hcju/isabelleps huggingface dataset:
verit_la_disequality: fixes a :: "'h"   and b :: "'h" shows "a = b \<or> \<not> a \<le> b \<or> \<not> b \<le> a"
gfp_the_while_option_lattice: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f"   and "finite UNIV" shows "gfp f = the (while_option (\<lambda>A. f A \<noteq> A) f top)"
unforwarded_witness: fixes a :: "nat"   and sb :: "'b memref list"   and X :: "nat set" assumes "a \<in> unforwarded_non_volatile_reads sb X" shows "\<exists>xs ys t v. sb = xs @ Read\<^sub>s\<^sub>b False a t v # ys \<and> a \<notin> outstanding_refs is_Write\<^sub>s\<^sub>b xs"
num_sylow_normalizer: fixes P :: "'a set" assumes "P \<in> subgroups_of_size (p ^ a)" shows "card (rcosets\<^bsub>G\<lparr>carrier := conjG.stabilizer P\<rparr>\<^esub> P) * p ^ a = card (conjG.stabilizer P)"
comm_ring_hom_axioms: fixes a :: "'b" shows "comm_ring_hom (\<lambda>p. poly p a)"
cond_spmf_alt: fixes p :: "'h spmf"   and A :: "'h set" shows "cond_spmf p A = mk_lossless (p \<upharpoonleft> A)"
elem_of_symcl_2: fixes a :: "'c"   and b :: "'c"   and Rel :: "('c \<times> 'c) set" assumes "(a, b) \<in> Rel" shows "(b, a) \<in> symcl Rel"
ground_n_entails_bot: shows "\<G>_Fset N \<Turnstile>G {BG}"
proj_psi_proj_k_le_exproj_k: fixes k :: "nat"   and l :: "nat" shows "tensor_P (proj_psi_l k) (proj_k l) \<le>\<^sub>L tensor_P (1\<^sub>m N) (proj_k l)"
ExI: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" assumes "\<exists>y. [\<phi> y in v]" shows "[\<^bold>\<exists>x. \<phi> x in v]"
ifbd_top_pres: shows "top_pres bd\<^sup>-\<^sub>\<F>"
perp2_sym: fixes P :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "P Perp2 A B C D" shows "P Perp2 C D A B"
edg_simps_1: fixes x :: "'v"   and v :: "'v vertex" shows "edg (Vertex x) v = (\<exists>y. v = Edge x y \<and> edge \<Delta> x y)"
trans: fixes \<delta> :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'a"   and s1 :: "'a"   and s2 :: "'a"   and s3 :: "'a" assumes "pre_RDR.less_eq \<delta> s1 s2"   and "pre_RDR.less_eq \<delta> s2 s3" shows "pre_RDR.less_eq \<delta> s1 s3"
bij_eq_rename: fixes h :: "'c \<Rightarrow> 'd"   and t :: "'d program" assumes "bij h" shows "t = rename h (inv (rename h) t)"
un_Node3_def: fixes tree :: "'a tree" shows "un_Node3 tree = (case tree of \<langle>x21, x22, x23\<rangle> \<Rightarrow> x23)"
fromMOD_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) Recursion.model" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.fromMOD Delta MOD \<equiv> \<lparr>igWls = \<lambda>s X'X. FixSyn.wls Delta s (fst X'X) \<and> gWls MOD s (snd X'X), igWlsAbs = \<lambda>us_s A'A. FixSyn.wlsAbs Delta us_s (fst A'A) \<and> gWlsAbs MOD us_s (snd A'A), igVar = \<lambda>xs x. (FixVars.Var xs x, gVar MOD xs x), igAbs = \<lambda>xs x X'X. (FixVars.Abs xs x (fst X'X), gAbs MOD xs x (fst X'X) (snd X'X)), igOp = \<lambda>delta iinp biinp. (FixVars.Op delta (lift fst iinp) (lift fst biinp), gOp MOD delta (lift fst iinp) (lift snd iinp) (lift fst biinp) (lift snd biinp)), igFresh = \<lambda>ys y X'X. fresh ys y (fst X'X) \<and> gFresh MOD ys y (fst X'X) (snd X'X), igFreshAbs = \<lambda>ys y A'A. freshAbs ys y (fst A'A) \<and> gFreshAbs MOD ys y (fst A'A) (snd A'A), igSwap = \<lambda>zs z1 z2 X'X. (FixVars.swap zs z1 z2 (fst X'X), gSwap MOD zs z1 z2 (fst X'X) (snd X'X)), igSwapAbs = \<lambda>zs z1 z2 A'A. (FixVars.swapAbs zs z1 z2 (fst A'A), gSwapAbs MOD zs z1 z2 (fst A'A) (snd A'A)), igSubst = \<lambda>ys Y'Y y X'X. (FixVars.subst ys (fst Y'Y) y (fst X'X), gSubst MOD ys (fst Y'Y) (snd Y'Y) y (fst X'X) (snd X'X)), igSubstAbs = \<lambda>ys Y'Y y A'A. (FixVars.substAbs ys (fst Y'Y) y (fst A'A), gSubstAbs MOD ys (fst Y'Y) (snd Y'Y) y (fst A'A) (snd A'A))\<rparr>"
