hcju/isabelleps huggingface dataset:
domD: fixes a :: "'b"   and m :: "'b \<Rightarrow> 'c option" assumes "a \<in> dom m" shows "\<exists>b. m a = \<lfloor>b\<rfloor>"
simps: fixes l :: "nat"   and i :: "nat"   and c :: "'c \<Rightarrow> bool Heap"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'c Heap"   and s :: "'c" shows "imp_for_down l i c f s = (let i = i - 1 in c s \<bind> (\<lambda>ctn. if ctn then f i s \<bind> (\<lambda>s. if l < i then imp_for_down l i c f s else return s) else return s))"
it_to_list_def: fixes it :: "'g \<Rightarrow> ('h \<Rightarrow> bool) \<Rightarrow> ('i \<Rightarrow> 'i list \<Rightarrow> 'i list) \<Rightarrow> 'j list \<Rightarrow> 'f"   and s :: "'g" shows "it_to_list it s \<equiv> it s (\<lambda>_. True) (\<lambda>x l. l @ [x]) []"
integral_fst': fixes f :: "'a \<times> 'b \<Rightarrow> 'c" assumes "integrable (M1 \<Otimes>\<^sub>M M2) f" shows "LINT x|M1. LINT y|M2. f (x, y) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) f"
typings_setD: fixes \<Gamma> :: "binding list"   and fs :: "(char list \<times> trm) list"   and fTs :: "(char list \<times> POPLmarkRecord.type) list"   and l :: "char list"   and T :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> fs [:] fTs"   and "(l, T) \<in> set fTs" shows "\<exists>t. fs\<langle>l\<rangle>\<^sub>? = \<lfloor>t\<rfloor> \<and> \<Gamma> \<turnstile> t : T"
disc_1: fixes x1 :: "'b" shows "is_Prim\<^sub>0 \<^bold>\<langle>x1\<^bold>\<rangle>\<^sub>0"
supteq_subset_WPO_NS: assumes "\<And>f k. set (\<sigma> (f, k)) = {0..<k}" shows "{\<unrhd>} \<subseteq> WPO_NS"
assms_2: shows "dst \<notin> E `` {v} - dom PRED"
dgrad_set_le_refl: fixes d :: "'c \<Rightarrow> nat"   and S :: "'c set" shows "dgrad_set_le d S S"
closure_ofE2: fixes a :: "'d"   and S :: "('d, 'e) eq_object_scheme"   and A :: "'d set" assumes "a \<in> closure_of\<^bsub>S\<^esub> A" obtains a' :: "'d" where "a \<in> carrier S" and "a' \<in> A" and "a .=\<^bsub>S\<^esub> a'"
ring_simprules_25: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "ring R"   and "x \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
dord_fun_lin: fixes ord :: "('a \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'c) \<Rightarrow> bool"   and s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c" assumes "ord s t \<or> ord t s" shows "dord_fun ord s t \<or> dord_fun ord t s"
fdisjnt_insert: fixes x :: "'a"   and N :: "'a fset"   and M :: "'a fset" assumes "x |\<notin>| N"   and "fdisjnt M N" shows "fdisjnt (finsert x M) N"
substn_subst_0: fixes t :: "Core.term"   and s :: "Core.term" shows "subst_bv1 t 0 s = subst_bv2 t 0 s"
ex_un_null: shows "\<exists>!n. \<forall>f. S.VV.comp n f = n \<and> S.VV.comp f n = n"
intros_1: fixes x :: "'a"   and A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "x \<in> A" shows "x \<in> directed_towards A r"
assms_3: shows "BIT_inv s x [x0, y0]"
defval_conf: fixes P :: "'b prog"   and T :: "ty" shows "P \<turnstile> default_val T :\<le> T"
rank_le_ncols: fixes A :: "(('b, 'c) vec, 'd) vec" shows "rank A \<le> ncols A"
af_subsequence_W: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set"   and n :: "nat" shows "af (\<phi> W\<^sub>n \<psi>) (prefix (Suc n) w) = af \<psi> (prefix (Suc n) w) or\<^sub>n (af \<phi> (prefix (Suc n) w) and\<^sub>n af (\<phi> W\<^sub>n \<psi>) (w [1 \<rightarrow> Suc n]))"
simps_3: fixes c :: "poly"   and s :: "tm" shows "islin (Eq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
