hcju/isabelleps huggingface dataset:
ucast_0_I: fixes x :: "'b word" assumes "x = 0" shows "ucast x = 0"
measure_bind: fixes f :: "'b \<Rightarrow> 'c measure"   and M :: "'b pmf"   and N :: "'c measure"   and X :: "'c set" assumes "measure_pmf.random_variable M (subprob_algebra N) f"   and "X \<in> sets N" shows "Sigma_Algebra.measure (measure_pmf M \<bind> f) X = measure_pmf.expectation M (\<lambda>x. Sigma_Algebra.measure (f x) X)"
finite_partJoin: fixes P :: "'a set set"   and Q :: "'a set set" assumes "finite P"   and "finite Q" shows "finite (partJoin P Q)"
dist_midpoint_3: fixes a :: "'a"   and b :: "'a" shows "dist (midpoint a b) a = dist a b / 2"
hyps: fixes s1 :: "nat"   and r1 :: "nat \<Rightarrow>f value option"   and s2 :: "nat"   and r2 :: "nat \<Rightarrow>f value option" assumes "executionally_equivalent e1 s1 r1 e2 s2 r2 (map (\<lambda>a. case a of (l, i, o) \<Rightarrow> (l, i)) t)" shows "accepts_trace e2 s2 r2 t = accepts_trace e1 s1 r1 t"
exec1_drop_suffix_aux: fixes cs12 :: "com list"   and s :: "state"   and cs1'2 :: "com list"   and s' :: "state" assumes "(cs12, s) \<rightarrow> (cs1'2, s')" shows "\<forall>cs1 cs2 cs1'. cs12 = cs1 @ cs2 \<and> cs1'2 = cs1' @ cs2 \<and> cs1 \<noteq> [] \<longrightarrow> (cs1, s) \<rightarrow> (cs1', s')"
healthy_bounded_byD2: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "real"   and P :: "'c \<Rightarrow> real" assumes "healthy t"   and "bounded_by b P"   and "sound P" shows "bounded_by b (t P)"
simps_2: fixes x :: "mreal" shows "mreal_max x MInfty = x"
ptensor_mat_carrier: fixes m1 :: "'b mat"   and m2 :: "'b mat" shows "tensor_P m1 m2 \<in> carrier_mat ps2_P.d0 ps2_P.d0"
IH: fixes i :: "nat" assumes "i < hd (dims A)" shows "cprank_max (prod_list (dims (subtensor A i))) (subtensor A i)"
compute_rflow_def: fixes cf :: "nat \<times> nat \<Rightarrow> int" shows "compute_rflow cf \<equiv> assert (RGraph c s t cf) \<bind> (\<lambda>_. return (flow_of_cf cf))"
finer_than_iff_nhds: fixes T1 :: "'c topology"   and T2 :: "'c topology" shows "(T1 finer_than T2) = ((\<forall>X. openin T2 X \<longrightarrow> openin T1 (X \<inter> topspace T1)) \<and> topspace T1 \<subseteq> topspace T2)"
fres_anti: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and z :: "'c \<times> 'd" assumes "less_eq_prod x y" shows "less_eq_prod (prod.fres z y) (prod.fres z x)"
entropy_chain_rule: fixes X :: "'key \<Rightarrow> 'a"   and Y :: "'key \<Rightarrow> 'b" assumes "simple_function (point_measure keys (\<lambda>x. ennreal (K x))) X"   and "simple_function (point_measure keys (\<lambda>x. ennreal (K x))) Y" shows "K.entropy_Pow (\<lambda>x. (X x, Y x)) = K.entropy_Pow X + K.conditional_entropy_Pow Y X"
comp_arr_ide: fixes a :: "'c"   and f :: "'c" assumes "C.ide a"   and "C.seq f a" shows "f \<cdot> a = f"
intro: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" assumes "bidel_arc G M a"   and "bidel_arc_head_isolated_axioms G M a" shows "bidel_arc_head_isolated G M a"
iT_Mult_inext: fixes n :: "nat"   and k :: "nat"   and I :: "nat set" shows "inext (n * k) (I \<otimes> k) = inext n I * k"
emeasure_eq_measure: fixes A :: "'a set" shows "emeasure M A = ennreal (prob A)"
list_ex_iff: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "list_ex P xs = Bex (set xs) P"
obtain_extensional_function_from_number_partition: fixes A :: "'c set"   and B :: "'d set"   and N :: "nat multiset" assumes "finite A"   and "finite B"   and "number_partition (card A) N"   and "size N \<le> card B" shows "\<exists>f\<in>A \<rightarrow>\<^sub>E B. image_mset card (mset_set ((\<lambda>b. {x \<in> A. f x = b}) ` B - {{}})) = N"
