hcju/isabelleps huggingface dataset:
contours_in_eval: fixes ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option"   and Q :: "contour \<Rightarrow> bool"   and \<beta> :: "nat \<Rightarrow> contour option"   and f :: "val" assumes "\<forall>b'\<in>contours_in_ve ve. Q b'"   and "\<forall>b'\<in>ran \<beta>. Q b'" shows "\<forall>b'\<in>contours_in_d (\<A> f \<beta> ve). Q b'"
rnddvd: fixes b :: "nat"   and a :: "nat" assumes "b dvd a" shows "roundup a b * b = a"
inv_in_homI: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "D.iso f"   and "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>D.inv f : b \<Rightarrow>\<^sub>D a\<guillemotright>"
lead_coeff_mult: fixes p :: "'b poly"   and q :: "'b poly" shows "lead_coeff (p * q) = lead_coeff p * lead_coeff q"
assms_2: shows "b \<star> \<mu> \<noteq> null"
not_le: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.linorder less_eq less" shows "(\<not> less_eq x y) = less y x"
finite_reduced_GB_finite: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "finite F" shows "finite (reduced_GB F)"
aa_loc: fixes x :: "'a"   and y :: "'a" shows "d (x \<cdot> d y) = d (x \<cdot> y)"
coeff_monom: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b"   and m' :: "nat \<Rightarrow>\<^sub>0 nat" shows "coeff (monom m a) m' = (a when m' = m)"
convex_hull_eq_empty: fixes S :: "'b set" shows "(convex hull S = {}) = (S = {})"
maps_def: fixes f :: "'h \<Rightarrow> 'g list"   and xs :: "'h list" shows "List.maps f xs = concat (map f xs)"
OPT2_A': fixes x :: "'a"   and y :: "'a"   and qs :: "'a list" assumes "x \<noteq> y"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom y])" shows "real (T\<^sub>p [x, y] qs (OPT2 qs [x, y])) = 1"
map_lasso_def: fixes f :: "'f \<Rightarrow> 'e"   and L :: "('f, 'g) lasso_scheme" shows "map_lasso f L \<equiv> \<lparr>lasso_reach = map f (lasso_reach L), lasso_va = f (lasso_va L), lasso_cysfx = map f (lasso_cysfx L)\<rparr>"
smooth_compat_restrict_chartD: fixes c1 :: "('c, 'd) chart"   and U :: "'c set"   and k :: "enat"   and c2 :: "('c, 'd) chart" assumes "domain c1 \<subseteq> U"   and "open U"   and "k-smooth_compat c1 (restrict_chart U c2)" shows "k-smooth_compat c1 c2"
TS_a'': fixes x :: "nat"   and y :: "nat"   and x0 :: "nat"   and y0 :: "nat"   and s :: "(nat list \<times> nat list) pmf"   and qs :: "nat list"   and h0 :: "nat list" assumes "x \<noteq> y"   and "{x, y} = {x0, y0}"   and "TS_inv s x [x0, y0]"   and "set qs \<subseteq> {x, y}"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom y])" shows "TS_inv (Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS h0)) s qs) (last qs) [x0, y0] \<and> T\<^sub>p_on_rand' (Partial_Cost_Model.embed (rTS h0)) s qs = 2"
diff_strict_right_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less a b" shows "less (minus a c) (minus b c)"
lborel_integral_real_affine: fixes c :: "real"   and f :: "real \<Rightarrow> 'd"   and t :: "real" assumes "c \<noteq> 0" shows "integral\<^sup>L lborel f = \<bar>c\<bar> *\<^sub>R (LBINT x. f (t + c * x))"
ide_char: fixes a :: "'s" shows "S.ide a = (S.arr a \<and> S.dom a = a \<and> S.cod a = a)"
interleave_set: fixes zs :: "'a list"   and xs :: "'a list"   and ys :: "'a list" assumes "zs \<in> xs \<otimes> ys" shows "set zs = set xs \<union> set ys"
blinfun_of_matrix_works: fixes f :: "'d \<Rightarrow>\<^sub>L 'e" shows "blinfun_of_matrix (\<lambda>i j. blinfun_apply f j \<bullet> i) = f"
reduced_word_forI_length: fixes as :: "'a list"   and A :: "'a set"   and a :: "'a" assumes "as \<in> lists A"   and "sum_list as = a"   and "length as = word_length A a" shows "reduced_word_for A a as"
