hcju/isabelleps huggingface dataset:
comp_associative: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "times (times x y) z = times x (times y z)"
Fejer_kernel_def: shows "Fejer_kernel \<equiv> \<lambda>n x. if n = 0 then 0 else (\<Sum>r<n. Dirichlet_kernel r x) / real n"
finite_subset: fixes A :: "'f set"   and B :: "'f set" assumes "A \<subseteq> B"   and "finite B" shows "finite A"
ide_compE: fixes g :: "'a"   and f :: "'a" assumes "Right.ide (Right.comp g f)" obtains "Right.seq g f" and "Right.seq f g" and "Right.comp g f = Right.dom f" and "Right.comp g f = Right.cod g"
Fvars_imp: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "Fvars (imp \<phi> \<chi>) = Fvars \<phi> \<union> Fvars \<chi>"
is_pseudonatural_equivalence: shows "pseudonatural_equivalence (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G F \<Phi>\<^sub>F \<sigma>'.map\<^sub>0 \<sigma>'.map\<^sub>1"
simps_1: fixes x21 :: "'a"   and x22 :: "'a"   and x23 :: "'b"   and y21a :: "'a"   and y22a :: "'a"   and y23a :: "'b" shows "(J_C.MkArr x21 x22 x23 = J_C.MkArr y21a y22a y23a) = (x21 = y21a \<and> x22 = y22a \<and> x23 = y23a)"
MGU_is_Unifier: fixes \<sigma> :: "('b \<times> 'b trm) list"   and t :: "'b trm"   and u :: "'b trm" assumes "MGU \<sigma> t u" shows "Unifier \<sigma> t u"
split_Write\<^sub>s\<^sub>b_in_outstanding_refs: fixes a :: "nat"   and xs :: "'b memref list" assumes "a \<in> outstanding_refs is_Write\<^sub>s\<^sub>b xs" shows "\<exists>sop volatile v ys zs A L R W. xs = ys @ Write\<^sub>s\<^sub>b volatile a sop v A L R W # zs"
hn_refine_def: fixes \<Gamma> :: "assn"   and c :: "'e Heap"   and \<Gamma>' :: "assn"   and R :: "'f \<Rightarrow> 'e \<Rightarrow> assn"   and m :: "'f nres" shows "hn_refine \<Gamma> c \<Gamma>' R m \<equiv> nofail m \<longrightarrow> <\<Gamma>> c <\<lambda>r. \<Gamma>' * (\<exists>\<^sub>Ax. R x r * \<up> (RETURN x \<le> m))>\<^sub>t"
e2tail_tail: fixes s :: "recf \<times> nat list \<times> nat list"   and ss :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "e2tail (encode_config (s # ss, rv)) = Partial_Recursive.list_encode (map encode_frame ss)"
a_get_child_nodes_def: fixes ptr :: "(_) object_ptr" shows "get_child_nodes ptr = invoke a_get_child_nodes_tups ptr ()"
ipurge_ref_empty: fixes v :: "'e"   and u :: "'e"   and I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and xs :: "'f list"   and X :: "'f set" assumes "v = u \<or> v \<in> sinks I D u xs"   and "\<And>x. x \<in> X \<Longrightarrow> (v, D x) \<in> I" shows "ipurge_ref I D u xs X = {}"
phils_phil: fixes N :: "nat"   and n :: "nat"   and e :: "dining_event"   and s :: "nat list" assumes "DiningPhilosophers N"   and "0 < n"   and "e \<in> DiningPhilosophers.phils_transitions N n s" shows "phil e < n"
smult_def: shows "smult \<equiv> map_fun id (map_fun id Abs_poly) (\<lambda>a p n. a * coeff p n)"
Derives1_bound: fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "Derives1 a i r b" shows "i < length a"