length_f: fixes xs :: "bool list"   and ys :: "bool list" assumes "length xs = len"   and "length ys = len" shows "length (f xs ys) = len"
mult_if_delta: fixes P :: "bool"   and q :: "'a" shows "(if P then 1::'a else (0::'a)) * q = (if P then q else (0::'a))"
continuous_image_subset_interior: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on S f"   and "inj_on f S"   and "DIM('b) \<le> DIM('a)" shows "f ` interior S \<subseteq> interior (f ` S)"
compose: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h' :: "'d \<Rightarrow> 'b"   and G :: "'d filter" assumes "f \<in> o[F](g)"   and "filterlim h' F G" shows "(\<lambda>x. f (h' x)) \<in> o[G](\<lambda>x. g (h' x))"
sorted_list_of_set_Un: fixes A :: "'a set"   and B :: "'a set" assumes "A \<lless> B"   and "finite A"   and "finite B" shows "sorted_list_of_set (A \<union> B) = sorted_list_of_set A @ sorted_list_of_set B"
node_comp_def: fixes i :: "nat"   and np :: "('c, 'd seq_action) automaton"   and R\<^sub>i :: "nat set" shows "\<langle>i : np : R\<^sub>i\<rangle> \<equiv> \<lparr>init = {NodeS i s R\<^sub>i |s. s \<in> init np}, trans = node_sos (automaton.trans np)\<rparr>"
lambda_system_strong_sum: fixes f :: "'a set \<Rightarrow> ennreal"   and a :: "'a set"   and A :: "nat \<Rightarrow> 'a set"   and n :: "nat" assumes "positive M f"   and "a \<in> M"   and "range A \<subseteq> lambda_system \<Omega> M f"   and "disjoint_family A" shows "(\<Sum>i = 0..<n. f (a \<inter> A i)) = f (a \<inter> \<Union> (A ` {0..<n}))"
rep_nat_pp_inverse: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat_pp :: "'a \<Rightarrow> (nat, nat) pp"   and abs_nat_pp :: "(nat, nat) pp \<Rightarrow> 'a"   and lex_comp' :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg' :: "'a \<Rightarrow> nat"   and x :: "'a" assumes "class.nat_pp_compare plus zero less_eq less rep_nat_pp abs_nat_pp lex_comp' deg'" shows "abs_nat_pp (rep_nat_pp x) = x"
B1: shows "l \<times> x = l' \<otimes> x"
weight_less_iff: fixes s :: "('f, 'c) Term.term"   and t :: "('f, 'c) Term.term" assumes "weight_less s t" shows "weight_le s t \<and> weight s < weight t"
subspace_scale: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'d set"   and x :: "'d"   and c :: "'c" assumes "module scale"   and "module.subspace scale S"   and "x \<in> S" shows "scale c x \<in> S"
induct: fixes P :: "(int \<times> int) \<times> int \<times> int \<Rightarrow> (int \<times> int) list \<Rightarrow> bool"   and a0 :: "(int \<times> int) \<times> int \<times> int"   and a1 :: "(int \<times> int) list" assumes "\<And>c\<^sub>0 c\<^sub>1. P (c\<^sub>0, c\<^sub>1) []"   and "\<And>c\<^sub>0 c\<^sub>1 uu_. P (c\<^sub>0, c\<^sub>1) [uu_]"   and "\<And>c\<^sub>0 c\<^sub>1 p\<^sub>0 v va. \<lbrakk>\<And>vb vc. \<lbrakk>vb = Time_Monad.val (take_tm 7 (v # va)); vc = Time_Monad.val (find_closest_bf_tm p\<^sub>0 vb); dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case vc of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))\<rbrakk> \<Longrightarrow> P (c\<^sub>0, c\<^sub>1) (v # va); \<And>vb vc. \<lbrakk>vb = Time_Monad.val (take_tm 7 (v # va)); vc = Time_Monad.val (find_closest_bf_tm p\<^sub>0 vb); \<not> dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case vc of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))\<rbrakk> \<Longrightarrow> P (p\<^sub>0, vc) (v # va)\<rbrakk> \<Longrightarrow> P (c\<^sub>0, c\<^sub>1) (p\<^sub>0 # v # va)" shows "P a0 a1"
set_fsga: shows "set (find_consistent_signs_at_roots (coprime_r (cast_rat_list fs)) (cast_rat_list fs)) = set (characterize_consistent_signs_at_roots (coprime_r (cast_rat_list fs)) (cast_rat_list fs))"