x: shows "x \<in> RF\<^sup>\<circ> S"
induct: fixes P :: "(Vars.var \<times> exp) list \<Rightarrow> bool"   and a0 :: "(Vars.var \<times> exp) list" assumes "P []"   and "\<And>v e binds. P (delete v binds) \<Longrightarrow> P ((v, e) # binds)" shows "P a0"
hmset_pair_decompose: fixes m1 :: "hmultiset"   and m2 :: "hmultiset" shows "\<exists>k n1 n2. m1 = k + n1 \<and> m2 = k + n2 \<and> (head_\<omega> n1 \<noteq> head_\<omega> n2 \<or> n1 = 0 \<and> n2 = 0)"
add_new_mmuaux'_def: shows "add_new_mmuaux' = add_new_mmuaux"
simps: fixes e :: "'c"   and a :: "'d"   and q :: "('c, 'd) BsSkewBinomialTree list" shows "deleteMin' (Element e a q) = (case findMin q of Element ey ay q1 \<Rightarrow> Element ey ay (meld q1 (deleteMin q)))"
compl_le_swap1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'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 y :: "'b"   and x :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "less_eq y (uminus x)" shows "less_eq x (uminus y)"
mult_cancel_right1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and c :: "'a"   and b :: "'a" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(c = times b c) = (c = zero \<or> b = one)"
abs_summable_on_union: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" assumes "f abs_summable_on A"   and "f abs_summable_on B" shows "f abs_summable_on A \<union> B"
chosen_right_adjoints_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" shows "chosen_right_adjoints V H \<a> \<i> src trg \<equiv> bicategory V H \<a> \<i> src trg"
diff_invariant_def: fixes I :: "'b \<Rightarrow> bool"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and G :: "'b \<Rightarrow> bool" shows "diff_invariant I f U S t\<^sub>0 G \<equiv> (\<Union> \<circ> \<P> (g_orbital f G U S t\<^sub>0)) {s. I s} \<subseteq> {s. I s}"
SReal_Int_Infinitesimal_zero: shows "\<real> \<inter> Infinitesimal = {0}"
outside_Un_outside_Un: fixes S :: "'a set"   and T :: "'a set"   and U :: "'a set" assumes "S \<inter> outside (T \<union> U) = {}" shows "outside (T \<union> U) \<subseteq> outside (T \<union> S)"
HoHV_def: fixes \<mu> :: "'a \<times> 'a \<times> 'a" shows "H.HoHV \<mu> \<equiv> if H.VVV.arr \<mu> then (fst \<mu> \<cdot> fst (snd \<mu>)) \<cdot> snd (snd \<mu>) else V.null"
flatten_set_group_mod_inj: shows "inj_on (\<lambda>U. SOME g. g \<in> U) (carrier (G Mod H))"
ide_is_iso: fixes a :: "'s" assumes "ide a" shows "local.iso a"
awalkE': fixes G :: "('b, 'c) pre_digraph"   and u :: "'b"   and p :: "'c list"   and v :: "'b" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v" obtains "set (pre_digraph.awalk_verts G u p) \<subseteq> verts G" and "set p \<subseteq> arcs G" and "pre_digraph.cas G u p v" and "hd (pre_digraph.awalk_verts G u p) = u" and "last (pre_digraph.awalk_verts G u p) = v" and "u \<in> verts G" and "v \<in> verts G"
r_def: shows "r = (p - 1) div 2 * ((q - 1) div 2)"
addrow_carrier: fixes ad :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mul :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and k :: "nat"   and l :: "nat"   and A :: "'b mat"   and n :: "nat"   and nc :: "nat" shows "(mat_addrow_gen ad mul a k l A \<in> carrier_mat n nc) = (A \<in> carrier_mat n nc)"
hyps_3: assumes "\<And>a1 a2. \<lbrakk>a1 \<in> A; a2 \<in> A; a1 \<noteq> a2\<rbrakk> \<Longrightarrow> keys (f a1) \<inter> keys (f a2) = {}" shows "keys (sum f A) = (\<Union>a\<in>A. keys (f a))"
simps_1: fixes G :: "'a list_graph" shows "implc_get_offending_flows [] G = []"
