hcju/isabelleps huggingface dataset:
delete_commutes: fixes xs :: "('c \<times> 'd \<times> bool) list"   and i1 :: "'c"   and i2 :: "'c" shows "delete xs i1 \<bind> (\<lambda>ys. delete ys i2) = delete xs i2 \<bind> (\<lambda>ys. delete ys i1)"
add_cancel_left_left: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and b :: "'k"   and a :: "'k" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(plus b a = a) = (b = zero)"
alt_group_card_carrier: fixes n :: "nat" assumes "2 \<le> n" shows "2 * card (carrier (alt_group n)) = fact n"
factorsE: fixes G :: "('c, 'd) monoid_scheme"   and fs :: "'c list"   and a :: "'c" assumes "factors G fs a" obtains "\<forall>f\<in>set fs. irreducible G f" and "foldr (\<otimes>\<^bsub>G\<^esub>) fs \<one>\<^bsub>G\<^esub> = a"
is_cparser_def: fixes p :: "char list \<Rightarrow> char list + 'a \<times> char list" shows "is_cparser p = (\<forall>s r x. p s = Inr (x, r) \<longrightarrow> length r < length s)"
remdups_fwd_acc_set: fixes Acc :: "'b set"   and xs :: "'b list" shows "set (remdups_fwd_acc Acc xs) = set xs - Acc"
collapse: fixes quat :: "quat" shows "Quat (quat.Re quat) (Im1 quat) (Im2 quat) (Im3 quat) = quat"
semigroup_add_hom_axioms: shows "semigroup_add_hom hom"
elims_2: fixes x :: "nat"   and xa :: "(nat \<times> nat) list"   and xb :: "nat" assumes "isPath x xa xb" obtains u :: "nat" and v :: "nat" where "x = u__" and "xa = []" and "xb = v__" and "u__ = v__"   | u :: "nat" and xc :: "nat" and y :: "nat" and p :: "(nat \<times> nat) list" and v :: "nat" where "x = u" and "xa = (xc, y) # p" and "xb = v" and "u = xc \<and> (xc, y) \<in> E \<and> isPath y p v"
two_chain_vertical_boundary_is_boundary_chain: fixes twoChain :: "(real \<times> real \<Rightarrow> real \<times> real) set" shows "boundary_chain (two_chain_vertical_boundary twoChain)"
fun_merge_def: fixes fs :: "('d \<Rightarrow> 'c) list"   and as :: "'d set list"   and a :: "'d" shows "fun_merge fs as a = (fs ! (LEAST i. i < length as \<and> a \<in> as ! i)) a"
m_x_asum: fixes a :: "'a"   and b :: "'a"   and m :: "nat" assumes "a \<in> carrier A"   and "b \<in> carrier A" shows "aSum A m (a \<plusminus> b) = aSum A m a \<plusminus> aSum A m b"
False: shows "set_spmf p \<inter> A \<noteq> {}"
comp_ide_self: fixes a :: "'a \<times> 'a" assumes "CC.ide a" shows "CC.comp a a = a"
c: fixes x :: "lam"   and z :: "lam" assumes "\<And>y.  x \<longrightarrow>\<^sub>\<beta> y \<Longrightarrow> P y z"   and "\<And>u.  z \<longrightarrow>\<^sub>\<beta> u \<Longrightarrow> P x u" shows "P x z"
system_needed_asses_include_action_needed_asses_i: fixes A :: "'d set"   and B :: "'d set"   and f :: "('d, 'e) fmap" assumes "A \<subseteq> B" shows "fmrestrict_set A (fmrestrict_set B f) = fmrestrict_set A f"
inf_l_def: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> b = a * (a r\<rightarrow> b)"
e_hd_def: fixes e :: "nat" shows "e_hd e \<equiv> if e = 0 then 0 else hd (Partial_Recursive.list_decode e)"
disjnt_sym: fixes A :: "'c set"   and B :: "'c set" assumes "disjnt A B" shows "disjnt B A"
upt_arbitrary_l: fixes i :: "nat"   and j :: "nat"   and l :: "nat list" shows "[i..<j] @ l = upt_tailrec i j l"
