hcju/isabelleps huggingface dataset:
take_all: fixes xs :: "'g list"   and n :: "nat" assumes "length xs \<le> n" shows "take n xs = xs"
assms: shows "eq_\<I>_gpv A \<I> gpv gpv'"
gamma2_as_euclid_space_fun: shows "gamma2 = (\<lambda>x. (a + (b - a) * x) *\<^sub>R j + (g1 (a + (b - a) * x), 0))"
simps: fixes r :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and s :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a1 :: "'f"   and a2 :: "'h" shows "(r OO s) a1 a2 = (\<exists>a b c. a1 = a \<and> a2 = c \<and> r a b \<and> s b c)"
elims: fixes x :: "'a Literal"   and y :: "'a" assumes "atom x = y" obtains A :: "'a" where "x = Pos Aa__" and "y = Aa__"   | A :: "'a" where "x = Neg A" and "y = A"
simps_11: fixes R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x2 :: "'h"   and y2 :: "'i" shows "rel_option R (Some x2) (Some y2) = R x2 y2"
has: shows "has_ell2_norm (\<lambda>i. c * x i)"
assms: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) \<tau>"
err_semilat_JType_esl: fixes wf_mb :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" assumes "wf_prog wf_mb P" shows "err_semilat (SemiType.esl P)"
intro: fixes M :: "'e pre_map"   and G :: "('f, 'e) pre_digraph" assumes "edge_succ M permutes arcs G"   and "\<And>v. \<lbrakk>v \<in> verts G; out_arcs G v \<noteq> {}\<rbrakk> \<Longrightarrow> cyclic_on (edge_succ M) (out_arcs G v)" shows "digraph_map_axioms G M"
cong_add_lcancel: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and uminus :: "'c \<Rightarrow> 'c"   and division_segment :: "'c \<Rightarrow> 'c"   and a :: "'c"   and x :: "'c"   and y :: "'c"   and n :: "'c" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo (plus a x) (plus a y) n = unique_euclidean_semiring.cong modulo x y n"
add: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "integrable M f"   and "P f"   and "integrable M g"   and "P g" shows "P (\<lambda>x. f x + g x)"
crsp_step_inc_pre: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and n :: "nat"   and asa :: "nat"   and lma :: "nat list" assumes "ly = layout_of ap"   and "crsp ly (as, lm) (s, l, r) ires"   and "abc_step_l (as, lm) (Some (Inc n)) = (asa, lma)" shows "\<exists>stp k. steps0 (Suc 0, l, r) (findnth n @ shift tinc_b (2 * n)) stp = (2 * n + 10, Bk # Bk # ires, <lma> @ Bk \<up> k) \<and> 0 < stp"
analyzed_in_def: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set" shows "analyzed_in t M \<equiv> \<forall>K R. Ana t = (K, R) \<and> (\<forall>k\<in>set K. M \<turnstile>\<^sub>c k) \<longrightarrow> (\<forall>r\<in>set R. M \<turnstile>\<^sub>c r)"
PScope_simp: fixes \<pi> :: "char list \<Rightarrow> com option"   and \<pi>' :: "char list \<Rightarrow> com option"   and p :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>: (PScope \<pi>' p, s) \<Rightarrow> t = \<pi>': (p, s) \<Rightarrow> t"
listassoc_wfactorsD: fixes as :: "'a list"   and bs :: "'a list"   and a :: "'a"   and b :: "'a" assumes "as [\<sim>] bs"   and "wfactors G as a"   and "wfactors G bs b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "a \<sim> b"
zmod_eq_0_iff: fixes m :: "int"   and d :: "int" shows "(m mod d = 0) = (\<exists>q. m = d * q)"
assms: fixes x :: "'b"   and T :: "PreSimplyTyped.type"   and M :: "'b trm" shows "A \<noteq> Fn x T M"
monomial_0D: fixes t :: "'e"   and c :: "'f" assumes "monomial c t = 0" shows "c = (0::'f)"
