hcju/isabelleps huggingface dataset:
assms_3: fixes a :: "atom"   and b :: "atom" assumes "sort_of a = sort_of b"   and "a \<noteq> b"   and "a \<in> S"   and "b \<in> S" shows "P (a \<rightleftharpoons> b)"
induct: fixes P :: "pol \<Rightarrow> pol \<Rightarrow> bool"   and a0 :: "pol"   and a1 :: "pol" assumes "\<And>a b. P (Pc a) (Pc b)"   and "\<And>c i P. (c \<noteq> 0 \<Longrightarrow> P P (Pc c)) \<Longrightarrow> P (Pc c) (Pinj i P)"   and "\<And>i P c. (c \<noteq> 0 \<Longrightarrow> P P (Pc c)) \<Longrightarrow> P (Pinj i P) (Pc c)"   and "\<And>c P i Q. \<lbrakk>c \<noteq> 0 \<Longrightarrow> P P (Pc c); c \<noteq> 0 \<Longrightarrow> P Q (Pc c)\<rbrakk> \<Longrightarrow> P (Pc c) (PX P i Q)"   and "\<And>P i Q c. \<lbrakk>c \<noteq> 0 \<Longrightarrow> P P (Pc c); c \<noteq> 0 \<Longrightarrow> P Q (Pc c)\<rbrakk> \<Longrightarrow> P (PX P i Q) (Pc c)"   and "\<And>x P y Q. \<lbrakk>x = y \<Longrightarrow> P P Q; \<lbrakk>x \<noteq> y; y < x\<rbrakk> \<Longrightarrow> P (Pinj (x - y) P) Q; \<lbrakk>x \<noteq> y; \<not> y < x\<rbrakk> \<Longrightarrow> P (Pinj (y - x) Q) P\<rbrakk> \<Longrightarrow> P (Pinj x P) (Pinj y Q)"   and "\<And>x P Q y R. \<lbrakk>x = 0 \<Longrightarrow> P P (PX Q y R); \<lbrakk>x \<noteq> 0; x = 1\<rbrakk> \<Longrightarrow> P (Pinj x P) Q; \<lbrakk>x \<noteq> 0; x = 1\<rbrakk> \<Longrightarrow> P R P; \<lbrakk>x \<noteq> 0; x \<noteq> 1\<rbrakk> \<Longrightarrow> P (Pinj x P) Q; \<lbrakk>x \<noteq> 0; x \<noteq> 1\<rbrakk> \<Longrightarrow> P R (Pinj (x - 1) P)\<rbrakk> \<Longrightarrow> P (Pinj x P) (PX Q y R)"   and "\<And>P x R y Q. \<lbrakk>y = 0 \<Longrightarrow> P (PX P x R) Q; \<lbrakk>y \<noteq> 0; y = 1\<rbrakk> \<Longrightarrow> P (Pinj y Q) P; \<lbrakk>y \<noteq> 0; y = 1\<rbrakk> \<Longrightarrow> P R Q; \<lbrakk>y \<noteq> 0; y \<noteq> 1\<rbrakk> \<Longrightarrow> P (Pinj y Q) P; \<lbrakk>y \<noteq> 0; y \<noteq> 1\<rbrakk> \<Longrightarrow> P R (Pinj (y - 1) Q)\<rbrakk> \<Longrightarrow> P (PX P x R) (Pinj y Q)"   and "\<And>P1 x P2 Q1 y Q2. \<lbrakk>P P1 Q1; P P2 Q2; P P1 (mkPinj 1 Q2); P Q1 (mkPinj 1 P2)\<rbrakk> \<Longrightarrow> P (PX P1 x P2) (PX Q1 y Q2)" shows "P a0 a1"
map_comp: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b"   and v :: "'c multiset" shows "image_mset g (image_mset f v) = image_mset (g \<circ> f) v"
f_Exec_Stream_Acc_LocalState__State_Idle_all: fixes l :: "nat"   and k :: "nat"   and localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d"   and i :: "nat"   and xs :: "'g message_af list" assumes "0 < l"   and "l \<le> k"   and "State_Idle localState output_fun trans_fun (localState c)"   and "\<And>c m. State_Idle localState output_fun trans_fun (localState c) \<Longrightarrow> State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun (m # NoMsg\<^bsup>l - Suc 0\<^esup>) c))"   and "i < length xs" shows "State_Idle localState output_fun trans_fun (f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun xs c ! i)"
exists_core_chain: fixes C :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) ring set" assumes "C \<in> Chains (relation_of (\<lesssim>) \<S>)" obtains C' :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) ring set" where "C' \<subseteq> extensions" and "C = law_restrict ` C'"
assms_2: shows "[\<^bold>\<exists>x. \<lparr>A!,x\<^sup>P\<rparr> in v]"
