hcju/isabelleps huggingface dataset:
SeqHaddP_SUCC_lemma: fixes y' :: "name"   and s :: "tm"   and j :: "tm"   and k :: "tm"   and y :: "tm" assumes "atom y' \<sharp> (s, j, k, y)" shows "{SeqHaddP s j (SUCC k) y} \<turnstile> SyntaxN.Ex y' (SeqHaddP s j k (Var y') AND y EQ SUCC (Var y'))"
subspace_mul: fixes S :: "'b set"   and x :: "'b"   and c :: "real" assumes "subspace S"   and "x \<in> S" shows "c *\<^sub>R x \<in> S"
finestPart: fixes X :: "'a set" shows "finestpart X = {{x} |x. x \<in> X}"
uniform_limit_integral_cbox: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b"   and g :: "'b \<Rightarrow> 'c"   and F :: "'a filter" assumes "uniform_limit (cbox a b) f g F"   and "\<And>n. continuous_on (cbox a b) (f n)"   and "F \<noteq> bot" obtains I :: "'a \<Rightarrow> 'c" and J :: "'c" where "\<And>n. (f n has_integral I n) (cbox a b)" and "(g has_integral J) (cbox a b)" and "(I \<longlongrightarrow> J) F"
simps_2: fixes t1 :: "'b Tree"   and x :: "'b"   and t2 :: "'b Tree" shows "setOf (T t1 x t2) = setOf t1 \<union> setOf t2 \<union> {x}"
emeasure_pmf_single: fixes M :: "'d pmf"   and x :: "'d" shows "emeasure (measure_pmf M) {x} = ennreal (pmf M x)"
bvars\<^sub>s\<^sub>s\<^sub>t_subst: fixes A :: "('e, 'd) stateful_strand_step list"   and \<delta> :: "'d \<Rightarrow> ('e, 'd) Term.term" shows "bvars\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>) = bvars\<^sub>s\<^sub>s\<^sub>t A"
a1: fixes n :: "nat" assumes "i \<le> n"   and "\<forall>m\<ge>i. m < n \<longrightarrow> P m" shows "P n"
prems_3: shows "ListMem h (state_list s0 (a # as))"
sterms_not_choice: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and q :: "('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>"   and "q \<in> sterms \<Gamma> p" shows "not_choice q"
assms: shows "c \<in>\<^sub>\<circ> \<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>I. \<Union>\<^sub>\<circ> ((\<lambda>aa\<in>\<^sub>\<circ>A a. set {\<langle>a, aa\<rangle>}) `\<^sub>\<circ> A a)) `\<^sub>\<circ> I)"
mask_AND_less_0: fixes x :: "'b word"   and n :: "nat"   and m :: "nat" assumes "x AND mask n = 0"   and "m \<le> n" shows "x AND mask m = 0"
dghm_proj_components_2: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and i :: "V" shows "\<pi>\<^sub>D\<^sub>G I \<AA> i\<lparr>ArrMap\<rparr> = (\<lambda>f\<in>\<^sub>\<circ>dg_prod I \<AA>\<lparr>Arr\<rparr>. f\<lparr>i\<rparr>)"
mult_isol: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "\<lparr>z\<rparr>x\<lparr>z \<cdot> y\<rparr>"
HFinite_Infinitesimal_not_zero: fixes x :: "'b star" assumes "x \<in> HFinite - Infinitesimal" shows "x \<noteq> 0"
in_prime_factors_imp_dvd: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "p \<in># factorial_semiring.prime_factorization zero times one normalize x" shows "dvd.dvd times p x"
isl_def: fixes sum :: "'d + 'e" shows "isl sum = (\<exists>x1. sum = Inl x1)"
that: fixes k :: "int" assumes "arccos (cos \<theta>) = \<bar>\<theta> - real_of_int k * (2 * pi)\<bar>" shows "thesis"
\<Sigma>_def: fixes insert_message :: "('b, 'a) insert_message" shows "\<Sigma> insert_message = (case insert_message of InsertMessage x1 x2 x3 x4 \<Rightarrow> x4)"