noetherian_ring_axioms_def: fixes R :: "('c, 'd) ring_scheme" shows "noetherian_ring_axioms R \<equiv> \<forall>I. ideal I R \<longrightarrow> (\<exists>A\<subseteq>carrier R. finite A \<and> I = Idl\<^bsub>R\<^esub> A)"
projective_family_def: fixes I :: "'b set"   and P :: "'b set \<Rightarrow> ('b \<Rightarrow> 'c) measure"   and M :: "'b \<Rightarrow> 'c measure" shows "projective_family I P M \<equiv> (\<forall>J H. J \<subseteq> H \<longrightarrow> finite H \<longrightarrow> H \<subseteq> I \<longrightarrow> P J = distr (P H) (Pi\<^sub>M J M) (\<lambda>f. restrict f J)) \<and> (\<forall>J. finite J \<longrightarrow> J \<subseteq> I \<longrightarrow> prob_space (P J))"
distinct_12: fixes x41 :: "'d"   and x42 :: "'c"   and x43 :: "('a, 'b, 'c, 'd) Tree"   and x31 :: "'d"   and x32 :: "'b" shows "tAct x41 x42 x43 \<noteq> tPred x31 x32"
induct: fixes P :: "'c list \<Rightarrow> 'c list \<Rightarrow> bool"   and a0 :: "'c list"   and a1 :: "'c list" assumes "\<And>x xs y ys. (x = y \<Longrightarrow> P xs ys) \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>uv_. P [] uv_"   and "\<And>uu_. P uu_ []" shows "P a0 a1"
approx_add_left_cancel: fixes d :: "'a star"   and b :: "'a star"   and c :: "'a star" assumes "d + b \<approx> d + c" shows "b \<approx> c"
col_fst_def: fixes A :: "'a mat" shows "col_fst A = vec (dim_row A) (\<lambda>i. A $$ (i, 0))"
blocks_nempty: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "bl \<noteq> {}"
perpendicular_to_x_axis: fixes H :: "circline" assumes "is_poincare_line H" shows "perpendicular_to_x_axis H = perpendicular x_axis H"
acyclic_subset: fixes s :: "('x \<times> 'x) set"   and r :: "('x \<times> 'x) set" assumes "acyclic s"   and "r \<subseteq> s" shows "acyclic r"
dsg3: fixes domain_op :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.domain_semigroup domain_op times" shows "domain_op (times (domain_op x) y) = times (domain_op x) (domain_op y)"
simps_1: fixes x1 :: "\<omega>"   and y1 :: "\<omega>" shows "(\<omega>\<upsilon> x1 = \<omega>\<upsilon> y1) = (x1 = y1)"
cong_1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b c one"
finite_grab: fixes N :: "nat set"   and n :: "nat" shows "finite (fst (grab N n))"
le_infI2: fixes b :: "'a"   and x :: "'a"   and a :: "'a" assumes "b \<le> x" shows "a \<cdot> b \<le> x"
uminus_eq: fixes x :: "'c" shows "- x = (if x \<le> bot then top else bot)"
Per_def: 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" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Per Bet Cong A B C \<equiv> \<exists>C'. Tarski_neutral_dimensionless.Midpoint Bet Cong B C C' \<and> Cong A C A C'"
zero_le: fixes s :: "'lbl" shows "(0::'lbl) \<le> s"
simps_2: fixes n :: "nat"   and R :: "('a \<times> 'a) set" shows "R ^^ Suc n = R ^^ n O R"
return_partition_def: fixes A :: "'a set"   and k :: "nat" shows "return_partition A k = A \<inter> (T ^^ k) --` A - (\<Union>i\<in>{0<..<k}. (T ^^ i) --` A)"
odiff_eq_iff: fixes z :: "V"   and x :: "V"   and y :: "V" assumes "z \<unlhd> x" shows "(odiff x z = y) = (x = z + y)"
prems_4: shows "\<Union> (get_offending_flows (m # M) \<lparr>nodes = V, edges = E\<rparr>) \<subseteq> X"
linear_def: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f" shows "linear s1 s2 f \<equiv> vector_space s1 \<and> vector_space s2 \<and> module_hom s1 s2 f"