\<gamma>0: fixes S :: "('a \<times> 'a set) set"   and h :: "'a \<Rightarrow> 'b" assumes "S tagged_partial_division_of cbox a b"   and "\<gamma>0 fine S"   and "h \<in> F" shows "(\<Sum>(x, K)\<in>S. norm (content K *\<^sub>R h x - integral K h)) < \<epsilon> / 2"
in_rel: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R3 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and a :: "('d, 'f, 'h) F2"   and b :: "('e, 'g, 'i) F2" shows "F2rel R1 R2 R3 a b = (\<exists>z. z \<in> F2in {(x, y). R1 x y} {(x, y). R2 x y} {(x, y). R3 x y} \<and> F2map fst fst fst z = a \<and> F2map snd snd snd z = b)"
df: shows "x + d *\<^sub>R (y - x) \<in> rel_frontier S"
SC_ZObis_imp_SC_Wbis: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com" assumes "PL_Indis indis"   and "PL_Indis.SC_ZObis tval aval indis c" shows "PL_Indis.SC_Wbis tval aval indis c"
top_unique: fixes a :: "'a" shows "(top \<le> a) = (a = top)"
head_del_arc: fixes a :: "'b" shows "head (del_arc a) = head G"
atms_def: fixes r :: "formula Regex.regex" shows "atms r = (\<Union>\<phi>\<in>regex.atms r. if safe_formula \<phi> then {\<phi>} else case \<phi> of Neg \<phi>' \<Rightarrow> {\<phi>'} | _ \<Rightarrow> {})"
g': shows "\<forall>x\<in>g. \<exists>xa\<in>s. x = f xa"
invar_trie_update: fixes t :: "('c, 'd) trie"   and ks :: "'c list"   and v :: "'d" assumes "invar_trie t" shows "invar_trie (update_trie ks v t)"
longest_prefix: fixes s :: "nat \<Rightarrow> nat option"   and j :: "nat"   and z :: "nat" assumes "total_cons s"   and "\<exists>t. eval (total_cons.r_prefixes s) [t, j] \<up>"   and "z = total_cons.longest_prefix s j" shows "(\<forall>x<z. eval (total_cons.r_psi s) [j, x] \<down>) \<and> (\<forall>x\<ge>z. eval (total_cons.r_psi s) [j, x] \<up>)"
permutes_natset_le: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "p permutes S"   and "\<forall>i\<in>S. p i \<le> i" shows "p = id"
SCRS3: fixes p :: "((int \<times> int) set set, int) Combinators list" assumes "singleCombinators p" shows "singleCombinators (rm_MT_rules C p)"
commute: fixes a :: "'b"   and b :: "'b" shows "a \<sqinter> b = b \<sqinter> a"
init_config_def: fixes c :: "('a, 'b, 'c) configuration_scheme" shows "init_config c \<equiv> \<forall>loc. c_imp c loc = {#}\<^sub>z \<and> c_work c loc = zmset_of (mset_set (set_antichain (frontier (c_pts c loc))))"
PiE_def: fixes S :: "'e set"   and T :: "'e \<Rightarrow> 'f set" shows "Pi\<^sub>E S T = Pi S T \<inter> extensional S"
fib_body_def: shows "fib_body \<equiv> \<Lambda> r. Nat_case\<cdot>1\<cdot>(Nat_case\<cdot>1\<cdot>(\<Lambda> n. r\<cdot>n + r\<cdot>(n + 1)))"
case_two_psi_only_prefix: fixes j :: "nat" assumes "\<exists>t. local.prefixes t j \<up>" shows "\<exists>y. (\<forall>x. 0 < x \<and> x < y \<longrightarrow> \<psi> j x \<down>= 0 \<or> \<psi> j x \<down>= 1) \<and> (\<forall>x\<ge>y. \<psi> j x \<up>)"
case_cong_weak: fixes M :: "nat"   and M' :: "nat"   and f1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a" assumes "M = M'" shows "(case M of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = (case M' of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x)"
