hcju/isabelleps huggingface dataset:
alphaAbs_weakestEx_imp_strongestEx: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X"   and "alphaAbs_ex_equal_or_qAFresh xs x X xs' x' X'" shows "alphaAbs_ex_distinct_qAFresh xs x X xs' x' X'"
fmember_implies_member: fixes e :: "'b"   and f :: "'b fset" assumes "e |\<in>| f" shows "e \<in> fset f"
SP_inv_def: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and I :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" shows "SP_inv M l I = (\<lambda>s0 r h. \<forall>s l1 n. (M, l, s, n, l1, r) \<in> Step \<longrightarrow> I s0 s h)"
not_cong_2143: 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 "\<not> Cong A B C D" shows "\<not> Cong B A D C"
F_def: shows "F = (\<lambda>x. \<bar>u 1 x\<bar> * indicat_real {x. K \<le> \<bar>u 1 x\<bar>} x)"
Sup_least: fixes A :: "('a \<times> 'b) set"   and z :: "'a \<times> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> less_eq_prod x z" shows "less_eq_prod (Sup_prod A) z"
collision_union: fixes A :: "real set"   and B :: "real set" shows "collision (A \<union> B) = (collision A \<or> collision B)"
list_del_hi: fixes t :: "'b tree"   and st :: "nat" assumes "braun t"   and "size t = st" shows "list (del_hi st t) = butlast (list t)"
trace_callee_resource_of_oracle2: fixes callee :: "'f \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'f) spmf"   and p :: "'f spmf"   and xs :: "('e \<times> 'd) list"   and x :: "'e" shows "trace_callee run_resource (map_spmf (resource_of_oracle callee) p) xs x = trace_callee callee p xs x"
p_preserves_inv_imps_work_sum: fixes c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t" assumes "next_propagate' c0 c1 loc t"   and "inv_imps_work_sum c0" shows "inv_imps_work_sum c1"
abelian_group_hom_axioms: shows "abelian_group_hom G H h"
injective_publicKey: fixes b :: "keymode"   and A :: "agent"   and c :: "keymode"   and A' :: "agent" assumes "publicKey b A = publicKey c A'" shows "b = c \<and> A = A'"
exec_step_ind_Invokestatic_NonStatic: fixes D :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and ins :: "bytecode"   and xt :: "ex_table"   and P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and m :: "jvm_method"   and n :: "nat"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "(D, b, Ts, T, mxs, mxl\<^sub>0, ins, xt) = method P C M"   and "P \<turnstile> C sees M, NonStatic :  Ts\<rightarrow>T = m in D" shows "exec_step_ind (StepI (Invokestatic C M n)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt IncompatibleClassChangeError\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
converse_rtranclpE: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and z :: "'d" assumes "r\<^sup>*\<^sup>* x z" obtains "x = z"   | y :: "'d" where "r x y" and "r\<^sup>*\<^sup>* y z"
mexec_eq_mexecd: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and t :: "'thread_id"   and xcp :: "'addr option"   and h :: "'heap"   and frs :: "'addr frame list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "\<Phi> \<turnstile> t:(xcp, h, frs) \<surd>" shows "mexec P t ((xcp, frs), h) = mexecd P t ((xcp, frs), h)"
P_mod_mult_const1: fixes I :: "'a set"   and J :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and d :: "'a" assumes "ideal S I"   and "ideal S J"   and "pol_coeff S (n, f)"   and "P_mod R S X I (polyn_expr R X n (n, f))"   and "d \<in> J" shows "P_mod R S X (I \<diamondsuit>\<^sub>r\<^bsub>S\<^esub> J) (polyn_expr R X n (n, f) \<cdot>\<^sub>r d)"