source: fixes S1 :: "'procS"   and S2 :: "'procS"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "(S1, S2) \<in> SRel" shows "SourceTerm S1 \<R>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> SourceTerm S2"
remove_prime_factor: fixes i :: "nat"   and n :: "nat"   and ps :: "nat list"   and m :: "nat"   and qs :: "nat list" assumes "remove_prime_factor i n ps = (m, qs)"   and "1 < i"   and "n \<noteq> 0" shows "\<exists>rs. qs = rs @ ps \<and> n = m * prod_list rs \<and> \<not> i dvd m \<and> set rs \<subseteq> {i}"
inverse_divide: 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 (divide a b) = divide b a"
bool_to_ternary_simp1: fixes X :: "bool" shows "(bool_to_ternary X = TernaryTrue) = X"
S_def: shows "S = calc_spread k\<^sub>0 ps"
set_plus_intro: fixes a :: "'a"   and C :: "'a set"   and b :: "'a"   and D :: "'a set" assumes "a \<in> C"   and "b \<in> D" shows "a + b \<in> C + D"
assms_1: shows "wf\<^sub>t\<^sub>r\<^sub>m s"
smc_SemiCAT_components_4: fixes \<alpha> :: "V" shows "smc_SemiCAT \<alpha>\<lparr>Cod\<rparr> = (\<lambda>\<FF>\<in>\<^sub>\<circ>all_smcfs \<alpha>. \<FF>\<lparr>HomCod\<rparr>)"
Inr_def: fixes b :: "V" shows "ZFC_Cardinals.Inr b \<equiv> \<langle>1, b\<rangle>"
divide_float_def: fixes a :: "('c, 'd) IEEE.float"   and b :: "('c, 'd) IEEE.float" shows "a / b = fdiv To_nearest a b"
random_serial_dictatorship_support: fixes agents' :: "'agent set"   and alts' :: "'alt set" assumes "finite agents"   and "finite alts"   and "agents' \<subseteq> agents"   and "alts' \<noteq> {}"   and "alts' \<subseteq> alts" shows "set_pmf (random_serial_dictatorship agents' alts' R) \<subseteq> alts'"
exp_cf3_upper_bound_pos: fixes x :: "real" assumes "0 \<le> x"   and "0 < numer_cf3 (- x)" shows "exp x \<le> exp_cf3 x"
enum_idx_bound: fixes x :: "'c" shows "enum_idx x < length enum_class.enum"
snoc_3: fixes c :: "('p, 's, 'm) configuration" shows "P c [] c"
r_1: shows "(r, v) = Max_by fst (vote_set (votes sa) {p})"
SKIPnot: fixes s :: "char list \<Rightarrow> int"   and p :: "nat"   and t :: "char list \<Rightarrow> int" shows "(\<not> (SKIP, s) \<Rightarrow> p \<Down> t) = (s \<noteq> t \<or> p \<noteq> Suc 0)"
monomials_Var: fixes x :: "nat" shows "monomials (Var x) = {monomial 1 x}"
affine_diff_divide: fixes S :: "'b set"   and k :: "real"   and x :: "'b"   and y :: "'b" assumes "affine S"   and "k \<noteq> 0"   and "k \<noteq> 1"   and "x \<in> S"   and "y /\<^sub>R (1 - k) \<in> S" shows "(x - y) /\<^sub>R k \<in> S"
ss_ss: fixes i :: "nat"   and q :: "nat" assumes "i \<le> q" shows "simplicial_simplex (Suc q) {x. x 0 \<in> {0..1} \<and> x \<circ> Suc \<in> standard_simplex q} (simp q i)"
ih: assumes "\<forall>(a, b)\<in>set pi. a \<sharp> x \<and> b \<sharp> x" shows "pi \<bullet> x = x"
latestAct_prop_2: fixes n :: "nat"   and nid :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>n'. latestAct_cond nid t n n'" shows "\<langle>nid \<leftarrow> t\<rangle>\<^bsub>n\<^esub> < n"
of_nat_1_eq_iff: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(one = semiring_1.of_nat one plus zero n) = (n = 1)"
len_hchop_right_left_border: fixes v :: "view"   and ts :: "traffic"   and c :: "cars"   and v1 :: "view"   and v2 :: "view" shows "len v ts c = ext v \<and> v=v1\<parallel>v2 \<longrightarrow> left (len v2 ts c) = left (ext v2)"