set_insort_on: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and xs :: "'c list" shows "set (insort_key_on r f x xs) = insert x (set xs)"
elims: fixes x :: "complex \<times> complex \<times> complex \<times> complex"   and y :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_inv x = y" obtains a :: "complex" and b :: "complex" and c :: "complex" and d :: "complex" where "x = (a, b, c, d)" and "y = (1 / (a * d - b * c)) *\<^sub>s\<^sub>m (d, - b, - c, a)"
less_sqmatrix_def: fixes A :: "('c, 'd) sqmatrix"   and B :: "('c, 'd) sqmatrix" shows "A < B \<equiv> A \<le> B \<and> A \<noteq> B"
Mutrec: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and n :: "nat"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "finite G"   and "card G = n"   and "verified G"   and "A \<in> G" shows "{} \<rhd> Call : A"
individually_rational_on_Cd: fixes ds :: "'b set"   and X :: "'a set"   and d :: "'b" assumes "ManiPref.individually_rational_on ds X" shows "ManiPref.Cd d X = ManiPref'.dX X d"
assms: fixes i :: "'a" shows "(f i has_derivative f' i) F"
ThrowReds: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>throw e,s\<rangle> \<rightarrow>* \<langle>throw e',s'\<rangle>"
above_upclosed: fixes x :: "'a"   and ao :: "'a option"   and y :: "'a" assumes "x \<in> above ao"   and "x < y" shows "y \<in> above ao"
eq_dff_def: fixes x :: "'b fract"   and y :: "'b fract" shows "(x =dff y) = (divides_ff x y \<and> divides_ff y x)"
integrable_on_localized_vector_derivative: fixes f :: "'a \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and a :: "real"   and b :: "real" shows "((\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b}) = ((\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..b})"
zerofix: fixes s :: "'b" shows "s \<circ> (0::'a) = (0::'a)"
simps_2: fixes l :: "('a, 'b) rbt"   and c :: "color"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "rbt_sorted (Node l c k v r) = ((\<forall>x\<in>rbt_set l. x < k) \<and> Ball (rbt_set r) ((<) k) \<and> rbt_sorted l \<and> rbt_sorted r)"
coset_mult_inv1: fixes M :: "'a set"   and x :: "'a"   and y :: "'a" assumes "M #> x \<otimes> inv y = M"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "M \<subseteq> carrier G" shows "M #> x = M #> y"
map_filter_M_def: fixes f :: "'d \<Rightarrow> ('a, 'b, 'c option) prog"   and xs :: "'d list" shows "map_filter_M f xs = map_M f xs \<bind> (\<lambda>ys_opts. filter_M (\<lambda>x. return (x \<noteq> None)) ys_opts \<bind> map_M (\<lambda>x. return (the x)))"
partition_on_set_mset_distinct: fixes A :: "'a multiset"   and P :: "'a multiset multiset"   and p :: "'a set"   and p' :: "'a set" assumes "partition_on_mset A P"   and "distinct_mset A"   and "p \<in># image_mset set_mset P"   and "p' \<in># image_mset set_mset P"   and "p \<noteq> p'" shows "p \<inter> p' = {}"
finite_chain_with3_def: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and X :: "'a set" shows "[[x .. y .. z]X] \<equiv> [[.. x .. y .. z ..]X] \<and> \<not> (\<exists>w\<in>X. [[w x y]] \<or> [[y z w]])"
ln_le_minus_one: fixes x :: "real" assumes "0 < x" shows "ln x \<le> x - 1"
sigma_finite_measure_count_space_countable: fixes A :: "'a set" assumes "countable A" shows "sigma_finite_measure (count_space A)"