double_inj_on: fixes A :: "nat set" shows "inj_on ((*) 2) A"
rev_bl_order_def: fixes F :: "bool"   and xs :: "bool list"   and ys :: "bool list" shows "rev_bl_order F xs ys = (length xs = length ys \<and> (xs = ys \<and> F \<or> (\<exists>n<length xs. drop (Suc n) xs = drop (Suc n) ys \<and> \<not> xs ! n \<and> ys ! n)))"
Ext_accessible: fixes P :: "pname" shows "tprg \<turnstile> Class Ext accessible_in P"
coeff_inverse: fixes x :: "'b poly" shows "Abs_poly (poly.coeff x) = x"
is_none_code_2: fixes x :: "'f" shows "Option.is_none \<lfloor>x\<rfloor> = False"
list_all2_conv_all_nth: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'e list"   and ys :: "'f list" shows "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>i<length xs. P (xs ! i) (ys ! i)))"
onl_oinvariant_weakenD: fixes A :: "('a \<times> ('b, 'c, 'd, 'e) seqp, 'f) automaton"   and S' :: "'a \<Rightarrow> 'a \<Rightarrow> 'f \<Rightarrow> bool"   and U' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<Gamma> :: "'d \<Rightarrow> ('b, 'c, 'd, 'e) seqp"   and P :: "'a \<times> 'e \<Rightarrow> bool"   and \<sigma> :: "'a"   and p :: "('b, 'c, 'd, 'e) seqp"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'e" assumes "A \<Turnstile> (S', U' \<rightarrow>) onl \<Gamma> P"   and "(\<sigma>, p) \<in> oreachable A S U"   and "l \<in> labels \<Gamma> p"   and "\<And>s s' a. S s s' a \<Longrightarrow> S' s s' a"   and "\<And>s s'. U s s' \<Longrightarrow> U' s s'" shows "P (\<sigma>, l)"
poly_roots_finite: fixes p :: "'a poly" assumes "p \<noteq> 0" shows "finite {x. poly p x = (0::'a)}"
iterate_to_list_def: fixes it :: "('a list \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a list \<Rightarrow> 'a list) \<Rightarrow> 'a list \<Rightarrow> 'a list" shows "iterate_to_list it = it (\<lambda>_. True) (#) []"
lcoeff_nonzero_deg: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "deg R p \<noteq> 0"   and "p \<in> carrier (UP R)" shows "coeff (UP R) p (deg R p) \<noteq> \<zero>\<^bsub>R\<^esub>"
lifting_lemma: fixes C :: "'a eclause"   and P :: "'a eclause set"   and S :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list"   and C' :: "'a literal set" assumes "derivable C P S \<sigma> Ground C'" shows "\<exists>D \<theta> \<eta>. derivable D P S \<theta> FirstOrder C' \<and> \<sigma> \<doteq> \<theta> \<lozenge> \<eta> \<and> trms_subsumes D C \<eta>"
csegment_midpoint_subset: fixes a :: "'c"   and b :: "'c" shows "closed_segment (midpoint a b) b \<subseteq> closed_segment a b"
lenext_total: fixes gts :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" assumes "gts ys xs \<or> gts xs ys \<or> ys = xs" shows "lenext gts ys xs \<or> lenext gts xs ys \<or> ys = xs"
iter_tensor_of_H_tensor_Id_is_HId: fixes n :: "nat" shows "Matrix.mat (2 ^ n) (2 ^ n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>n\<^esub>  j / sqrt 2 ^ n)) \<Otimes> Quantum.Id 1 = Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if i mod 2 = j mod 2 then (- 1) ^ i div 2 \<cdot>\<^bsub>n\<^esub>  j div 2 / sqrt 2 ^ n else 0))"
R2_def: shows "R2 = {(I1, I2). \<exists>c1 c1' c2 c2' b b'. I1 = [if b then c1 else c2 fi] \<and> I2 = [if b' then c1' else c2' fi] \<and> [c1] \<approx>\<^bsub>d\<^esub> [c1'] \<and> [c2] \<approx>\<^bsub>d\<^esub> [c2'] \<and> ([c1] \<approx>\<^bsub>d\<^esub> [c2] \<or> [c1'] \<approx>\<^bsub>d\<^esub> [c2'])}"
