hcju/isabelleps huggingface dataset:
Diff_triv: fixes A :: "'c set"   and B :: "'c set" assumes "A \<inter> B = {}" shows "A - B = A"
rep_eq: fixes x :: "'a set"   and xa :: "('a \<times> 'a) set" shows "set_of_rel (Rel x xa) = Id_on x \<union> xa"
branch_instr_sub1_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and instr_name :: "sparc_operation" assumes "low_equal s1 s2" shows "branch_instr_sub1 instr_name s1 = branch_instr_sub1 instr_name s2"
configuration_token_succeeds: fixes x :: "nat"   and q :: "'a ltl_prop_equiv_quotient"   and n :: "nat"   and y :: "nat" assumes "x \<in> configuration q n"   and "y \<in> configuration q n" shows "token_succeeds x = token_succeeds y"
Asm: fixes a :: "triple"   and A :: "triple set" assumes "a \<in> A" shows "A |\<turnstile> {a}"
path_strategies_eventually_constant: fixes G :: "('c, 'd) ParityGame_scheme"   and S :: "'c set"   and p :: "Player"   and good :: "'c \<Rightarrow> ('c \<Rightarrow> 'c) set"   and r :: "(('c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> 'c)) set"   and P :: "'c llist" assumes "WellOrderedStrategies G S p good r"   and "\<not> lfinite P"   and "lset P \<subseteq> S"   and "Digraph.valid_path G P"   and "ParityGame.path_conforms_with_strategy G p P (WellOrderedStrategies.well_ordered_strategy G S good r)" shows "\<exists>n. \<forall>m\<ge>n. WellOrderedStrategies.path_strategies good r P $ n = WellOrderedStrategies.path_strategies good r P $ m"
gen_mHom_eq: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme"   and H :: "'e set"   and f :: "'e \<Rightarrow> 'i"   and g :: "'e \<Rightarrow> 'i" assumes "R module M"   and "R module N"   and "generator R M H"   and "f \<in> mHom R M N"   and "g \<in> mHom R M N"   and "\<forall>h\<in>H. f h = g h" shows "f = g"
exhaust_disc: fixes pr_operation' :: "pr_operation'" obtains "is_RELABEL' pr_operation'"   | "is_NONSAT_PUSH' pr_operation'"   | "is_SAT_PUSH' pr_operation'"
compet_MTF': fixes s0 :: "'b list"   and rs :: "'b list" assumes "distinct s0" shows "real (T_on MTF s0 rs) \<le> 2 * real (T_opt s0 rs)"
assms_1: shows "c = ((Pr n f g, xs, ls) # fs, rv)"
B'_def: shows "B' \<equiv> (*\<^sub>C) \<i> ` B"
det_fst_Gauss_Jordan_PA_eq_fst_Gauss_Jordan_det_P: fixes A :: "(('b, 'c) vec, 'c) vec" shows "fst (Gauss_Jordan_det_P A) = det (fst (Gauss_Jordan_PA A))"
assms: assumes "snd (rep_nat_term u) = snd (rep_nat_term v)" shows "to1 u v = to2 u v"
le_rexp_refl: fixes r :: "'b rexp" shows "le_rexp r r"
apfst_def: fixes f :: "'i \<Rightarrow> 'k" shows "apfst f = map_prod f id"
finite_edges: shows "finite {(loc1, s, loc2). s \<in>\<^sub>A summary loc1 loc2}"
intros_1: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'g"   and jviewInit :: "'b \<Rightarrow> 'g \<Rightarrow> 'f"   and jviewIncr :: "'b \<Rightarrow> 'g \<Rightarrow> 'f \<Rightarrow> 'f"   and s :: "'e"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'h list"   and envTrans :: "'h \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and jp :: "'b \<Rightarrow> ('g, 'd, 'i) Protocol" assumes "IncrEnvironment jkbp jview envObs jviewInit jviewIncr"   and "s \<in> set envInit" shows "tInit s \<in> IncrEnvironment.jpTraces envInit envAction envTrans envObs jp"