vcomp_can: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and t :: "'b bicategorical_language.term"   and u :: "'b bicategorical_language.term"   and v :: "'b bicategorical_language.term" assumes "bicategory V H \<a> \<i> src trg"   and "bicategorical_language.Ide V src trg t"   and "bicategorical_language.Ide V src trg u"   and "bicategorical_language.Ide V src trg v"   and "bicategorical_language.Nmlize V t = bicategorical_language.Nmlize V u"   and "bicategorical_language.Nmlize V u = bicategorical_language.Nmlize V v" shows "V (bicategory.can V H \<a> \<i> src trg v u) (bicategory.can V H \<a> \<i> src trg u t) = bicategory.can V H \<a> \<i> src trg v t"
let3: fixes l :: "nat"   and binds :: "((nat \<times> char list) \<times> lambda) list"   and c' :: "call"   and x :: "lambda" assumes "call.Let l binds c' \<in> calls x" shows "fst ` set binds \<subseteq> vars x"
steps_append: fixes qs :: "'a list"   and as :: "'b list"   and step :: "'c \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c"   and s :: "'c"   and qs' :: "'a list"   and as' :: "'b list" assumes "length qs = length as" shows "On_Off.steps step s (qs @ qs') (as @ as') = On_Off.steps step (On_Off.steps step s qs as) qs' as'"
ide_compE: fixes g :: "'s"   and f :: "'s" assumes "S.ide (g \<cdot> f)" obtains "S.seq g f" and "S.seq f g" and "g \<cdot> f = S.dom f" and "g \<cdot> f = S.cod g"
L_def: shows "L = {p \<in> set ys. fst p < fst m}"
quasinorm_intersection_subset_1: fixes N1 :: "'c quasinorm"   and N2 :: "'c quasinorm" shows "N1 \<inter>\<^sub>N N2 \<subseteq>\<^sub>N N1"
elem_cong_l: fixes a :: "'a"   and a' :: "'a"   and A :: "'a set" assumes "a \<in> carrier L"   and "a' \<in> carrier L"   and "A \<subseteq> carrier L"   and "a' .= a"   and "a .\<in> A" shows "a' .\<in> A"
tseytin_assmt_backlookup: fixes J :: "'a formula"   and F :: "'a formula" assumes "J \<in> set (subformulae F)" shows "(the (map_of (map (\<lambda>(a, b). (b, a)) (tseytin_assmt F)) J), J) \<in> set (tseytin_assmt F)"
assms: shows "((=) ===>\<^sub>T R) g f"
positive_flip_kauff_mat: shows "kauff_mat left_over = kauff_mat right_over"
assms_1: shows "\<tau>s \<turnstile>\<^sub>\<tau> t : \<tau>1"
that: fixes f :: "nat \<Rightarrow> nat" assumes "\<forall>i. f (Suc i) icd\<^bsup>\<pi>\<^esup>\<rightarrow> f i"   and "range f = {i. \<forall>j>i. j cd\<^bsup>\<pi>\<^esup>\<rightarrow> i}"   and "\<nexists>i. f 0 cd\<^bsup>\<pi>\<^esup>\<rightarrow> i" shows "thesis"
eqpoll_def: fixes A :: "'d set"   and B :: "'e set" shows "A \<approx> B \<equiv> \<exists>f. bij_betw f A B"
span_mono: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'c set" assumes "module scale"   and "A \<subseteq> B" shows "module.span scale A \<subseteq> module.span scale B"
sams_lea123_suma2__lea: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a"   and G' :: "'a"   and H' :: "'a"   and I' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LeA Bet Cong A B C A' B' C'"   and "Tarski_neutral_dimensionless.SAMS Bet Cong A' B' C' D E F"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I"   and "Tarski_neutral_dimensionless.SumA Bet Cong A' B' C' D E F G' H' I'" shows "Tarski_neutral_dimensionless.LeA Bet Cong G H I G' H' I'"
zero_neq_one_ow_axioms_def: fixes one :: "'a"   and zero :: "'a" shows "zero_neq_one_ow_axioms one zero \<equiv> zero \<noteq> one"
collapse: fixes option :: "'b option" assumes "option \<noteq> None" shows "\<lfloor>the option\<rfloor> = option"
wf_flatten_PLUS: fixes X :: "'b rexp set"   and n :: "nat" assumes "finite X" shows "local.wf n (PLUS (sorted_list_of_set X)) = (\<forall>r\<in>X. local.wf n r)"