Abs_permutation_inject: fixes x :: "'b \<Rightarrow> 'b"   and y :: "'b \<Rightarrow> 'b" assumes "x \<in> {f. bij f}"   and "y \<in> {f. bij f}" shows "(Abs_permutation x = Abs_permutation y) = (x = y)"
assms_4: shows "Q_I_pos = projectQuery I (filterQuery I Qp)"
singleton_iff: fixes b :: "'g"   and a :: "'g" shows "(b \<in> {a}) = (b = a)"
simps_8: fixes p :: "('a, 'b) form" shows "params (Forall p) = params p"
view_determination: fixes \<sigma> :: "'b"   and u :: "'a"   and \<rho> :: "'b"   and v :: "'a" assumes "put \<sigma> u = put \<rho> v" shows "u = v"
dg_Par_Cod_app: fixes \<alpha> :: "V"   and x :: "V" assumes "arr_Par \<alpha> x" shows "dg_Par \<alpha>\<lparr>Cod\<rparr>\<lparr>x\<rparr> = x\<lparr>ArrCod\<rparr>"
eq_matI: fixes B :: "'b mat"   and A :: "'b mat" assumes "\<And>i j. \<lbrakk>i < dim_row B; j < dim_col B\<rbrakk> \<Longrightarrow> A $$ (i, j) = B $$ (i, j)"   and "dim_row A = dim_row B"   and "dim_col A = dim_col B" shows "A = B"
ws_progD: fixes C :: "cname"   and c :: "class" assumes "class C = Some c"   and "C \<noteq> Object"   and "ws_prog" shows "is_class (super c) \<and> (super c, C) \<notin> subcls1\<^sup>+"
step_Normal_elim_cases_6: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e"   and u :: "('e, 'd, 'f) com \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Cond b c1 c2, Normal s) \<rightarrow> u" obtains "u = (c1, Normal s)" and "s \<in> b"   | "u = (c2, Normal s)" and "s \<notin> b"
maxsimpchain_append_reduce1: fixes xs :: "'a set list"   and ys :: "'a set list" assumes "gallery (xs @ ys)" shows "gallery xs"
select_convs_1: fixes getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and more :: "'a" shows "getF \<lparr>getF = getF, getM = getM, \<dots> = more\<rparr> = getF"
simps_5: fixes e1 :: "expr"   and e2 :: "expr" shows "\<lbrace>Multiplication e1 e2\<rbrace> = \<lbrace>e1\<rbrace> * \<lbrace>e2\<rbrace>"
preprm_ext_compose_left: fixes P :: "('b \<times> 'b) list"   and S :: "('b \<times> 'b) list"   and \<pi> :: "('b \<times> 'b) list" assumes "P =p S" shows "preprm_compose \<pi> P =p preprm_compose \<pi> S"
assms: shows "(u \<longlongrightarrow> ereal l) F"
simps_3: shows "cospan_shape.Dom cospan_shape.TT = cospan_shape.TT"
nextMax_def: fixes x :: "nat set" shows "nextMax x \<equiv> if x = {} then 0 else Suc (Max x)"
set_restriction_inext_eq: fixes interval_fun :: "nat set \<Rightarrow> nat set"   and n :: "nat"   and I :: "nat set" assumes "set_restriction interval_fun"   and "n \<in> interval_fun I"   and "inext n I \<in> interval_fun I" shows "inext n (interval_fun I) = inext n I"
lapprox_rat_nonneg: fixes x :: "int"   and y :: "int"   and n :: "nat" assumes "0 \<le> x"   and "0 \<le> y" shows "0 \<le> real_of_float (lapprox_rat n x y)"
algebraic_int_of_int: fixes m :: "int" shows "algebraic_int (of_int m)"
k_invariant_empt: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'g"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'h"   and jviewInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'g"   and jviewIncr :: "'b \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> 'g"   and simf :: "'e Trace \<Rightarrow> 'i"   and simRels :: "'b \<Rightarrow> ('i \<times> 'i) set"   and simVal :: "'i \<Rightarrow> 'c \<Rightarrow> bool"   and simAbs :: "'j \<Rightarrow> 'i set"   and simObs :: "'b \<Rightarrow> 'j \<Rightarrow> 'h"   and simInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'j"   and simTrans :: "'b \<Rightarrow> 'j \<Rightarrow> 'j list"   and simAction :: "'b \<Rightarrow> 'j \<Rightarrow> 'd list"   and aOps :: "('k, 'j, 'd list) MapOps"   and tOps :: "('l, 'j \<times> 'h, 'j) MapOps"   and a :: "'b" assumes "AlgorithmForAgent jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal simAbs simObs simInit simTrans simAction aOps tOps" shows "AlgorithmForAgent.k_invariant jkbp envInit envAction envTrans envVal jview simf simAbs simObs simTrans simAction aOps tOps a (AlgorithmForAgent.k_empt aOps tOps)"