saturate_eq: fixes X :: "('b \<times> real) nonzero set" shows "Proj -` Proj ` X = (\<Union>c\<in>UNIV - {0}. (*\<^sub>R) c ` X)"
state_repc: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" shows "state (repc cfg) = reps (state cfg)"
update_convs_5: fixes ExtAgrK' :: "rat option \<Rightarrow> rat option"   and NonceS :: "rat option"   and IntMapK :: "rat option"   and ExtMapK :: "rat option"   and IntAgrK :: "rat option"   and ExtAgrK :: "rat option"   and more :: "'a" shows "ExtAgrK_update ExtAgrK' \<lparr>NonceS = NonceS, IntMapK = IntMapK, ExtMapK = ExtMapK, IntAgrK = IntAgrK, ExtAgrK = ExtAgrK, \<dots> = more\<rparr> = \<lparr>NonceS = NonceS, IntMapK = IntMapK, ExtMapK = ExtMapK, IntAgrK = IntAgrK, ExtAgrK = ExtAgrK' ExtAgrK, \<dots> = more\<rparr>"
seq_simps_2: fixes x :: "'c" shows "seq\<cdot>x\<cdot>\<bottom> = \<bottom>"
trcl_uncons_cases: fixes c :: "'f"   and e :: "'g"   and w :: "'g list"   and c' :: "'f"   and S :: "('f \<times> 'g \<times> 'f) set" assumes "(c, e # w, c') \<in> trcl S" obtains ch :: "'f" where "(c, e, ch) \<in> S" and "(ch, w, c') \<in> trcl S"
domainI: fixes x :: "'b" assumes "x \<sim> x" shows "x \<in> domain"
usedBy_bop: fixes x :: "'e \<Longrightarrow> 'f"   and u :: "('g, 'f) uexpr"   and v :: "('h, 'f) uexpr"   and f :: "'g \<Rightarrow> 'h \<Rightarrow> 'i" assumes "x \<natural> u"   and "x \<natural> v" shows "x \<natural> bop f u v"
le_infE: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<le> inf a b" obtains "x \<le> a" and "x \<le> b"
mcont_case_resumption: fixes lub :: "'g set \<Rightarrow> 'g"   and ord :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'f option \<Rightarrow> 'g"   and g :: "'h \<Rightarrow> ('i \<Rightarrow> ('f, 'h, 'i) Resumption.resumption) \<Rightarrow> ('f, 'h, 'i) Resumption.resumption \<Rightarrow> 'g" assumes "mcont (flat_lub None) option_ord lub ord f"   and "\<And>out. mcont resumption.lub_fun resumption.le_fun lub ord (\<lambda>c. g out c (Resumption.resumption.Pause out c))"   and "class.ccpo lub ord (mk_less ord)"   and "\<And>x. ord (f None) x" shows "mcont resumption_lub resumption_ord lub ord (\<lambda>r. case r of Resumption.resumption.Done x \<Rightarrow> f x | Resumption.resumption.Pause out c \<Rightarrow> g out c r)"
a0n_defs_1: shows "a0n \<equiv> \<lparr>init = a0n_init, trans = a0n_trans, obs = id\<rparr>"
skl1_inv4_def: shows "skl1_inv4 \<equiv> {s. \<forall>Rb A B gnx Ni. 0 < signalsResp s (Commit A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>) \<longrightarrow> guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress s Rb = Some {xny, xni, xnr, xgnx, xgny, xsk, xEnd} \<and> sklvl1.guessed_frame Rb xgnx = Some gnx \<and> sklvl1.guessed_frame Rb xni = Some Ni}"
list_all2_Cons2: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'e list"   and y :: "'f"   and ys :: "'f list" shows "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
af_composition: fixes g :: "('f \<Rightarrow> 'd) af"   and f :: "('e \<Rightarrow> 'f) af"   and x :: "'e af" shows "pure (\<circ>) \<diamondop> g \<diamondop> f \<diamondop> x = g \<diamondop> (f \<diamondop> x)"