GEN_OP_tag_def: fixes P :: "'a" shows "GEN_OP_tag P \<equiv> P"
to_dual_ord_inverse: fixes y :: "'b" assumes "y \<in> UNIV" shows "of_dual_ord (to_dual_ord y) = y"
comm_neutral: fixes a :: "'d" shows "a * (1::'d) = a"
hyps_3: shows "P \<parallel> Q \<Longrightarrow>\<^sub>\<tau> P'' \<parallel> Q"
solution: fixes s :: "'s"   and D :: "'s pmf" assumes "s \<in> S1"   and "D \<in> K s" shows "(\<Sum>t\<in>S. pmf D t * x t) \<le> x s"
sub_subgroup_dvd_card: fixes H :: "'a set"   and J :: "'a set" assumes "subgroup H G"   and "subgroup J G"   and "J \<subseteq> H" shows "card J dvd card H"
subst_cls_union: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and C :: "'b literal multiset"   and D :: "'b literal multiset"   and \<sigma> :: "'c" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "substitution_ops.subst_cls subst_atm (C + D) \<sigma> = substitution_ops.subst_cls subst_atm C \<sigma> + substitution_ops.subst_cls subst_atm D \<sigma>"
valid_def: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "\<upsilon> X \<equiv> \<lambda>\<tau>. if X \<tau> = \<bottom> \<tau> then false \<tau> else true \<tau>"
supset_lbound_of_def: fixes x :: "'b set"   and y :: "'b set"   and b :: "'b set" shows "dual_order.lbound_of x y b \<equiv> order.greater_eq b x \<and> order.greater_eq b y"
trace_core_eq_complete: fixes core1 :: "('a, 'b, 'c, 'd, 'e, 'f) core"   and core2 :: "('g, 'b, 'c, 'd, 'e, 'f) core"   and E :: "'b set"   and IA :: "'c set"   and IU :: "'d set"   and p :: "'a spmf"   and q :: "'g spmf" assumes "trace_core_eq core1 core2 E IA IU p q" obtains S :: "'a spmf \<Rightarrow> 'g spmf \<Rightarrow> bool" where "S p q" and "\<And>p q e. \<lbrakk>S p q; e \<in> E\<rbrakk> \<Longrightarrow> weight_spmf (p \<bind> (\<lambda>s. cpoke core1 s e)) = weight_spmf (q \<bind> (\<lambda>s. cpoke core2 s e))" and "\<And>p q e. \<lbrakk>S p q; e \<in> E\<rbrakk> \<Longrightarrow> S (mk_lossless (p \<bind> (\<lambda>s. cpoke core1 s e))) (mk_lossless (q \<bind> (\<lambda>s. cpoke core2 s e)))" and "\<And>p q ia. \<lbrakk>S p q; ia \<in> IA\<rbrakk> \<Longrightarrow> p \<bind> (\<lambda>s1. map_spmf fst (cfunc_adv core1 s1 ia)) = q \<bind> (\<lambda>s2. map_spmf fst (cfunc_adv core2 s2 ia))" and "\<And>p q ia oa. \<lbrakk>S p q; ia \<in> IA\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (p \<bind> (\<lambda>s1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (q \<bind> (\<lambda>s2. cfunc_adv core2 s2 ia)) oa)" and "\<And>p q iu. \<lbrakk>S p q; iu \<in> IU\<rbrakk> \<Longrightarrow> p \<bind> (\<lambda>s1. map_spmf fst (cfunc_usr core1 s1 iu)) = q \<bind> (\<lambda>s2. map_spmf fst (cfunc_usr core2 s2 iu))" and "\<And>p q iu ou. \<lbrakk>S p q; iu \<in> IU\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (p \<bind> (\<lambda>s1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q \<bind> (\<lambda>s2. cfunc_usr core2 s2 iu)) ou)"
Int_commute: fixes A :: "'f set"   and B :: "'f set" shows "A \<inter> B = B \<inter> A"