optimize_matches_option_generic: fixes rs :: "'c rule list"   and P :: "'c match_expr \<Rightarrow> bool"   and f :: "'c match_expr \<Rightarrow> 'c match_expr option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and s :: "state"   and t :: "state" assumes "\<forall>r\<in>set rs. P (get_match r)"   and "\<And>m m'. \<lbrakk>P m; f m = Some m'\<rbrakk> \<Longrightarrow> matches \<gamma> m' p = matches \<gamma> m p"   and "\<And>m. \<lbrakk>P m; f m = None\<rbrakk> \<Longrightarrow> \<not> matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>optimize_matches_option f rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
kth_res_equals_subseq: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "\<exists>n. is_subseq_of s (filtered_seq s (kth_res_equals k n)) \<and> (\<forall>m. filtered_seq s (kth_res_equals k n) m k = n)"
ideal_generated_def: fixes S :: "'a set" shows "ideal_generated S = \<Inter> {I. ideal I \<and> S \<subseteq> I}"
scaleR_nonneg_nonneg: fixes a :: "real"   and x :: "'e" assumes "0 \<le> a"   and "(0::'e) \<le> x" shows "(0::'e) \<le> a *\<^sub>R x"
is_nth_powerE: fixes n :: "nat"   and x :: "'b" assumes "is_nth_power n x" obtains y :: "'b" where "x = y ^ n"
comp_null_2: fixes f :: "'a" shows "f \<star> H.null = H.null"
prj_chine_assoc_1: shows "cods.Prj\<^sub>1 \<cdot> cods.chine_assoc = cods.Prj\<^sub>1\<^sub>1"
Lim_null: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and net :: "'c filter" shows "(f \<longlongrightarrow> l) net = ((\<lambda>x. f x - l) \<longlongrightarrow> (0::'d)) net"
add_mono_thms_linordered_field_5: fixes i :: "'b"   and j :: "'b"   and k :: "'b"   and l :: "'b" assumes "i < j \<and> k < l" shows "i + k < j + l"
simps_1: fixes vs :: "'a set"   and s :: "('a, 'b) fmap" shows "vars_change [] vs s = []"
add_eq_0_iff_both_eq_0: fixes x :: "'a multiset"   and y :: "'a multiset" shows "(x + y = \<Empt>) = (x = \<Empt> \<and> y = \<Empt>)"
well_valued0_prepend: fixes xs :: "'b list"   and t :: "'b tree"   and ys :: "'b list"   and as :: "'b list" assumes "length xs \<le> l t - 1"   and "length ys = length xs"   and "well_shaped t"   and "well_valued0 (xs @ as) t" shows "well_valued0 (ys @ as) t"
intro: fixes F :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "overrider F"   and "idem_overrider_axioms F" shows "idem_overrider F"
no_self_loop: shows "\<forall>u. (u, u) \<notin> E"
field_differentiable_Gamma: fixes z :: "'a"   and A :: "'a set" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "Gamma field_differentiable at z within A"
divide_nonneg_neg: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less y zero" shows "less_eq (divide x y) zero"
econc_mono3: fixes m :: "('c + 'd) nres"   and m' :: "('c + 'd) nres"   and E :: "('e \<times> 'c) set"   and R :: "('f \<times> 'd) set" assumes "m \<le> m'" shows "\<Down>\<^sub>E E R m \<le> \<Down>\<^sub>E E R m'"
strictly_maximal_literal_def: fixes C :: "'a eclause"   and L :: "'a literal"   and \<sigma> :: "('a \<times> 'a trm) list" shows "strictly_maximal_literal C L \<sigma> = (\<forall>x\<in>cl_ecl C - {L}. (subst_lit x \<sigma>, subst_lit L \<sigma>) \<in> lit_ord)"
id_rad_invim: fixes I :: "'a set" assumes "ideal R I" shows "rad_ideal R I = rInvim R (R /\<^sub>r I) (pj R I) (nilrad (R /\<^sub>r I))"
par_right_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Par C D" shows "A B Par D C"
continuous_zeta: fixes s :: "complex"   and A :: "complex set" assumes "s \<noteq> 1" shows "continuous (at s within A) zeta"