dbm_entry_int: fixes x :: "real DBMEntry" assumes "x \<noteq> \<infinity>"   and "get_const x \<in> \<int>" shows "\<exists>d. x = Le (real_of_int d) \<or> x = Lt (real_of_int d)"
zero_le_square: fixes 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 abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "less_eq zero (times a a)"
inheritedEmpty: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool" assumes "inherited subs P" shows "P {}"
eq_id_iff: fixes f :: "'e \<Rightarrow> 'e" shows "(\<forall>x. f x = x) = (f = id)"
contour_integral_linepath_same_Im: fixes a :: "complex"   and b :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "Im a = Im b"   and "Re a < Re b" shows "contour_integral (linepath a b) f = integral {Re a..Re b} (\<lambda>x. f (complex_of_real x + complex_of_real (Im a) * \<i>))"
contract_recovery: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "K \<subseteq> (K \<div> \<phi>) \<oplus> \<phi>"
Subset_refl: fixes H :: "fm set"   and t :: "tm" shows "H \<turnstile> t SUBS t"
not_inj: fixes x :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(not x \<tau> = not y \<tau>) = (x \<tau> = y \<tau>)"
mset_pos_supset: fixes A :: "'a zmultiset" shows "A \<subseteq>#\<^sub>z zmset_of (mset_pos A)"
finite_int_iff_bounded: fixes S :: "int set" shows "finite S = (\<exists>k. abs ` S \<subseteq> {..<k})"
intro: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme" assumes "h \<in> ring_hom R S" shows "ring_hom_cring_axioms R S h"
bound_compare_defs_9: fixes c :: "'a"   and b :: "'a option" shows "c \<ge>\<^sub>u\<^sub>b b = \<unrhd>\<^sub>u\<^sub>b (<) c b"
open_Union: fixes "open" :: "'c set \<Rightarrow> bool"   and K :: "'c set set" assumes "class.topological_space open"   and "\<forall>S\<in>K. open S" shows "open (\<Union> K)"
intro: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Exit :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j" assumes "Postdomination sourcenode targetnode kind valid_edge Entry Exit"   and "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "StandardControlDependencePDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit"
lex_ord_pair_EqI: fixes xs :: "('a \<times> 'd) list"   and ys :: "('a \<times> 'e) list"   and f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> order option" assumes "oalist_inv_raw xs"   and "oalist_inv_raw ys"   and "\<And>k. k \<in> fst ` set xs \<union> fst ` set ys \<Longrightarrow> f k (lookup_pair xs k) (lookup_pair ys k) = Some Eq" shows "lex_ord_pair f xs ys = Some Eq"
satisfies_properties_R_def: fixes p :: "real poly"   and qs :: "real poly list"   and subsets :: "(nat list \<times> nat list) list"   and signs :: "rat list list"   and matrix :: "rat mat" shows "satisfies_properties_R p qs subsets signs matrix = (all_list_constr_R subsets (length qs) \<and> well_def_signs (length qs) signs \<and> distinct signs \<and> satisfy_equation_R p qs subsets signs \<and> invertible_mat matrix \<and> matrix = M_mat_R signs subsets \<and> set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs)"
semiring_simprules_4: fixes R :: "('a, 'b) ring_scheme" assumes "semiring R" shows "\<one>\<^bsub>R\<^esub> \<in> carrier R"
image_mset_add_mset: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and M :: "'c multiset" shows "image_mset f (add_mset a M) = add_mset (f a) (image_mset f M)"
strict_trans2: fixes b :: "'b word"   and a :: "'b word"   and c :: "'b word" assumes "b <s a"   and "c \<le>s b" shows "c <s a"
minus_subgroup: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and a :: "'b" assumes "class.group_add minus plus zero uminus"   and "A \<in> group_add.subgroup plus zero uminus"   and "uminus a \<in> A" shows "a \<in> A"
