hcju/isabelleps huggingface dataset:
det_h: shows "det (M_mat signs subsets) \<noteq> 0"
can_coherence: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "par f g"   and "can f"   and "can g" shows "f = g"
has_field_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list" shows "P \<turnstile> C has F:T in D \<equiv> \<exists>FDTs. P \<turnstile> C has_fields FDTs \<and> map_of FDTs (F, D) = \<lfloor>T\<rfloor>"
qfree_asubst_peps: fixes rcs :: "real \<times> real list"   and a :: "atom" shows "qfree (asubst\<^sub>+ rcs a)"
uniform_limit_sequentially_iff: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and S :: "'c set"   and l :: "'c \<Rightarrow> 'd" shows "uniform_limit S f l sequentially = (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. dist (f n x) (l x) < e)"
closure_eq_rel_interior_eq: fixes S1 :: "'b set"   and S2 :: "'b set" assumes "convex S1"   and "convex S2" shows "(closure S1 = closure S2) = (rel_interior S1 = rel_interior S2)"
wtE_not_grdLit_unprot: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and x :: "Preliminaries.var" assumes "Ik.wtE \<xi>"   and "\<not> unprot (GE.tpOfV x)" shows "\<not> GE.satL \<xi> (grdLit x)"
poly_card_roots_less: fixes a :: "real"   and p :: "real poly" shows "card {x. x < a \<and> poly p x = 0} = count_roots_below p a - (if poly p a = 0 \<and> p \<noteq> 0 then 1 else 0)"
tiTable_ack_init0: fixes req :: "nat \<Rightarrow> reqType list"   and a1 :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and st :: "nat \<Rightarrow> GatewayStatus"   and b :: "nat \<Rightarrow> ECall_Info list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list" assumes "tiTable_SampleT req a1 stop lose (fin_inf_append [init_state] st) b ack i1 vc st"   and "req 0 = []" shows "ack 0 = [init_state]"
case_prod_map_prod: fixes h :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and x :: "'e \<times> 'f" shows "(case map_prod f g x of (x, xa) \<Rightarrow> h x xa) = (case x of (l, r) \<Rightarrow> h (f l) (g r))"
ctor2_def: shows "ctor2 = unfold2 (F1map id dtor1 dtor2) (F2map id dtor1 dtor2)"
connex_subset: fixes B :: "'a set" assumes "B \<subseteq> A" shows "connex B (\<sqsubseteq>)"
systemLOC_def: fixes x :: "chanID"   and i :: "AbstrLevelsID" shows "systemLOC x i \<equiv> (\<exists>C1\<in>AbstrLevel i. x \<in> IN C1) \<and> (\<exists>C2\<in>AbstrLevel i. x \<in> OUT C2)"
assms_4: fixes s2 :: "'s2" assumes "bad2 s2" shows "callee_invariant_on oracle1 (\<lambda>s1. bad1 s1 \<and> X_bad s1 s2) \<I>"
rtrancl_into_trancl2: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>*" shows "(a, c) \<in> r\<^sup>+"
vc: shows "vc C (\<lambda>s. enat k * Q s)"
inv_prod_rel_eq: fixes P :: "('c \<times> 'a) set"   and Q :: "('d \<times> 'b) set" shows "(P \<times>\<^sub>r Q)\<inverse> = P\<inverse> \<times>\<^sub>r Q\<inverse>"
pred_eqI: fixes P :: "'a Predicate.pred"   and Q :: "'a Predicate.pred" assumes "\<And>w. pred.eval P w = pred.eval Q w" shows "P = Q"
fMin_in: fixes A :: "'b fset" assumes "A \<noteq> {||}" shows "fMin A |\<in>| A"
map_PP_insort: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and xs :: "('x \<Rightarrow>\<^sub>0 nat) list" shows "map PP (pp_pm.ordered_powerprod_lin.insort x xs) = ordered_powerprod_lin.insort (PP x) (map PP xs)"