cases: fixes a :: "msg"   and H :: "msg set" assumes "a \<in> synth H" obtains X :: "msg" where "a = X__" and "X__ \<in> H"   | agt :: "agent" where "a = msg.Agent agt"   | n :: "nat" where "a = Number n"   | X :: "msg" where "a = Hash Xa__" and "Xa__ \<in> synth H"   | X :: "msg" and Y :: "msg" where "a = \<lbrace>Xb__, Y\<rbrace>" and "Xb__ \<in> synth H" and "Y \<in> synth H"   | X :: "msg" and K :: "key" where "a = Crypt K X" and "X \<in> synth H" and "Key K \<in> H"
ceq: fixes ceq :: "('a \<Rightarrow> 'a \<Rightarrow> bool) option"   and eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.ceq ceq"   and "ceq = Some eq" shows "eq = (=)"
cc_restr_perm: fixes p :: "perm"   and S :: "Vars.var set"   and G :: "CoCalls" assumes "supp p \<sharp>* S"   and "finite S" shows "(p \<bullet> G) G|` S = G G|` S"
simple_match_ip_Drop: fixes d :: "'a word" assumes "simple_match_ip (src m) s" shows "(runFw s d c (SimpleRule m Drop # rs) = Decision FinalAllow) = (\<not> simple_match_ip (dst m) d \<and> runFw s d c rs = Decision FinalAllow)"
c_manifold_axioms: fixes k :: "enat" shows "c_manifold charts_eucl k"
sr: fixes x :: "real" assumes "poly (charpoly A) x = 0" shows "x \<le> 1"
star_irr: fixes x :: "'a" shows "1\<^sub>\<sigma> \<le> x\<^sup>\<star>"
l3_inv1_init: shows "init l3 \<subseteq> l3_inv1"
vals3: fixes l :: "lambda"   and p :: "lambda" assumes "L l \<in> vals p" shows "l \<in> lambdas p"
y'z_3: shows "z = \<pi>_Fbcc y'z"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "gt_wt a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> wt t >\<^sub>p wt s)"
ofilter_ordLess: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "Well_order r"   and "ofilter r A" shows "(A \<subset> Field r) = (Restr r A <o r)"
persistent_wclause_in_P_if_persistent_clause_in_P: fixes C :: "'a literal multiset" assumes "C \<in> Liminf_llist (lmap P_of_state (lmap state_of_wstate Sts))" shows "\<exists>i. (C, i) \<in> Liminf_llist (lmap (set_mset \<circ> wP_of_wstate) Sts)"
mat_of_row_def: fixes y :: "'b vec" shows "mat_of_row y = mat 1 (dim_vec y) (\<lambda>ij. y $ snd ij)"
ex_productE: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'b \<Rightarrow> 'a" assumes "\<exists>a. has_as_product J D a" obtains a :: "'a" and \<pi> :: "'b \<Rightarrow> 'a" where "product_cone J (\<cdot>) D a \<pi>"
openin_subtopology_refl: fixes U :: "'c topology"   and V :: "'c set" shows "openin (subtopology U V) V = (V \<subseteq> topspace U)"
the_trivial_conv: fixes v :: "'a"   and vs :: "'a list" shows "the_trivial v vs = (if \<exists>v'\<in>set vs. v' \<noteq> v \<and> set vs - {v'} \<subseteq> {v} then Some (THE v'. v' \<in> set vs \<and> v' \<noteq> v \<and> set vs - {v'} \<subseteq> {v}) else None)"
irreducible\<^sub>d_def: fixes p :: "'c poly" shows "irreducible\<^sub>d p = (0 < degree p \<and> (\<forall>q r. degree q < degree p \<longrightarrow> degree r < degree p \<longrightarrow> p \<noteq> q * r))"
swap_cols_rows_block_dims_2: fixes i :: "nat"   and j :: "nat"   and A :: "'b mat" shows "dim_col (??.Jordan_Normal_Form_Existence.swap_cols_rows_block i j A) = dim_col A"
join_dual: fixes L :: "('c, 'd) gorder_scheme"   and p :: "'c"   and q :: "'c" shows "p \<squnion>\<^bsub>inv_gorder L\<^esub> q = p \<sqinter>\<^bsub>L\<^esub> q"
that: fixes a :: "'a" assumes "p = monom a 0"   and "a \<noteq> (0::'a)" shows "thesis"
A1: shows "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>"