CD_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "Bossy.CD_on ds X = (\<Union>d\<in>ds. Bossy.Cd d X)"
count_poly_roots': fixes p :: "'b poly"   and x :: "'b" shows "count (poly_roots p) x = (if p = 0 then 0 else order x p)"
succ_vars_id: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state" assumes "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s" shows "tvars (\<T> s) = tvars (\<T> s')"
cinner_zero_left: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner zero x = 0"
allvar_cases: shows "Denotational_Semantics.allvars = allrvars \<union> alldvars"
funpow_add': fixes m :: "nat"   and n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "(f ^^ (m + n)) x = (f ^^ m) ((f ^^ n) x)"
MaxR_opt_eq_fold': fixes A :: "'a set" shows "MaxR_opt A = Finite_Set.fold MaxR_f None A"
merge_blindable: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option" assumes "merge_on UNIV h bo m" shows "merge_on UNIV (hash_blindable h) (blinding_of_blindable h bo) (merge_blindable h m)"
antisym_r: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq>\<^bsub>r\<^esub> y"   and "y \<sqsubseteq>\<^bsub>r\<^esub> x"   and "x \<in> A"   and "y \<in> A" shows "x = y"
while_C_def: fixes c :: "'a control_state_scheme \<Rightarrow> bool"   and B :: "'a control_state_scheme \<Rightarrow> (unit \<times> 'a control_state_scheme) option" shows "_while_C c B \<equiv> \<lambda>\<sigma>. if exec_stop \<sigma> then Some ((), \<sigma>) else (_while_SE (\<lambda>\<sigma>. \<not> exec_stop \<sigma> \<and> c \<sigma>) B ;- unset_break_status) \<sigma>"
induct: fixes x1 :: "('fun, 'var) Term.term set"   and x2 :: "nat"   and x3 :: "('fun, 'var) Term.term"   and P :: "('fun, 'var) Term.term set \<Rightarrow> nat \<Rightarrow> ('fun, 'var) Term.term \<Rightarrow> bool" assumes "\<langle>x1; x2\<rangle> \<turnstile>\<^sub>n x3"   and "\<And>t M. t \<in> M \<Longrightarrow> P M 0 t"   and "\<And>T f M steps. \<lbrakk>length T = arity f; public f; \<And>t. t \<in> set T \<Longrightarrow> \<langle>M; steps t\<rangle> \<turnstile>\<^sub>n t; \<And>t. t \<in> set T \<Longrightarrow> P M (steps t) t\<rbrakk> \<Longrightarrow> P M (Suc (Max (insert 0 (steps ` set T)))) (Fun f T)"   and "\<And>M n t K T steps t\<^sub>i. \<lbrakk>\<langle>M; n\<rangle> \<turnstile>\<^sub>n t; P M n t; Ana t = (K, T); \<And>k. k \<in> set K \<Longrightarrow> \<langle>M; steps k\<rangle> \<turnstile>\<^sub>n k; \<And>k. k \<in> set K \<Longrightarrow> P M (steps k) k; t\<^sub>i \<in> set T\<rbrakk> \<Longrightarrow> P M (Suc (Max (insert n (steps ` set K)))) t\<^sub>i" shows "P x1 x2 x3"
mem_minus_sum_multTr4: fixes A :: "'a set"   and B :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "A \<subseteq> carrier R"   and "ideal R B" shows "(\<forall>j\<le>n. f j \<in> set_mult R A B) \<longrightarrow> -\<^sub>a \<Sigma>\<^sub>e R f n \<in> sum_mult R A B"
interchange_columns_transpose: fixes A :: "(('d, 'f) vec, 'e) vec"   and a :: "'e"   and b :: "'e" shows "interchange_columns (Finite_Cartesian_Product.transpose A) a b = Finite_Cartesian_Product.transpose (interchange_rows A a b)"
pax4: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "|~ \<box>[P]_v = (_actrans P v \<and> \<circle>\<box>[P]_v)"
4: shows "card {x. regular x \<and> x \<le> - - h \<and> x \<le> - choose_arc h \<and> x \<le> - choose_arc h\<^sup>T} < card {x. regular x \<and> x \<le> - - h}"
ma_0: shows "0 = real_of (ma_of_rat 0)"
prime_field_finite_field_ops_int: shows "prime_field_gen (finite_field_ops_int p) mod_ring_rel p"