elims: fixes x :: "nat \<times> nat list"   and xa :: "nat"   and y :: "nat" assumes "mv_box_stage3 x xa = y" obtains as :: "nat" and lm :: "nat list" and m :: "nat" where "x = (as, lm)" and "xa = m" and "y = (if as = 1 then 3 else if as = 2 then 2 else if as = 0 then 1 else 0)"
pdec'_1: fixes e :: "nat" shows "pdec1' e = pdec1 e"
rel_G_Grp_weak: fixes l1 :: "'n \<Rightarrow> 'u"   and l2 :: "'o \<Rightarrow> 'v"   and co1 :: "'p \<Rightarrow> 'w"   and co2 :: "'q \<Rightarrow> 'x"   and contra1 :: "'y \<Rightarrow> 'r"   and contra2 :: "'z \<Rightarrow> 's" shows "rel_G (Grp UNIV l1) (Grp UNIV l2) (Grp UNIV co1) (Grp UNIV co2) (Grp UNIV contra1)\<inverse>\<inverse> (Grp UNIV contra2)\<inverse>\<inverse> = Grp UNIV (map_G l1 l2 co1 co2 contra1 contra2)"
preserves_equivalence: fixes f :: "'a"   and g :: "'a"   and \<eta> :: "'a"   and \<epsilon> :: "'a" assumes "internal_equivalence f g \<eta> \<epsilon>" shows "S.internal_equivalence (S.UP f) (S.UP g) (S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit (src f)) (S.inv (UP.unit (trg f)) \<cdot>\<^sub>S S.UP \<epsilon> \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g))"
assms_2: fixes x :: "'a" shows "(1::'a) + x\<^sup>\<star> \<cdot> x = x\<^sup>\<star>"
dirichlet_prod_completely_multiplicative_left: fixes k :: "nat"   and f :: "nat \<Rightarrow> complex"   and h :: "nat \<Rightarrow> complex" assumes "0 < k"   and "completely_multiplicative_function f"   and "multiplicative_function h"   and "\<And>p. prime p \<Longrightarrow> f p \<noteq> 0 \<and> f p \<noteq> h p" shows "dirichlet_prod f (\<lambda>k. moebius_mu k * h k) k = f k * (\<Prod>p\<in>prime_factors k. 1 - h p / f p)"
assms: shows "c |\<in>| A |\<union>| B"
vimage_eq_rcoset_1: fixes f :: "'a \<Rightarrow> 'd"   and H :: "'a set"   and a :: "'a" assumes "weak_group_morphism f H G"   and "a \<in> carrier G" shows "{b \<in> carrier G. f b = f a} = H #> a"
plus_monotone: fixes s :: "('x, nat) pp"   and t :: "('x, nat) pp"   and u :: "('x, nat) pp" assumes "ord s t" shows "ord (s + u) (t + u)"
map_pmf_cong: fixes p :: "'b pmf"   and q :: "'b pmf"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "p = q"   and "\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x" shows "map_pmf f p = map_pmf g q"
efor_def: fixes lb :: "int"   and ub :: "int"   and f :: "int \<Rightarrow> 'd \<Rightarrow> ('c + 'd) nres"   and \<sigma> :: "'d" shows "efor lb ub f \<sigma> \<equiv> ebind (EASSERT (lb \<le> ub)) (\<lambda>_. ebind (EWHILET (\<lambda>(i, \<sigma>). i < ub) (\<lambda>(i, \<sigma>). ebind (f i \<sigma>) (\<lambda>\<sigma>. ERETURN (i + 1, \<sigma>))) (lb, \<sigma>)) (\<lambda>(uu_, \<sigma>). ERETURN \<sigma>))"
orthogonal_complement_antimono: fixes B :: "'b set"   and A :: "'b set" assumes "B \<subseteq> A" shows "orthogonal_complement A \<subseteq> orthogonal_complement B"
alpha_alphaInp: fixes delta :: "'h"   and qinp :: "'d \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option"   and qbinp :: "'e \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option"   and delta' :: "'h"   and qinp' :: "'d \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option"   and qbinp' :: "'e \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option" shows "(qOp delta qinp qbinp #= qOp delta' qinp' qbinp') = (delta = delta' \<and> (qinp, qinp') \<in> alphaInp \<and> (qbinp, qbinp') \<in> alphaBinp)"