total_quasi_ordered_subset: fixes B :: "'a set" assumes "B \<subseteq> A" shows "total_quasi_ordered_set B (\<sqsubseteq>)"
asIMOD_igWlsBinp: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> (('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> ('varSort, 'b) sAbs) option" assumes "sWlsNE SEM" shows "igWlsBinp (asIMOD SEM) delta binp = ((\<forall>val. liftAll (\<lambda>X. sWlsVal SEM val \<or> X val = undefined) binp) \<and> (\<forall>val. sWlsVal SEM val \<longrightarrow> sWlsBinp SEM delta (lift (\<lambda>X. X val) binp)))"
congs_length: fixes F\<^sub>1 :: "'a list"   and F\<^sub>2 :: "'a list" assumes "F\<^sub>1 \<cong> F\<^sub>2" shows "|F\<^sub>2| = |F\<^sub>1|"
asubg_whole: fixes A :: "('c, 'd) aGroup_scheme" assumes "aGroup A" shows "A +> carrier A"
A: shows "\<forall>\<^sub>F x in F. f x \<noteq> 0"
ind_hyp: assumes "\<And>l. l \<in> set ls \<Longrightarrow> distinct l \<and> sorted l"   and "\<And>l. l \<in> set (merge l1 l2 # acc) \<Longrightarrow> distinct l \<and> sorted l" shows "distinct (merge_list (merge l1 l2 # acc) ls) \<and> sorted (merge_list (merge l1 l2 # acc) ls) \<and> set (merge_list (merge l1 l2 # acc) ls) = set (concat ((merge l1 l2 # acc) @ ls))"
ae_filter_eq_bot_iff: fixes M :: "'b measure" shows "(ae_filter M = bot) = (emeasure M (space M) = 0)"
subsumption_preserves_redundancy: fixes C :: "'a Literal set"   and S :: "'a Literal set set"   and D :: "'a Literal set" assumes "redundant C S"   and "subsumes C D" shows "redundant D S"
implies_sup: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "x \<leadsto> z \<le> y \<leadsto> z \<leadsto> (x \<squnion> y \<leadsto> z)"
opeUU0: shows "openin (top_of_set U) U0"
in_keys_timesE: fixes v :: "'c"   and p :: "'c \<Rightarrow>\<^sub>0 'd"   and q :: "'c \<Rightarrow>\<^sub>0 'd" assumes "v \<in> keys (p * q)" obtains t :: "'c" and u :: "'c" where "t \<in> keys p" and "u \<in> keys q" and "v = t + u"
hcomp_in_hom\<^sub>P\<^sub>B\<^sub>H_1: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a"   and \<nu> :: "'a"   and c :: "'a" assumes "arr \<mu> \<and> a \<in> sources \<mu> \<and> b \<in> targets \<mu>"   and "arr \<nu> \<and> b \<in> sources \<nu> \<and> c \<in> targets \<nu>" shows "hseq \<nu> \<mu> \<and> a \<in> sources (\<nu> \<star> \<mu>) \<and> c \<in> targets (\<nu> \<star> \<mu>)"
yun_wrel_pderiv: fixes F :: "int poly"   and c :: "rat"   and f :: "rat poly" assumes "yun_wrel F c f" shows "yun_wrel (pderiv F) c (pderiv f)"
vsubstEnv_idEnv: fixes xs :: "'varSort"   and y :: "'var"   and x :: "'var" shows "idEnv &[y // x]_xs = idEnv [x \<leftarrow> Var xs y]_xs"
\<nu>: shows "arr \<nu> \<and> \<guillemotleft>\<nu> : local.dom \<nu> \<Rightarrow> cod \<nu>\<guillemotright> \<and> local.iso \<nu> \<and> \<guillemotleft>local.inv \<nu> : cod \<nu> \<Rightarrow> local.dom \<nu>\<guillemotright>"
tauChainPar2: fixes Q :: "ccs"   and Q' :: "ccs"   and P :: "ccs" assumes "Q \<Longrightarrow>\<^sub>\<tau> Q'" shows "P \<parallel> Q \<Longrightarrow>\<^sub>\<tau> P \<parallel> Q'"
assms_1: shows "QD('d\<^sub>1) = QD('d\<^sub>2)"