borel_measurable_times_ennreal: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and g :: "'b \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
enat_diff_diff_eq: fixes k :: "enat"   and m :: "enat"   and n :: "enat" shows "k - m - n = k - (m + n)"
strategyproofness_1: shows "(pmf (sds R8) d < pmf (sds R26) a \<or> pmf (sds R8) d + pmf (sds R8) b < pmf (sds R26) a + pmf (sds R26) c) \<or> pmf (sds R8) d = pmf (sds R26) a \<and> pmf (sds R8) d + pmf (sds R8) b = pmf (sds R26) a + pmf (sds R26) c"
effect_raiseE: fixes x :: "String.literal"   and h :: "heap"   and h' :: "heap"   and r :: "'d" assumes "effect (raise x) h h' r" obtains "False"
real_non_neg_mat_def: fixes A :: "((complex, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "real_non_neg_mat A \<equiv> \<forall>a\<in>elements_mat_h A. a \<in> \<real> \<and> 0 \<le> Re a"
fun_value_in_range: fixes f :: "ZF"   and x :: "ZF" assumes "isFun f"   and "Elem x (HOLZF.Domain f)" shows "Elem (f \<acute> x) (HOLZF.Range f)"
permute_atom_def: fixes p :: "perm"   and a :: "atom" shows "p \<bullet> a = Rep_perm p a"
run_step_return_imp_lfilled: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and es :: "e list"   and s' :: "s"   and vs' :: "v list"   and res :: "v list" assumes "run_step d i (s, vs, es) = (s', vs', RSReturn res)" shows "s = s' \<and> vs = vs' \<and> (\<exists>n lfilled es_c. Lfilled_exact n lfilled (vs_to_es res @ [$Return] @ es_c) es)"
callee_invariant_extend_state_oracle_const: fixes "oracle" :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf"   and I :: "'a \<Rightarrow> bool" shows "callee_invariant \<dagger>oracle (\<lambda>(s', s). I s')"
black_dequeue_Mark: fixes b :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state"   and p :: "'f process_name"   and ws :: "('g, 'h, 'e) mem_store_action list"   and r :: "'e"   and fl :: "bool" shows "black b (s(sys := s sys\<lparr>heap := (sys_heap s)(r := map_option (obj_mark_update (\<lambda>_. fl)) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)\<rparr>)) = (black b s \<and> b \<noteq> r \<or> b = r \<and> fl = sys_fM s \<and> obj_at (\<lambda>s. True) r s \<and> \<not> grey r s)"
component_refl: fixes F :: "'c program" shows "F \<le> F"
IH: assumes "\<forall>a\<in>set fs. is_lit_plus a" shows "is_disj \<^bold>\<Or>fs"
cm_all'_def: fixes c0 :: "('a, 't, 'b) Propagate.configuration_scheme"   and \<Delta> :: "('a \<times> 't) zmultiset" shows "cm_all' c0 \<Delta> = Finite_Set.fold (\<lambda>(loc, t) c. apply_cm c loc t (zcount \<Delta> (loc, t))) c0 (set_zmset \<Delta>)"
simps_5: fixes b :: "bexp"   and C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" shows "preT (IF b THEN C\<^sub>1 ELSE C\<^sub>2) e = (\<lambda>s. if bval b s then preT C\<^sub>1 e s else preT C\<^sub>2 e s)"
assms_2: shows "\<forall>n w. prices Mkt asset n w \<noteq> 0"
dnso1: fixes x :: "'a" shows "x + d x \<cdot> x = d x \<cdot> x"
hk: shows "homeomorphism (ball (0::'a) 1) UNIV h k"
nSym: fixes P :: "pi"   and Q :: "pi" assumes "\<not> P \<equiv>\<^sub>e Q" shows "\<not> Q \<equiv>\<^sub>e P"
cases: fixes a1 :: "'d set"   and a2 :: "'d set"   and a3 :: "'d set"   and a4 :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set" assumes "((a1, a2), a3, a4) \<in> br_step \<delta>" obtains q :: "'d" and W :: "'d set" and Q' :: "'d set" and Q :: "'d set" and W' :: "'d set" where "a1 = Q" and "a2 = W" and "a3 = Q'" and "a4 = W'" and "q \<in> W" and "Q' = Q \<union> br_dsq \<delta> q (Q, W)" and "W' = W - {q} \<union> (br_dsq \<delta> q (Q, W) - Q)"