solution_s: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) set"   and s :: "'s"   and v :: "(nat, 'a) mapping" assumes "checked_s t as s"   and "solution_s s = v" shows "\<langle>v\<rangle> \<Turnstile>\<^sub>t t \<and> \<langle>v\<rangle> \<Turnstile>\<^sub>a\<^sub>s (snd ` as)"
add_pdevs_def: shows "add_pdevs = binop_pdevs (+)"
simps: fixes r :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d" shows "Rangep r a = (\<exists>aa b. a = b \<and> r aa b)"
is_Msg_def: fixes x :: "'b" shows "is_Msg x \<equiv> \<not> is_NoMsg x"
isTrue_iff_bprv_PPf: fixes n :: "'b"   and \<phi> :: "'c" assumes "n \<in> num"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "isTrue (PPf n \<langle>\<phi>\<rangle>) = bprv (PPf n \<langle>\<phi>\<rangle>)"
rt_strictly_fresher_def': fixes i :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" shows "(rt1 \<sqsubset>\<^bsub>i\<^esub> rt2) = (nsqn\<^sub>r (the (rt1 i)) < nsqn\<^sub>r (the (rt2 i)) \<or> nsqn\<^sub>r (the (rt1 i)) = nsqn\<^sub>r (the (rt2 i)) \<and> \<pi>\<^sub>5 (the (rt2 i)) < \<pi>\<^sub>5 (the (rt1 i)))"
compact_convex_hull: fixes S :: "'b set" assumes "compact S" shows "compact (convex hull S)"
ring_simprules_25: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "ring R"   and "x \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
t12sub: shows "{flow0 x t1--flow0 x t2} \<subseteq> {a<--<b}"
expl_cond_expect_borel_measurable: fixes Y :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and X :: "'c \<Rightarrow> real" assumes "disc_fct Y"   and "point_measurable M (space N) Y" shows "expl_cond_expect M Y X \<in> borel_measurable M"
mod_exp_nat_aux_correct: fixes m :: "nat"   and y :: "nat"   and x :: "nat"   and n :: "nat" shows "mod_exp_nat_aux m y x n mod m = x ^ n * y mod m"
that: fixes x :: "'a" assumes "x \<in> U"   and "S = connected_component_set U x" shows "thesis"
lsubgroup_def: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup" shows "lgroup.lsubgroup plus zero uminus inf sup = {A \<in> group_add.subgroup plus zero uminus. \<forall>a b. a \<in> A \<and> b \<in> A \<longrightarrow> inf a b \<in> A \<and> sup a b \<in> A}"
that: fixes f :: "(real, 'n) vec \<Rightarrow> (real, 'n) vec" assumes "orthogonal_transformation f"   and "det (matrix f) = 1"   and "f a = b" shows "thesis"
prems_1: shows "\<forall>x y. x \<le> f x y \<and> y \<le> f x y"
left_uniqueD: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and z :: "'c"   and y :: "'b" assumes "left_unique A"   and "A x z"   and "A y z" shows "x = y"
stabilizer_def: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set"   and m :: "'e" assumes "group_action G \<phi> M" shows "group_action.stabilizer G \<phi> m = {g \<in> carrier G. \<phi> g m = m}"
fresh_Vs: fixes p :: "perm"   and Vs :: "name set"   and a :: "atom" assumes "quote_perm p Vs"   and "a \<sharp> p" shows "a \<sharp> Vs"
shadow_root_delete_get_2: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and object_ptr :: "(_) object_ptr" assumes "delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr h = Some h'"   and "object_ptr \<noteq> cast shadow_root_ptr" shows "get object_ptr h' = get object_ptr h"
min_alg1_def: fixes s1 :: "('e, 'd, 'f) F1 \<Rightarrow> 'd"   and s2 :: "('e, 'd, 'f) F2 \<Rightarrow> 'f" shows "min_alg1 s1 s2 = (\<Union>i\<in>Field SucFbd. fst (min_algs s1 s2 i))"
