hcju/isabelleps huggingface dataset:
cases: fixes x :: "'a list \<times> 'a list" obtains acc :: "'a list" and xs :: "'a list" where "x = (acc, xs)"
length_us: fixes y :: "nat list"   and i :: "nat" shows "length (\<^bold>u y i) = length y"
TESL_sem_decreases_head: fixes \<phi> :: "'b TESL_atomic"   and \<Phi> :: "'b TESL_atomic list" shows "\<lbrakk>\<lbrakk> \<phi> # \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L \<subseteq> \<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
normalize_def: fixes xs :: "('c, 'd) bintree option list" shows "Binomial_Queue.normalize xs = rev (normalize' (rev xs))"
subspaceI: fixes Sa :: "('a \<Rightarrow> real) set" assumes "Sa \<subseteq> diff_fun_space"   and "0 \<in> Sa"   and "\<And>x y. \<lbrakk>x \<in> Sa; y \<in> Sa\<rbrakk> \<Longrightarrow> x + y \<in> Sa"   and "\<And>c x. x \<in> Sa \<Longrightarrow> c *\<^sub>R x \<in> Sa" shows "sub.diff_fun_space.m1.subspace Sa"
coplanar_perm_3: 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 D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet A C D B"
perm_pi_simp_2: fixes pi :: "(name \<times> name) list"   and x :: "'a" shows "rev pi \<bullet> pi \<bullet> x = x"
\<delta>_prodI: fixes qs1 :: "'a list"   and qs2 :: "'b list"   and q1 :: "'a"   and l :: "'c"   and \<delta>1 :: "('a, 'c) ta_rule set"   and q2 :: "'b"   and \<delta>2 :: "('b, 'c) ta_rule set" assumes "length qs1 = length qs2"   and "q1 \<rightarrow> l qs1 \<in> \<delta>1"   and "q2 \<rightarrow> l qs2 \<in> \<delta>2" shows "(q1, q2) \<rightarrow> l zip qs1 qs2 \<in> \<delta>_prod \<delta>1 \<delta>2"
GreenThm_typeI_typeII_divisible_region_finite_holes: fixes s :: "(real \<times> real) set"   and i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain_typeI :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and two_chain_typeII :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and C :: "real \<times> real \<Rightarrow> real \<times> real" assumes "green_typeI_typeII_chain s i j F two_chain_typeI two_chain_typeII"   and "\<forall>(k, \<gamma>)\<in>boundary C. valid_path \<gamma>"   and "only_vertical_division (boundary C) two_chain_typeI"   and "only_horizontal_division (boundary C) two_chain_typeII"   and "s = cubeImage C" shows "integral (cubeImage C) (\<lambda>x. partial_vector_derivative (\<lambda>x. F x \<bullet> j) i x - partial_vector_derivative (\<lambda>x. F x \<bullet> i) j x) = one_chain_line_integral F {i, j} (boundary C)"
DD: shows "dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0} = DIM('a)"
rep_eq: fixes x :: "'b \<Rightarrow> 'd \<Rightarrow> 'c"   and xa :: "('b, 'd) mapping" shows "rep (map_values x xa) = (\<lambda>xb. map_option (x xb) (rep xa xb))"
xi_le1: fixes i :: "'n" shows "\<parallel>x $ i\<parallel> \<le> 1"
\<delta>_def: shows "\<delta> = Min {h' e' - h_minus_aux j e' |e'. e' \<in> set (cycle_edges C)}"
gen_dfsT_def: fixes gds :: "('b, 'a) gen_dfs_struct"   and V0 :: "'b set" shows "gen_dfs_defs.gen_dfsT gds V0 \<equiv> gds_init gds \<bind> WHILE\<^sub>T (gen_dfs_defs.gen_cond gds V0) (gen_dfs_defs.gen_step gds V0)"
simps_3: fixes M :: "'a"   and I :: "('a, 'b, 'c) input" shows "inputLength (M\<lparr>I) = inputLength' I"
word_leq_minus_one_le: fixes y :: "'a word"   and x :: "'a word" assumes "y \<noteq> 0"   and "x \<le> y - 1" shows "x < y"
NEXT_def: fixes s :: "'b sparc_state" shows "NEXT s \<equiv> case execute_instruction () s of (x, True) \<Rightarrow> None | (x, False) \<Rightarrow> Some (snd x)"