is_GB_cofactor_boundI_subset_zero: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and b :: "nat" assumes "F \<subseteq> {0}" shows "pp_pm.is_GB_cofactor_bound F b"
exec_instr_stk_offer: fixes ins :: "'a instr list"   and pc :: "nat"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'b"   and stk :: "'a val list"   and loc :: "'a val list"   and C :: "String.literal"   and M :: "String.literal"   and frs :: "'a frame list"   and ta' :: "('a, 'c, 'b) jvm_thread_action"   and xcp' :: "'a option"   and h' :: "'b"   and stk' :: "'a val list"   and loc' :: "'a val list"   and pc' :: "nat"   and addr2thread_id :: "'a \<Rightarrow> 'c"   and thread_id2addr :: "'c \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'b"   and allocate :: "'b \<Rightarrow> htype \<Rightarrow> ('b \<times> 'a) set"   and typeof_addr :: "'b \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'b \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'b \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'c"   and stk'' :: "'a val list" assumes "check_instr' (ins ! pc) P h stk loc C M pc frs"   and "(ta', xcp', h', (stk', loc', C, M, pc') # frs) \<in> JVM_heap_base.exec_instr addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write (ins ! pc) P t h stk loc C M pc frs" shows "(ta', xcp', h', (stk' @ stk'', loc', C, M, pc') # frs) \<in> JVM_heap_base.exec_instr addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write (ins ! pc) P t h (stk @ stk'') loc C M pc frs"
sig_red_rtrancl_sing_regI: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(sig_red sing_reg top_tail F)\<^sup>*\<^sup>* p q" shows "(sig_red (\<preceq>\<^sub>t) top_tail F)\<^sup>*\<^sup>* p q"
exhaust_sel: fixes llist :: "'c llist" obtains "llist = LNil"   | "llist = LCons (lhd llist) (ltl llist)"
is_set_to_sorted_listE: fixes ordR :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Rk :: "('h \<times> 'g) set"   and Rs :: "('h \<times> 'g) set \<Rightarrow> ('i \<times> 'g set) set"   and tsl :: "'i \<Rightarrow> 'h list"   and s :: "'i"   and s' :: "'g set" assumes "is_set_to_sorted_list ordR Rk Rs tsl"   and "(s, s') \<in> \<langle>Rk\<rangle>Rs" obtains l' :: "'g list" where "(tsl s, l') \<in> \<langle>Rk\<rangle>list_rel" and "RETURN l' \<le> it_to_sorted_list ordR s'"
neg_one_less_numeral: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (uminus one) (numeral.numeral one plus m)"
V_valid_def: fixes v :: "'b V_rec" shows "V_valid v \<equiv> VN_disjoint v \<and> VC_disjoint v \<and> NC_disjoint v"
append_same_eq: fixes ys :: "'f list"   and xs :: "'f list"   and zs :: "'f list" shows "(ys @ xs = zs @ xs) = (ys = zs)"
qe_dlo_def: shows "qe_dlo = DLO\<^sub>e.lift_dnfeq_qe qe_dlo\<^sub>1"
i_f_join_i_expand_iT_Mult: fixes k :: "nat"   and I :: "nat set"   and f :: "nat \<Rightarrow> 'b message_af" assumes "0 < k"   and "finite I" shows "f \<odot> k \<Join>\<^sub> (I \<otimes> k) = f \<Join>\<^sub> I"
interval_of_uminus: fixes a :: "'a" shows "interval_of (- a) = - interval_of a"