att_and: fixes As :: "'a attree list"   and s :: "'a set \<times> 'a set" shows "(\<turnstile>As \<oplus>\<^sub>\<and>\<^bsup>s\<^esup>) = (case As of [] \<Rightarrow> fst s \<subseteq> snd s | [a] \<Rightarrow> \<turnstile>a \<and> attack a = s | a # ab # list \<Rightarrow> \<turnstile>a \<and> fst (attack a) = fst s \<and> \<turnstile>ab # list \<oplus>\<^sub>\<and>\<^bsup>(snd (attack a), snd s)\<^esup>)"
transpose_rowvector: fixes v :: "('b, 'd) vec" shows "Finite_Cartesian_Product.transpose (rowvector v) = columnvector v"
p_eq_0_imp: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set"   and cfg :: "'a cfg" assumes "Reachability_Problem K S S1 S2"   and "cfg \<in> Finite_Markov_Decision_Process.valid_cfg K S"   and "Reachability_Problem.p K S S1 S2 (state cfg) = 0" shows "Reachability_Problem.v K S S1 S2 cfg = 0"
tminus_sum_mono: fixes S :: "'a set"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "finite S" shows "sum f S \<ominus> sum g S \<le> (\<Sum>x\<in>S. f x \<ominus> g x)"
finite_Collect_subsets: fixes A :: "'c set" assumes "finite A" shows "finite {B. B \<subseteq> A}"
assms: fixes s :: "'state"   and s1 :: "'state"   and vl :: "'value list"   and vl1 :: "'value list" assumes "reachNT s"   and "reach s1"   and "\<Delta> s vl s1 vl1" shows "iaction (disjAll \<Delta>s) s vl s1 vl1 \<or> (vl \<noteq> [] \<or> vl1 = []) \<and> reaction (disjAll \<Delta>s) s vl s1 vl1"
IH: shows "(\<Sum>k\<le>mm. (of_nat mm + a gchoose k) * x ^ k * y ^ (mm - k)) = (\<Sum>k\<le>mm. (- a gchoose k) * (- x) ^ k * (x + y) ^ (mm - k))"
returned_missing_cd_or_loop: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and k :: "nat"   and n :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "\<pi> 0 = \<pi>' 0"   and "\<nexists>k'. cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'"   and "\<forall>n'. \<pi>' n' \<noteq> local.return"   and "\<pi> n = local.return" obtains i :: "nat" and i' :: "nat" where "i < k" and "cs\<^bsup>\<pi>\<^esup> i = cs\<^bsup>\<pi>'\<^esup> i'" and "\<pi> (Suc i) \<noteq> \<pi>' (Suc i')" and "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> i \<or> (\<forall>j'>i'. j' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> i')"
type_preservation: fixes e :: "Syntax.term"   and e' :: "Syntax.term"   and \<tau> :: "ty" assumes "\<lless>[], e\<ggreater> I\<rightarrow> \<lless>[], e'\<ggreater>"   and "{$$} \<turnstile>\<^sub>W e : \<tau>" shows "{$$} \<turnstile>\<^sub>W e' : \<tau>"
simps_10: fixes i :: "int"   and c :: "int"   and e :: "Cooper.num" shows "iszlfm (NDvd i (CN 0 c e)) = (0 < c \<and> 0 < i \<and> numbound0 e)"
std_axioms: shows "std Fstd_empty Fstd_get Fstd_add Fstd_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy>"
nop_axioms_def: fixes A :: "V"   and n :: "V"   and f :: "V" shows "nop_axioms A n f \<equiv> n \<in>\<^sub>\<circ> \<omega> \<and> \<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n \<and> \<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A"