abs_eq: fixes x :: "8 word" shows "uint_of_uint8 (Abs_uint8 x) = Abs_uint (ucast x)"
openin_rel_interior: fixes S :: "'a set" shows "openin (top_of_set (affine hull S)) (rel_interior S)"
cases: 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 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"   and a :: "'e Trace" assumes "IncrEnvironment jkbp jview envObs jviewInit jviewIncr"   and "IncrEnvironment.jpTracesp envInit envAction envTrans envObs jp a" obtains s :: "'e" where "a = tInit s__" and "s__ \<in> set envInit"   | t :: "'e Trace" and eact :: "'h" and aact :: "'b \<Rightarrow> 'd" and s :: "'e" where "a = t \<leadsto> s" and "IncrEnvironment.jpTracesp envInit envAction envTrans envObs jp t" and "eact \<in> set (envAction (tLast t))" and "\<And>a. aact a \<in> set (pAct (jp a) (IncrEnvironment.runJP envObs jp t a))" and "s = envTrans eact aact (tLast t)"
Sup_spec_def: fixes X :: "'c set" shows "Sup_spec X = SPEC (\<lambda>r. \<forall>x\<in>X. x \<le> r)"
simps_1: fixes x :: "name"   and y :: "name"   and s :: "lam" shows "Var x[y::=s] = (if x = y then s else Var x)"
hypnat_add_diff_inverse: fixes m :: "nat star"   and n :: "nat star" assumes "\<not> m < n" shows "n + (m - n) = m"
app_def: fixes f :: "ZF"   and x :: "ZF" shows "f \<acute> x \<equiv> THE y. Opair x y |\<in>| f"
atm_of_uminus: fixes L :: "'b literal" shows "atm_of (- L) = atm_of L"
bij_betw_set_is_empty: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B"   and "card A \<noteq> card B" shows "{f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} = {}"
lift_opt_def: fixes m :: "'d option"   and e :: "'c" shows "lift_opt m e \<equiv> case m of None \<Rightarrow> Inl e | Some x \<Rightarrow> Inr x"
safe_max_empty: fixes X :: "'b set" shows "(safe_max X = None) = (X = {})"
set_fst_eq: shows "fst ` {(t, X). (t, X) \<in> set (x # xs) \<and> as \<in> Z X t} = fst ` {(t, X). (t, X) \<in> set xs \<and> as \<in> Z X t} \<union> (if as \<in> Z X t then {t} else {})"
S_intersections_in_S_2: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" assumes "p \<noteq> 0"   and "q \<noteq> 0"   and "proj2_abs p \<noteq> proj2_abs q"   and "proj2_abs p \<in> hyp2" shows "S_intersection2 p q \<in> S"
suffix_imp_sublist: fixes xs :: "'a list"   and ys :: "'a list" assumes "suffix xs ys" shows "sublist xs ys"
valuation_aux_1: fixes M :: "object atom formula set"   and \<A> :: "object atom \<Rightarrow> bool" assumes "\<forall>\<phi>\<in>close_world M. \<A> \<Turnstile> \<phi>" shows "\<A> = valuation M"
F_preserves_objects: fixes AA :: "('d, 'e, 'f) category_scheme"   and BB :: "('g, 'h, 'i) category_scheme"   and F :: "('d, 'e, 'g, 'h) functor" assumes "Functor F : AA \<longrightarrow> BB" shows "F \<^bsub>\<o>\<^esub> \<in> Ob\<^bsub>AA\<^esub> \<rightarrow> Ob\<^bsub>BB\<^esub>"
InvariantEquivalentZLAfterApplyLearn: fixes state :: "State"   and F0 :: "Literal list list" assumes "InvariantEquivalentZL (getF state) (getM state) F0"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "getConflictFlag state" shows "let state' = applyLearn state in InvariantEquivalentZL (getF state') (getM state') F0"