le_opt_refl: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b option" assumes "order r" shows "x \<sqsubseteq>\<^bsub>Opt.le r\<^esub> x"
linear_surjective_imp_injective: fixes f :: "'b \<Rightarrow> 'c" assumes "linear (*a) (*b) f"   and "surj f"   and "vs2.dim UNIV = vs1.dim UNIV" shows "inj f"
IH: assumes "recfn a f" shows "f \<simeq> wrap_Id f n"
that: fixes ms'' :: "'node list"   and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow>\<^sub>\<tau> (ms'',s'')"   and "S,f \<turnstile> (ms'',s'') =as'\<Rightarrow>\<^sub>\<tau> (ms',s')" shows "thesis"
kth_obs_cs_missing_cs: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and k :: "nat"   and i :: "nat"   and i' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "is_kth_obs \<pi> k i"   and "is_kth_obs \<pi>' k i'"   and "cs\<^bsup>\<pi>\<^esup> i \<noteq> cs\<^bsup>\<pi>'\<^esup> i'" obtains l :: "nat" and j :: "nat" where "j \<le> i" and "is_kth_obs \<pi> l j" and "\<nexists>j'. cs\<^bsup>\<pi>\<^esup> j = cs\<^bsup>\<pi>'\<^esup> j'"   | l' :: "nat" and j' :: "nat" where "j' \<le> i'" and "is_kth_obs \<pi>' l' j'" and "\<nexists>j. cs\<^bsup>\<pi>\<^esup> j = cs\<^bsup>\<pi>'\<^esup> j'"
Inf_greatest: fixes A :: "('c \<times> 'd) set"   and z :: "'c \<times> 'd" assumes "\<And>x. x \<in> A \<Longrightarrow> less_eq_prod z x" shows "less_eq_prod z (Inf_prod A)"
H: shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"
assms_4: shows "\<not> (\<bar>l\<bar> = \<infinity> \<and> \<bar>m\<bar> = \<infinity>)"
step: shows "P @ P' \<turnstile> (i, s, stk) \<rightarrow> (i', s'', stk'')"
in_segment_1: fixes x :: "'b"   and a :: "'b"   and b :: "'b" shows "(x \<in> closed_segment a b) = (\<exists>u\<ge>0. u \<le> 1 \<and> x = (1 - u) *\<^sub>R a + u *\<^sub>R b)"
reduction_def: fixes q :: "nat"   and \<A> :: "'c_a \<Rightarrow> (unit, 'call + 'guess, 'ret + unit) gpv"   and c_a :: "'c_a" shows "reduction q \<A> c_a = lift_spmf (spmf_of_set {..<q}) \<bind> (\<lambda>j_star. inline_stop reduction_oracle (\<A> c_a) (Inr j_star) \<bind> (\<lambda>(uu_, s). Generative_Probabilistic_Value.Done (projl s)))"
mono_gamma_c: fixes C1 :: "(char list \<Rightarrow> 'av) option acom"   and C2 :: "(char list \<Rightarrow> 'av) option acom" assumes "C1 \<le> C2" shows "\<gamma>\<^sub>c C1 \<le> \<gamma>\<^sub>c C2"
append_step1I: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list"   and xs :: "'a list"   and vs :: "'a list"   and us :: "'a list" assumes "step1 r ys xs \<and> vs = us \<or> ys = xs \<and> step1 r vs us" shows "step1 r (ys @ vs) (xs @ us)"
elements_mat_h_def: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "elements_mat_h A = range (\<lambda>(i, j). A $h i $h j)"
accept_def: fixes w :: "nat \<Rightarrow> 'L" shows "accept w \<equiv> \<exists>r. is_run r \<and> w = L \<circ> r"
step_Normal_elim_cases_9: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e"   and u :: "('e, 'd, 'f) com \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (DynCom c, Normal s) \<rightarrow> u" obtains "u = (c s, Normal s)"
completionE: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> {uu_. \<exists>S N N'. uu_ = S \<union> N \<and> S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N'}" obtains S :: "'b set" and N :: "'b set" and N' :: "'b set" where "A = S \<union> N" and "N \<subseteq> N'" and "N' \<in> null_sets M" and "S \<in> sets M"
descartes_roots_test_def: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "descartes_roots_test a b p = nat (changes (coeffs (fcompose p [:a, b:] [:1, 1:])))"
tensor_P_dim: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and dims :: "nat list"   and R :: "nat"   and K :: "nat"   and A :: "'a mat"   and B :: "'a mat" assumes "grover_state_sig n f dims R K" shows "partial_state2.ptensor_mat dims (grover_state_sig.vars1 n) (grover_state_sig.vars2 n) A B \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G"