CONS_wrt_phi_eq_CONS: shows "CONS_wrt \<phi> = CONS"
set_del_ops: fixes oper :: "'a operation"   and rects :: "'a rectangle list" shows "(oper \<in> set (del_ops rects)) = (op_idx oper < length rects \<and> oper = del_op rects (op_idx oper))"
assms_1: shows "wf\<^sub>s\<^sub>s\<^sub>t \<A>"
JF_cind: fixes R1 :: "('d JF1 \<times> 'd JF1) set"   and R2 :: "('d JF2 \<times> 'd JF2) set" assumes "sbis UNIV UNIV dtor1 dtor2 R1 R2" shows "R1 \<subseteq> Id \<and> R2 \<subseteq> Id"
Invariants_func_is_invariant: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "f \<in> borel_measurable Invariants"   and "x \<in> space M" shows "f (T x) = f x"
subdigraph_axioms_def: fixes \<BB> :: "V"   and \<CC> :: "V" shows "subdigraph_axioms \<BB> \<CC> \<equiv> (\<forall>a. a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr> \<longrightarrow> a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>) \<and> (\<forall>a b f. f : a \<mapsto>\<^bsub>\<BB>\<^esub> b \<longrightarrow> f : a \<mapsto>\<^bsub>\<CC>\<^esub> b)"
that: fixes A :: "'a set" assumes "open A"   and "x \<in> A"   and "f ` A \<subseteq> B" shows "thesis"
invertible_inverse_closed: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c" assumes "monoid M composition unit"   and "monoid.invertible M composition unit u"   and "u \<in> M" shows "monoid.inverse M composition unit u \<in> M"
median_5_aux: fixes x1 :: "'b"   and x2 :: "'b"   and x4 :: "'b"   and x5 :: "'b"   and x3 :: "'b" assumes "x1 \<le> x2"   and "x4 \<le> x5"   and "x1 \<le> x4" shows "median_5_aux x1 x2 x3 x4 x5 = median [x1, x2, x3, x4, x5]"
le_ASSERT_iff: fixes S :: "'ma"   and \<Phi> :: "bool"   and S' :: "'ma" shows "(S \<le> bind (ASSERT \<Phi>) (\<lambda>_. S')) = (\<Phi> \<longrightarrow> S \<le> S')"
min_max_distrib2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.linorder less_eq less" shows "ord.min less_eq a (ord.max less_eq b c) = ord.max less_eq (ord.min less_eq a b) (ord.min less_eq a c)"
WCR_SN_on_imp_CR_on: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "WCR r"   and "SN_on r A" shows "CR_on r A"
min_key_val_alt: fixes xs :: "'x"   and ko :: "'o" assumes "fst (list_of_oalist xs) \<noteq> []" shows "min_key_val ko xs = local.hd (reorder ko xs)"
borel_eq_atLeastAtMost: shows "borel = sigma UNIV (range (\<lambda>(a, b). {a..b}))"
aFa: shows "a \<in> F \<^bsub>\<o>\<^esub> A"
substEnv_preserves_freshEnv_aux: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rho"   and "good Y"   and "freshEnv zs z rho"   and "fresh zs z Y"   and "zs \<noteq> ys \<or> z \<noteq> y" shows "freshEnv zs z (rho &[Y / y]_ys)"
Units_closed: fixes x :: "'a mod_ring poly" assumes "x \<in> Units R" shows "x \<in> carrier R"
that: fixes e :: "real" assumes "0 < e"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>G\<in>\<G>. ball x e \<subseteq> G" shows "thesis"
