :: "('b, 'c) Combinators" assumes "\<And>a. a \<in> set z \<Longrightarrow> first_bothNet x \<noteq> first_bothNet a"   and "a \<in> set (separate z)" shows "first_bothNet x \<noteq> first_bothNet a"
Cons_3: shows "\<forall>ca. ca el c # F'' \<longrightarrow> clauseTautology ca"
K_finite: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and s :: "'a" assumes "Finite_Markov_Decision_Process K S"   and "s \<in> S" shows "finite (K s)"
distinct_14: fixes x81 :: "poly"   and x82 :: "nat"   and x83 :: "poly"   and x1 :: "int \<times> int" shows "CN x81 x82 x83 \<noteq> C x1"
not_model_if_exists_unmodeled_singleton_clause: fixes F :: "'b formula"   and L :: "'b literal"   and \<nu> :: "'b \<Rightarrow> bool" assumes "is_cnf F"   and "{L} \<in> cnf F"   and "\<not> lit_semantics \<nu> L" shows "\<not> \<nu> \<Turnstile> F"
taylor_expansion_def: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and p :: "nat \<Rightarrow> 'c" shows "taylor_expansion R a p = Cring_Poly.compose R p (X_poly_plus R a)"
finite_Diff_insert: fixes A :: "'a set"   and a :: "'a"   and B :: "'a set" shows "finite (A - insert a B) = finite (A - B)"
return_time_exivl: fixes P :: "'a set"   and x :: "'a" assumes "returns_to P x"   and "closed P" shows "return_time P x \<in> existence_ivl0 x"
fsafe: shows "fsafe (seq2fml (\<Gamma>, \<Delta>))"
sorted_insort_insert_key: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and x :: "'b" assumes "sorted (map f xs)" shows "sorted (map f (insort_insert_key f x xs))"
rel_map_1: fixes Sb :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'d \<Rightarrow> 'a"   and x :: "'d pmf"   and y :: "'b pmf" shows "rel_pmf Sb (map_pmf i x) y = rel_pmf (\<lambda>x. Sb (i x)) x y"
extremum: fixes a :: "'b::order_bot" shows "\<bottom> \<le> (a::'b::order_bot)"
ag_eq_sol2: fixes A :: "('e, 'f) aGroup_scheme"   and a :: "'e"   and x :: "'e"   and b :: "'e" assumes "aGroup A"   and "a \<in> carrier A"   and "x \<in> carrier A"   and "b \<in> carrier A"   and "x \<plusminus>\<^bsub>A\<^esub> a = b" shows "x = b \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> a"
ifthenelse_def: fixes p :: "'a"   and x :: "'a"   and y :: "'a" shows "if p then x else y fi = t p \<cdot> x + !p \<cdot> y"
top_1: shows "top_on\<^sub>c C1 (- vars (strip C2))"
coboundedI2: fixes c :: "'c"   and b :: "'c"   and a :: "'c" assumes "c \<le> b" shows "c \<le> a \<squnion> b"
SES_valid_def: fixes SES :: "('c, 'd) SES_rec" shows "SES_valid SES \<equiv> s0_is_state SES \<and> ses_inputs_are_events SES \<and> ses_outputs_are_events SES \<and> ses_inputs_outputs_disjoint SES \<and> correct_transition_relation SES"
AE_pos: shows "AE \<omega> in Pi\<^sub>M UNIV (\<lambda>n. exponential (R n)). \<forall>i. 0 < \<omega> i"
\<nu>\<kappa>_def: shows "\<nu>\<kappa> \<equiv> map_fun id make\<kappa> Some"
pref_exp_le: fixes u :: "'b list"   and m :: "nat"   and n :: "nat" assumes "u \<noteq> \<epsilon>"   and "u \<^sup>@ m \<le>p u \<^sup>@ n" shows "m \<le> n"
mod_diff_left_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and c :: "'d"   and b :: "'d" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (minus (modulo a c) b) c = modulo (minus a b) c"
disj_assoc: fixes P :: "bool"   and Q :: "bool"   and R :: "bool" shows "((P \<or> Q) \<or> R) = (P \<or> Q \<or> R)"
inverse_arrows_sym: fixes f :: "'a"   and g :: "'a" shows "A.inverse_arrows f g = A.inverse_arrows g f"
notxbeforez: shows "\<not> x < z in s_TS init h \<sigma> (length (as @ [x] @ bs @ [x]) + k + 1)"
strict_trans: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "lt b a"   and "lt c b" shows "lt c a"
zip_append1: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'e list" shows "xs @ ys || zs = (xs || take (length xs) zs) @ (ys || drop (length xs) zs)"
advice_sets_element_subfrmlsn_1: fixes xs :: "'a ltln list"   and ys :: "'a ltln list"   and \<phi> :: "'a ltln" assumes "(xs, ys) \<in> set (advice_sets \<phi>)" shows "set xs \<subseteq> subformulas\<^sub>\<mu> \<phi>"
elims_2: fixes x :: "('d, 'e) fmap"   and xa :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list" assumes "sat_precond_as x xa" obtains s :: "('d, 'e) fmap" where "x = s__" and "xa = []"   | s :: "('d, 'e) fmap" and a :: "('d, 'e) fmap \<times> ('d, 'e) fmap" and as :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list" where "x = s" and "xa = a # as" and "fst a \<subseteq>\<^sub>f s \<and> sat_precond_as (state_succ s a) as"