compute_lapprox_rat: fixes prec :: "nat"   and x :: "int"   and y :: "int" shows "lapprox_rat prec x y = (if y = 0 then 0 else if 0 \<le> x then if 0 < y then lapprox_posrat prec (nat x) (nat y) else - rapprox_posrat prec (nat x) (nat (- y)) else if 0 < y then - rapprox_posrat prec (nat (- x)) (nat y) else lapprox_posrat prec (nat (- x)) (nat (- y)))"
singlPart_inj_on: fixes f :: "'c \<Rightarrow> 'd"   and I :: "'c set" shows "inj_on ((`) f) (singlPart I) = inj_on f I"
sum_to_poly: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(\<Sum>x\<in>A. [:f x:]) = [:sum f A:]"
Suc_lessI: fixes m :: "nat"   and n :: "nat" assumes "m < n"   and "Suc m \<noteq> n" shows "Suc m < n"
preserves_hhom: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "\<guillemotleft>F \<mu> : F.map\<^sub>0 (src\<^sub>C \<mu>) \<rightarrow>\<^sub>D F.map\<^sub>0 (trg\<^sub>C \<mu>)\<guillemotright>"
coclop_closure: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "coclop f" shows "(x \<in> range f) = (f x = x)"
Perp_in_perm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D" shows "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D \<and> Tarski_neutral_dimensionless.PerpAt Bet Cong X B A C D \<and> Tarski_neutral_dimensionless.PerpAt Bet Cong X A B D C \<and> Tarski_neutral_dimensionless.PerpAt Bet Cong X B A D C \<and> Tarski_neutral_dimensionless.PerpAt Bet Cong X C D A B \<and> Tarski_neutral_dimensionless.PerpAt Bet Cong X C D B A \<and> Tarski_neutral_dimensionless.PerpAt Bet Cong X D C A B \<and> Tarski_neutral_dimensionless.PerpAt Bet Cong X D C B A"
eq_fold: fixes g :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "sum g A = Finite_Set.fold ((+) \<circ> g) (0::'a) A"
SN_I: fixes A :: "('b \<times> 'b) set" assumes "\<And>a. SN_on A {a}" shows "SN A"
trace_set_traces: fixes P :: "'a process" shows "trace_set (traces P)"
ext: fixes g :: "'a"   and f :: "'a" assumes "L.comp g f \<noteq> L.null" shows "L.seq g f"
c_def: shows "c = (\<lambda>k. ((- 1) ^ Suc k * bernoulli (Suc k) / fact (Suc k)) *\<^sub>R (fs k (real_of_int b) - fs k (real_of_int a)))"
app_last: fixes t :: "dB"   and ts :: "dB list"   and u :: "dB" shows "(t \<degree>\<degree> ts) \<degree> u = t \<degree>\<degree> (ts @ [u])"
less: fixes i :: "nat" assumes "i < length xs" shows "inv \<pi> i < length xs"
inv_solve_right: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<in> carrier V"   and "b \<in> carrier V"   and "c \<in> carrier V" shows "(a = b \<oplus>\<^bsub>V\<^esub> \<ominus>\<^bsub>V\<^esub> c) = (b = a \<oplus>\<^bsub>V\<^esub> c)"
pole_in_disc: fixes M :: "moebius"   and a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex"   and z :: "complex_homo"   and H :: "ocircline" assumes "M = mk_moebius a b c d"   and "c \<noteq> 0"   and "a * d - b * c \<noteq> 0"   and "is_pole M z"   and "z \<in> disc H" shows "\<not> pos_oriented (moebius_ocircline M H)"
intro: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b" assumes "\<And>a. f z a = a"   and "\<And>a. f (inverse a) a = z" shows "group_axioms f z inverse"
prop_atoms_nested_prop_atoms: fixes \<phi> :: "'b ltln" shows "prop_atoms \<phi> \<subseteq> nested_prop_atoms \<phi>"
nn_integral_mono: fixes M :: "'a measure"   and u :: "'a \<Rightarrow> ennreal"   and v :: "'a \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x" shows "integral\<^sup>N M u \<le> integral\<^sup>N M v"
d_repr: shows "d_repr (IArray.of_fun (\<lambda>i. if i = 0 then 1 else d\<mu>_impl fs_init !! (i - 1) !! (i - 1)) (Suc m)) fs_init"