SN_preserved: fixes t1 :: "lam"   and t2 :: "lam" assumes "SN t1"   and " t1 \<longrightarrow>\<^sub>\<beta> t2" shows "SN t2"
list_set_rel_sv: fixes R :: "('a \<times> 'b) set" assumes "single_valued R" shows "single_valued (\<langle>R\<rangle>list_set_rel)"
simps_2: fixes h :: "'b \<Rightarrow> int"   and t1 :: "'b Tree"   and x :: "'b"   and t2 :: "'b Tree" shows "sortedTree h (T t1 x t2) = (sortedTree h t1 \<and> (\<forall>l\<in>setOf t1. h l < h x) \<and> (\<forall>r\<in>setOf t2. h x < h r) \<and> sortedTree h t2)"
Sup_inf_eq_bot_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and B :: "'a set"   and a :: "'a" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "(inf (Sup B) a = bot) = (\<forall>b\<in>B. inf b a = bot)"
obind_def: fixes f :: "'d \<Rightarrow> 'f option"   and g :: "'f \<Rightarrow> 'd \<Rightarrow> 'e option" shows "f |>> g \<equiv> \<lambda>s. case f s of None \<Rightarrow> None | Some x \<Rightarrow> g x s"
always_eventually_always: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<diamond>\<box>\<diamond>P = \<box>\<diamond>P"
cis_mult_cmod_id: fixes x :: "complex" shows "cis (arg x) * complex_of_real (cmod x) = x"
vrange_atE: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<phi> i\<lparr>ArrMap\<rparr>)" obtains a :: "V" where "a \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" and "\<phi> i\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> = b"
rep_eq: fixes x :: "'d \<Rightarrow>\<^sub>0 'e"   and xa :: "'d \<Rightarrow>\<^sub>0 'e" shows "lookup (x - xa) = (\<lambda>k. lookup x k - lookup xa k)"
hyps_3: shows "otherwith S {i} (orecvmsg R) \<sigma> \<sigma>' a"
simp: shows "local.quadratic_hensel_loop j = (if j \<le> 1 then (p, S1, T1, D1, H1) else if even j then case local.quadratic_hensel_loop (j div 2) of (q, S, T, D, H) \<Rightarrow> let qq = q * q in case local.quadratic_hensel_step q S T D H of (S', T', D', H') \<Rightarrow> (qq, S', T', D', H') else case local.quadratic_hensel_loop (j div 2 + 1) of (q, S, T, D, H) \<Rightarrow> case local.quadratic_hensel_step q S T D H of (S', T', D', H') \<Rightarrow> let qq = q * q; pj = qq div p; down = poly_mod.Mp pj in (pj, down S', down T', down D', down H'))"
rev_finite_subset: fixes B :: "'f set"   and A :: "'f set" assumes "finite B"   and "A \<subseteq> B" shows "finite A"
cfg_onD_action: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> MDP.cfg_on s" shows "action cfg \<in> K s"
elims: fixes x :: "('a, 'b) ta_rule"   and y :: "'a set" assumes "rule_states x = y" obtains q :: "'a" and l :: "'b" and qs :: "'a list" where "x = q \<rightarrow> l qs" and "y = insert q (set qs)"
mult_ac_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * b * c = a * (b * c)"
simps_48: fixes x6 :: "(char list \<times> trm) list"   and x41 :: "trm"   and x42 :: "trm" shows "Rcd x6 \<noteq> x41 \<bullet> x42"
Sum1_1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi" assumes "P \<Longrightarrow>a<\<nu>x> \<prec> P'" shows "P \<oplus> Q \<Longrightarrow>a<\<nu>x> \<prec> P'"
table_mem_sub: fixes v :: "val"   and v' :: "val"   and t :: "(val \<times> val) list" assumes "(v, v') \<in> set t" shows "Tf t <:: T v \<rightarrow> T v'"
mult_vec_zero_right: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "p ** 0 = 0"
forest_components_equivalence: fixes x :: "'a" assumes "x \<sqinter> id x \<le> top" shows "(top \<le> top \<sqinter> top \<and> top \<sqinter> top \<sqinter> (top \<sqinter> top) \<le> top \<sqinter> top) \<and> local.comp_inf.symmetric (top \<sqinter> top)"