assms_1: shows "(P, !Q) \<in> bangRel Rel"
left_neutral: fixes a :: "'c" shows "top \<sqinter> a = a"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (mul_instr_sub1 instr_name result s1)) \<and> t2 = snd (fst (mul_instr_sub1 instr_name result s2))"
union_ac_4: fixes x :: "'a"   and y :: "'a"   and M :: "'a multiset" shows "x, y, M = y, x, M"
that: fixes gs :: "'a list" assumes "set gs \<subseteq> generate G A"   and "cycle gs"   and "is_idirprod (generate G A) (\<lambda>g. generate G {g}) (set gs)"   and "successively (dvd) (map ord gs)"   and "card (set gs) \<le> card A" shows "thesis"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and signs :: "signs" shows "P (case signs of Pos \<Rightarrow> f1 | Neg \<Rightarrow> f2) = ((signs = Pos \<longrightarrow> P f1) \<and> (signs = Neg \<longrightarrow> P f2))"
sup_subset_mset_count: fixes A :: "'b multiset"   and B :: "'b multiset"   and x :: "'b" shows "count (A \<union># B) x = max (count A x) (count B x)"
inter_set_filter: fixes A :: "'a set"   and xs :: "'a list" shows "A \<inter> set xs = set (filter (\<lambda>x. x \<in> A) xs)"
pmf_bind_spmf_None: fixes p :: "'d spmf"   and f :: "'d \<Rightarrow> 'c spmf" shows "pmf (p \<bind> f) None = pmf p None + (LINT x|measure_spmf p. pmf (f x) None)"
simps_4: fixes c :: "int"   and e :: "Ferrack.num" shows "isrlfm (NEq (CN 0 c e)) = (0 < c \<and> numbound0 e)"
PO_refines_def: fixes R :: "('e \<times> 'f) set"   and Ta :: "('e, 'g) TS_scheme"   and Tc :: "('f, 'h) TS_scheme" shows "PO_refines R Ta Tc \<equiv> init Tc \<subseteq> R `` init Ta \<and> {R} TS.trans Ta, TS.trans Tc {> R}"
rep_eq: fixes x :: "('a, 'c, 'b, 'd) gen_ssa_cfg_wf" shows "gen_ssa_wf_inEdges x = gen_inEdges (Rep_gen_ssa_cfg_wf x)"
finfun_apply: fixes x :: "'c \<Rightarrow>f 'd" shows "finfun_apply x \<in> finfun"
weak_tau_modality_def: fixes x :: "('idx, 'pred, 'act) formula" shows "weak_tau_modality x \<equiv> Disj (map_bset (tau_steps x) (Abs_bset UNIV))"
g_def: fixes x :: "'c" shows "g x = (if Regions.isIntv (I x) then Lt (real_of_int (- int (intv_const (I x)))) else if Regions.isConst (I x) then Le (real_of_int (- int (intv_const (I x)))) else Lt (real_of_int (- int (k x))))"
sup_assoc: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.semilattice_sup sup less_eq less" shows "sup (sup x y) z = sup x (sup y z)"
extend_indets_power: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and n :: "nat" shows "extend_indets (p ^ n) = extend_indets p ^ n"
eventually_conj_iff: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and F :: "'c filter" shows "(\<forall>\<^sub>F x in F. P x \<and> Q x) = (eventually P F \<and> eventually Q F)"
mask_eq_0_eq_x: fixes x :: "'b word"   and w :: "'b word" shows "(x && w = 0) = (x && ~~ w = x)"
add_edge_wf: fixes G :: "'b graph"   and v :: "'b"   and v' :: "'b" assumes "wf_graph G" shows "wf_graph (FiniteGraph.add_edge v v' G)"