simple_append: fixes p\<^sub>1 :: "('b \<times> 'b) list"   and p\<^sub>2 :: "('b \<times> 'b) list" shows "simple (p\<^sub>1 @ p\<^sub>2) = (simple p\<^sub>1 \<and> simple p\<^sub>2 \<and> uedge ` set p\<^sub>1 \<inter> uedge ` set p\<^sub>2 = {})"
sussmann_open_mapping: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and f' :: "'a \<Rightarrow> 'b"   and g' :: "'b \<Rightarrow> 'a"   and T :: "'a set" assumes "open S"   and "continuous_on S f"   and "x \<in> S"   and "(f has_derivative f') (at x)"   and "bounded_linear g'"   and "f' \<circ> g' = id"   and "T \<subseteq> S"   and "x \<in> interior T" shows "f x \<in> interior (f ` T)"
F_def: shows "F \<equiv> \<lambda>x. sqrt (4 / x - 1) * x - 2 * arctan (sqrt (4 / x - 1) * (x - 2) / (x - 4))"
assms_1: shows "f \<in> L F (\<lambda>x. t x (g x))"
weave_complementary_nthss: fixes A :: "nat set"   and xs :: "'a list" shows "weave A (nths xs A) (nths xs (- A)) = xs"
QuotientI: fixes Abs :: "'b \<Rightarrow> 'a"   and Rep :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and T :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>a. Abs (Rep a) = a"   and "\<And>a. R (Rep a) (Rep a)"   and "\<And>r s. R r s = (R r r \<and> R s s \<and> Abs r = Abs s)"   and "T = (\<lambda>x y. R x x \<and> Abs x = y)" shows "Quotient R Abs Rep T"
compat_different: fixes cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and i :: "nat"   and x :: "'Var" assumes "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "i < length cms\<^sub>1"   and "x \<in> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "mem\<^sub>1 x \<noteq> mem\<^sub>2 x \<and> dma mem\<^sub>1 x = Low \<and> x \<notin> \<C>"
path_simps_3: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and la :: "'b list"   and lb :: "'b list"   and w :: "'b" shows "path E u (la @ lb) w = (\<exists>v. path E u la v \<and> path E v lb w)"
euclidean_dist_l2: fixes x :: "'b"   and y :: "'b" shows "dist x y = L2_set (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"
of_int_poly_i_def: fixes ops :: "'b arith_ops_record"   and f :: "int poly" shows "of_int_poly_i ops f = map (arith_ops_record.of_int ops) (coeffs f)"
intra_path_Append: fixes n :: "'node"   and as :: "'edge list"   and n'' :: "'node"   and as' :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^sub>\<iota>* n''"   and "n'' -as'\<rightarrow>\<^sub>\<iota>* n'" shows "n -as @ as'\<rightarrow>\<^sub>\<iota>* n'"
s_0_word: fixes w :: "word32" shows "get_S (w AND 4294967167) = 0"
brec_construct_witness_correct: fixes Qm :: "('a, 'b tree) HashMap.hashmap"   and r :: "('a, 'b) ta_rule" assumes "hm.invar Qm" shows "brec_construct_witness Qm r = construct_witness (hm.\<alpha> Qm) r"
assms_2: shows "{x} \<union> \<Union> S = {a..b}"
sorted_filter': fixes l :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "sorted l" shows "sorted (filter P l)"
rel_gpv''_lossless_gpvD2: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and gpv :: "('g, 'i, 'k) gpv"   and gpv' :: "('h, 'j, 'l) gpv"   and \<I>' :: "('j, 'l) \<I>"   and \<I> :: "('i, 'k) \<I>" assumes "rel_gpv'' A C R gpv gpv'"   and "lossless_gpv \<I>' gpv'"   and "rel_\<I> C R \<I> \<I>'" shows "lossless_gpv \<I> gpv"
Mem: fixes x :: "name" assumes "x mem xvec" shows "x \<in> supp M"
Constrains_Un: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set"   and B :: "'b set"   and B' :: "'b set" assumes "F \<in> A Co A'"   and "F \<in> B Co B'" shows "F \<in> A \<union> B Co A' \<union> B'"