Suc: fixes j' :: "nat" shows "real_of_float (lb n ((F ^^ j') s) (f j') x) \<le> horner F G n ((F ^^ j') s) (f j') (real_of_float x) \<and> horner F G n ((F ^^ j') s) (f j') (real_of_float x) \<le> real_of_float (ub n ((F ^^ j') s) (f j') x)"
FAddSA_def: fixes G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option"   and SSA :: "'d \<times> ('d, 'e, 'f) seqauto" shows "(G [f+] SSA) = (let (S, SA) = SSA in if S \<in> dom G \<and> S \<notin> States SA then G ++ [S \<mapsto> insert SA (the (G S))] ++ EmptyMap (States SA) else G)"
exE_realizer: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b \<times> 'a"   and Q :: "'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" assumes "P (snd p) (fst p)"   and "\<And>x y. P y x \<Longrightarrow> Q (f x y)" shows "Q (let (x, y) = p in f x y)"
subgroup_imp_group: fixes H :: "'a set" assumes "subgroup H (add_monoid R)" shows "Group.group (add_monoid R\<lparr>carrier := H\<rparr>)"
wf_mstate_msteps: fixes \<pi> :: "(String.literal \<times> event_data list) prefix"   and R :: "event_data list set"   and st :: "('a, 'b) mstate"   and v :: "event_data option list"   and \<pi>' :: "(String.literal \<times> event_data list) prefix"   and X :: "(nat \<times> event_data option list set) list \<times> ('a, 'b) mstate"   and i :: "nat" assumes "wf_mstate \<phi> \<pi> R st"   and "mem_restr R v"   and "\<pi> \<le> \<pi>'"   and "X = msteps (pdrop (plen \<pi>) \<pi>') st" shows "wf_mstate \<phi> \<pi>' R (snd X) \<and> ((i, v) \<in> flatten_verdicts (fst X)) = ((i, v) \<in> M \<pi>' - M \<pi>)"
call_result: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and X :: "final_decision"   and rrs :: "'c rule list" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some rs"   and "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z Decision X" shows "\<Gamma>,\<gamma>,p\<turnstile> Rule m (Call chain) # rrs \<Rightarrow>\<^sub>z Decision X"
vs_def: shows "vs = {((r, a), v) |r a v. ((r, a), v) \<in> map_graph (\<lambda>(x, y). v_hist x y) \<and> a \<in> S}"
wb_lens_weak: fixes x :: "'c \<Longrightarrow> 'd" assumes "wb_lens x" shows "weak_lens x"
bsinleton_eq_bdoubleton_1: fixes x :: "'d"   and y :: "'d"   and z :: "'d" shows "(bsingleton x = bdoubleton y z) = (x = y \<and> y = z)"
preallocated_hext: fixes h :: "'heap"   and h' :: "'heap" assumes "preallocated h"   and "h \<unlhd> h'" shows "preallocated h'"
assms: fixes \<tau> :: "'a"   and \<tau>' :: "'a" assumes "P \<tau> = P \<tau>'"   and "Q \<tau> = Q \<tau>'"   and "R \<tau> = R \<tau>'" shows "S \<tau> = S \<tau>'"
digit_to_char_def: fixes n :: "nat" shows "digit_to_char n = (if n = 0 then CHR ''0'' else if n = 1 then CHR ''1'' else if n = 2 then CHR ''2'' else if n = 3 then CHR ''3'' else if n = 4 then CHR ''4'' else if n = 5 then CHR ''5'' else if n = 6 then CHR ''6'' else if n = 7 then CHR ''7'' else if n = 8 then CHR ''8'' else if n = 9 then CHR ''9'' else CHR ''X'')"
less_eq_natural_def: shows "(\<le>) \<equiv> map_fun nat_of_natural (map_fun nat_of_natural id) (\<le>)"
integral_component_eq: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and k :: "'b" assumes "f integrable_on S" shows "integral S (\<lambda>x. f x \<bullet> k) = integral S f \<bullet> k"
space_subgroup_props_3: fixes n :: "nat"   and E :: "'a set"   and v1 :: "'a"   and v2 :: "'a" assumes "dimension n K E"   and "v1 \<in> E"   and "v2 \<in> E" shows "v1 \<oplus> v2 \<in> E"