label_stab_map_bij_betw_vertices: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "label_wrt C0 \<phi>"   and "fixespointwise \<phi> C0" shows "bij_betw (\<lambda>v. {w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>. w \<rightarrow> \<phi> v = v}) (\<Union> X) (\<Union> (CoxeterComplex.TheComplex (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})))"
ochoiceT1: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and p' :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat"   and q :: "('e, 'f, 'g, 'h) seqp" assumes "((\<sigma>, p), a, \<sigma>', p') \<in> oseqp_sos \<Gamma> i" shows "((\<sigma>, p \<oplus> q), a, \<sigma>', p') \<in> oseqp_sos \<Gamma> i"
collapse_3: fixes event :: "('a, 'b, 'c) event" assumes "isRecv event" shows "Recv (getId event) (occurs_on event) (partner event) (un_Recv4 event) (un_Recv5 event) (getMsg event) = event"
triangulized: fixes A :: "'a mat"   and n :: "nat"   and r' :: "'a"   and A' :: "'a mat" assumes "mute_fun mf"   and "det_selection_fun sel_fun"   and "A \<in> carrier_mat n n"   and "local.triangulize A = (r', A')" shows "upper_triangular A'"
EqForLVar_axioms: shows "EqForLVar eq_idx_for_lvar"
at_least_at_most_closed: fixes a :: "'a"   and b :: "'a" shows "\<lbrace>a..b\<rbrace>\<^bsub>L\<^esub> \<subseteq> carrier L"
gcd_proj2_if_dvd: fixes b :: "'b"   and a :: "'b" assumes "b dvd a" shows "gcd a b = normalize b"
goodM_gives_correct_PG: fixes MsgPInitRes :: "msg"   and LID_M :: "nat"   and xid :: "msg"   and cc :: "msg"   and cm :: "msg"   and P :: "agent"   and EKj :: "nat"   and M :: "agent"   and evs :: "event list" assumes "MsgPInitRes = \<lbrace>Number LID_M, xid, cc, cm, cert P EKj onlyEnc (priSK RCA)\<rbrace>"   and "Crypt (priSK M) (Hash MsgPInitRes) \<in> parts (knows Spy evs)"   and "evs \<in> set_pur"   and "M \<notin> bad" shows "\<exists>j trans. P = PG j \<and> Notes M \<lbrace>Number LID_M, Agent P, trans\<rbrace> \<in> set evs"
l3_partial_secrecy_def: shows "l3_partial_secrecy \<equiv> {s. dy_fake_msg (l3_state.bad s) (ik s \<inter> payload) (local.abs (ik s)) \<inter> secret s = {}}"
cases: fixes x :: "('e set \<times> 'e set) list \<times> (('g, 'h, 'e) prot_fun, 'f) Term.term \<times> (('g, 'h, 'e) prot_fun, 'f) Term.term" obtains TI :: "('e set \<times> 'e set) list" and xa :: "'f" and y :: "'f" where "x = (TIa__, Var xa, Var y)"   | TI :: "('e set \<times> 'e set) list" and f :: "('g, 'h, 'e) prot_fun" and T :: "(('g, 'h, 'e) prot_fun, 'f) Term.term list" and g :: "('g, 'h, 'e) prot_fun" and S :: "(('g, 'h, 'e) prot_fun, 'f) Term.term list" where "x = (TI, Fun f T, Fun g S)"   | uu_ :: "('e set \<times> 'e set) list" and v :: "('g, 'h, 'e) prot_fun" and va :: "(('g, 'h, 'e) prot_fun, 'f) Term.term list" and vb :: "'f" where "x = (uu___, Fun v__ va__, Var vb__)"   | uu_ :: "('e set \<times> 'e set) list" and vb :: "'f" and v :: "('g, 'h, 'e) prot_fun" and va :: "(('g, 'h, 'e) prot_fun, 'f) Term.term list" where "x = (uu_, Var vb, Fun v va)"
rep_eq: fixes x :: "nat"   and xa :: "nat_int" shows "(x \<^bold>\<in> xa) = (x \<in> Rep_nat_int xa)"
boundedI: fixes b :: "'e multiset"   and a :: "'e multiset"   and c :: "'e multiset" assumes "b \<subseteq># a"   and "c \<subseteq># a" shows "b \<union># c \<subseteq># a"
finsum_cong': fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a vec"   and f :: "'b \<Rightarrow> 'a vec" assumes "A = B"   and "g \<in> B \<rightarrow> carrier_vec n"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "finsum V f A = finsum V g B"
confluentI: fixes R :: "('a \<times> 'a) set" assumes "\<And>x y y'. \<lbrakk>(x, y) \<in> R; (x, y') \<in> R\<rbrakk> \<Longrightarrow> \<exists>z. (y, z) \<in> R \<and> (y', z) \<in> R" shows "Joinable.confluent R"