summset_add_mset: fixes a :: "'a vec"   and M :: "'a vec multiset" assumes "a \<in> carrier_vec n"   and "set_mset M \<subseteq> carrier_vec n" shows "M.summset (add_mset a M) = a + M.summset M"
zip_commute: fixes xs :: "'b list"   and ys :: "'c list" shows "zip xs ys = map2 (\<lambda>x y. (y, x)) ys xs"
y: shows "y \<in> aform_err e ((c, xs), d)"
expr_has_density_cexpr_sound_aux: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "expr"   and t :: "pdf_type"   and vs :: "nat list"   and vs' :: "nat list"   and \<delta> :: "cexpr"   and f :: "cexpr" assumes "\<Gamma> \<turnstile> e : t"   and "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c e \<Rightarrow> f"   and "cdens_ctxt_invar vs vs' \<Gamma> \<delta>"   and "free_vars e \<subseteq> set vs \<union> set vs'" shows "dens_ctxt_\<alpha> (vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> (\<lambda>x xa. ennreal (eval_cexpr f x xa)) \<and> is_density_expr (vs, vs', \<Gamma>, \<delta>) t f"
sqr_pos: fixes x :: "real" assumes "x \<noteq> 0" shows "0 < x * x"
priK_in_initState: fixes b :: "keymode"   and A :: "agent" shows "Key (privateKey b A) \<in> initState A"
path_connected_punctured_convex: fixes S :: "'a set"   and a :: "'a" assumes "convex S"   and "aff_dim S \<noteq> 1" shows "path_connected (S - {a})"
Wbis_def: shows "Wbis \<equiv> bis Wretr"
lfp_greatest: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'b \<Rightarrow> 'b"   and A :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>u. less_eq (f u) u \<Longrightarrow> less_eq A u" shows "less_eq A (complete_lattice.lfp Inf less_eq f)"
not_distinct_decomp: assumes "\<not> distinct l" shows "\<exists>xs ys zs y. l = xs @ [y] @ ys @ [y] @ zs"
add_gr_0: fixes m :: "nat"   and n :: "nat" shows "(0 < m + n) = (0 < m \<or> 0 < n)"
atLeastAtMost_borel: fixes a :: "'b"   and b :: "'b" shows "{a..b} \<in> sets borel"
simps_1: shows "local.map (J.MkIde False) = a0"
add_le_imp_le_right: fixes a :: "'b zmultiset"   and c :: "'b zmultiset"   and b :: "'b zmultiset" assumes "a + c \<subseteq>#\<^sub>z b + c" shows "a \<subseteq>#\<^sub>z b"
homotopic_paths_imp_homotopic_loops: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" assumes "homotopic_paths s p q"   and "pathfinish p = pathstart p"   and "pathfinish q = pathstart p" shows "homotopic_loops s p q"
simulation_refl: shows "simulation Id (stutter_extend B) (stutter_extend B)"
a1: shows "i < dim_row (\<langle>|v\<rangle>| * A\<^sup>\<dagger>)"
sup_bot_left: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and bot :: "'a"   and x :: "'a" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup bot x = x"
count_Bag: fixes xs :: "('b, nat) alist" shows "count (Bag xs) = count_of (alist.impl_of xs)"
hierauto_def: shows "hierauto = {(D, F, E, G) |D F E G. HierAuto D F E G}"
gr\<I>_1: shows "ground (subst_range \<I>)"
simps_1: fixes k :: "nat"   and c :: "nat" shows "shift k c TNat = TNat"
keys_mult: fixes f :: "'b \<Rightarrow>\<^sub>0 'c"   and g :: "'b \<Rightarrow>\<^sub>0 'c" shows "keys (f * g) \<subseteq> {a + b |a b. a \<in> keys f \<and> b \<in> keys g}"