elims_1: fixes x :: "('i \<times> constraint) list"   and xa :: "'i set"   and xb :: "'a ns_constraint list \<times> 's"   and y :: "bool" assumes "invariant_cs x xa xb = y" obtains cs :: "('i \<times> constraint) list" and J :: "'i set" and ds :: "'a ns_constraint list" and s :: "'s" where "x = cs" and "xa = J" and "xb = (ds, s)" and "y = (ds = map snd (to_ns cs) \<and> invariant_nsc (to_ns cs) J s)"
has_integral_combine_tagged_division_topdown: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and p :: "('a \<times> 'a set) set" assumes "f integrable_on cbox a b"   and "p tagged_division_of cbox a b" shows "(f has_integral (\<Sum>(x, K)\<in>p. integral K f)) (cbox a b)"
ord_inj_mem: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e"   and a :: "'a" assumes "Order E"   and "ord_inj D E f"   and "a \<in> carrier D" shows "f a \<in> carrier E"
step_Normal_elim_cases_9: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'e \<Rightarrow> 'e"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Basic f # cs, css, Normal s) \<rightarrow> u" obtains "u = (cs, css, Normal (f s))"
Map_cod: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "A_B.arr f" shows "A_B.Map (A_B.cod f) = A_B.Cod f"
is_distinguishing_formula_def: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and x :: "('a, 'c, 'd) formula"   and P :: "'b"   and Q :: "'b" assumes "indexed_nominal_ts TYPE('a) satisfies transition" shows "indexed_nominal_ts.is_distinguishing_formula satisfies transition x P Q \<equiv> nominal_ts.valid satisfies transition P x \<and> \<not> nominal_ts.valid satisfies transition Q x"
pair_return_pmf2: fixes x :: "'d pmf"   and y :: "'e" shows "pair_pmf x (return_pmf y) = map_pmf (\<lambda>x. (x, y)) x"
length_sort: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" assumes "class.linorder less_eq less" shows "length (linorder.sort_key less_eq f xs) = length xs"
Un_iff: fixes c :: "'d"   and A :: "'d set"   and B :: "'d set" shows "(c \<in> A \<union> B) = (c \<in> A \<or> c \<in> B)"
rank_lt: fixes a :: "hf"   and b :: "hf" assumes "a \<^bold>\<in> b" shows "rank a < rank b"
has_field_derivative_eq_has_derivative_blinfun: fixes f :: "'a \<Rightarrow> 'a"   and f' :: "'a"   and x :: "'a"   and U :: "'a set" shows "(f has_field_derivative f') (at x within U) = (f has_derivative blinfun_apply (blinfun_mult_right f')) (at x within U)"
winding_number_unique_loop: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex"   and n :: "complex" assumes "path \<gamma>"   and "z \<notin> path_image \<gamma>"   and "pathfinish \<gamma> = pathstart \<gamma>"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>p. valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and> (\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * n" shows "winding_number \<gamma> z = n"
preserves_dom: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "D.dom (E.map f) = E.map (\<F>C.dom f)"
assms: fixes T :: "LambdaType.type" assumes "\<And>T1 T2. T = T1 \<Rightarrow> T2 \<Longrightarrow> P T1"   and "\<And>T1 T2. T = T1 \<Rightarrow> T2 \<Longrightarrow> P T2" shows "P T"
OadmitFO_OProd_simps: fixes \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm"   and ODE1 :: "('c + 'b, 'd) ODE"   and ODE2 :: "('c + 'b, 'd) ODE"   and U :: "('d + 'd) set" shows "OadmitFO \<sigma> (OProd ODE1 ODE2) U = (OadmitFO \<sigma> ODE1 U \<and> OadmitFO \<sigma> ODE2 U)"