exec_Call_body_aux: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and s :: "('d, 'f) xstate"   and t :: "('d, 'f) xstate" assumes "\<Gamma> p = Some bdy" shows "\<Gamma>\<turnstile> \<langle>Call p,s\<rangle> \<Rightarrow> t = \<Gamma>\<turnstile> \<langle>bdy,s\<rangle> \<Rightarrow> t"
ln: shows "cs\<^bsup>\<pi>\<^esup> m = cs\<^bsup>\<pi>'\<^esup> m' \<and> cs\<^bsup>\<pi>\<^esup> n = cs\<^bsup>\<pi>'\<^esup> n' \<and> n' dd\<^bsup>\<pi>',v\<^esup>\<rightarrow> l' \<and> l' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> m'"
hyps: fixes lm2 :: "nat list"   and lm3 :: "nat list"   and lm4 :: "nat list" assumes "n \<le> aa - ba"   and "ba < aa"   and "length lm1 = ba"   and "length lm2 = aa - ba - n"   and "length lm3 = n" shows "{\<lambda>nl. nl = lm1 @ 0 \<up> n @ lm2 @ lm3 @ lm4} mv_boxes aa ba n {\<lambda>nl. nl = lm1 @ lm3 @ lm2 @ 0 \<up> n @ lm4}"
sat_dualD_1: fixes y :: "'a vec"   and A :: "'a mat"   and c :: "'a vec" assumes "y \<in> sat_dual A c" shows "[y\<^sub>v*A]=c"
int_pow_consistent: fixes H :: "'a set"   and x :: "'a"   and n :: "int" assumes "subgroup H (G\<lparr>carrier := generate G {a}\<rparr>)"   and "x \<in> H" shows "x [^]\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub> n = x [^]\<^bsub>G\<lparr>carrier := generate G {a}, carrier := H\<rparr>\<^esub> n"
simps_2: fixes e :: "'a"   and a :: "'b"   and r :: "nat"   and ts :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.tree_invar (SkewBinomialHeapStruc.Node e a (Suc r) ts) = (\<exists>e1 a1 ts1 e2 a2 ts2 e' a'. SkewBinomialHeapStruc.tree_invar (SkewBinomialHeapStruc.Node e1 a1 r ts1) \<and> SkewBinomialHeapStruc.tree_invar (SkewBinomialHeapStruc.Node e2 a2 r ts2) \<and> (SkewBinomialHeapStruc.Node e a (Suc r) ts = SkewBinomialHeapStruc.link (SkewBinomialHeapStruc.Node e1 a1 r ts1) (SkewBinomialHeapStruc.Node e2 a2 r ts2) \<or> SkewBinomialHeapStruc.Node e a (Suc r) ts = SkewBinomialHeapStruc.skewlink e' a' (SkewBinomialHeapStruc.Node e1 a1 r ts1) (SkewBinomialHeapStruc.Node e2 a2 r ts2)))"
elims: fixes x :: "int \<times> (int \<times> int) \<times> int \<times> int"   and xa :: "int \<times> (int \<times> int) \<times> int \<times> int"   and xb :: "int"   and xc :: "(int \<times> int) list"   and y :: "int \<times> (int \<times> int) \<times> int \<times> int" assumes "combine_code x xa xb xc = y" obtains \<delta>\<^sub>L :: "int" and p\<^sub>0\<^sub>L :: "int \<times> int" and p\<^sub>1\<^sub>L :: "int \<times> int" and \<delta>\<^sub>R :: "int" and p\<^sub>0\<^sub>R :: "int \<times> int" and p\<^sub>1\<^sub>R :: "int \<times> int" and l :: "int" and ps :: "(int \<times> int) list" where "x = (\<delta>\<^sub>L, p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L)" and "xa = (\<delta>\<^sub>R, p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R)" and "xb = l" and "xc = ps" and "y = (let (\<delta>, c\<^sub>0, c\<^sub>1) = if \<delta>\<^sub>L < \<delta>\<^sub>R then (\<delta>\<^sub>L, p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) else (\<delta>\<^sub>R, p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) in Let (filter (\<lambda>p. (fst p - l)\<^sup>2 < \<delta>) ps) (find_closest_pair_code (\<delta>, c\<^sub>0, c\<^sub>1)))"