evalRecvArgs_pState_inv: fixes prog :: "program"   and p :: "pState"   and rargs :: "recvArg list"   and xs :: "integer list"   and g :: "gState\<^sub>I" assumes "pState_inv prog p" shows "pState_inv prog (snd (evalRecvArgs rargs xs g p))"
p_def: fixes s :: "'b"   and t :: "'b"   and n :: "nat" shows "K2.p s t n = K2.T.prob s {\<omega> \<in> space (K2.T s). (s ## \<omega>) !! n = t}"
hd_construct: fixes q :: "('c, 'd) pq" assumes "\<not> PQ.is_empty q" shows "hd (pq.alist_of q) = (PQ.min q, the (PQ.priority q (PQ.min q)))"
bal_inorder: fixes n :: "nat"   and xs :: "'b list"   and t :: "'b tree"   and zs :: "'b list" assumes "n \<le> length xs"   and "bal n xs = (t, zs)" shows "xs = inorder t @ zs \<and> size t = n"
delivered: shows "(msgs c i \<noteq> [] \<and> hd (msgs c i) = m) \<and> (msgs c' i = [] \<or> hd (msgs c' i) \<noteq> m)"
simps: fixes p :: "real poly"   and qs :: "real poly list"   and m :: "rat mat"   and subs :: "(nat list \<times> nat list) list"   and signs :: "rat list list" shows "reduce_system_R p (qs, m, subs, signs) = reduction_step_R m signs subs (solve_for_lhs_R p qs subs m)"
lget_proc: shows "lget_proc \<equiv> lift_get_proc get_proc Main"
vors_2: shows "vs1 = between (vertices f) ram1 ram2"
F_is_faithful: shows "faithful_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F"
big_trans: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> O[F](g)"   and "g \<in> O[F](h)" shows "f \<in> O[F](h)"
bij_betw_partition_of_equiv_with_k_classes: fixes A :: "'a set"   and k :: "nat" shows "bij_betw ((//) A) {R. equiv A R \<and> card (A // R) = k} {P. partition_on A P \<and> card P = k}"
ar_coset_subsetD: fixes I :: "'a set"   and a :: "'a"   and x :: "'a" assumes "ideal R I"   and "a \<in> carrier R"   and "x \<in> a \<uplus>\<^bsub>R\<^esub> I" shows "x \<in> carrier R"
countable_insert_iff: fixes x :: "'f"   and A :: "'f set" shows "countable (insert x A) = countable A"
balanced: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "int (card ps) = \<t>" shows "int (\<B>' index ps) = \<Lambda>\<^sub>t'"
xor_def: fixes x :: "'a"   and y :: "'a" shows "x \<oplus> y = (x \<squnion> y) \<sqinter> - (x \<sqinter> y)"
dual_fun_def: fixes f :: "'b \<Rightarrow> 'b" shows "dual_fun f = uminus \<circ> f \<circ> uminus"
induct: fixes P :: "('b \<times> 'c) list \<Rightarrow> bool"   and a0 :: "('b \<times> 'c) list" assumes "P []"   and "\<And>p ps. P (AList.delete (fst p) ps) \<Longrightarrow> P (p # ps)" shows "P a0"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "hhom\<^sub>C.in_hom f a b = (hhom\<^sub>C.arr a \<and> hhom\<^sub>C.arr b \<and> hhom\<^sub>C.arr f \<and> \<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>)"
sup_assoc: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" shows "x \<union># y \<union># z = x \<union># (y \<union># z)"
abs_zero: fixes abs :: "'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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs zero = zero"
DFT_lower: fixes m :: "nat"   and a :: "nat \<Rightarrow> complex"   and i :: "nat" shows "DFT (2 * m) a i = DFT m (\<lambda>i. a (2 * i)) i + FFT.root (2 * m) ^ i * DFT m (\<lambda>i. a (2 * i + 1)) i"
FinDimVectorSpace_def: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set" shows "FinDimVectorSpace smult V \<equiv> VectorSpace smult V \<and> FinDimVectorSpace_axioms smult V"