one_app_congL: fixes t1 :: "lam"   and t2 :: "lam"   and s :: "lam" assumes " t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2" shows " App t1 s \<longrightarrow>\<^sub>\<beta>\<^sup>* App t2 s"
trace_parallel_plan_sas_plus_length_gt_one_if: fixes k :: "nat"   and I :: "'c \<Rightarrow> 'd option"   and \<psi> :: "('c, 'd) sas_plus_operator list list" assumes "k < length (trace_parallel_plan_sas_plus I \<psi>) - 1" shows "1 < length (trace_parallel_plan_sas_plus I \<psi>)"
hyps: fixes n :: "nat" shows "map_index' n f xs = map2 f [n..<n + length xs] xs"
abs_in_Basis_absE: fixes y :: "'e" assumes "\<bar>y\<bar> \<in> Basis" obtains "\<bar>y\<bar> = y"   | "\<bar>y\<bar> = - y"
inj\<pi>: shows "inj_on \<pi> (elts \<alpha>)"
suminf_scaleR_left: fixes X :: "nat \<Rightarrow> real"   and x :: "'b" assumes "summable X" shows "(\<Sum>n. X n) *\<^sub>R x = (\<Sum>n. X n *\<^sub>R x)"
unit: fixes y :: "'a"   and M :: "'a trm" shows "substitutes Unit y M Unit"
measurable_compose_countable_restrict: fixes P :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and M :: "'e measure"   and Q :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "countable {i. P i}"   and "f \<in> M \<rightarrow>\<^sub>M count_space UNIV"   and "\<And>i. P i \<Longrightarrow> Measurable.pred M (Q i)" shows "Measurable.pred M (\<lambda>x. P (f x) \<and> Q (f x) x)"
substT_notIn: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and x :: "'a"   and t1 :: "'b"   and t2 :: "'b" assumes "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst"   and "x \<in> var"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "x \<notin> FvarsT t1" shows "substT t1 t2 x = t1"
dist_les_1: fixes cfun :: "Value \<rightarrow> Value"   and discr' :: "bool discr" shows "Fn\<cdot>cfun \<notsqsubseteq> B\<cdot>discr'"
ma: shows "(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows) \<le> ma"
hyps: fixes xs :: "'a llist" assumes "enat n < llength xs" shows "lhd (ldropn n xs) = lnth xs n"
assms_2: shows "(\<lambda>n. v n - u n) \<longlonglongrightarrow> (0::'a)"
bet_suma__sams: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "A B C D E F SumA G H I"   and "Bet G H I" shows "SAMS A B C D E F"
lnth_0: fixes x :: "'c"   and xs :: "'c llist" shows "lnth (LCons x xs) 0 = x"
union_univ: shows "(\<Union>y\<in>columns A. {i. y = column i A}) = UNIV"
orderI: fixes a :: "'d upred"   and b :: "'d upred" assumes "a = (a \<or> b)" shows "a \<sqsubseteq> b"
finite_measure_Diff: fixes A :: "'c set"   and p :: "'c spmf"   and B :: "'c set" assumes "A \<in> sets (measure_spmf p)"   and "B \<in> sets (measure_spmf p)"   and "B \<subseteq> A" shows "Sigma_Algebra.measure (measure_spmf p) (A - B) = Sigma_Algebra.measure (measure_spmf p) A - Sigma_Algebra.measure (measure_spmf p) B"
ctr: fixes ws :: "'b stream list" shows "stranspose ws = map shd ws ## stranspose (map stl ws)"
assms: shows "x \<in> RF\<^sup>\<circ> T"
trans_closure_of_TRel_refl_cond: fixes TP :: "'procT"   and TQ :: "'procT"   and TRel :: "('procT \<times> 'procT) set" assumes "(TP, TQ) \<in> (TRel \<union> _Collect (T1, T2) (\<exists>S. T1 = \<lbrakk>S\<rbrakk> \<and> T2 = \<lbrakk>S\<rbrakk>))\<^sup>+" shows "(TP, TQ) \<in> TRel\<^sup>*"
assms: shows "set {a, b} \<in>\<^sub>\<circ> Vset \<alpha>"
supp_def: fixes x :: "'d" shows "supp x = {a. infinite {b. [(a, b)] \<bullet> x \<noteq> x}}"
assms: fixes p :: "Vars.var list" assumes "p \<in> ps" shows "pathCard p \<sqsubseteq> ce"