term_variants_pred_dense: fixes P :: "'c set"   and fs :: "'c list"   and u :: "('c, 'd) Term.term"   and t :: "('c, 'd) Term.term"   and g :: "'c"   and Q :: "'c set"   and gs :: "'c list" assumes "term_variants_pred (\<lambda>x. if x \<in> P then fs else []) u t"   and "g \<in> Q"   and "g \<in> set gs" shows "\<exists>s. term_variants_pred (\<lambda>x. if x \<in> P then gs else []) u s \<and> term_variants_pred (\<lambda>x. if x \<in> Q then fs else []) s t"
carr_dsumag_def: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" shows "carr_dsumag I A = {f. finiteHom I A f}"
ifex_ite_induct: fixes P :: "'a ifex \<Rightarrow> 'a ifex \<Rightarrow> 'a ifex \<Rightarrow> bool"   and i :: "'a ifex"   and t :: "'a ifex"   and e :: "'a ifex" assumes "\<And>i t e. \<lbrakk>lowest_tops [i, t, e] = None; i = Trueif\<rbrakk> \<Longrightarrow> P i t e"   and "\<And>i t e. \<lbrakk>lowest_tops [i, t, e] = None; i = Falseif\<rbrakk> \<Longrightarrow> P i t e"   and "\<And>i t e a. \<lbrakk>\<And>val. P (restrict_top i a val) (restrict_top t a val) (restrict_top e a val); lowest_tops [i, t, e] = Some a\<rbrakk> \<Longrightarrow> P i t e" shows "P i t e"
simps_3: fixes x1 :: "'f"   and x21 :: "'g"   and x22 :: "'h \<Rightarrow> ('f, 'g, 'h) env option" shows "Val x1 \<noteq> Env x21 x22"
word_bitwise_m1_simps_2: fixes x :: "'b word" shows "- 1 AND x = x"
slice_def: fixes "from" :: "nat"   and to :: "nat"   and list :: "'b list" shows "slice from to list = take (to - from) (drop from list)"
dc2c_principal: shows "dc2c (principal_dchar n) = principal_char G"
R_finrank_def: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set" assumes "R_scalar_mult R" shows "R_scalar_mult.R_finrank R smult M = (\<exists>n. \<forall>ms. set ms \<subseteq> M \<and> R_scalar_mult.R_lin_independent R smult ms \<longrightarrow> length ms \<le> n)"
trss_drop_all_context: fixes s :: "'a list"   and c :: "'a list multiset"   and w :: "('b, 'c) label list"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((s, c), w, s', c') \<in> trcl (trss fg)" obtains csp :: "'a list multiset" where "c' = csp + c" and "((s, {#}), w, s', csp) \<in> trcl (trss fg)"
neg_equal_iff_equal: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "(\<ominus>\<^bsub>R\<^esub> x = \<ominus>\<^bsub>R\<^esub> y) = (x = y)"
path_split_general: fixes xs :: "'a list"   and ys :: "'a list" assumes "\<forall>zs. xs \<noteq> ys @ zs" obtains j :: "nat" and zs :: "'a list" where "xs = take j ys @ zs" and "j < length ys" and "\<forall>k>j. \<forall>zs'. xs \<noteq> take k ys @ zs'"
next_Plus2: fixes fsm1 :: "('b, 'c) fsm_scheme"   and fsm2 :: "('b, 'd) fsm_scheme"   and st2 :: "hf"   and x :: "'b"   and st' :: "hf" shows "next (Plus fsm1 fsm2) (HF.Inr st2) x st' = (\<exists>st2'. st' = HF.Inr st2' \<and> next fsm2 st2 x st2')"
nxt: fixes q :: "hf"   and x :: "'a" assumes "q \<in> states Accessible_dfa" shows "nxt Accessible_dfa q x \<in> states Accessible_dfa"
composite_simps_1: shows "src (f \<star> g) = src g"
finished_discovered: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme" assumes "DFS_invar G param s" shows "dom (finished s) \<subseteq> dom (discovered s)"