encode_operator_effects_is_cnf_ii: fixes t :: "nat"   and \<Pi> :: "('b, 'c) strips_problem_scheme"   and op :: "'b strips_operator" shows "is_cnf \<^bold>\<And>map (\<lambda>v. \<^bold>\<not> (Atom (Operator t (index (\<Pi>\<^sub>\<O>) op))) \<^bold>\<or> \<^bold>\<not> (Atom (State (Suc t) (index (\<Pi>\<^sub>\<V>) v)))) (delete_effects_of op)"
o3: fixes s1 :: "lam"   and s2 :: "lam"   and a :: "name" assumes " s1 \<longrightarrow>\<^sub>1 s2" shows " Lam [a].s1 \<longrightarrow>\<^sub>1 Lam [a].s2"
C: shows "(Asset n, \<lbrace>Num 2, PubKey B\<rbrace>) \<in> s"
synthetic_div_unique_lemma: fixes c :: "'b"   and p :: "'b poly"   and a :: "'b" assumes "smult c p = pCons a p" shows "p = 0"
orderI: fixes a :: "'b"   and b :: "'b" assumes "a = sup_class.sup a b" shows "b \<le> a"
Z_inj_def: shows "local.Z_inj = {f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> inj_on f {0..<n}} \<times> {\<pi>. \<pi> permutes {0..<n}}"
actks: fixes t :: "nat \<Rightarrow> cnf"   and kid :: "'kid"   and n :: "nat"   and p :: "PROB" assumes "t \<in> arch"   and "\<parallel>kid\<parallel>\<^bsub>t n\<^esub>"   and "p = prob kid"   and "p \<in> ksop (\<sigma>\<^bsub>kid\<^esub>t n)" shows "(\<exists>n'\<ge>n. \<parallel>kid\<parallel>\<^bsub>t n'\<^esub> \<and> (p, solve p) = ksns (\<sigma>\<^bsub>kid\<^esub>t n') \<and> (\<forall>n''\<ge>n. n'' < n' \<longrightarrow> \<parallel>kid\<parallel>\<^bsub>t n''\<^esub>)) \<or> (\<forall>n'\<ge>n. \<parallel>kid\<parallel>\<^bsub>t n'\<^esub> \<and> (p, solve p) \<noteq> ksns (\<sigma>\<^bsub>kid\<^esub>t n'))"
continuous_at_of_ereal: fixes x0 :: "ereal" assumes "\<bar>x0\<bar> \<noteq> \<infinity>" shows "isCont real_of_ereal x0"
JN_constrains: fixes i :: "'c"   and I :: "'c set"   and F :: "'c \<Rightarrow> 'd program"   and A :: "'d set"   and B :: "'d set" assumes "i \<in> I" shows "(JOIN I F \<in> A co B) = (\<forall>i\<in>I. F i \<in> A co B)"
set_subseqs_eq: fixes ys :: "'c list" shows "set (subseqs ys) = {xs. subseq xs ys}"
cond_b_def: fixes a :: "nat list"   and b :: "nat list"   and xs :: "nat list" shows "cond_b a b xs = (\<forall>k\<le>length a. take k a \<bullet> take k xs \<le> b \<bullet> map (max_y_impl a b (take k xs)) [0..<length b])"
M1_mult_M1: shows "M1 * M1 = M1"
forall_\<nu>_def: shows "forall \<equiv> forall\<^sub>\<nu>"
imirror_Max: fixes I :: "nat set" assumes "finite I" shows "Max (imirror I) = Max I"
extremum_strict: fixes a :: "'b word" shows "\<not> a < 0"
l'_def_1: shows "Lfilled k l es''' lfilledk'"
adjacent_half_chamber_system_image: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set"   and D :: "'b set" assumes "ThinChamberComplexFolding X f"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "D \<in> ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X"   and "C \<sim> D" shows "f ` D = C"
surjective: fixes r :: "'b alpha_d_scheme" shows "r = \<lparr>ok = ok r, \<dots> = more r\<rparr>"
space_extend_measure: fixes G :: "'e \<Rightarrow> 'd set"   and I :: "'e set"   and \<Omega> :: "'d set"   and \<mu> :: "'e \<Rightarrow> ennreal" assumes "G ` I \<subseteq> Pow \<Omega>" shows "space (extend_measure \<Omega> I G \<mu>) = \<Omega>"
ladder_i_0_bound: fixes \<alpha> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<gamma> :: "symbol list" assumes "LeftDerivationLadder \<alpha> D L \<gamma>" shows "ladder_i L 0 < length \<alpha>"