i_Exec_Comp_Stream_Acc_Output__eq_Msg_iSince_conv: fixes k :: "nat"   and m :: "'a message_af"   and s :: "nat \<Rightarrow> 'a message_af"   and output_fun :: "'b \<Rightarrow> 'a message_af"   and trans_fun :: "'c message_af \<Rightarrow> 'b \<Rightarrow> 'b"   and input :: "nat \<Rightarrow> 'c message_af"   and c :: "'b"   and t :: "nat" assumes "0 < k"   and "m \<noteq> NoMsg"   and "s = output_fun \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = m) = (s t2 = NoMsg. t2 \<S> t1 [t * k\<dots>,k - Suc 0]. s t1 = m)"
finite_reachable_advance: fixes E :: "('b \<times> 'b) set"   and v0 :: "'b"   and v :: "'b" assumes "finite (E\<^sup>* `` {v0})"   and "(v0, v) \<in> E\<^sup>*" shows "finite (E\<^sup>* `` {v})"
value_h: shows "value_flow \<Delta> h = value_flow \<Delta> g"
assms: fixes x :: "'a" shows "norm (f x) \<le> b * norm x"
z_def: fixes n :: "nat" shows "z n = (any, replicate n False)"
hpsafe_Sequence_simps: fixes a :: "('a, 'b, 'c) hp"   and b :: "('a, 'b, 'c) hp" shows "hpsafe (a ;; b) = (hpsafe a \<and> hpsafe b)"
p_id_term: fixes x :: "('a \<times> 'a set) set" assumes "x \<cdot> 1\<^sub>\<pi> = x" shows "x \<subseteq> 1\<^sub>\<pi>"
word_plus_and_or: fixes x :: "'b word"   and y :: "'b word" shows "(x AND y) + (x OR y) = x + y"
fold_Cons_rev: fixes xs :: "'c list" shows "fold (#) xs = (@) (rev xs)"
A_ub2: fixes t :: "'b tree"   and a :: "'b" assumes "bst t"   and "a \<in> set_tree t" shows "A a t \<le> log \<alpha> (real (size1 t) / 2) + 1"
NSLIM_def: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'b"   and L :: "'c" shows "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L = (\<forall>x. x \<noteq> star_of a \<and> x \<approx> star_of a \<longrightarrow> (*f* f) x \<approx> star_of L)"
downset_set_iso: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "mono (ord.downset_set less_eq)"
cong: fixes vid1 :: "'c"   and pid1 :: "'b"   and pid2 :: "'b"   and pid3 :: "'b" shows "ids.DCaxiom vid1 pid1 pid2 pid3 \<equiv> ids.DCaxiom vid1 pid1 pid2 pid3"
comm_monoid_set_def: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" shows "comm_monoid_set f z \<equiv> comm_monoid f z"
loop_back: fixes A :: "abc_inst list"   and gap :: "'a list"   and xs :: "nat list"   and n :: "nat"   and m :: "nat"   and x :: "nat"   and anything :: "nat list" assumes "length A = length gap + 4"   and "length xs = n" shows "\<exists>stp. abc_steps_l (length A, xs @ m # 0 # x # anything) (A @ [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gap + 4)]) stp = (0, xs @ m # x # 0 # anything)"
graph_extD1: fixes H :: "'c set"   and h :: "'c \<Rightarrow> real"   and H' :: "'c set"   and h' :: "'c \<Rightarrow> real"   and x :: "'c" assumes "graph H h \<subseteq> graph H' h'"   and "x \<in> H" shows "h x = h' x"
PNT1_imp_PNT1': assumes "\<pi> \<sim>[at_top] (\<lambda>x. x / ln x)" shows "(\<lambda>x. ln (\<pi> x)) \<sim>[at_top] ln"
exhaust: fixes A :: "'c llist set" obtains "\<forall>x\<in>A. lnull x"   | "\<not> (\<forall>x\<in>A. lnull x)"
CFG_call_edge_length: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and ins :: "'c list"   and outs :: "'c list" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Def Use ParamDefs ParamUses"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "(p, ins, outs) \<in> set procs" shows "length fs = length ins"