diamond: fixes M :: "lam"   and M1 :: "lam"   and M2 :: "lam" assumes " M \<longrightarrow>\<^sub>1 M1"   and " M \<longrightarrow>\<^sub>1 M2" shows "\<exists>M3.  M1 \<longrightarrow>\<^sub>1 M3 \<and>  M2 \<longrightarrow>\<^sub>1 M3"
spectrum_def: fixes A :: "'a mat" shows "spectrum A = Collect (eigenvalue A)"
c1_notStuck: shows "\<Gamma>|\<^bsub>M\<^esub>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"
finprod_mono_neutral_cong_right: fixes B :: "'c set"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a" assumes "finite B"   and "A \<subseteq> B"   and "\<And>i. i \<in> B - A \<Longrightarrow> g i = \<zero>"   and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"   and "g \<in> B \<rightarrow> carrier R" shows "finsum R g B = finsum R h A"
induct: fixes x1 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and x2 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and P :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> bool" assumes "x1 \<Rightarrow> x2"   and "\<And>\<Gamma> e x S. P (\<Gamma>, App e x, S) (\<Gamma>, e, Arg x # S)"   and "\<And>\<Gamma> y e x S. P (\<Gamma>, Lam [y]. e, Arg x # S) (\<Gamma>, e[y::=x], S)"   and "\<And>\<Gamma> x e S. map_of \<Gamma> x = Some e \<Longrightarrow> P (\<Gamma>, Var x, S) (delete x \<Gamma>, e, Upd x # S)"   and "\<And>x \<Gamma> e S. \<lbrakk>x \<notin> domA \<Gamma>; isVal e\<rbrakk> \<Longrightarrow> P (\<Gamma>, e, Upd x # S) ((x, e) # \<Gamma>, e, S)"   and "\<And>\<Delta> \<Gamma> S e. \<lbrakk>atom ` domA \<Delta> \<sharp>* \<Gamma>; atom ` domA \<Delta> \<sharp>* S\<rbrakk> \<Longrightarrow> P (\<Gamma>, Let \<Delta> e, S) (\<Delta> @ \<Gamma>, e, S)"   and "\<And>\<Gamma> scrut e1 e2 S. P (\<Gamma>, scrut ? e1 : e2, S) (\<Gamma>, scrut, Alts e1 e2 # S)"   and "\<And>\<Gamma> b e1 e2 S. P (\<Gamma>, Bool b, Alts e1 e2 # S) (\<Gamma>, if b then e1 else e2, S)" shows "P x1 x2"
X_GF: shows "\<forall>\<psi>\<in>X. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>[Y]\<^sub>\<mu>)"
iT_Div_cut_le: fixes k :: "nat"   and I :: "nat set"   and t :: "nat" assumes "0 < k" shows "(I \<oslash> k) \<down>\<le> t = I \<down>\<le> (t * k + (k - Suc 0)) \<oslash> k"
qmpt_factorI: fixes proj :: "'a \<Rightarrow> 'c"   and M2 :: "'c measure"   and T2 :: "'c \<Rightarrow> 'c" assumes "proj \<in> quasi_measure_preserving M M2"   and "AE x in M. proj (T x) = T2 (proj x)"   and "qmpt M2 T2" shows "qmpt_factor proj M2 T2"
seqsim_prev_nextnat: fixes s :: "nat \<Rightarrow> 'b" shows "s \<approx> s |\<^sub>s nextnat s - 1"
path_conforms_LNil: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" shows "path_conforms_with_strategy p LNil \<sigma>"
dom_tensor: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "arr f"   and "arr g" shows "local.dom (f \<otimes> g) = local.dom f \<otimes> local.dom g"
apply_matrix_add: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a matrix"   and b :: "'a matrix" assumes "\<forall>x y. f (x + y) = f x + f y"   and "f (0::'a) = (0::'a)" shows "apply_matrix f (a + b) = apply_matrix f a + apply_matrix f b"
vs: fixes j :: "nat" assumes "j < dim_row B'" shows "R (vs j) (vs' j)"
admits_diagonal_reduction_imp_exists_algorithm_is_SNF_all: fixes m :: "nat"   and n :: "nat" assumes "\<forall>A\<in>carrier_mat m n. admits_diagonal_reduction A" shows "\<exists>algorithm. \<forall>A\<in>carrier_mat m n. is_SNF A (algorithm A)"