unit_disc_fix_f_circline_indirect: fixes M :: "moebius"   and f :: "complex_homo \<Rightarrow> complex_homo"   and H :: "circline" assumes "unit_disc_fix M"   and "f = moebius_pt M \<circ> conjugate" shows "unit_disc_fix_f_circline f H = (moebius_circline M \<circ> conjugate_circline) H"
type_definition_ennreal: shows "type_definition enn2ereal e2ennreal' {x. 0 \<le> x}"
fst_conv: fixes x1 :: "'k"   and x2 :: "'l" shows "fst (x1, x2) = x1"
Infinitesimal_approx: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> Infinitesimal"   and "y \<in> Infinitesimal" shows "x \<approx> y"
simps_33: fixes x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt"   and x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "While x61 x62 \<noteq> x31@[x32]"
simps_2: fixes a :: "'a list"   and list :: "'a list list" shows "flatten (a # list) = a @ flatten list"
st_proj2_continuous: shows "continuous_on (UNIV - {bot_sphere}) st_proj2"
min_dnf_min_set: fixes \<phi> :: "'b ltln" shows "min_set (min_dnf \<phi>) = min_dnf \<phi>"
OUTfromVCorrect1_data18: shows "OUTfromVCorrect1 data18"
bot: fixes s :: "'a" shows "\<bottom> \<preceq> s"
assms_1: shows "Complete_Partial_Order.chain (ord_option ord) Y"
segment_a'_a_neD: assumes "segment G.face_cycle_succ rev.a' a' \<noteq> {}" shows "segment G.face_cycle_succ rev.a' a' \<in> rev.H.face_cycle_sets"
simps_18: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "Skip \<noteq> x31@[x32]"
psi_l_dim: fixes l :: "nat" shows "psi_l l \<in> carrier_vec N"
that: fixes X1 :: "('a \<times> 'b \<Rightarrow>\<^sub>L 'a) set"   and X2 :: "('a \<times> 'b \<Rightarrow>\<^sub>L 'a) set" assumes "X0 = X1 \<union> X2"   and "flowsto X1 T CX Y"   and "flowsto X2 T CX Z" shows "thesis"
assms_2: assumes "a \<in>\<^sub>\<circ> f i" shows "P"
sumZero: fixes P :: "pi" shows "P \<oplus> \<zero> \<sim>\<^sup>s\<^sub>l P"
solves_ode_continuous_on: fixes y :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "real set"   and X :: "'b set" assumes "(y solves_ode f) T X" shows "continuous_on T y"
less_eq_option_None_code: fixes x :: "'b option" shows "(None \<le> x) = True"
Mertens: fixes n :: "nat" assumes "n \<noteq> 0" shows "\<bar>(\<Sum>p | prime p \<and> p \<le> n. ln (real p) / real p) - ln (real n)\<bar> \<le> 7"
intros_3: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<equiv>\<^sub>e Q"   and "Q \<equiv>\<^sub>e R" shows "P \<equiv>\<^sub>e R"
lookup_monomial_If: fixes k :: "'a"   and v :: "'b" shows "lookup (monomial v k) = (\<lambda>k'. if k = k' then v else (0::'b))"
map_poly_1': fixes f :: "'b \<Rightarrow> 'a" assumes "f (1::'b) = (1::'a)" shows "map_poly f 1 = 1"
Polys_closed_sum_list: fixes xs :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list"   and X :: "'a set" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<in> P[X]" shows "sum_list xs \<in> P[X]"
closure_of_subset_Int: fixes X :: "'b topology"   and S :: "'b set" shows "topspace X \<inter> S \<subseteq> X closure_of S"
op_set_to_sorted_list_def: fixes ordR :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a set" shows "op_set_to_sorted_list ordR s \<equiv> SPEC (\<lambda>l. set l = s \<and> distinct l \<and> sorted_wrt ordR l)"
dual_linorder: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "class.linorder (\<lambda>x y. less_eq y x) (\<lambda>x y. less y x)"