gfp_trans_unfold: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>t u. \<lbrakk>le_utrans t u; \<And>P. unitary P \<Longrightarrow> unitary (t P); \<And>P. unitary P \<Longrightarrow> unitary (u P)\<rbrakk> \<Longrightarrow> le_utrans (T t) (T u)"   and "\<And>t P. \<lbrakk>\<And>Q. unitary Q \<Longrightarrow> unitary (t Q); unitary P\<rbrakk> \<Longrightarrow> unitary (T t P)" shows "equiv_utrans (gfp_trans T) (T (gfp_trans T))"
trans: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "xs <~~> ys"   and "ys <~~> zs" shows "xs <~~> zs"
nextl_mono: fixes Q' :: "hf set"   and Q :: "hf set"   and u :: "'a list" assumes "Q' \<subseteq> Q" shows "ST.nextl Q' u \<subseteq> ST.nextl Q u"
module_hom_with_axioms_def: fixes plus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and plus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'d \<Rightarrow> 'f" shows "module_hom_with_axioms plus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2 f \<equiv> (\<forall>b1 b2. f (plus\<^sub>M\<^sub>_\<^sub>1 b1 b2) = plus\<^sub>M\<^sub>_\<^sub>2 (f b1) (f b2)) \<and> (\<forall>r b. f (scale\<^sub>1 r b) = scale\<^sub>2 r (f b))"
wset_final_okD: fixes ws :: "'d \<Rightarrow> 'e wait_set_status option"   and ts :: "'d \<Rightarrow> ('x \<times> 'f \<Rightarrow>f nat) option"   and t :: "'d"   and w :: "'e wait_set_status" assumes "wset_final_ok ws ts"   and "ws t = \<lfloor>w\<rfloor>" shows "\<exists>x ln. ts t = \<lfloor>(x, ln)\<rfloor> \<and> \<not> final x"
reduction_doesnt_break_things_invertibility_R: fixes subsets :: "(nat list \<times> nat list) list"   and signs :: "rat list list"   and p :: "real poly"   and qs :: "real poly list" assumes "length subsets = length signs"   and "invertible_mat (M_mat_R signs subsets)"   and "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs"   and "satisfy_equation_R p qs subsets signs" shows "invertible_mat (get_matrix_R (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs)))"
image_map_upd: fixes x :: "'c"   and A :: "'c set"   and m :: "'c \<Rightarrow> 'd option"   and y :: "'d" assumes "x \<notin> A" shows "m(x \<mapsto> y) ` A = m ` A"
integral_map_pmf: fixes g :: "'e \<Rightarrow> real"   and M :: "'e pmf"   and f :: "real \<Rightarrow> real" shows "measure_pmf.expectation (map_pmf g M) f = measure_pmf.expectation M (\<lambda>x. f (g x))"
PPs_closed_except': fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set"   and Y :: "'b set" assumes "t \<in> .[X]" shows "except t Y \<in> .[X - Y]"
k_th_has_derivative: shows "((\<lambda>x. nth_derivative k f x 1) has_derivative (\<lambda>x. 0)) (at 0)"
usappconst_def: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and f :: "char" shows "usappconst \<sigma> U f \<equiv> case SConst \<sigma> f of None \<Rightarrow> Aterm (Const f) | Some r \<Rightarrow> if FVT r \<inter> U = {} then Aterm r else undeft"
not_arg_cong_Inr: fixes x :: "'c"   and y :: "'c" assumes "x \<noteq> y" shows "Inr x \<noteq> Inr y"
phiCount: fixes g :: "'g" shows "other.phiCount g = card (dom (phis' g))"
AE_integrable_fst: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "integrable (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). f x y)" shows "AE x in M1. integrable M2 (f x)"
converse_rtranclE2: fixes xa :: "'c"   and xb :: "'d"   and za :: "'c"   and zb :: "'d"   and r :: "(('c \<times> 'd) \<times> 'c \<times> 'd) set" assumes "((xa, xb), za, zb) \<in> r\<^sup>*" obtains "(xa, xb) = (za, zb)"   | a :: "'c" and b :: "'d" where "((xa, xb), a, b) \<in> r" and "((a, b), za, zb) \<in> r\<^sup>*"