\<phi>_eq: fixes x :: "real" assumes "x \<in> {(real_of_int i' - 1) / 2 ^ (l' + 1)..(real_of_int i' + 1) / 2 ^ (l' + 1)}" shows "\<phi> (l, i) x = 2 ^ (l + 1) * x - (real_of_int i - 1)"
Zero_not_eq_One: shows "quiver.Zero \<noteq> quiver.One"
4: shows "inconsist i (f \<triangleright> (n + m)) \<down>"
is_interval_cc: fixes b :: "'a"   and a :: "'a" shows "is_interval {b..a}"
interpret_others_1: fixes f :: "atom fm"   and xs :: "'a list" shows "QEdlo_ex.interpret (AllQ f) xs = (\<forall>x. QEdlo_ex.interpret f (x # xs))"
ord_eq_le_trans: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd"   and c :: "'c \<times> 'c \<Rightarrow> 'd" assumes "a = b"   and "b \<preceq> c" shows "a \<preceq> c"
suffixes_rev: fixes xs :: "'a list" shows "suffixes (rev xs) = map rev (prefixes xs)"
permute_zero: fixes x :: "'b \<Rightarrow> 'b" shows "atom_fun_permute.permute_fun 0 x = x"
h: shows "|~ P \<longrightarrow> \<circle>P \<or> \<circle>Q"
integral_fst: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "integrable (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). f x y)" shows "LINT x|M1. integral\<^sup>L M2 (f x) = LINT (x, y)|(M1 \<Otimes>\<^sub>M M2). f x y"
filterlim_of_int_at_top: fixes f :: "real \<Rightarrow> 'a"   and F :: "'a filter" assumes "filterlim f F at_top" shows "LIM x at_top. f (real_of_int x) :> F"
Let_def: fixes s :: "'c"   and f :: "'c \<Rightarrow> 'b" shows "Let s f \<equiv> f s"
2: shows "a \<in> allAllocations ({seller} \<union> N) (set \<Omega>)"
variation_congr: fixes x :: "real"   and x' :: "real"   and y :: "real"   and y' :: "real" assumes "sgn x = sgn x'"   and "sgn y = sgn y'" shows "variation x y = variation x' y'"
bres_interchange: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "less_eq_prod (times_prod' (prod.bres x y) z) (prod.bres x (times_prod' y z))"
aligned\<^sub>e_encode: fixes e :: "nat"   and M :: "nat" shows "local.aligned\<^sub>e e (local.encode e M)"
span_superset: fixes S :: "'b set" shows "S \<subseteq> m1.span S"
finite_Bounded_vec_Max: fixes A :: "'b vec set"   and n :: "nat" assumes "A \<subseteq> carrier_vec n"   and "finite A" shows "A \<subseteq> Bounded_vec (Max {\<bar>a $ i\<bar> |a i. a \<in> A \<and> i < n})"
prems_2: fixes x :: "'a" shows "(x \<in> T) = (x \<in> T') \<or> f x = TTree.empty"
simps_1: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and x :: "'a"   and xs :: "'a list" shows "not_MT C (x # xs) = (if dom (C x) = {} then not_MT C xs else True)"
a2: shows " \<theta><\<Gamma>> \<turnstile> t1 : \<theta><T1> "
connected_component_of_subset: fixes S :: "'b set"   and x :: "'b"   and y :: "'b"   and T :: "'b set" assumes "connected_component S x y"   and "S \<subseteq> T" shows "connected_component T x y"
4: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ B) (Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ Abs \<tau> (B $ Bv 0))"
hyps_1: fixes a :: "coname"   and M :: "trm"   and M' :: "trm" assumes "<a>:M \<in> \<parallel><A>\<parallel>"   and "M \<longrightarrow>\<^sub>a* M'" shows "<a>:M' \<in> \<parallel><A>\<parallel>"
4: fixes g :: "'t \<Rightarrow>\<^sub>0 'b" assumes "g \<in> M" shows "lt (q'' g \<odot> g) \<preceq>\<^sub>t v"
simps_3: fixes lev :: "nat"   and u :: "Term_Class.term"   and t :: "Term_Class.term" shows "replace_bound lev (\<Lambda> u) t = \<Lambda> replace_bound (lev + 1) u t"
shift_0: fixes xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list" shows "shift 0 xt = xt"
I_def: shows "I = (\<lambda>s. invsL s \<and> invs s\<down>)"
X: shows "X \<in> sets (dens_ctxt_measure ({}, V', \<Gamma>, \<lambda>_. 1) \<rho> \<bind> f)"