intersecttwo: fixes A :: "'a set set"   and A' :: "'a set set"   and x :: "'a set"   and C :: "'a set" assumes "A \<in> \<A>"   and "A' \<in> \<A>"   and "x \<in> A \<inter> A'"   and "C \<in> A \<inter> A'"   and "chamber C" shows "\<exists>f. ChamberComplexIsomorphism A A' f \<and> fixespointwise f x \<and> fixespointwise f C"
finestpartSubset: fixes X :: "'b set"   and Y :: "'b set" assumes "X \<subseteq> Y" shows "finestpart X \<subseteq> finestpart Y"
measurable_cong_sets: fixes M :: "'e measure"   and M' :: "'e measure"   and N :: "'f measure"   and N' :: "'f measure" assumes "sets M = sets M'"   and "sets N = sets N'" shows "M \<rightarrow>\<^sub>M N = M' \<rightarrow>\<^sub>M N'"
elims_3: fixes x :: "'a Point"   and xa :: "'a Point"   and xb :: "'a Point" assumes "\<not> collinear x xa xb" obtains p :: "'a Point" and q :: "'a Point" and r :: "'a Point" where "x = p" and "xa = q" and "xb = r" and "\<nexists>\<alpha> \<beta>. \<alpha> + \<beta> = (1::'a) \<and> positionVector p = \<alpha> ** positionVector q \<oplus> \<beta> ** positionVector r"
triple_schroeder_p: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" shows "(x * y * z \<le> - w) = ((x[[w]]) * z\<^sup>T \<le> - y)"
simps: fixes range :: "'a set"   and f :: "'b \<Rightarrow> 'a option"   and m :: "'b" shows "rorc.rnd_oracle range f m = (case f m of None \<Rightarrow> spmf_of_set range \<bind> (\<lambda>r. return_spmf (r, f(m \<mapsto> r))) | Some r \<Rightarrow> return_spmf (r, f))"
pairwise_Suc: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b stream"   and i :: "nat" assumes "PTA_Reachability.pairwise P xs" shows "P (xs !! i) (xs !! Suc i)"
simps_1: fixes var :: "nat"   and p :: "real mpoly" shows "substNegInfinity var (Eq p) = allZero p var"
invertible_mat_smult_mat: fixes A :: "'b mat"   and k :: "'b" assumes "invertible_mat A"   and "k dvd (1::'b)" shows "invertible_mat (k \<cdot>\<^sub>m A)"
simps_3: fixes OP :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and N :: "'a"   and r :: "'a"   and v :: "'a"   and va :: "'a list" shows "rexp_of_list OP N (r # v # va) = OP r (rexp_of_list OP N (v # va))"
eval\<Pi>\<^sub>3_inject: fixes x :: "\<Pi>\<^sub>3"   and y :: "\<Pi>\<^sub>3" shows "(eval\<Pi>\<^sub>3 x = eval\<Pi>\<^sub>3 y) = (x = y)"
restrict_count_space: fixes B :: "'b set"   and A :: "'b set" shows "restrict_space (count_space B) A = count_space (A \<inter> B)"
FI_match: fixes p :: "assn"   and q :: "assn"   and m :: "(assn \<times> assn) list"   and ps :: "assn"   and up :: "assn"   and qs :: "assn"   and uq :: "assn"   and f :: "assn" assumes "p \<Longrightarrow>\<^sub>A q"   and "FI ((p, q) # m) (ps * up) (qs * uq) SLN SLN f" shows "FI m (ps * p) (qs * q) up uq f"
case_cong: fixes pratt :: "pratt"   and pratt' :: "pratt"   and f1 :: "nat \<Rightarrow> 'a"   and g1 :: "nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and g2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a" assumes "pratt = pratt'"   and "\<And>x1. pratt' = Prime x1 \<Longrightarrow> f1 x1 = g1 x1"   and "\<And>x21 x22 x23. pratt' = Triple x21 x22 x23 \<Longrightarrow> f2 x21 x22 x23 = g2 x21 x22 x23" shows "(case pratt of Prime x1 \<Rightarrow> f1 x1 | Triple x21 x22 x23 \<Rightarrow> f2 x21 x22 x23) = (case pratt' of Prime x1 \<Rightarrow> g1 x1 | Triple x21 x22 x23 \<Rightarrow> g2 x21 x22 x23)"