four_composable_arrows_of_spans_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data"   and \<pi> :: "'b arrow_of_spans_data"   and \<rho> :: "'b arrow_of_spans_data" shows "four_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<pi> \<rho> \<equiv> ((span_bicategory C prj0 prj1 \<and> arrow_of_spans C \<mu>) \<and> arrow_of_spans C \<nu> \<and> arrow_of_spans C \<pi> \<and> arrow_of_spans C \<rho>) \<and> (two_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<and> two_composable_arrows_of_spans C prj0 prj1 \<nu> \<pi>) \<and> two_composable_arrows_of_spans C prj0 prj1 \<pi> \<rho> \<and> three_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<pi> \<and> three_composable_arrows_of_spans C prj0 prj1 \<nu> \<pi> \<rho>"
STA_Suc: fixes A :: "'c set"   and n :: "nat"   and branch :: "(('d, 'c) fm list \<times> 'c) list" assumes "A, n \<turnstile> branch" shows "A, Suc n \<turnstile> branch"
ub1: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "Semilat A r f"   and "x \<in> A"   and "y \<in> A" shows "x \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y"
aezfun_setspanD: fixes x :: "('a, 'b) aezfun"   and A :: "'b set" assumes "x \<in> aezfun_setspan A" shows "supp (aezfun x) \<subseteq> A"
antisym: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "less_eq x y"   and "less_eq y x" shows "x = y"
residue_primroot_bij_betw_primroots: fixes m :: "nat"   and g :: "nat" assumes "1 < m"   and "residue_primroot m g" shows "bij_betw (\<lambda>i. g ^ i mod m) (totatives (totient m)) {g \<in> totatives m. residue_primroot m g}"
chain_map_simplicial_cone: fixes r :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and g :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"   and p :: "nat"   and q :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int"   and v :: "nat \<Rightarrow> real" assumes "simplicial_simplex r S g"   and "simplicial_chain p (standard_simplex q) c"   and "v \<in> standard_simplex q"   and "q \<le> r" shows "chain_map (Suc p) g (simplicial_cone p v c) = simplicial_cone p (g v) (chain_map p g c)"
right_cancel: fixes b :: "'d"   and a :: "'d"   and c :: "'d" shows "(b + a = c + a) = (b = c)"
call: shows "calls1 es'' = \<lfloor>(a, M, vs)\<rfloor>"
PAR: fixes xs :: "'param"   and p :: "'varSort" shows "|varsOf xs p| <o |UNIV|"
steps_Clos_equiv: fixes x :: "nat"   and y :: "nat"   and rel :: "(nat \<times> nat) set"   and n :: "nat" shows "((x, y) \<in> steps rel n) = ((x, y) \<in> Clos rel n)"
less_imp_le: fixes x :: "'a multiset"   and y :: "'a multiset" assumes "x \<subset># y" shows "x \<subseteq># y"
min_less_iff_disj: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(min x y < z) = (x < z \<or> y < z)"
range_vars_alt_def: fixes s :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "range_vars s \<equiv> fv\<^sub>s\<^sub>e\<^sub>t (subst_range s)"
invertibleI: fixes A :: "(('c, 'e) vec, 'd) vec"   and B :: "(('c, 'd) vec, 'e) vec" assumes "A ** B = mat (1::'c)"   and "B ** A = mat (1::'c)" shows "invertible A"
simps_17: fixes s :: "nat \<Rightarrow> floatarith"   and f :: "float" shows "subst_floatarith s (Num f) = Num f"
tendsto_iff: fixes dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'e"   and l :: "'e"   and F :: "'f filter" assumes "class.metric_space dist uniformity open" shows "filterlim f (topological_space.nhds open l) F = (\<forall>e>0. \<forall>\<^sub>F x in F. dist (f x) l < e)"
REP_in_Map: fixes A :: "('a, 'a set) Maps.arr" assumes "Maps.arr A" shows "Maps.REP A \<in> Maps.Map A"
mult_single: fixes k :: "'a"   and a :: "'b"   and l :: "'a"   and b :: "'b" shows "monomial a k * monomial b l = monomial (a * b) (k + l)"