pcr_word_def: shows "pcr_word \<equiv> (=) OO cr_word"
assms: shows "\<not> validate_ground_clause (int_clset S) C"
group_sizes_min: fixes x :: "nat" assumes "x \<in> group_sizes" shows "1 \<le> x"
path_component_of_refl: fixes X :: "'b topology"   and x :: "'b" shows "path_component_of X x x = (x \<in> topspace X)"
reachable_awalkI: fixes u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "H.awalk u p v" shows "u \<rightarrow>\<^sup>*\<^bsub>app_iso hom G\<^esub> v"
assms_5: fixes it :: "'u set"   and \<sigma> :: "'\<sigma>" assumes "it \<subseteq> \<alpha> m"   and "it \<noteq> {}"   and "\<not> c \<sigma>"   and "I it \<sigma>"   and "\<forall>k\<in>it. \<forall>j\<in>\<alpha> m - it. k \<le> j" shows "P \<sigma>"
strip_guards_bseq: fixes F :: "'f set"   and c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com" shows "strip_guards F (bseq c1 c2) = bseq (strip_guards F c1) (strip_guards F c2)"
eAbs_simp3: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model"   and xs :: "'varSort"   and x :: "'var" shows "eAbs MOD xs x ERR = ERR"
fss_nonz: shows "\<forall>f\<in>set fss. f \<noteq> 0"
cyclic_group_def: fixes G :: "('a, 'b) monoid_scheme" shows "cyclic_group G \<equiv> \<exists>x\<in>carrier G. subgroup_generated G {x} = G"
assms_2: shows "MaxR.MaxR_opt (linord_of_list r) D = Some x"
f_Exec_Stream_Init_take: fixes n :: "nat"   and trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d" shows "f_Exec_Comp_Stream_Init trans_fun xs c \<down> Suc n = f_Exec_Comp_Stream_Init trans_fun (xs \<down> n) c"
pred_inject_2: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" shows "pred_err P (OK a) = P a"
ordered_keys_Mapping: fixes t :: "('c, 'd) RBT.rbt" shows "Mapping.ordered_keys (RBT_Mapping.Mapping t) = RBT.keys t"
coeff_const: fixes a :: "'a"   and i :: "nat" shows "coeff [:a:] i = (if i = 0 then a else (0::'a))"
I_prime: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "primeideal I R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "a \<otimes>\<^bsub>R\<^esub> b \<in> I" shows "a \<in> I \<or> b \<in> I"
conjugate_conjr2: fixes y :: "'a" shows "conjugate (\<lambda>x. x \<rhd> y) = (\<lhd>) y"
rel_return_pmf: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'c"   and x2 :: "'d" shows "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2"
tail_def: fixes s :: "nat \<Rightarrow> 'b" shows "tail s \<equiv> s |\<^sub>s 1"
inducts: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "\<And>list. P (TVar list)"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P ty1 \<rightarrow> ty2" shows "P ty"
set_eqI: fixes A :: "'k set"   and B :: "'k set" assumes "\<And>x. (x \<in> A) = (x \<in> B)" shows "A = B"
SeqConstP_imp_SubstTermP: fixes s :: "tm"   and kk :: "tm"   and c :: "tm"   and t :: "tm"   and w :: "name" shows "{SeqConstP s kk c, TermP t} \<turnstile> SubstTermP \<guillemotleft>Var w\<guillemotright> t c c"
simps_26: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "char list \<Rightarrow> 'a"   and x5 :: "char list" shows "(case Class x5 of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x) = f5 x5"
m_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<oplus> y = y \<oplus> x"
rep_eq: fixes x :: "'a" shows "(*\<^sub>V) (cblinfun_mult_left x) = (\<lambda>y. y * x)"
brauns_correct: fixes i :: "nat"   and xs :: "'b list"   and k :: "nat" assumes "i < min (length xs) (2 ^ k)" shows "braun_list (brauns k xs ! i) (take_nths i k xs)"