strict_assoc': fixes f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr"   and h :: "('a S.E.term, 'a) S.arr" assumes "S.ide f"   and "S.ide g"   and "S.ide h"   and "S.src f = S.trg g"   and "S.src g = S.trg h" shows "S.ide (S.\<a>' f g h)"
card_Diff_insert: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" assumes "finite A"   and "a \<in> A"   and "a \<notin> B" shows "card (A - insert a B) = card (A - B) - 1"
lam_simps_1: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>f : local.prod a b \<rightarrow> c\<guillemotright>" shows "arr (\<Lambda> a b c f)"
eOp_nchotomy: fixes einp :: "'index \<Rightarrow> 'c withERR option"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym"   and ebinp :: "'bindex \<Rightarrow> 'd withERR option" shows "(\<exists>inp binp. einp = OKI inp \<and> igWlsInp MOD delta inp \<and> ebinp = OKI binp \<and> igWlsBinp MOD delta binp) \<or> eOp MOD delta einp ebinp = ERR"
vec_vec_Tensor_length: fixes x :: "'a list"   and y :: "'a list" shows "length (vec_vec_Tensor x y) = length x * length y"
in_set_conv_decomp: fixes x :: "'b"   and xs :: "'b list" shows "(x \<in> ran\<^sub>l xs) = (\<exists>ys zs. xs = ys @ x # zs)"
inject: fixes x21 :: "'c"   and x22 :: "'c llist"   and y21 :: "'c"   and y22 :: "'c llist" shows "(LCons x21 x22 = LCons y21 y22) = (x21 = y21 \<and> x22 = y22)"
lres_left_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x / z \<le> y / z"
intros_7: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and e0s :: "e list"   and ts :: "t list"   and t2s :: "t list"   and es :: "e list"   and n :: "nat" assumes "\<S>\<bullet>\<C> \<turnstile> e0s : ts _> t2s"   and "\<S>\<bullet>\<C>\<lparr>label := [ts] @ label \<C>\<rparr> \<turnstile> es : [] _> t2s"   and "length ts = n" shows "\<S>\<bullet>\<C> \<turnstile> [Label n e0s es] : [] _> t2s"
A1: fixes x :: "i"   and y :: "i" shows "(x\<box> \<cong> \<box>y \<^bold>\<leftarrow> E (x \<^bold>\<cdot> y)) \<^bold>\<and> (E (x \<^bold>\<cdot> y) \<^bold>\<leftarrow> x\<box> \<cong> \<box>y)"
CSP2_R_commute: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "CSP2 (R P) = R (CSP2 P)"
pp_first_pp_fusion: fixes x :: "'b ppath"   and y :: "'b ppath" assumes "pp_last x = pp_first y" shows "pp_first (pp_fusion x y) = pp_first x"
eq_Nil_null: fixes xs :: "'k list" shows "(xs = []) = List.null xs"
simps_2: fixes a :: "'a Bonk_Schramm_extension_unfolded"   and b :: "'a Bonk_Schramm_extension_unfolded" shows "map_aux (middle a b) = suc wo_aux ({map_aux a} \<union> {map_aux b})"
valid_path_rev_aux_Append: fixes cs :: "'edge list"   and as' :: "'edge list"   and as :: "'edge list" assumes "valid_path_rev_aux cs as'"   and "valid_path_rev_aux (upd_rev_cs cs as') as" shows "valid_path_rev_aux cs (as @ as')"
state_at_0_2: fixes i :: "nat" shows "gap i 0 = 1"
transpose_minus: fixes A :: "('c, 'd) sq_matrix"   and B :: "('c, 'd) sq_matrix" shows "Square_Matrix.transpose (A - B) = Square_Matrix.transpose A - Square_Matrix.transpose B"