new_tree_order: shows "h' \<turnstile> to_tree_order (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr) \<rightarrow>\<^sub>r new_tree_order"
Open: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>l a[b] \<prec> P'"   and "a \<noteq> b" shows "<\<nu>b>P \<Longrightarrow>\<^sub>l a<\<nu>b> \<prec> P'"
alpha_Tree_permute_rep_commute: fixes p :: "perm"   and t\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha>" shows "p \<bullet> rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha> =\<^sub>\<alpha> rep_Tree\<^sub>\<alpha> (p \<bullet> t\<^sub>\<alpha>)"
disc_2: fixes xs :: "'c llist"   and P :: "'c \<Rightarrow> bool" assumes "\<not> lnull xs"   and "P (lhd xs)" shows "\<not> lnull (ltakeWhile P xs)"
assms: shows "rt1 \<sqsubset>\<^bsub>dip\<^esub> rt2"
assms: shows "(x, y) = split_n_m_x (dim_vec b) (dim_vec c) X"
hyps: assumes "\<not> distinct ws" shows "\<exists>xs y ys zs. ws = xs @ y # ys @ y # zs \<and> distinct (ys @ [y])"
distinguishing_weak_formula_is_weak: fixes P :: "'state"   and Q :: "'state" assumes "\<not> P \<equiv>\<cdot> Q" shows "weak_formula (distinguishing_weak_formula P Q)"
le: fixes x :: "'c" assumes "x \<noteq> a" shows "dist (g x) m \<le> dist (f x) l"
invv: shows "inner_prod v v = complex_of_real (\<Sum>i = 0..<m. cmod (conjugate (v $ i) * v $ i))"
CallRedsObj: 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"   and M :: "char list"   and es :: "char list exp list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(es),s\<rangle> \<rightarrow>* \<langle>e'\<bullet>M(es),s'\<rangle>"
of_int_1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus 1 = one"
g: shows "(g \<longlongrightarrow> g a) (at_left a)"
simps_1: fixes x :: "'f"   and \<sigma> :: "'f \<Rightarrow> ('d, 'e) Term.term" shows "Var x \<cdot> \<sigma> = \<sigma> x"
ev: shows "\<forall>\<^sub>F s in F. (x s, t s) \<in> Sigma X existence_ivl0"
\<psi>_naturality_2: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "D.inv (\<tau>.\<psi> (C.cod \<mu>)) \<cdot>\<^sub>D (\<sigma>'.map\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D \<tau>.\<psi> (C.dom \<mu>) = F \<mu>"
map_empty: fixes empty :: "'a"   and update :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and lookup :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and invar :: "'a \<Rightarrow> bool" assumes "Map empty update delete lookup invar" shows "lookup empty = Map.empty"
card_length: fixes xs :: "'b list" shows "order.greater_eq (length xs) (card (set xs))"
simps_2: fixes x2 :: "'i"   and y2 :: "'i" shows "(Inr x2 = Inr y2) = (x2 = y2)"
matchs_reorder': fixes fps :: "(char list \<times> pat) list"   and fTs :: "(char list \<times> POPLmarkRecord.type) list"   and \<Delta> :: "binding list"   and fs :: "(char list \<times> trm) list"   and ts :: "trm list" assumes "\<turnstile> fps [:] fTs \<Rightarrow> \<Delta>"   and "\<forall>(l, U)\<in>set fTs. \<exists>u. fs\<langle>l\<rangle>\<^sub>? = \<lfloor>u\<rfloor>"   and "\<turnstile> fps [\<rhd>] map (\<lambda>(l, T). (l, the (fs\<langle>l\<rangle>\<^sub>?))) fTs \<Rightarrow> ts" shows "\<turnstile> fps [\<rhd>] fs \<Rightarrow> ts"