map_to_nat_correct_4: fixes s :: "('a, unit) HashMap.hashmap" assumes "hs.invar s" shows "hm.invar (hh_map_to_nat.map_to_nat s)"
map_filter_finite: fixes m :: "'a \<Rightarrow> 'b option"   and P :: "'a \<Rightarrow> bool" assumes "finite (dom m)" shows "finite (dom (map_filter P m))"
Bossy_CH_BCh: fixes X :: "'a set" shows "Bossy.CH X = BCh (Xh d1h1) X \<union> BCh (Xh d1h3) X"
subgroup_imp_group: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set" assumes "Group.group G"   and "subgroup H G" shows "Group.group (G\<lparr>carrier := H\<rparr>)"
surj_iff: fixes f :: "'e \<Rightarrow> 'd" shows "surj f = (f \<circ> inv f = id)"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "ext ext"   and "ext_compat_cons_axioms ext" shows "ext_compat_cons ext"
blinfun_apply_cases: fixes y :: "'d \<Rightarrow> 'e" assumes "y \<in> {f. bounded_linear f}" obtains x :: "'d \<Rightarrow>\<^sub>L 'e" where "y = blinfun_apply x"
orbit_cong: fixes t :: "'b"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "t \<in> orbit f t"   and "\<And>s. s \<in> orbit f t \<Longrightarrow> g s = f s" shows "orbit g t = orbit f t"
rderiv_lderiv: fixes as :: "'a"   and r :: "'b rexp" shows "rderiv as r = REV (lderiv as (REV r))"
simps_17: fixes x1 :: "'a" shows "vars_hd (Var x1) = {x1}"
base_list_and: fixes Sji :: "'a set list"   and li :: "'a set"   and s :: "'a set" assumes "Sji \<noteq> []"   and "tl Sji \<noteq> []"   and "Sji ! 0 = li"   and "Sji ! (length Sji - 1) = s"   and "\<And>i. i < length Sji - 1 \<Longrightarrow> \<turnstile>\<N>\<^bsub>(Sji ! i, Sji ! Suc i)\<^esub>" shows "\<turnstile>map (\<lambda>i. \<N>\<^bsub>(Sji ! i, Sji ! Suc i)\<^esub>) [0..<length Sji - Suc 0] \<oplus>\<^sub>\<and>\<^bsup>(li, s)\<^esup>"
foo29: fixes a :: "((int \<times> int) set set, int) Combinators"   and b :: "((int \<times> int) set set, int) Combinators" assumes "dom (C a) \<noteq> {}"   and "dom (C a) \<inter> dom (C b) = {}" shows "a \<noteq> b"
perm_supp_eq: fixes p :: "perm"   and x :: "'b" assumes "supp p \<sharp>* x" shows "p \<bullet> x = x"
span1: shows "span {y. a \<bullet> y = 0} \<subseteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
axioms_1: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" assumes "reducedly_dynamically_complete_calculus Bot Inf entails Red_I Red_F" shows "calculus Bot Inf entails Red_I Red_F"
homotopy_equivalent_space_imp_isomorphic_reduced_homology_groups: fixes X :: "'a topology"   and Y :: "'b topology"   and p :: "int" assumes "X homotopy_equivalent_space Y" shows "reduced_homology_group p X \<cong> reduced_homology_group p Y"
longest_prefix: fixes j :: "nat"   and z :: "nat" assumes "\<exists>t. local.prefixes t j \<up>"   and "z = longest_prefix j" shows "(\<forall>x<z. \<psi> j x \<down>) \<and> (\<forall>x\<ge>z. \<psi> j x \<up>)"
update_strict_mono: fixes f :: "'b \<Rightarrow> 'b" assumes "\<And>x. x < f x" shows "strict_mono_state (State_Monad.update f)"
cong_exp_iff_simps_10: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'c \<times> 'c"   and divmod_step :: "num \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c \<times> 'c"   and m :: "num"   and q :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus (num.Bit0 m)) (numeral.numeral one plus (num.Bit0 q)) = modulo (numeral.numeral one plus (num.Bit1 n)) (numeral.numeral one plus (num.Bit0 q))) = False"
rank_match: shows "vec_space.rank (dim_row (take_rows_from_matrix (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))) (rows_to_keep (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))))) (take_rows_from_matrix (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))) (rows_to_keep (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))))) = dim_row (take_rows_from_matrix (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))) (rows_to_keep (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))))"