t: shows "eval (Mn 2 (Cn 3 r_less [recf.Id 3 2, Cn 3 r_length [Cn 3 r_prefixes [recf.Id 3 0, recf.Id 3 1]]])) [j, x] \<down>= t"
mem_val_w32_10_mem_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and a :: "word36" assumes "mem_equal s1 s2 a" shows "mem_val_w32 10 a s1 = mem_val_w32 10 a s2"
P_L: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (imp (PP \<langle>\<phi>L \<phi>\<rangle>) (PP \<langle>\<phi>\<rangle>))"
t_fourtimes_append: fixes ires :: "cell list"   and rs :: "nat"   and n :: "nat" shows "\<exists>stp ln rn. steps0 (Suc 0 + length (t_wcode_main_first_part @ shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, Bk # Bk # ires, Oc \<up> Suc rs @ Bk \<up> n) ((t_wcode_main_first_part @ shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) @ shift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)]) stp = (Suc t_fourtimes_len + length (t_wcode_main_first_part @ shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)]) div 2, Bk \<up> ln @ Bk # Bk # ires, Oc \<up> Suc (4 * rs) @ Bk \<up> rn)"
do_log_cases: fixes op :: "lop"   and v :: "v"   and e :: "exp0" obtains "do_log op v e = None"   | v' :: "v" where "do_log op v e = Some (Val v')"   | "do_log op v e = Some (Exp e)"
simps_2: fixes r :: "'a language"   and s :: "'a language" shows "\<dd> (StarLR r s) = (\<lambda>a. StarLR (\<dd> (Times r (Plus One s)) a) s)"
token_fails_def: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and x :: "nat" shows "mojmir_def.token_fails \<Sigma> \<delta> q\<^sub>0 w F x = (\<exists>n. semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n) \<and> semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n \<notin> F)"
hyps_1: fixes as :: "'a list list"   and as' :: "'a list list"   and bs :: "'a list list"   and bs' :: "'a list list" assumes "k = length as"   and "concat as = concat as'"   and "concat bs = concat bs'"   and "as \<in> lists (- {[]})"   and "bs \<in> lists (- {[]})"   and "strict_sorted (interact as bs)"   and "length bs \<le> length as"   and "length as \<le> Suc (length bs)"   and "as' \<in> lists (- {[]})"   and "bs' \<in> lists (- {[]})"   and "strict_sorted (interact as' bs')"   and "length bs' \<le> length as'"   and "length as' \<le> Suc (length bs')"   and "length as = length as'"   and "length bs = length bs'" shows "as = as' \<and> bs = bs'"
elts_vinsert: fixes x :: "V"   and y :: "V" shows "elts (vinsert x y) = insert x (elts y)"
piecewise_differentiable_compose: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and g :: "'b \<Rightarrow> 'c" assumes "f piecewise_differentiable_on S"   and "g piecewise_differentiable_on f ` S"   and "\<And>x. finite (S \<inter> f -` {x})" shows "g \<circ> f piecewise_differentiable_on S"
comp_null_1: fixes f :: "(bool, 'arr list) concrete_category.arr" shows "null \<cdot> f = null"
carrier_matD_2: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "dim_col A = nc"
comp_pp_semi_commute: fixes x :: "'a"   and y :: "'a" shows "x * - - y \<le> - - (x * y)"
in_set_zip_lengthE: fixes x :: "nat"   and y :: "'a"   and xs :: "'a list" assumes "(x, y) \<in> set (zip [0..<length xs] xs)" obtains "x < length xs" and "xs ! x = y"
freess_app: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and xs :: "'b list"   and ys :: "'b list" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.freess frees (xs @ ys) = pre_term.freess frees xs |\<union>| pre_term.freess frees ys"
continuous_cosh: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "continuous F f" shows "continuous F (\<lambda>x. cosh (f x))"