setcompr_eq_image: fixes f :: "'i \<Rightarrow> 'h"   and P :: "'i \<Rightarrow> bool" shows "{f x |x. P x} = f ` {x. P x}"
suc_k_lte_t: shows "Suc k \<le> t"
approximation_preproc_floatarith_12: fixes x :: "real" shows "tan x = sin x / cos x"
HPhase1or2ReadThen_HInv4d_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc"   and q :: "Proc" assumes "HPhase1or2ReadThen s s' p d r"   and "HInv4d s q"   and "p \<noteq> q" shows "HInv4d s' q"
atLeast0LessThan: fixes n :: "nat" shows "{0..<n} = {..<n}"
seq_conv_if: fixes x :: "'b" shows "seq\<cdot>x = (if x = \<bottom> then \<bottom> else ID)"
bot_unique: fixes a :: "'a" shows "(top \<le> a) = (a = top)"
ik_append_subset_1: fixes S :: "('a, 'b) strand_step list"   and S' :: "('a, 'b) strand_step list" shows "ik\<^sub>s\<^sub>t S \<subseteq> ik\<^sub>s\<^sub>t (S @ S')"
E: shows "0 < Min (offs_set_prev ns (x # xs) index key mi ma i)"
relabel_preserve_other: fixes u :: "nat"   and v :: "nat" assumes "u \<noteq> v" shows "relabel_effect f l u v = l v"
surjective: fixes r :: "('c, 'd) alpha_rp_scheme" shows "r = \<lparr>ok = ok r, wait = wait r, tr = tr r, ref = ref r, \<dots> = alpha_rp.more r\<rparr>"
comm_monoid_axioms: shows "comm_monoid (\<and>*) \<box>"
acyclic_single_valued_finite: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "acyclic r"   and "single_valued r"   and "(x, y) \<in> r\<^sup>*" shows "finite (r \<inter> {a. (x, a) \<in> r\<^sup>*} \<times> {b. (b, y) \<in> r\<^sup>*})"
doesnt_modify_def: fixes c :: "'Com\<^sub>C"   and x :: "'Var\<^sub>C" shows "conc.doesnt_modify c x = (\<forall>mds mem c' mds' mem'. \<langle>c, mds, mem\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c', mds', mem'\<rangle>\<^sub>C \<longrightarrow> mem x = mem' x \<and> dma\<^sub>C mem x = dma\<^sub>C mem' x)"
natural_transformation_\<rr>: shows "natural_transformation (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) R S.map \<rr>"
ideal_sumTr1: fixes A :: "'a set"   and B :: "'a set" assumes "ideal R A"   and "ideal R B" shows "A \<minusplus> B = \<Inter> {J. ideal R J \<and> A \<union> B \<subseteq> J}"
partition_factors_def: fixes xs :: "('b \<times> nat) list" shows "partition_factors xs = (let n = sum_list (map snd xs) div 2 in case partition_factors_main n xs of ([], []) \<Rightarrow> ([], []) | ([], [x]) \<Rightarrow> ([], [x]) | ([], x # y # ys) \<Rightarrow> ([x], y # ys) | ([x], b) \<Rightarrow> ([x], b) | (x # y # ys, []) \<Rightarrow> ([x], y # ys) | (x # y # ys, ad # listb) \<Rightarrow> (x # y # ys, ad # listb))"
times_Measurement_System_ext_def: fixes x :: "\<lparr>unit_sys :: 'c, \<dots> :: 'd\<rparr>"   and y :: "\<lparr>unit_sys :: 'c, \<dots> :: 'd\<rparr>" shows "x \<cdot> y = \<lparr>unit_sys = unit, \<dots> = mmore x \<cdot> mmore y\<rparr>"
elims_2: fixes x :: "Runs.var set option"   and xa :: "Runs.var" assumes "in_progress x xa" obtains S :: "Runs.var set" and x :: "Runs.var" where "xa__ = Some S" and "xa = x" and "x \<in> S"
integrate_insert_final_step: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list"   and m :: "('a, 'b) insert_message"   and pm :: "'a extended"   and sm :: "'a extended"   and l :: "nat" assumes "integrate_insert_commute_insert M a s m"   and "integrate_insert_commute_insert.invariant a s m pm sm"   and "idx s pm = Inr l"   and "idx s sm = Inr (Suc l)" shows "integrate_insert_commute_insert.is_certified_associated_string' M a m (Inr (take l s @ to_woot_char m # drop l s))"
