hcju/isabelleps huggingface dataset:
siso_coind: fixes phi :: "('test, 'atom) com \<Rightarrow> bool"   and c :: "('test, 'atom) com" assumes "phi c"   and "\<And>c s c' s'. \<lbrakk>phi c; (c, s) \<rightarrow>c (c', s')\<rbrakk> \<Longrightarrow> phi c' \<or> siso c'"   and "\<And>c s t c' s'. \<lbrakk>phi c; s \<approx> t; (c, s) \<rightarrow>c (c', s')\<rbrakk> \<Longrightarrow> \<exists>t'. s' \<approx> t' \<and> (c, t) \<rightarrow>c (c', t')"   and "\<And>c s t s'. \<lbrakk>phi c; s \<approx> t; (c, s) \<rightarrow>t s'\<rbrakk> \<Longrightarrow> \<exists>t'. s' \<approx> t' \<and> (c, t) \<rightarrow>t t'" shows "siso c"
wt_env_mono: fixes is_lub :: "ty \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   and P :: "'b J_prog"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'b expr"   and T :: "ty"   and E' :: "String.literal \<Rightarrow> ty option" assumes "is_lub,P,E \<turnstile> e :: T"   and "E \<subseteq>\<^sub>m E'" shows "is_lub,P,E' \<turnstile> e :: T"
irrefl: fixes a :: "nat" shows "\<not> (a dvd a \<and> a \<noteq> a)"
IH: assumes "A \<inter> B = {}" shows "Pi_pmf (A \<union> B) dflt p = map_pmf (\<lambda>a. case a of (f, g) \<Rightarrow> \<lambda>x. if x \<in> A then f x else g x) (pair_pmf (Pi_pmf A dflt p) (Pi_pmf B dflt p))"
Tarski_2D_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "Tarski_2D Bet Cong \<equiv> Tarski_neutral_dimensionless Bet Cong \<and> Tarski_2D_axioms Bet Cong"
rPar1: fixes \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* C"   and "distinct (bn \<alpha>)" shows "Prop C \<Psi> (P \<parallel> !P) (\<alpha> \<prec> P' \<parallel> !P)"
word_mod_2p_is_mask: fixes n :: "nat"   and x :: "'b word" assumes "0 < 2 ^ n" shows "x mod 2 ^ n = x AND mask n"
Rep_rat_inverse: fixes x :: "rat" shows "Abs_rat (Rep_rat x) = x"
zmod_zminus2_not_zero: fixes k :: "int"   and l :: "int" assumes "k mod - l \<noteq> 0" shows "k mod l \<noteq> 0"
less_eq_st_rep_iff: fixes r1 :: "(char list \<times> 'a) list"   and r2 :: "(char list \<times> 'a) list" shows "local.less_eq_st_rep r1 r2 = (\<forall>x. fun_rep r1 x \<le> fun_rep r2 x)"
quot_dbfm_fresh: fixes s :: "atom"   and A :: "dbfm" shows "s \<sharp> quot_dbfm A"
Knaster_Tarski_idem_inf_eq: fixes L :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "weak_complete_lattice L"   and "Mono\<^bsub>L\<^esub> f"   and "Idem\<^bsub>L\<^esub> f"   and "f \<in> carrier L \<rightarrow> carrier L"   and "A \<subseteq> fps L f" shows "Lattice.inf (fpl L f) A .=\<^bsub>L\<^esub> f (Lattice.inf L A)"
uniformly_continuous_on_minus: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "uniformly_continuous_on s f" shows "uniformly_continuous_on s (\<lambda>x. - f x)"
cont_def: fixes f :: "'a set \<Rightarrow> 'b set" shows "cont f = (\<forall>S. chain S \<longrightarrow> f (\<Union> (range S)) = (\<Union>n. f (S n)))"
simps: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and a1 :: "nat exp list"   and a2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and a3 :: "nat exp list"   and a4 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list" shows "P \<turnstile>\<^sub>1 \<langle>a1,a2\<rangle> [\<Rightarrow>] \<langle>a3,a4\<rangle> = ((\<exists>s. a1 = [] \<and> a2 = s \<and> a3 = [] \<and> a4 = s) \<or> (\<exists>e s\<^sub>0 v s\<^sub>1 es es' s\<^sub>2. a1 = e # es \<and> a2 = s\<^sub>0 \<and> a3 = Val v # es' \<and> a4 = s\<^sub>2 \<and> P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle> \<and> P \<turnstile>\<^sub>1 \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>) \<or> (\<exists>e s\<^sub>0 e' s\<^sub>1 es. a1 = e # es \<and> a2 = s\<^sub>0 \<and> a3 = throw e' # es \<and> a4 = s\<^sub>1 \<and> P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>))"
bin_cat_eq_push_bit_add_take_bit: fixes n :: "nat"   and l :: "int"   and k :: "int" shows "concat_bit n l k = push_bit n k + take_bit n l"
k_invariantAOD: fixes ec :: "'rep"   and ec' :: "'rep"   and A :: "('ma, 'mt) AlgState" assumes "k_isNode ec"   and "k_isNode ec'"   and "simAbs ec' = simAbs ec"   and "k_invariant A" shows "lookup aOps (aActs A) ec = lookup aOps (aActs A) ec'"
support_def: fixes z :: "'d"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "support z A f = {x \<in> A. f x \<noteq> z}"
assms: fixes M :: "real" shows "\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. M \<le> Gromov_product_at a (u m) (u n)"
bits_div_by_0: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and one :: "'g"   and divide :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and modulo :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bit :: "'g \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'g" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide a zero = zero"
inverse_mult_distrib: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "inverse (times a b) = times (inverse a) (inverse b)"
frontier_convex_hull_cases: fixes S :: "'a set" assumes "\<not> affine_dependent S" shows "frontier (convex hull S) = (if card S \<le> DIM('a) then convex hull S else \<Union> {convex hull (S - {x}) |x. x \<in> S})"
cbelow_on_first_below: fixes A :: "'c set"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "'c"   and rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "cbelow_on A ord z rel a b" shows "ord a z"
assms_4: shows "a \<star> f \<cong> a \<star> g"
lsteps_cong: fixes ys :: "'a list"   and xs :: "'a list"   and f :: "'a \<Rightarrow> nat"   and g :: "'a \<Rightarrow> nat"   and l :: "nat"   and l' :: "nat"   and u :: "nat"   and v :: "nat" assumes "ys = xs"   and "\<And>x. x \<in> set xs \<Longrightarrow> f x = g x"   and "l = l'" shows "lsteps xs f l u v = lsteps ys g l' u v"
gb_schema_direct_def: fixes sel :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and bs0 :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list"   and data0 :: "'g" shows "gb_schema_direct sel ap ab compl bs0 data0 = (let data = (length bs0, data0); bs1 = fst (add_indices (bs0, data0) (0, data0)); bs = ab [] [] bs1 data in map (\<lambda>(f, uu_, d). (f, d)) (gb_schema_aux sel ap ab compl [] (count_rem_components bs, data) bs (ap [] [] [] bs1 data)))"
assms_2: shows "case_nat t \<Gamma> \<turnstile>\<^sub>c e' : t'"