assms_3: shows "c \<in>\<^sub>\<circ> dg_prod J \<AA>\<lparr>Obj\<rparr>"
subl: shows "{flow0 x t1<--<flow0 x t2} \<subseteq> {a<--<flow0 x t2}"
assms: shows "s' = snd (fst (add_instr_sub1 instr_name result rs1_val operand2 s)) \<and> get_S (cpu_reg_val PSR s) = 0"
asubg_inc_zero: fixes H :: "'a set" assumes "M +> H" shows "\<zero> \<in> H"
rtrancl_into_trancl1: fixes a :: "'ap"   and b :: "'ap"   and r :: "('ap \<times> 'ap) set"   and c :: "'ap" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>+"
cardl: shows "card (obs_ids \<pi> \<inter> {..<j}) = l"
stepB: shows "(!p \<cdot> l + y + p \<cdot> r + !p \<cdot> r)\<^sup>\<infinity> = (y + !p \<cdot> l + r \<cdot> !p)\<^sup>\<infinity> \<cdot> (r \<cdot> p)\<^sup>\<infinity>"
nonzero_Basis: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and u :: "'c" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis" shows "u \<noteq> zero"
SC_case: fixes l :: "nat list" shows "SC l < ack 1 (sum_list l)"
fail_tgt: shows "target (io_init @ [io_last] || tr2 @ [fst FAIL] || tr1_init @ [snd FAIL]) (initial M2, initial M1) = FAIL"
PiE_dflt_empty_iff: fixes A :: "'b set"   and dflt :: "'c"   and B :: "'b \<Rightarrow> 'c set" shows "(PiE_dflt A dflt B = {}) = (\<exists>x\<in>A. B x = {})"
le_anti_symmetry: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Le Bet Cong A B C D"   and "Tarski_neutral_dimensionless.Le Bet Cong C D A B" shows "Cong A B C D"
continuous_rhs_comp: fixes S :: "'b set"   and g :: "'b \<Rightarrow> real"   and h :: "'b \<Rightarrow> 'a" assumes "continuous_on S g"   and "continuous_on S h"   and "g ` S \<subseteq> T"   and "h ` S \<subseteq> X" shows "continuous_on S (\<lambda>x. f (g x) (h x))"
rel_conversep: fixes R1 :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_prod R1\<inverse>\<inverse> R2\<inverse>\<inverse> = (rel_prod R1 R2)\<inverse>\<inverse>"
poly_root_factor_1: fixes r :: "'a"   and q :: "'a poly"   and k :: "'a" shows "(poly ([:r, 1::'a:] * q) k = (0::'a)) = (k = - r \<or> poly q k = (0::'a))"
length_append: fixes xs :: "'d list"   and ys :: "'d list" shows "length (xs @ ys) = length xs + length ys"
mtf20: fixes x :: "'a"   and xs :: "'a list" shows "mtf2 0 x xs = xs"
mono: fixes x :: "'b" shows "heap.mono_body (\<lambda>f. U (F (C f)) x)"
X_to_Un_supp_eqvt: fixes pi :: "('b \<times> 'b) list"   and X :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pi \<bullet> X_to_Un_supp X = X_to_Un_supp (pi \<bullet> X)"
par_by_merge_alt_def: fixes P :: "('d, 'f) urel"   and M :: "(('d, 'f, 'g) mrg, 'e) urel"   and Q :: "('d, 'g) urel" shows "P \<parallel>\<^bsub>M\<^esub> Q = (\<lceil>P\<rceil>\<^sub>0 \<and> \<lceil>Q\<rceil>\<^sub>1 \<and> $\<Sigma>\<^sub><\<acute> =\<^sub>u $\<^bold>v) ;; M"
sel_2: fixes z :: "complex" shows "Im (cnj z) = - Im z"
hcomp_closed: fixes f :: "'a"   and g :: "'a" assumes "arr f \<and> src f = a \<and> trg f = a"   and "arr g \<and> src g = a \<and> trg g = a"   and "trg f = src g" shows "hseq g f \<and> src (g \<star> f) = a \<and> trg (g \<star> f) = a"