distrib_left_numeral: fixes v :: "num"   and b :: "'b"   and c :: "'b" shows "numeral v * (b + c) = numeral v * b + numeral v * c"
configuration_rev_step'': fixes x :: "nat"   and n :: "nat" assumes "x \<in> configuration q\<^sub>0 (Suc n)" shows "x = Suc n \<or> (\<exists>q'. q\<^sub>0 = \<delta> q' (w n) \<and> x \<in> configuration q' n)"
Ex_finite_integrable_function: shows "\<exists>ha\<in>borel_measurable (density M h). integral\<^sup>N (density M h) ha \<noteq> \<infinity> \<and> (\<forall>x\<in>space (density M h). 0 < ha x \<and> ha x < \<infinity>)"
ex1I: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "\<exists>!x. P x"
bounds: shows "x \<in> {0..<2 ^ p + 5}"
basic2: shows "\<forall>u v w. u = v \<oplus> w \<longrightarrow> v \<ominus> u = ~ w"
commute: fixes a :: "'k multiset"   and b :: "'k multiset" shows "a \<inter># b = b \<inter># a"
stream_defined: fixes xs :: "'a LList" shows "stream\<cdot>xs \<noteq> \<bottom>"
J_mono: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m" shows "J n \<subseteq> J m"
subgraph_def2: fixes G\<^sub>1 :: "('d, 'e) labeled_graph"   and G\<^sub>2 :: "('d, 'e) labeled_graph" assumes "graph G\<^sub>1"   and "graph G\<^sub>2" shows "subgraph G\<^sub>1 G\<^sub>2 = (vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2 \<and> edges G\<^sub>1 \<subseteq> edges G\<^sub>2)"
simps_3: fixes sys :: "'b \<Rightarrow> ('a, 'b) process_term"   and T1 :: "('a, 'b) process_term"   and T2 :: "('a, 'b) process_term" shows "CH1 sys (CH T1 T2) = T1"
closed_segment_subset: fixes x :: "'a"   and S :: "'a set"   and y :: "'a" assumes "x \<in> S"   and "y \<in> S"   and "convex S" shows "closed_segment x y \<subseteq> S"
wpQ_skip: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "wp\<^sub>Q SKIP Q = (\<lambda>s. eSuc (Q s))"
stacks_rewrite: fixes cs :: "'edge list"   and m :: "'node"   and rs :: "'edge list"   and ms :: "'node list" assumes "valid_call_list cs m"   and "valid_return_list rs m"   and "\<forall>i<length rs. rs ! i \<in> get_return_edges (cs ! i)"   and "length rs = length cs"   and "ms = targetnodes rs" shows "\<forall>i<length cs. call_of_return_node (ms ! i) (sourcenode (cs ! i))"
norm_inverse: fixes a :: "'a" shows "norm (inverse a) = inverse (norm a)"
Cons_1: fixes s :: "('St, 'U, 'C) state" assumes "isState s" shows "\<exists>sl. wffp sl \<and> list_all isState sl \<and> hd sl = s \<and> map getGMUserCom (tl sl) = ucl"
prems_1: shows "insert_var x \<Gamma> t \<turnstile>\<^sub>c \<integral>\<^sub>c e' \<partial>t'' : t'"
valid_refs_inv_def: shows "valid_refs_inv = (\<lambda>s. \<forall>x xa. mut_m.reachable x xa s \<or> grey_reachable xa s \<longrightarrow> obj_at (\<lambda>s. True) xa s)"
ipurge_ref_aux_subset: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list"   and X :: "'b set" shows "ipurge_ref_aux I D U xs X \<subseteq> X"
obeh_def: fixes S :: "('b, 'a) spec" shows "obeh S \<equiv> map (obs S) ` beh S"
fair_def: fixes Ns :: "'f set llist" shows "fair Ns = (Inf_from (Liminf_llist Ns) \<subseteq> Sup_llist (lmap Red_I Ns))"
eval_member: fixes xq :: "'a seq" shows "member xq = eval (pred_of_seq xq)"