ltrm_of_sum_diff_degree: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "deg R q < deg R p" shows "monom P ((p \<oplus>\<^bsub>P\<^esub> q) (deg R (p \<oplus>\<^bsub>P\<^esub> q))) (deg R (p \<oplus>\<^bsub>P\<^esub> q)) = monom P (p (deg R p)) (deg R p)"
elims_3: fixes wtFsym :: "'e \<Rightarrow> bool"   and wtPsym :: "'f \<Rightarrow> bool"   and arOf :: "'e \<Rightarrow> 'g list"   and resOf :: "'e \<Rightarrow> 'g"   and intT :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and intF :: "'e \<Rightarrow> 'h list \<Rightarrow> 'h"   and intP :: "'f \<Rightarrow> 'h list \<Rightarrow> bool"   and x :: "Preliminaries.var \<Rightarrow> 'h"   and xa :: "('e, 'f) atm" assumes "Struct wtFsym wtPsym arOf resOf intT intF intP"   and "\<not> Struct.satA intF intP x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'h" and T1 :: "'e trm" and T2 :: "'e trm" where "x = \<xi>'__" and "xa = Eq T1 T2" and "Struct.int intF \<xi>'__ T1 \<noteq> Struct.int intF \<xi>'__ T2"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'h" and p :: "'f" and Tl :: "'e trm list" where "x = \<xi>" and "xa = Pr p Tl" and "\<not> intP p (map (Struct.int intF \<xi>) Tl)"
finite_F: fixes m :: "'a" shows "finite (gbg_F (to_gbg_ext m))"
restrict_topci_def: fixes p :: "nat"   and vr :: "nat"   and vl :: "bool"   and bdd :: "bddi" shows "restrict_topci p vr vl bdd = case_ifexici (return p) (return p) (\<lambda>v te ee. return (if v = vr then if vl then te else ee else p)) p bdd"
elims_2: fixes x :: "'b tree" assumes "is_root x" obtains hp :: "'b tree" where "x = hp" and "case hp of \<langle>\<rangle> \<Rightarrow> True | \<langle>l, x, r\<rangle> \<Rightarrow> r = \<langle>\<rangle>"
measure_zero_union: fixes s :: "'c set"   and M :: "'c pmf"   and t :: "'c set" assumes "s \<in> measure_pmf.events M"   and "t \<in> measure_pmf.events M"   and "measure_pmf.prob M t = 0" shows "measure_pmf.prob M (s \<union> t) = measure_pmf.prob M s"
simps: fixes x :: "nat"   and y :: "nat" shows "bezw x y = (if y = 0 then (1, 0) else (snd (bezw y (x mod y)), fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int (x div y)))"
and_not_mask: fixes w :: "'a word"   and n :: "nat" shows "w && ~~ mask n = w >> n << n"
LSC_Contract: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset" assumes "F, F, \<Gamma> \<Rightarrow>\<^sub>n" shows "F, \<Gamma> \<Rightarrow>\<^sub>n"
wpQ_WhileTrue: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "bval b s" shows "wp\<^sub>Q c (wp\<^sub>Q (WHILE b DO c) Q) s + 1 \<le> wp\<^sub>Q (WHILE b DO c) Q s"
simps_1: fixes l :: "'b" shows "translation (A_Lbl l) = LG {(l, 0, 1)} {0, 1}"
ordLeq_iff_ordLess_or_ordIso: fixes r :: "('f \<times> 'f) set"   and r' :: "('g \<times> 'g) set" shows "(r \<le>o r') = (r <o r' \<or> r =o r')"
cgenideal_self: fixes R :: "('c, 'd) ring_scheme"   and i :: "'c" assumes "ring R"   and "i \<in> carrier R" shows "i \<in> PIdl\<^bsub>R\<^esub> i"
subgraph_app_isoI: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and H :: "('a, 'b) pre_digraph" assumes "digraph_isomorphism hom"   and "subgraph H G" shows "subgraph (app_iso hom H) (app_iso hom G)"
x_in: shows "x \<in> set (characterize_consistent_signs_at_roots p (qs1 @ qs2))"