two_power_eq: fixes n :: "nat"   and m :: "nat" assumes "n < LENGTH('b)"   and "m < LENGTH('b)" shows "(2 ^ n = 2 ^ m) = (n = m)"
overlap_reflexivity: fixes x :: "i" assumes "PM" shows "O x x"
init_inv: fixes root :: "(att \<times> char list, att, nat) env" assumes "init users \<Midarrow>bs\<Rightarrow> root" shows "Q root"
fps_deriv_mult: fixes f :: "'a fps"   and g :: "'a fps" shows "fps_deriv (f * g) = f * fps_deriv g + fps_deriv f * g"
Predict_subset_\<pi>: fixes k :: "nat"   and X :: "item set"   and T :: "(symbol \<times> character list) set" shows "Predict k X \<subseteq> \<pi> k T X"
HR'_is_R': shows "(\<mu> \<rho>''. (\<mu> \<rho>'. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<mu> \<rho>''. \<rho>' ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>''))(x := Abs_cfun (ESem_syn e)\<cdot>(\<mu> \<rho>''. \<rho>' ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>''))) ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>'') = (\<mu> \<rho>'. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<mu> \<rho>''. \<rho>' ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>''))(x := Abs_cfun (ESem_syn e)\<cdot>(\<mu> \<rho>''. \<rho>' ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>'')))"
proj_psi_is_projection: shows "proj_psi * proj_psi = proj_psi"
clop_alt: fixes f :: "'b \<Rightarrow> 'b" shows "clop f = (\<forall>x y. (x \<le> f y) = (f x \<le> f y))"
assms_3: shows "\<forall>t. (M' \<union> N \<turnstile> t) = (M' \<union> N \<union> D \<turnstile>\<^sub>c t)"
le_zero_eq: fixes n :: "'c multiset" shows "(n \<subseteq># {#}) = (n = {#})"
eq_app_right_finite_index_imp_dfa: fixes L :: "'b list set" assumes "finite (UNIV // eq_app_right L)" obtains M :: "'b dfa" where "dfa M" and "dfa.language M = L" and "card (states M) = card (UNIV // eq_app_right L)"
card3: shows "card (derangements (S - {x})) = count_derangements (Suc n)"
dom_map_of_conv_image_fst: fixes xys :: "('c \<times> 'd) list" shows "dom (table_of xys) = fst ` set xys"
Var_inj: fixes xs :: "'varSort"   and x :: "'var"   and ys :: "'varSort"   and y :: "'var" shows "(Var xs x = Var ys y) = (xs = ys \<and> x = y)"
has_snapshotted: fixes t :: "('a, 'b, 'c) event llist"   and p :: "'a" assumes "cotrace init t" shows "ps (coS t (some_snapshot t p)) p \<noteq> None \<and> enat (some_snapshot t p) \<le> llength t"
comp_cod_arr: fixes f :: "'t"   and b :: "'t" assumes "S'.arr f"   and "S'.cod f = b" shows "b \<cdot>\<acute> f = f"
rep_eq: fixes x :: "'a set"   and xa :: "('a, 'b) ffun" shows "pfun_of (x \<lhd>\<^sub>f xa) = x \<lhd>\<^sub>p pfun_of xa"
covector_conv_vector: fixes v :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_bounded_idempotent_semiring.covector v = matrix_bounded_idempotent_semiring.vector (v\<^sup>t)"
connected_segment: fixes x :: "'b"   and y :: "'b" shows "connected (closed_segment x y)"
le_sup_iff: fixes sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'f"   and z :: "'f" assumes "class.semilattice_sup sup less_eq less" shows "less_eq (sup x y) z = (less_eq x z \<and> less_eq y z)"
ofilter_ordLeq: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "Well_order r"   and "ofilter r A" shows "Restr r A \<le>o r"
ConjI: fixes C :: "triple set"   and A :: "triple set" assumes "\<forall>c\<in>C. A |\<turnstile> {c}" shows "A |\<turnstile> C"
x_prop: shows "\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0"
in_Q: shows "Q\<^sub>x \<in> Q \<and> Q\<^sub>z \<in> Q"