normalize_simps_6: fixes b :: "'d set"   and c :: "('d, 'e, 'f) com" shows "Language.normalize (While b c) = While b (Language.normalize c)"
11: shows "w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = bot"
Collect_bex_eq: fixes A :: "'c set"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
If: fixes P :: "state \<Rightarrow> bool"   and b :: "state \<Rightarrow> bool"   and c :: "com"   and Q :: "state \<Rightarrow> bool"   and d :: "com" assumes "\<turnstile>\<^sub>t {\<lambda>s. P s \<and> b s} c {Q}"   and "\<turnstile>\<^sub>t {\<lambda>s. P s \<and> \<not> b s} d {Q}" shows "\<turnstile>\<^sub>t {P} IF b THEN c ELSE d {Q}"
assms_3: shows "negchecks_model \<I> D' X F F'"
option_while'_inj: fixes s :: "'b option"   and s' :: "'b option"   and C :: "'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'b option"   and s'' :: "'b option" assumes "(s, s') \<in> option_while' C B"   and "(s, s'') \<in> option_while' C B" shows "s' = s''"
restrict_Pi_cancel: fixes x :: "'f \<Rightarrow> 'g"   and I :: "'f set"   and A :: "'f \<Rightarrow> 'g set" shows "(restrict x I \<in> Pi I A) = (x \<in> Pi I A)"
lookup_add: fixes f :: "'e \<Rightarrow>\<^sub>0 'd"   and g :: "'e \<Rightarrow>\<^sub>0 'd"   and k :: "'e" shows "lookup (f + g) k = lookup f k + lookup g k"
env_restr_mono2: fixes S2 :: "'a set"   and S1 :: "'a set"   and m :: "'a \<Rightarrow> 'b" assumes "S2 \<subseteq> S1" shows "m f|` S2 \<sqsubseteq> m f|` S1"
snoc_suffix_snoc: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list"   and y :: "'a" shows "suffix (xs @ [x]) (ys @ [y]) = (x = y \<and> suffix xs ys)"
mHom_eq: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g"   and g :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N"   and "g \<in> mHom R M N"   and "\<forall>m\<in>carrier M. f m = g m" shows "f = g"
eq_OO: fixes R :: "'h \<Rightarrow> 'i \<Rightarrow> bool" shows "(=) OO R = R"
Eats_Subset_E: fixes x :: "tm"   and z :: "tm"   and y :: "tm"   and H :: "fm set"   and C :: "fm" assumes "insert (x SUBS z) (insert (y IN z) H) \<turnstile> C" shows "insert (Eats x y SUBS z) H \<turnstile> C"
incl_def: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b" assumes "set_category S \<AA>" shows "set_category.incl S \<AA> f = (partial_magma.arr S f \<and> set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f) \<subseteq> set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.cod S f) \<and> f = set_category_given_img.mkArr S (set_category.some_img S \<AA>) \<AA> (set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f)) (set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.cod S f)) (\<lambda>x. x))"
r: shows "norm x < r \<and> ereal r < fps_conv_radius f"
calc_poincare_distance_def: fixes u :: "complex_homo"   and i1 :: "complex_homo"   and v :: "complex_homo"   and i2 :: "complex_homo" shows "calc_poincare_distance u i1 v i2 = \<bar>ln (Re_cross_ratio u i1 v i2)\<bar>"
snd_Gauss_Jordan_upt_k_iarrays_PA: fixes A :: "'b iarray iarray"   and k :: "nat" shows "snd (Gauss_Jordan_upt_k_iarrays_PA A k) = Gauss_Jordan_upt_k_iarrays A k"
tarski_absolute_space_axioms_def: fixes B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "tarski_absolute_space_axioms B \<equiv> (\<forall>a b. B a b a \<longrightarrow> a = b) \<and> (\<forall>a b c p q. B a p c \<and> B b q c \<longrightarrow> (\<exists>x. B p x b \<and> B q x a)) \<and> (\<forall>X Y. (\<exists>a. \<forall>x y. x \<in> X \<and> y \<in> Y \<longrightarrow> B a x y) \<longrightarrow> (\<exists>b. \<forall>x y. x \<in> X \<and> y \<in> Y \<longrightarrow> B x b y))"
neighborhoods_def: fixes x :: "'b" shows "neighborhoods x \<equiv> {U. is_open U \<and> x \<in> U}"