A_wf_trms: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (N2 (unlabel A))"
not_eq_order_implies_strict: fixes a :: "'a"   and b :: "'a"   and ko :: "'a key_order" assumes "a \<noteq> b"   and "le_of_key_order ko a b" shows "lt_of_key_order ko a b"
strict_implies_order: fixes a :: "'c"   and b :: "'c" assumes "a < b" shows "a \<le> b"
ord_1: shows "AE x in M. f x \<le> g x"
simps_1: fixes f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g" shows "foldr f [] = id"
strong_Nullstellensatz: fixes X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "\<I> (\<V> F) = \<surd>ideal F"
smallo_powr: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real"   and p :: "real" assumes "f \<in> o[F](g)"   and "0 < p" shows "(\<lambda>x. \<bar>f x\<bar> powr p) \<in> o[F](\<lambda>x. \<bar>g x\<bar> powr p)"
11: shows "bprv (PPf p \<langle>neg \<phi>R\<rangle>)"
Cons_1: fixes i :: "nat"   and j :: "nat" assumes "xs ! i = x"   and "xs ! j = x"   and "i < j"   and "j < length xs"   and "distinct xs" shows "False"
ind_def: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" shows "ind P n = (if 0 < n \<and> P n then 1::'a else (0::'a))"
modal_distr: fixes \<phi> :: "i \<Rightarrow> bool"   and \<psi> :: "i \<Rightarrow> bool" assumes "universal (\<lambda>w. (r) w \<^bold>\<rightarrow> \<phi> \<^bold>\<rightarrow> \<psi>)" shows "\<lfloor>\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<psi>\<rfloor>"
13: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
nprv_negE0: fixes \<phi> :: "'fmla"   and F :: "'fmla set"   and \<psi> :: "'fmla" assumes "neg \<phi> \<in> F"   and "nprv F \<phi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv F \<psi>"
face_cycle_set_altdef: fixes a :: "'b" shows "G.face_cycle_set a = {(G.face_cycle_succ ^^ n) a |n. True}"
RelStay: fixes R :: "pi"   and S :: "pi"   and c :: "name" assumes "(R, S) \<in> Rel" shows "([c\<frown>c]R, S) \<in> Rel"
partition_on_def: fixes A :: "'c set"   and P :: "'c set set" shows "partition_on A P = (\<Union> P = A \<and> disjoint P \<and> {} \<notin> P)"
induct: fixes x1 :: "'b rexp"   and x2 :: "'b rexp"   and P :: "'b rexp \<Rightarrow> 'b rexp \<Rightarrow> bool" assumes "ACI x1 x2"   and "\<And>r. P r r"   and "\<And>r s. \<lbrakk>ACI r s; P r s\<rbrakk> \<Longrightarrow> P s r"   and "\<And>r s t. \<lbrakk>ACI r s; P r s; ACI s t; P s t\<rbrakk> \<Longrightarrow> P r t"   and "\<And>r1 s1 r2 s2. \<lbrakk>ACI r1 s1; P r1 s1; ACI r2 s2; P r2 s2\<rbrakk> \<Longrightarrow> P (Plus r1 r2) (Plus s1 s2)"   and "\<And>r1 s1 r2 s2. \<lbrakk>ACI r1 s1; P r1 s1; ACI r2 s2; P r2 s2\<rbrakk> \<Longrightarrow> P (Times r1 r2) (Times s1 s2)"   and "\<And>r s. \<lbrakk>ACI r s; P r s\<rbrakk> \<Longrightarrow> P (Star r) (Star s)"   and "\<And>r s t. P (Plus (Plus r s) t) (Plus r (Plus s t))"   and "\<And>r s. P (Plus r s) (Plus s r)"   and "\<And>r. P (Plus r r) r" shows "P x1 x2"
no_rewritten_fermat4: fixes c :: "int" shows "\<nexists>a b. a ^ 4 + b ^ 4 = c\<^sup>2 \<and> a * b * c \<noteq> 0 \<and> odd a \<and> coprime a b"
U_nonneg: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and y :: "'a" shows "0 \<le> MC_syntax.U K x y"
core_greatest: fixes m :: "'a set"   and a :: "'a set" assumes "m open"   and "m \<subseteq> a" shows "m \<subseteq> core a"
mod_eq_imp_diff_mod_0: fixes a :: "nat"   and m :: "nat"   and b :: "nat" assumes "a mod m = b mod m" shows "(b - a) mod m = 0"
face_of_aff_dim_lt: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "T face_of S"   and "T \<noteq> S" shows "aff_dim T < aff_dim S"
F2bd'_Cnotzero: fixes r2 :: "('d \<times> 'd) set" shows "(bd_F2 +c r2, czero) \<notin> ordIso \<and> Card_order (bd_F2 +c r2)"