inf_option_def: fixes x :: "'b option"   and y :: "'b option" shows "x \<sqinter> y = (case x of None \<Rightarrow> None | Some x \<Rightarrow> case y of None \<Rightarrow> None | Some y \<Rightarrow> Some (x \<sqinter> y))"
card_2_iff: fixes S :: "'c set" shows "(card S = 2) = (\<exists>x y. S = {x, y} \<and> x \<noteq> y)"
prems_2: shows "set (a # fs) \<subseteq> carrier G"
conjug_add_exp: fixes u :: "'b list"   and v :: "'b list"   and k :: "nat" assumes "u \<sim> v" shows "u \<^sup>@ k \<sim> v \<^sup>@ k"
thetaAtm_sym: fixes atm :: "'b" shows "sym (thetaAtm atm)"
1: fixes u :: "'v" shows "foreach_impl_body u = (\<lambda>(v, d) (Qi, \<pi>i). let qiv = Q_lookup Qi v; \<pi>iv = M_lookup \<pi>i v in if v \<noteq> r \<and> (qiv \<noteq> None \<or> \<pi>iv = None) \<and> enat d < enat_of_option qiv then (Q_update v d Qi, M_update v u \<pi>i) else (Qi, \<pi>i))"
inject_2: fixes x2 :: "'a"   and y2 :: "'a" shows "(ENV x2 = ENV y2) = (x2 = y2)"
offs_next_zero_cons_eq: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and x :: "'c"   and ns :: "nat list"   and mi :: "'d"   and ma :: "'d"   and i :: "nat"   and xs :: "'c list"   and ub :: "nat" assumes "index key x (length ns) mi ma = i"   and "offs_num (length ns) (x # xs) index key mi ma 0 = 0"   and "offs_set_prev ns (x # xs) index key mi ma i \<noteq> {}" shows "offs_next (ns[i := Suc (ns ! i)]) ub xs index key mi ma 0 = offs_next ns ub (x # xs) index key mi ma 0"
Upair_def: fixes a :: "ZF"   and b :: "ZF" shows "Upair a b \<equiv> Repl (Power (Power Empty)) (\<lambda>x. if x = Empty then a else b)"
enum_all: shows "enum_all = (\<lambda>P. all_n_lists enum_all_b (\<lambda>bs. P (the \<circ> map_of (zip enum_a bs))) (length enum_a))"
hyps: assumes "xs \<noteq> []" shows "min_list_param rel xs \<in> set xs"
manyof_def: fixes cs :: "char list" shows "manyof cs = many (\<lambda>c. c \<in> set cs)"
scale_cancel_right: fixes a :: "real"   and x :: "'a"   and b :: "real" shows "(a *\<^sub>R x = b *\<^sub>R x) = (a = b \<or> x = (0::'a))"
IH_2: fixes Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm" assumes "(X, Y) \<in> qSwapped" shows "\<forall>X' zs z1 z2. Y #= X' \<longrightarrow> Y #[[z1 \<and> z2]]_zs #= X' #[[z1 \<and> z2]]_zs"
eqpoll_Fpow: fixes A :: "'b set" assumes "infinite A" shows "Fpow A \<approx> A"
strong_induct: fixes P :: "'a \<Rightarrow> trm \<Rightarrow> bool"   and z :: "'a"   and trm :: "trm" assumes "\<And>vrs z. P z (Var vrs)"   and "\<And>vrs trm ty z. \<lbrakk>vrs \<sharp> z; vrs \<sharp> ty; \<And>z. P z trm\<rbrakk> \<Longrightarrow> P z (\<lambda>vrs:ty. trm)"   and "\<And>tyvrs trm ty z. \<lbrakk>tyvrs \<sharp> z; tyvrs \<sharp> ty; \<And>z. P z trm\<rbrakk> \<Longrightarrow> P z (\<lambda>tyvrs<:ty. trm)"   and "\<And>trm1 trm2 z. \<lbrakk>\<And>z. P z trm1; \<And>z. P z trm2\<rbrakk> \<Longrightarrow> P z (trm1 \<cdot> trm2)"   and "\<And>trm ty z. (\<And>z. P z trm) \<Longrightarrow> P z (trm \<cdot>\<^sub>\<tau> ty)" shows "P z trm"