iwf_globalize: fixes t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and \<Gamma> :: "'form fset"   and c :: "'form"   and "is" :: "nat list"   and f :: "'var \<Rightarrow> 'var" assumes "local_iwf t (\<Gamma> \<turnstile> c)" shows "plain_iwf (globalize is f t) (renameLCs f |`| \<Gamma> \<turnstile> renameLCs f c)"
simple_rotations_fix_unique: fixes r :: "Vertex \<Rightarrow> Vertex"   and v :: "Vertex"   and w :: "Vertex" assumes "r \<in> simple_rotations"   and "r \<noteq> id"   and "r v = v"   and "r w = w" shows "v = w"
NSsuminf_def: fixes f :: "nat \<Rightarrow> real" shows "NSsuminf f = (THE s. f NSsums s)"
pmf_spmf_of_pmf_None: fixes p :: "'c pmf" shows "pmf (spmf_of_pmf p) None = 0"
N1: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<turnstile> F \<longrightarrow> G" shows "|~ \<circle>F \<longrightarrow> \<circle>G"
x_2: shows "2 *\<^sub>R a - x = x"
progress_LN_return: fixes j :: "nat"   and lholed :: "Lholed"   and es :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and tvs :: "t list" assumes "Lfilled j lholed [$Return] es"   and "\<S>\<bullet>\<C> \<turnstile> es : [] _> ts"   and "return \<C> = Some tvs" shows "\<exists>lholed' vs \<C>'. Lfilled j lholed' (vs @ [$Return]) es \<and> \<S>\<bullet>\<C>' \<turnstile> vs : [] _> tvs \<and> const_list vs"
l2_defs_4: shows "l2 \<equiv> \<lparr>init = l2_init, trans = l2_trans, obs = id\<rparr>"
div_free_CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P: fixes A :: "'b set" shows "D (CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P A) = {}"
HFail_valueChosen: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and v :: "InputsOrNi" assumes "HFail s s' q"   and "valueChosen s v"   and "v \<in> Inputs" shows "valueChosen s' v"
list_induct2': fixes P :: "'a list \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list" assumes "P [] []"   and "\<And>x xs. P (x # xs) []"   and "\<And>y ys. P [] (y # ys)"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)" shows "P xs ys"
lnull_stakeWhile: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c stream" shows "lnull (stakeWhile P xs) = (\<not> P (shd xs))"
stable_on_stable_pair_on: fixes ds :: "'b set"   and X :: "'a set" assumes "stable_on ds X" shows "stable_pair_on ds (XD_smallest ds X, XH_largest ds X)"
ex1_iff_ex_Uniq: fixes P :: "'a \<Rightarrow> bool" shows "(\<exists>!x. P x) = ((\<exists>x. P x) \<and> (\<exists>\<^sub>\<le>\<^sub>1 x. P x))"
simps_1: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f" shows "cond_action_oks s t [] = True"
Conf_HAStates2: fixes S :: "'d"   and ST :: "('d, 'e, 'f) status" assumes "S \<in> Conf ST" shows "S \<in> HAStates (HA ST)"
HPhase1or2ReadThen_HInv4a_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 "HInv4a s q"   and "p \<noteq> q" shows "HInv4a s' q"
strict_lunit': fixes f :: "'a" assumes "ide f" shows "\<l>\<^sup>-\<^sup>1[f] = f"
ltl_prop_equiv_quotient_restricted_to_P_finite: fixes P :: "'c ltl set" assumes "finite P" shows "finite {Abs \<phi> |\<phi>. nested_propos \<phi> \<subseteq> P}"
symmatrix_preserve: fixes M :: "(('a, 'b) vec, 'b) vec"   and N :: "(('a, 'b) vec, 'b) vec" assumes "symmatrix M" shows "symmatrix (N ** M ** Finite_Cartesian_Product.transpose N)"
ARR_def: fixes f :: "'c local.term set" shows "ARR f \<equiv> f \<noteq> {} \<and> (\<forall>t. t \<in> f \<longrightarrow> f = {u. local.equiv t u})"