pair_measureI: fixes x :: "'c set"   and A :: "'c measure"   and y :: "'d set"   and B :: "'d measure" assumes "x \<in> sets A"   and "y \<in> sets B" shows "x \<times> y \<in> sets (A \<Otimes>\<^sub>M B)"
Int_convex_hull_insert_rel_exterior: fixes C :: "'b set"   and T :: "'b set"   and z :: "'b"   and S :: "'b set" assumes "convex C"   and "T \<subseteq> C"   and "z \<in> rel_interior C"   and "disjnt S (rel_interior C)" shows "S \<inter> convex hull insert z T = S \<inter> convex hull T"
vector_space_pair_on_def: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'e set"   and scale\<^sub>1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e" shows "vector_space_pair_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2 \<equiv> vector_space_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 \<and> vector_space_on U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2"
compare_double_simps_3: fixes x :: "double"   and y :: "double" shows "(0 < compare_double x y) = (\<not> Code_Double.is_nan x \<and> Code_Double.is_nan y \<or> fcompare_double x y = ccode.Gt)"
Hausdorff_space_def: fixes X :: "'b topology" shows "Hausdorff_space X \<equiv> \<forall>x y. x \<in> topspace X \<and> y \<in> topspace X \<and> x \<noteq> y \<longrightarrow> (\<exists>U V. openin X U \<and> openin X V \<and> x \<in> U \<and> y \<in> V \<and> disjnt U V)"
set_map0_3: fixes f1 :: "'h \<Rightarrow> 'l"   and f2 :: "'i \<Rightarrow> 'm"   and f3 :: "'j \<Rightarrow> 'k" shows "F2set \<circ> Fmap f1 f2 f3 = (`) f3 \<circ> F2set"
subcls_direct1: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and D :: "qtname" assumes "class G C = Some c"   and "C \<noteq> Object"   and "D = super c" shows "G\<turnstile>C\<preceq>\<^sub>C D"
assms_1: shows "acyclicP_on (fmran' xm) P"
inj_on_verts_sccs: shows "inj_on verts H.sccs"
prems_3: shows "set (v # vs) \<inter> V' = {}"
pred_set: fixes P :: "'d \<Rightarrow> bool" shows "list_all P = (\<lambda>x. Ball (set x) P)"
zero_neutral: fixes x :: "'a"   and y :: "'a" shows "x + y + (0::'a) = x + y"
Cons: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" assumes "\<Gamma>\<turnstile>c \<down> s"   and "\<forall>t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t \<longrightarrow> \<Gamma>\<turnstile>cs,css \<Down> t" shows "\<Gamma>\<turnstile>c # cs,css \<Down> s"
rep_eq: shows "generator local.upto_generator = (\<lambda>n. n \<le> bound, \<lambda>n. (n, n + 1))"
has_right_adjoint_functor: shows "right_adjoint_functor C D G'"
graph_def: shows "graph = (\<lambda>f. {(x, f x) |x. x \<in> UNIV})"
iT_Plus_neg_iWeakSince_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<B> t2 I \<oplus>- k. Q t2) = (P (t1 - k). t1 \<B> t2 I \<down>\<ge> k. Q (t2 - k))"
eq_strict_1: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and x :: "'a" assumes "class.Eq_strict eq" shows "eq\<cdot>x\<cdot>\<bottom> = \<bottom>"
shortest_path_least: fixes s :: "'v"   and p :: "enat list"   and u :: "'v" assumes "path s p u" shows "\<delta> s u \<le> sum_list p"