simps_2: fixes F :: "'b set \<Rightarrow> 'a"   and x :: "char list"   and a :: "aexp"   and M :: "'b acom set" shows "lift F (x ::= a) M = x ::= a {F (post ` M)}"
IH: assumes "i \<le> n"   and "j \<le> n"   and "k \<le> n"   and "i'' \<le> i"   and "j'' \<le> j" shows "fw m n k i j i' j' \<le> fw m n k i'' j'' i' j'"
le_iff_sup: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<sqinter> y = y)"
assms_4: shows "\<lbrace>r \<and> p\<rbrace> S \<lbrace>q\<rbrace>\<^sub>u"
of_bool_odd_eq_mod_2: fixes a :: "'a" shows "of_bool (odd a) = a mod (2::'a)"
igSwapAbsIPresIGWlsAbs_def: fixes MOD :: "('i, 'j, 'varSort, 'sort, 'k, 'l, 'm, 'n, 'o) model_scheme" shows "igSwapAbsIPresIGWlsAbs MOD \<equiv> \<forall>zs z1 z2 us s A. isInBar (us, s) \<and> igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD (us, s) (igSwapAbs MOD zs z1 z2 A)"
comm_monoid_list_set_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" shows "comm_monoid_list_set f z \<equiv> comm_monoid_list f z \<and> comm_monoid_set f z"
except_minus: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and q :: "'a \<Rightarrow>\<^sub>0 'b"   and S :: "'a set" shows "except (p - q) S = except p S - except q S"
homologous_rel_def: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set" shows "homologous_rel p X S \<equiv> \<lambda>a b. singular_relboundary p X S (a - b)"
C_unfoldl: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.C_algebra plus times one zero less_eq less star" shows "plus one (times x (star x)) = star x"
HPhase1or2ReadThen_HInv5: 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 "HInv5_inner s q"   and "Inv2c s"   and "HInv4c s p"   and "HInv4a s p" shows "HInv5_inner s' q"
synthetic_div_unique: fixes p :: "'c poly"   and c :: "'c"   and q :: "'c poly"   and r :: "'c" assumes "p + Polynomial.smult c q = pCons r q" shows "r = poly p c \<and> q = synthetic_div p c"
intro: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e" assumes "module s1"   and "module s2"   and "module_hom_axioms s1 s2 f" shows "module_hom s1 s2 f"
is_derivation_0: fixes p :: "'a" shows "is_derivation 0 p"
induct: fixes x1 :: "'b list"   and x2 :: "'b rexp"   and x3 :: "'b val"   and P :: "'b list \<Rightarrow> 'b rexp \<Rightarrow> 'b val \<Rightarrow> bool" assumes "x1 \<in> x2 \<rightarrow> x3"   and "P [] One Void"   and "\<And>c. P [c] (Atom c) (Atm c)"   and "\<And>s r1 v r2. \<lbrakk>s \<in> r1 \<rightarrow> v; P s r1 v\<rbrakk> \<Longrightarrow> P s (Plus r1 r2) (val.Left v)"   and "\<And>s r2 v r1. \<lbrakk>s \<in> r2 \<rightarrow> v; P s r2 v; s \<notin> lang r1\<rbrakk> \<Longrightarrow> P s (Plus r1 r2) (val.Right v)"   and "\<And>s1 r1 v1 s2 r2 v2. \<lbrakk>s1 \<in> r1 \<rightarrow> v1; P s1 r1 v1; s2 \<in> r2 \<rightarrow> v2; P s2 r2 v2; \<nexists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang r1 \<and> s\<^sub>4 \<in> lang r2\<rbrakk> \<Longrightarrow> P (s1 @ s2) (Times r1 r2) (Seq v1 v2)"   and "\<And>s1 r v s2 vs. \<lbrakk>s1 \<in> r \<rightarrow> v; P s1 r v; s2 \<in> Star r \<rightarrow> Stars vs; P s2 (Star r) (Stars vs); flat v \<noteq> []; \<nexists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang r \<and> s\<^sub>4 \<in> lang (Star r)\<rbrakk> \<Longrightarrow> P (s1 @ s2) (Star r) (Stars (v # vs))"   and "\<And>r. P [] (Star r) (Stars [])" shows "P x1 x2 x3"