simps_9: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "\<lbrace>\<^bold>\<a>\<^bold>[t, u, v\<^bold>]\<rbrace> = D.assoc \<lbrace>t\<rbrace> \<lbrace>u\<rbrace> \<lbrace>v\<rbrace>"
powr_add: fixes x :: "'b"   and a :: "'b"   and b :: "'b" shows "x powr (a + b) = x powr a * x powr b"
word_sub_less_iff: fixes x :: "'b word"   and y :: "'b word" shows "(x < x - y) = (x < y)"
CARD_mod_ring: shows "CARD('b mod_ring) = CARD('b)"
elims_1: fixes x :: "('b \<times> bal) tree"   and y :: "bool" assumes "avl x = y" obtains "x = \<langle>\<rangle>" and "y"   | l :: "('b \<times> bal) tree" and a :: "'b" and b :: "bal" and r :: "('b \<times> bal) tree" where "x = \<langle>l, (a, b), r\<rangle>" and "y = ((case b of Lh \<Rightarrow> height l = height r + 1 | Bal \<Rightarrow> height r = height l | Rh \<Rightarrow> height r = height l + 1) \<and> avl l \<and> avl r)"
gval_fold_not_invalid_all_valid_contra: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" assumes "\<exists>g\<in>set G. gval g s = invalid" shows "gval (fold gAnd G (Bc True)) s = invalid"
RP_derivation_imp_derive_derivation: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" assumes "chain (\<leadsto>RP) Sts" shows "chain (\<rhd>L) (lmap lclss_of_state Sts)"
bounded_iff: fixes A :: "'c DBMEntry set"   and x :: "'c DBMEntry" assumes "finite A"   and "A \<noteq> {}" shows "(x \<preceq> linordered_monoid.Min A) = (\<forall>a\<in>A. x \<preceq> a)"
approx_slp_sing_eq_None_iff: fixes p :: "nat"   and b :: "floatarith"   and xs :: "((real \<times> real pdevs) \<times> real) list" shows "(approx_slp p [b] xs = None) = (approx_floatarith p b xs = None)"
between_cong: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and C :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A C B"   and "Cong A C A B" shows "C = B"
Field_natLeq_on: fixes n :: "nat" shows "Field (natLeq_on n) = {x. x < n}"
closure_eq_empty: fixes S :: "'c set" shows "(closure S = {}) = (S = {})"
netmap_add_disjoint2: fixes s1 :: "'d net_state"   and s2 :: "'d net_state"   and \<sigma> :: "nat \<Rightarrow> 'd" assumes "\<forall>i\<in>net_ips s1 \<union> net_ips s2. the ((netmap s1 ++ netmap s2) i) = \<sigma> i" shows "\<forall>i\<in>net_ips s2. the (netmap s2 i) = \<sigma> i"
arr_iff_ARR: fixes f :: "'c local.term set" shows "arr f = ARR f"
21: shows "least (Suc ` A) \<in> Suc ` A"
intros: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat" assumes "(\<pi>, n) \<in> scp"   and "\<pi> n \<in> dom local.att" shows "(\<pi>, n) \<in> scop"
Id_refine: fixes S :: "'e nres" shows "S \<le> \<Down> Id S"
convex_on_def: fixes S :: "'b set"   and f :: "'b \<Rightarrow> real" shows "Convex.convex_on S f = (\<forall>x\<in>S. \<forall>y\<in>S. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
small: shows "LLL_invariant False i fs \<or> \<bar>fs.gs.\<mu> i (i - 1)\<bar> \<le> 1 / 2"
tr_ss_map: fixes T :: "x \<Rightarrow> x option"   and s_s'_list :: "(s \<times> s) list"   and a :: "s"   and b :: "s" shows "tr_ss_f T (map fst s_s'_list) = map snd s_s'_list \<and> (\<forall>x\<in>set s_s'_list. case x of (x, xa) \<Rightarrow> tr_s T x xa) \<and> (a, b) \<in> set s_s'_list \<longrightarrow> tr_s T a (tr_s_f T a)"
mat_help: shows "M_mat [[1], [- 1]] [[], [0]] = mat_of_rows_list 2 [[1, 1], [1, - 1]]"
weak_reduction_simulation_impl_OSou: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"   and "\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel\<^sup>*"   and "weak_reduction_simulation (Rel\<inverse>) (STCal Source Target)" shows "operational_sound (TRel\<^sup>*)"