Conj_def: fixes xset :: "('b, 'c, 'd) formula set['b]" shows "Conj xset = Abs_formula (Conj\<^sub>\<alpha> (map_bset Rep_formula xset))"
apx_def: fixes Z :: "('c \<Rightarrow> real) set" shows "Approx\<^sub>\<beta> Z \<equiv> \<Inter> {S. \<exists>U M. S = \<Union> U \<and> U \<subseteq> \<R>\<^sub>\<beta> \<and> Z \<subseteq> S \<and> (S = [M]\<^bsub>v,n\<^esub> \<and> dbm_int M n) \<and> beta_interp.normalized M}"
simps: fixes s1 :: "('a, 'b, 'c) SLin_state"   and s2 :: "('a, 'b, 'c) SLin_state" shows "P23 (s1, s2) = (\<not> initialized s2 \<longrightarrow> pendingSeqs s1 \<subseteq> pendingSeqs (f (s1, s2)))"
prime_power_dvd_multD: 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 p :: "'a"   and n :: "nat"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "comm_semiring_1.prime_elem times one zero p"   and "dvd.dvd times (power.power one times p n) (times a b)"   and "0 < n"   and "\<not> dvd.dvd times p a" shows "dvd.dvd times (power.power one times p n) b"
chain_mono: fixes Y :: "nat \<Rightarrow> 'c"   and i :: "nat"   and j :: "nat" assumes "chain Y"   and "i \<le> j" shows "Y i \<sqsubseteq> Y j"
assms_3: shows "ps (S t (i + 1)) p \<noteq> None"
assms_2: shows "wa_precise_refine WB WA \<alpha>2"
has_contour_integral_neg: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and g :: "real \<Rightarrow> complex" assumes "(f has_contour_integral i) g" shows "((\<lambda>x. - f x) has_contour_integral - i) g"
select_convs_2: fixes types :: "tf list"   and funcs :: "nat list"   and tab :: "nat option"   and mem :: "nat option"   and globs :: "nat list"   and more :: "'a" shows "inst.funcs \<lparr>types = types, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = funcs"
rPar1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>Q) P \<alpha> P'"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<alpha>"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* C"   and "distinct (bn \<alpha>)"   and "bn \<alpha> \<sharp>* Q"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>Q"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) \<alpha> (P' \<parallel> Q)"
S_def: shows "S = (SIGMA p:{p. prime p \<and> real p \<le> x}. {i. 0 < i \<and> real (p ^ i) \<le> x})"
key_hyp: shows "satisfies_properties_R p qs (get_subsets_R (reduce_system_R p (qs, base_case_info_R))) (get_signs_R (reduce_system_R p (qs, base_case_info_R))) (get_matrix_R (reduce_system_R p (qs, base_case_info_R)))"
closure_insert: fixes x :: "'c"   and s :: "'c set" shows "closure (insert x s) = insert x (closure s)"
eval_fps_exp: fixes c :: "'a"   and z :: "'a" shows "eval_fps (fps_exp c) z = exp (c * z)"
add_all_deducible_sound: fixes R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and S :: "'at Literal set set" assumes "Sound R" shows "entails_formula S (add_all_deducible_clauses R S)"
hom_mult: fixes x :: "'d"   and y :: "'d" shows "[:x * y:] = [:x:] * [:y:]"
inject: fixes x2 :: "'a"   and y2 :: "'a" shows "(Msg x2 = Msg y2) = (x2 = y2)"
prems_2: shows "s2 (l @ [a])\<Longrightarrow>\<^bsub>SES\<^esub> sn"
g: shows "g \<midarrow>f a\<rightarrow> l"
eq: shows "topspace X - Collect P = topspace X \<inter> {x. \<not> P x}"
mono: fixes x :: "'b" shows "coset.mono_body (\<lambda>f. U (F (C f)) x)"
nth_append_length: fixes xs :: "'b list"   and x :: "'b"   and ys :: "'b list" shows "(xs \<cdot> x # ys) ! \<^bold>|xs\<^bold>| = x"
cases: fixes x :: "real_alg_2" obtains n :: "nat" and rai :: "int poly \<times> rat \<times> rat" where "x = Irrational n rai"   | r :: "rat" where "x = Rational r"