exec_1_iff: fixes \<sigma> :: "jvm_state"   and \<sigma>' :: "jvm_state"   and P :: "jvm_prog" shows "P \<turnstile> \<sigma> -jvm\<rightarrow>\<^sub>1 \<sigma>' = (exec (P, \<sigma>) = \<lfloor>\<sigma>'\<rfloor>)"
eql3: shows "st.ptensor_mat (H_k k) (1\<^sub>m st.d2) = st.ptensor_mat (H_k k) (st1.ptensor_mat (1\<^sub>m 2) (1\<^sub>m st4.d2))"
MRList2: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and c :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "x \<in> dom (Cp a)" shows "applied_rule_rev Cp x (c @ b @ [a]) = \<lfloor>a\<rfloor>"
bdd_aboveI2: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'f set"   and f :: "'f \<Rightarrow> 'e"   and M :: "'e" assumes "class.preorder less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) M" shows "preorder.bdd_above less_eq (f ` A)"
cong_scalar_right: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and b :: "'c"   and c :: "'c"   and a :: "'c"   and d :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a" shows "unique_euclidean_semiring.cong modulo (times b d) (times c d) a"
that: fixes W :: "'a set"   and f' :: "'a \<Rightarrow> 'c" assumes "A \<subseteq> W"   and "W \<subseteq> carrier"   and "open W"   and "diff_fun k (charts_submanifold W) f'"   and "\<And>x. x \<in> A \<Longrightarrow> f x = f' x" shows "thesis"
exhaust: fixes y :: "stack_elem" obtains x11 :: "exp" and x12 :: "exp" where "y = Alts x11 x12"   | x2 :: "Vars.var" where "y = Arg x2"   | x3 :: "Vars.var" where "y = Upd x3"   | x4 :: "Vars.var" where "y = Dummy x4"
interleave_Nil2: fixes xs :: "'a list" shows "xs \<otimes> [] = {xs}"
simps_2: fixes ok :: "nat"   and f' :: "'b list"   and y :: "'b"   and r' :: "'b list" shows "exec (Rev ok [] f' [y] r') = App ok f' (y # r')"
simps: fixes P :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'d" shows "someFunc P x = Eps (P x)"
simple_integral_mult: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal"   and c :: "ennreal" assumes "simple_function M f" shows "\<integral>\<^sup>S x. c * f x \<partial>M = c * integral\<^sup>S M f"
mat_mult_left_right_inverse: fixes A :: "'c mat"   and n :: "nat"   and B :: "'c mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "A * B = 1\<^sub>m n" shows "B * A = 1\<^sub>m n"
simps_4: fixes b :: "bexp"   and C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "vc (IF b THEN C\<^sub>1 ELSE C\<^sub>2) Q = (vc C\<^sub>1 Q \<and> vc C\<^sub>2 Q)"
const_in_carrier: fixes t :: "'a \<times> nat" assumes "t \<in> set (dense_repr p)" shows "[fst t] \<in> carrier (K [X])"
banach_steinhaus: fixes f :: "'e \<Rightarrow> 'c \<Rightarrow>\<^sub>L 'd" assumes "\<And>x. bounded (range (\<lambda>n. f n *\<^sub>v x))" shows "bounded (range f)"
IH_1: shows "vars_of (t1 \<lhd> \<sigma>) = \<Union> {V. \<exists>x. x \<in> vars_of t1 \<and> V = vars_of (Var x \<lhd> \<sigma>)}"
cexpr_typing_integrate_vars: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and vs :: "nat list" assumes "\<Gamma> \<turnstile>\<^sub>c e : REAL" shows "\<Gamma> \<turnstile>\<^sub>c integrate_vars \<Gamma> vs e : REAL"
2: shows "homeomorphism (cbox c d) (cbox u v) f2 g2"
is_category: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "product_category C1 C2" shows "category (product_category.comp C1 C2)"
inf_zer: shows "\<forall>x\<in>{r<..p}. a * x\<^sup>2 + b * x + c = 0"