P_def: shows "P \<equiv> \<lambda>stepSs. let ((s, r), S) = shd stepSs; ((s', uu_), S') = shd (stl stepSs) in S \<in> structure \<and> \<not> sat S s \<and> descent (s, S) r (s', S')"
Rep_num1_cases: fixes y :: "unit" assumes "y \<in> UNIV" obtains x :: "1" where "y = Rep_num1 x"
group_hom_isoI: assumes "\<forall>x\<in>carrier \<F>\<^bsub>{()}\<^esub>. int.lift f x = \<one>\<^bsub>int_group\<^esub> \<longrightarrow> x = \<one>\<^bsub>\<F>\<^bsub>{()}\<^esub>\<^esub>"   and "int.lift f ` carrier \<F>\<^bsub>{()}\<^esub> = carrier int_group" shows "int.lift f \<in> Group.iso \<F>\<^bsub>{()}\<^esub> int_group"
transf_def: fixes n :: "nat"   and input :: "nat list" shows "transf n input \<equiv> n # input"
valid_weak_tau_modality_iff_tau_steps: fixes P :: "'state"   and x :: "('idx, 'pred, 'act) formula" shows "P \<Turnstile> weak_tau_modality x = (\<exists>n. P \<Turnstile> tau_steps x n)"
fold_def: fixes f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and A :: "'f set" shows "Finite_Set.fold f z A = (if finite A then THE y. fold_graph f z A y else z)"
g: shows "g \<in> L \<rightarrow>\<^sub>M N"
in_Under_above: fixes a :: "'c"   and r :: "('c \<times> 'c) set" assumes "a \<in> Field r" shows "a \<in> Under r (above r a)"
hyps_6: shows " t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2"
lin_independent_replace1_by_lincomb: fixes us :: "'v list"   and v :: "'v"   and vs :: "'v list"   and as :: "'f list"   and b :: "'f"   and bs :: "'f list" assumes "set us \<subseteq> V"   and "v \<in> V"   and "set vs \<subseteq> V"   and "R_lin_independent (us @ v # vs)"   and "length as = length us"   and "b \<noteq> (0::'f)" shows "R_lin_independent ((as @ b # bs) \<bullet>\<cdot> (us @ v # vs) # us @ vs)"
set_empty_def: shows "set_empty = (\<lambda>_. {})"
lm100: fixes x :: "'a"   and Y :: "'a set"   and X :: "'a set" assumes "x \<in> Y - X" shows "0 \<in> (X <|| Y) `` {x}"
g_independent: shows "\<forall>a m. a \<in> G \<longrightarrow> g (f m) a = g m a"
that: fixes b :: "real" assumes "b\<^sub>\<real> = a" shows "thesis"
cast_cast_below2: fixes A :: "'a defl"   and B :: "'a defl"   and x :: "'a" assumes "A \<sqsubseteq> B" shows "cast\<cdot>B\<cdot>(cast\<cdot>A\<cdot>x) = cast\<cdot>A\<cdot>x"
countable_Int1: fixes A :: "'b set"   and B :: "'b set" assumes "countable A" shows "countable (A \<inter> B)"
axioms_1: 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" assumes "green_typeI_typeII_chain s i j F two_chain_typeI two_chain_typeII" shows "R2 i j"
that: fixes as :: "'a list" assumes "prefix (as @ [x]) xs"   and "x \<notin> set as" shows "thesis"
elims_3: fixes x :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "\<not> BinomialHeapStruc.rank_invar x" obtains t :: "('a, 'b) BinomialHeapStruc.BinomialTree" and t' :: "('a, 'b) BinomialHeapStruc.BinomialTree" and bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" where "x = t # t' # bq" and "\<not> (BinomialHeapStruc.rank t < BinomialHeapStruc.rank t' \<and> BinomialHeapStruc.rank_invar (t' # bq))"
insert_dom_inj_on: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "inj_on f {0..<n}"   and "i < Suc n"   and "j < Suc n"   and "f ` {0..<n} = {0..<Suc n} - {j}" shows "inj_on (insert_dom f i j) {0..<Suc n}"