AB: shows "(A, B) \<in> \<langle>nat_rel, fully_unsorted_poly_rel O mset_poly_rel\<rangle>fmap_rel"
exp_times_self_eq_iff: fixes w :: "real"   and x :: "real" shows "(w * exp w = x) = (- exp (- 1) \<le> x \<and> (w = Lambert_W x \<or> x < 0 \<and> w = Lambert_W' x))"
clop_iso_var: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "clop f"   and "x \<le> y" shows "f x \<le> f y"
chine_simps_1: shows "C.arr \<mu>\<nu>\<pi>.chine"
k_mkAlgAuto_mkAutoSim_equiv: fixes t :: "'s Trace"   and a :: "'a" assumes "t \<in> jkbpC" shows "simAbs (runJP k_mkAlgAuto t a) = simAbs (runJP mkAutoSim t a)"
right_cancel: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "group f z inverse" shows "(f b a = f c a) = (b = c)"
k_invariant_step_old_aux: fixes x :: "'rep"   and ec :: "'rep"   and X :: "'rep list"   and Y :: "'mt"   and obs :: "'obs" assumes "k_isNode x"   and "k_isNode ec"   and "simAbs ec \<noteq> simAbs x" shows "lookup tOps (foldr (transUpdate x) X Y) (ec, obs) = lookup tOps Y (ec, obs)"
continuation_lemma: fixes A :: "'a list set"   and B :: "'a list set" assumes "regular A"   and "infinite B" shows "\<exists>x\<in>B. \<exists>y\<in>B. x \<noteq> y \<and> x \<approx>A y"
NS_Says_Spy_staticSecret: fixes B :: "agent"   and X :: "msg"   and evs :: "event list"   and m :: "msg"   and A :: "agent" assumes "Says Spy B X \<in> set evs"   and "m \<in> parts {X}"   and "m \<in> staticSecret A"   and "evs \<in> ns_public" shows "A \<in> bad"
the_def: fixes A :: "'a pred" shows "Predicate.the A = (THE x. eval A x)"
vfsequence_vlrestriction_vcard: shows "xs \<restriction>\<^sup>l\<^sub>\<circ> vcard xs = xs"
inject_10: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and y101 :: "expr"   and y102 :: "char list option"   and y103 :: "char list"   and y104 :: "expr list" shows "(Call x101 x102 x103 x104 = Call y101 y102 y103 y104) = (x101 = y101 \<and> x102 = y102 \<and> x103 = y103 \<and> x104 = y104)"
not_less: fixes x :: "'c DBMEntry"   and y :: "'c DBMEntry" shows "(\<not> x \<prec> y) = (y \<preceq> x)"
top_def: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a" assumes "class.finite_lattice_complete Inf Sup inf less_eq less sup bot top" shows "top = semilattice_sup.Sup_fin sup UNIV"
cdi_iff_no_strict_pd: fixes i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" shows "(i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k) = (is_path \<pi> \<and> k < i \<and> \<pi> i \<noteq> local.return \<and> (\<forall>j\<in>{k..i}. (\<pi> k, \<pi> j) \<notin> pdt))"
intros_2: fixes s :: "'a"   and Tr :: "'a \<Rightarrow> 'b set"   and Up :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and s\<^sub>0 :: "'a"   and e :: "'b" assumes "s \<in> \<RR> Tr Up s\<^sub>0"   and "e \<in> Tr s" shows "Up s e \<in> \<RR> Tr Up s\<^sub>0"
in_mono: fixes A :: "'k set"   and B :: "'k set"   and x :: "'k" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
ivl_integral_has_vderiv_on_subset_segment: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and c :: "real" assumes "continuous_on (closed_segment a b) f"   and "c \<in> closed_segment a b" shows "((\<lambda>u. ivl_integral c u f) has_vderiv_on f) (closed_segment a b)"
read_tmps_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option)" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "\<forall>i\<in>read_tmps sb. i < t"
paint_in_tree: fixes k :: "'a"   and x :: "'b"   and c :: "color"   and t :: "('a, 'b) rbt" shows "((k, x) \<in> set (entries (paint c t))) = ((k, x) \<in> set (entries t))"
seqsim_refl: fixes s :: "nat \<Rightarrow> 'b" shows "s \<approx> s"
length_insortTree: fixes t :: "'a tree"   and ts :: "'a tree list" shows "length (insortTree t ts) = length ts + 1"
inj_on_iff_eq_card: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite A" shows "inj_on f A = (card (f ` A) = card A)"
sigma_algebra_iff2: fixes \<Omega> :: "'b set"   and M :: "'b set set" shows "sigma_algebra \<Omega> M = (M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>s\<in>M. \<Omega> - s \<in> M) \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> \<Union> (range A) \<in> M))"
balE: fixes c :: "'a"   and T :: "'a list"   and c' :: "'a" assumes "bal c T c'" obtains "trace c T c'" and "\<forall>c'\<in>set T. stack c \<lesssim> stack c'" and "stack c' = stack c"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "one_hom hom"   and "one_hom_1_axioms hom" shows "one_hom_1 hom"