mojmir_accept_iff_rabin_accept_rank: fixes i :: "nat" shows "(finite \<MM>.fail \<and> finite (\<MM>.merge i) \<and> infinite (\<MM>.succeed i)) = accepting_pair\<^sub>R \<MM>.\<delta>\<^sub>\<R> \<MM>.q\<^sub>\<R> (\<MM>.fail\<^sub>R \<union> \<MM>.merge\<^sub>R i, \<MM>.succeed\<^sub>R i) w"
image_set_diff: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'d set" assumes "inj f" shows "f ` (A - B) = f ` A - f ` B"
leg0_simps_3: shows "trg t\<^sub>0 = src t"
simps_4: fixes prec :: "nat"   and x :: "nat"   and a :: "floatarith"   and vs :: "float interval option list" shows "isDERIV_approx prec x (Inverse a) vs = (isDERIV_approx prec x a vs \<and> (case approx prec a vs of None \<Rightarrow> False | Some ivl \<Rightarrow> 0 < lower ivl \<or> upper ivl < 0))"
emeasure_eq_measure: fixes M :: "'a pmf"   and A :: "'a set" shows "emeasure (measure_pmf M) A = ennreal (measure_pmf.prob M A)"
card_insert_if: fixes A :: "'d set"   and x :: "'d" assumes "finite A" shows "card (insert x A) = (if x \<in> A then card A else Suc (card A))"
i: shows "i < length (\<sigma> (f, Suc (length bef + length aft)))"
hyps_free_vertices_distinct: fixes v :: "'d"   and v' :: "'d"   and pth :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" assumes "terminal_path v v' pth" shows "distinct (map fst (map fst pth) @ [v'])"
scene_override_idem: fixes X :: "'a scene"   and s :: "'a" assumes "idem_scene X" shows "s \<oplus>\<^sub>S s on X = s"
OclIncluding_def: fixes x :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "x->including\<^sub>S\<^sub>e\<^sub>t(y) = (\<lambda>\<tau>. if (\<delta> x) \<tau> = true \<tau> \<and> (\<upsilon> y) \<tau> = true \<tau> then Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<lfloor>\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (x \<tau>)\<rceil>\<rceil> \<union> {y \<tau>}\<rfloor>\<rfloor> else invalid \<tau>)"
height_0_imp_sibling_id: fixes t :: "'a tree"   and a :: "'a" assumes "height t = 0" shows "sibling t a = a"
continuous_on_homotopic_join_lemma: fixes p :: "real \<Rightarrow> real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> real \<Rightarrow> 'b" assumes "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. p (fst y) (snd y))"   and "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. q (fst y) (snd y))"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish (p t) = pathstart (q t)" shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. (p (fst y) +++ q (fst y)) (snd y))"
P_def: shows "P = \<Prod>((min_int_poly \<circ> \<beta>) ` X)"
induct: fixes x :: "'d"   and A :: "('e, 'f, 'd, 'g) FSM_scheme"   and P :: "'d \<Rightarrow> bool" assumes "x \<in> nodes A"   and "\<And>p. p = initial A \<Longrightarrow> P p"   and "\<And>p a. \<lbrakk>p \<in> nodes A; P p; snd a \<in> succ A (fst a) p\<rbrakk> \<Longrightarrow> P (snd a)" shows "P x"
iprev_imirror_inext_conv': fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "n \<in> I" shows "iprev (mirror_elem n I) (imirror I) = mirror_elem (inext n I) I"
subdiv_step_axioms: shows "subdiv_step I rev_I H rev_H u v w uv uw vw"