simps_2: fixes f :: "funty" shows "V (TFun f) = VFun (Vf f)"
full_rank_imp_is_basis2: fixes A :: "(('b, 'c) vec, 'd) vec" assumes "rank A = ncols A" shows "vec.independent (columns A) \<and> vec.span (columns A) = col_space A \<and> card (columns A) = ncols A"
weakly_bisimilar_equivp: shows "equivp (\<approx>\<cdot>)"
two_composable_identity_arrows_of_spans_axioms: shows "two_composable_identity_arrows_of_spans (\<cdot>) prj0 prj1 g f"
set_mset_sup: fixes A :: "'a multiset"   and B :: "'a multiset" shows "set_mset (A \<union># B) = set_mset A \<union> set_mset B"
le_less: fixes x :: "'k multiset"   and y :: "'k multiset" shows "(x \<subseteq># y) = (x \<subset># y \<or> x = y)"
add_arc_in: fixes a :: "'a \<times> 'a" assumes "a \<in> parcs G" shows "G.add_arc a = with_proj G"
sympD: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d"   and a :: "'d" assumes "symp r"   and "r b a" shows "r a b"
fmeasurable_eq_sets: shows "fmeasurable (density M h) = sets (density M h)"
intro: fixes bind :: "'f \<Rightarrow> ('g \<Rightarrow> 'f) \<Rightarrow> 'f"   and pause :: "'d \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> 'f" assumes "\<And>out c f. bind (pause out c) f = pause out (\<lambda>i. bind (c i) f)" shows "monad_resumption_axioms bind pause"
pow_eq_id: fixes x :: "'c"   and n :: "nat" assumes "x \<in> carrier H" shows "(x [^]\<^bsub>H\<^esub> n = \<one>\<^bsub>H\<^esub>) = (H.ord x dvd n)"
prod_cases4: fixes y :: "'d \<times> 'e \<times> 'f \<times> 'g" obtains a :: "'d" and b :: "'e" and c :: "'f" and d :: "'g" where "y = (a, b, c, d)"
that: fixes i' :: "nat"   and j' :: "nat" assumes "i' < j'"   and "i' < length xs"   and "j' < length xs"   and "filter P xs ! i = xs ! i'"   and "filter P xs ! j = xs ! j'" shows "thesis"
gb_red_aux_not_zero: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "0 \<notin> set (gb_red_aux bs ps)"
fmrestrict_set_add_distrib: fixes A :: "'a set"   and m :: "('a, 'b) fmap"   and n :: "('a, 'b) fmap" shows "fmrestrict_set A (m ++\<^sub>f n) = fmrestrict_set A m ++\<^sub>f fmrestrict_set A n"
cross7_simps_14: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a + (b - c) = a + b - c"
exE_some: fixes P :: "'a \<Rightarrow> bool"   and c :: "'a" assumes "Ex P"   and "c \<equiv> Eps P" shows "P c"
UP_mapsto: shows "SC.UP \<in> UNIV \<rightarrow> SC.Univ"
in_set_permutations: fixes xs :: "'b list"   and ys :: "'b list" assumes "distinct xs" shows "(ys \<in> set (permutations xs)) = (distinct ys \<and> set xs = set ys)"
get_snd_lens: fixes x :: "'c"   and y :: "'b" shows "get\<^bsub>snd\<^sub>L\<^esub> (x, y) = y"
valid_rga: shows "rga_ops (pre @ suf @ [x])"
simps_2: fixes a :: "'b"   and x :: "'b"   and xs :: "'b list" shows "position a (x # xs) = (if a = x then 1 else Suc (position a xs))"
1: shows "inj_on (strict_from_inj k g) {0..<k}"
singular_chain_diff: fixes p :: "nat"   and X :: "'c topology"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X a"   and "singular_chain p X b" shows "singular_chain p X (a - b)"