union_disjoint: assumes "finite (A - {x})"   and "finite {x}"   and "(A - {x}) \<inter> {x} = {}" shows "sum g (A - {x} \<union> {x}) = sum g (A - {x}) + sum g {x}"
append_child_heap_is_wellformed_preserved: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "type_wf h"   and "known_ptrs h"   and "heap_is_wellformed h"   and "h \<turnstile> append_child ptr child \<rightarrow>\<^sub>h h'" shows "heap_is_wellformed h'"
monad_left_unit: fixes x :: "'e"   and f :: "'e \<rightarrow> 'c\<cdot>'d" shows "return\<cdot>x \<bind> f = f\<cdot>x"
assms_2: shows "mat (row_length M1) (length M1) M1"
atMost_eq_iff: fixes x :: "'a"   and y :: "'a" shows "({..x} = {..y}) = (x = y)"
stirling_sum_2_conv_stirling_sum'_1: fixes m :: "nat"   and x :: "'a" assumes "0 < m"   and "x \<noteq> (0::'a)" shows "stirling_sum' 1 m x = (1::'a) / x + (1::'a) / ((2::'a) * x\<^sup>2) + stirling_sum 2 m x"
bigtheta_trans1: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Omega>[F](g)"   and "g \<in> \<Theta>[F](h)" shows "f \<in> \<Omega>[F](h)"
cases: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and \<epsilon> :: "ennreal" assumes "hindrance_by \<Gamma> f \<epsilon>" obtains a :: "'a" where "a \<in> A \<Gamma>" and "a \<notin> \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)" and "d_OUT f a < weight \<Gamma> a" and "\<epsilon> < weight \<Gamma> a - d_OUT f a"
BRANCH_UN: fixes b :: "'f"   and f :: "'i \<Rightarrow> 'j \<Rightarrow> ('f + 'g + 'h, 'i) Old_Datatype.node set" shows "BRANCH b (\<lambda>d. \<Union> (range (f d))) = (\<Union>x. BRANCH b (\<lambda>d. f d x))"
cart2_append1_inj: fixes p :: "proj2"   and q :: "proj2" assumes "z_non_zero p"   and "cart2_append1 p = cart2_append1 q" shows "p = q"
child_def: fixes p :: "(nat \<times> int) list"   and dir :: "dir"   and d :: "nat" shows "child p dir d = p[d := (lv p d + 1, 2 * ix p d + Grid_Point.sgn dir)]"
idx_intro_ext: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list"   and i :: "nat" assumes "integrate_insert_commute M a s"   and "i < length (ext_ids s)" shows "idx s (ext_ids s ! i) = Inr i"
elims: fixes x :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and xa :: "'d"   and xb :: "nat \<Rightarrow> 'e"   and xc :: "nat"   and y :: "'d \<times> 'e \<times> 'd" assumes "run\<^sub>t x xa xb xc = y" obtains \<delta> :: "'d \<Rightarrow> 'e \<Rightarrow> 'd" and q\<^sub>0 :: "'d" and w :: "nat \<Rightarrow> 'e" and i :: "nat" where "x = \<delta>" and "xa = q\<^sub>0" and "xb = w" and "xc = i" and "y = (run \<delta> q\<^sub>0 w i, w i, run \<delta> q\<^sub>0 w (Suc i))"
relcomp_Image: fixes X :: "('b \<times> 'c) set"   and Y :: "('c \<times> 'a) set"   and Z :: "'b set" shows "(X O Y) `` Z = Y `` X `` Z"
inf_left_idem: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_inf inf less_eq less" shows "inf x (inf x y) = inf x y"
tau_om2: fixes x :: "'a" shows "\<tau> x\<^sup>\<omega> = \<tau> x"
nds_invars_def: fixes nds :: "('b, 'c) node_scheme set" shows "nds_invars nds \<equiv> inj_on name nds \<and> finite nds \<and> expand_init \<notin> name ` nds \<and> (\<forall>nd\<in>nds. finite (old nd) \<and> incoming nd \<subseteq> insert expand_init (name ` nds))"
simps_5: fixes fv :: "'e \<Rightarrow> 'd set"   and r :: "'e regex" shows "fv_regex fv (Star r) = fv_regex fv r"