no_messages_if_no_channel: fixes t :: "('a, 'b, 'c) event list"   and cid :: "nat"   and i :: "nat" assumes "trace init t final"   and "channel cid = None" shows "msgs (S t i) cid = []"
wf_not_refl: fixes r :: "('b \<times> 'b) set"   and a :: "'b" assumes "wf r" shows "(a, a) \<notin> r"
weakMonotonic: fixes P :: "ccs"   and A :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and B :: "(ccs \<times> ccs) set" assumes "P \<leadsto><A> Q"   and "A \<subseteq> B" shows "P \<leadsto><B> Q"
hmem_hinsert: fixes a :: "hf"   and b :: "hf"   and c :: "hf" shows "(a \<^bold>\<in> c \<triangleleft> b) = (a = b \<or> a \<^bold>\<in> c)"
ex_new_if_finite: fixes A :: "'a set" assumes "infinite UNIV"   and "finite A" shows "\<exists>a. a \<notin> A"
ComSecTyping_single_is_sound: fixes c :: "('exp, 'id) MWLsCom" assumes "\<turnstile>\<^bsub>\<C>\<^esub> c"   and "unique_PPc c" shows "WHATWHERE_Secure [c]"
monoid_vec_def: fixes ty :: "'a itself"   and n :: "nat" shows "monoid_vec ty n \<equiv> \<lparr>carrier = carrier_vec n, mult = (+), one = 0\<^sub>v n\<rparr>"
size_Union_distinct_cart_prod_filter: fixes M :: "'a multiset"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b multiset"   and c :: "nat" assumes "distinct_mset M"   and "\<And>p. p \<in># M \<Longrightarrow> size (filter_mset (P p) B) = c" shows "size (\<Sum>p\<in>#M. {#p#} \<times># filter_mset (P p) B) = size M * c"
set_not_emptyI: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> S" shows "S \<noteq> {}"
arc_to_ends_def: fixes G :: "('b, 'c) pre_digraph"   and e :: "'c" shows "arc_to_ends G e \<equiv> (tail G e, head G e)"
isGB_I_adds_lt: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "G \<subseteq> dgrad_p_set d m"   and "\<And>f. \<lbrakk>f \<in> pmdl G; f \<in> dgrad_p_set d m; f \<noteq> 0\<rbrakk> \<Longrightarrow> \<exists>g\<in>G. g \<noteq> 0 \<and> lt g adds\<^sub>t lt f" shows "is_Groebner_basis G"
u_simps_4: shows "local.dom (tab\<^sub>0 r) = tab\<^sub>0 r"
count_list_eq_0_iff: fixes xs :: "'a list"   and x :: "'a" shows "(count_list xs x = 0) = (x \<notin> set xs)"
goal_eval: shows "sinvar \<lparr>nodes = V, edges = E - {e}\<rparr> nP"
wf_cdecl_supD: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal"   and r :: "fdecl list \<times> 'a option mdecl list" assumes "wf_cdecl wf_md P (C, D, r)"   and "C \<noteq> Object" shows "is_class P D"
mset_insert: fixes insert :: "'b \<Rightarrow> 'b tree \<Rightarrow> 'b tree"   and del_min :: "'b tree \<Rightarrow> 'b tree"   and q :: "'b tree"   and x :: "'b" assumes "Heap insert del_min"   and "heap q" shows "mset_tree (insert x q) = {#x#} + mset_tree q"
elims: fixes x :: "real"   and xa :: "real \<times> real \<times> real \<times> real"   and y :: "real \<times> real \<times> real \<times> real" assumes "x *\<^sub>s\<^sub>v\<^sub>4 xa = y" obtains k :: "real" and a :: "real" and b :: "real" and c :: "real" and d :: "real" where "x = k" and "xa = (a, b, c, d)" and "y = (k * a, k * b, k * c, k * d)"
enat_ord_simps_2: fixes m :: "nat"   and n :: "nat" shows "(enat m < enat n) = (m < n)"