ub_sgn: shows "\<forall>x\<ge>ub. \<forall>p\<in>set ps. sgn (poly p x) = sgn_pos_inf p"
prems_2: shows "add_mset a M \<in> multisets A"
ipurge_ref_swap: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'b list"   and X :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "ipurge_ref I D u xs {x \<in> X. P x} = {x \<in> ipurge_ref I D u xs X. P x}"
fg_def: shows "fg = (\<lambda>w. f w * g w)"
intros_2: fixes p :: "'node"   and R :: "'node set"   and p' :: "'node"   and Sl :: "'node set"   and q :: "'node" assumes "not_blocked p R p'"   and "p' \<in> W"   and "Sl \<in> slices p'"   and "\<forall>q\<in>Sl \<inter> W. q \<notin> R \<and> \<not> blocking_min R q"   and "q \<in> Sl" shows "not_blocked p R q"
scalar_right_zero: fixes v :: "'a list"   and nn :: "nat" shows "scalar_prod v (vec0 nn) = (0::'a)"
module_hom_ident: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "module scale" shows "module_hom scale scale (\<lambda>x. x)"
af_ite_simps_1: fixes P :: "bool"   and w :: "'a set list" shows "af (if P then true\<^sub>n else false\<^sub>n) w = (if P then true\<^sub>n else false\<^sub>n)"
terminalE: fixes b :: "'a"   and a :: "'a" assumes "A.terminal b"   and "A.ide a" obtains f :: "'a" where "\<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright>" and "\<And>f'. \<guillemotleft>f' : a \<rightarrow>\<^sub>A b\<guillemotright> \<Longrightarrow> f' = f"
simps_1: fixes x :: "'v"   and y :: "'v" shows "h_plus 0 (x, y) = (if x = source \<Delta> then g (x, y) else 0)"
simps: fixes \<phi> :: "'a ltl" shows "\<P> \<phi> = (\<Delta>\<^sub>\<times> (\<lambda>\<chi>. semi_mojmir_def.step \<Sigma> \<up>af\<^sub>G (Abs (theG \<chi>))), \<iota>\<^sub>\<times> (\<^bold>G (G \<phi>)) (\<lambda>\<chi>. semi_mojmir_def.initial (Abs (theG \<chi>))), {local.combine_pairs' {\<^bold>\<upharpoonleft>\<^sub>\<chi> (mojmir_to_rabin_def.fail\<^sub>R \<Sigma> \<up>af\<^sub>G (Abs (theG \<chi>)) {q. \<G> \<Turnstile>\<^sub>P Rep q} \<union> mojmir_to_rabin_def.merge\<^sub>R \<up>af\<^sub>G (Abs (theG \<chi>)) {q. \<G> \<Turnstile>\<^sub>P Rep q} (\<pi> \<chi>), mojmir_to_rabin_def.succeed\<^sub>R \<up>af\<^sub>G (Abs (theG \<chi>)) {q. \<G> \<Turnstile>\<^sub>P Rep q} (\<pi> \<chi>)) |\<chi>. \<chi> \<in> \<G>} |\<G> \<pi>. \<G> \<subseteq> \<^bold>G (G \<phi>) \<and> G \<phi> \<in> \<G> \<and> (\<forall>\<chi>. \<pi> \<chi> < semi_mojmir_def.max_rank \<Sigma> \<up>af\<^sub>G (Abs (theG \<chi>)))})"
WhileCondThrow\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and e' :: "nat exp"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and c :: "nat exp" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
olv_ref_rel_def: shows "olv_ref_rel \<equiv> {(sa, sc). opt_obsv_state.next_round sc = v_state.next_round sa \<and> opt_obsv_state.decisions sc = v_state.decisions sa \<and> last_obs sc = map_option snd \<circ> process_mru (obsv_state.obs sa)}"
submonoid_inverse_closed: fixes u :: "'a \<Rightarrow> 'a" assumes "transformation.sub.invertible u"   and "u \<in> Translations" shows "transformation.symmetric.inverse u \<in> Translations"
r: shows "\<forall>x y. R x y \<longrightarrow> (\<forall>a b. R a b \<longrightarrow> S x a = T y b)"
monoE: fixes g :: "'d"   and f :: "'d"   and f' :: "'d" assumes "D.mono g"   and "D.seq g f"   and "D.seq g f'"   and "g \<cdot>\<^sub>D f = g \<cdot>\<^sub>D f'" shows "f' = f"