domains_comp: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" assumes "CopxC.seq g f" shows "CopxC.domains (g \<odot> f) = CopxC.domains f"
r_2: shows "Sigma_Algebra.measure M {x. infdist x S = r} = 0"
hoaret: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P p Q,A"
state_def: fixes n :: "nat"   and v :: "complex Matrix.mat" shows "state n v \<equiv> dim_col v = 1 \<and> dim_row v = 2 ^ n \<and> \<parallel>Matrix.col v 0\<parallel> = 1"
index_mult_mat_vec: fixes i :: "nat"   and A :: "'b mat"   and v :: "'b vec" assumes "i < dim_row A" shows "(A *\<^sub>v v) $ i = row A i \<bullet> v"
assms_5: shows "S' \<subseteq> \<Union> (range f)"
Quotient_fset: shows "Quotient list_eq abs_fset rep_fset cr_fset"
exhaust: fixes y :: "obj_tag" obtains x1 :: "qtname" where "y = CInst x1"   | x21 :: "ty" and x22 :: "int" where "y = Arr x21 x22"
that: fixes x :: "'a"   and xs' :: "'a list"   and y :: "'b"   and ys' :: "'b list" assumes "xs = x # xs'"   and "ys = y # ys'"   and "xy = (x, y)"   and "xys = zip xs' ys'" shows "thesis"
lossless_R1: fixes msgs :: "bool \<times> bool"   and \<sigma> :: "bool" shows "lossless_spmf (R1 msgs \<sigma>)"
remdups_filter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "remdups (List.filter P xs) = List.filter P (remdups xs)"
simps_5: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "not\<^sub>n (\<phi> and\<^sub>n \<psi>) = not\<^sub>n \<phi> or\<^sub>n not\<^sub>n \<psi>"
nth_Cons': fixes x :: "'k"   and xs :: "'k list"   and n :: "nat" shows "(x # xs) ! n = (if n = 0 then x else xs ! (n - 1))"
poly_dvd_1: fixes p :: "'c poly" shows "(p dvd 1) = (degree p = 0 \<and> coeff p 0 dvd (1::'c))"
seqI': fixes f :: "('b \<Rightarrow> 'a, 'b \<Rightarrow> 'a) concrete_category.arr"   and a :: "('b \<Rightarrow> 'a, 'b \<Rightarrow> 'a) concrete_category.arr"   and b :: "('b \<Rightarrow> 'a, 'b \<Rightarrow> 'a) concrete_category.arr"   and g :: "('b \<Rightarrow> 'a, 'b \<Rightarrow> 'a) concrete_category.arr"   and c :: "('b \<Rightarrow> 'a, 'b \<Rightarrow> 'a) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] c\<guillemotright>" shows "J_C.seq g f"
borel_eq_sigmaI1: fixes X :: "'c set set"   and F :: "'d \<Rightarrow> 'c set"   and A :: "'d set" assumes "borel = sigma UNIV X"   and "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (F ` A))"   and "\<And>i. i \<in> A \<Longrightarrow> F i \<in> sets borel" shows "borel = sigma UNIV (F ` A)"
geometric_series_id: fixes a :: "'a"   and b :: "'a"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "a [^] Suc n \<ominus> b [^] Suc n = (a \<ominus> b) \<otimes> geometric_series n a b"
csppa_Formal_in_in_slice: 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 Exit :: "'b"   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 x :: "nat"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and m :: "'b"   and i :: "nat"   and S :: "'b SDG_node set" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "x < length fs"   and "Formal_in (m, x + i) \<in> S" shows "SDG.csppa m S i fs ! x = fs ! x"