hcju/isabelleps huggingface dataset:
nonzero_coeffs_simps_2: fixes xs :: "'a"   and n :: "'b"   and a :: "('a \<times> 'b) multiset" shows "nonzero_coeffs (add_mset (xs, n) a) = (nonzero_coeffs a \<and> n \<noteq> (0::'b))"
STL4Edup: fixes A :: "'a \<Rightarrow> bool"   and sigma :: "behavior"   and F :: "behavior \<Rightarrow> bool"   and G :: "behavior \<Rightarrow> bool" assumes "sigma \<Turnstile> \<box>A"   and "sigma \<Turnstile> \<box>F"   and "\<turnstile> F \<and> \<box>A \<longrightarrow> G" shows "sigma \<Turnstile> \<box>G"
chain_def: fixes A :: "'d set set"   and C :: "'d set set" shows "subset.chain A C = (C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. (\<subset>)\<^sup>=\<^sup>= x y \<or> (\<subset>)\<^sup>=\<^sup>= y x))"
tree_map_of_works: fixes pres :: "('a \<times> 'b) list"   and x :: "'a" shows "lookup (tree_map_of' empty pres) x = AList_Upd_Del.map_of pres x"
transpos_ij_2: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i \<noteq> j" shows "transpos i j j = i"
B: shows "of_real_hom.mat_hom A \<in> carrier_mat n n"
Step: shows "cfg2 \<turnstile> msga \<mapsto> cfg3"
F_unfold: fixes i :: "name" shows "Fi i = (if i \<in> insert i Vs then Var (pi \<bullet> i) else Var i)"
lsnd_\<alpha>_def: shows "ListSetImpl_NotDist.lsnd_\<alpha> \<equiv> set"
call_cont: fixes lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'b" shows "cont (fun_lub lub) (fun_ord ord) lub ord (\<lambda>f. f t)"
cnf_of_operator_effect_encoding_contains_delete_effect_clause_if: fixes \<Pi> :: "'b strips_problem"   and k :: "nat"   and t :: "nat"   and op :: "'b strips_operator"   and v :: "'b" assumes "is_valid_problem_strips \<Pi>"   and "k < t"   and "op \<in> set (\<Pi>\<^sub>\<O>)"   and "v \<in> set (delete_effects_of op)" shows "{(Operator k (index (\<Pi>\<^sub>\<O>) op))\<inverse>, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<inverse>} \<in> cnf (encode_all_operator_effects \<Pi> (\<Pi>\<^sub>\<O>) t)"
line_is_pair_img: shows "{p. \<exists>x. x \<in> cbox a b \<and> p = (x, g2 x)} = (\<lambda>x. (x, g2 x)) ` cbox a b"
oconf_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and obj :: "char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set" shows "P,h \<turnstile> obj \<surd> \<equiv> let (C, S) = obj in (\<forall>Cs. Subobjs P C Cs \<longrightarrow> (\<exists>!fs'. (Cs, fs') \<in> S)) \<and> (\<forall>Cs fs'. (Cs, fs') \<in> S \<longrightarrow> Subobjs P C Cs \<and> (\<exists>fs Bs ms. class P (last Cs) = \<lfloor>(Bs, fs, ms)\<rfloor> \<and> P,h \<turnstile> fs' (:\<le>) map_of fs))"
restr_can_restrict_env: fixes r :: "C"   and e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "(\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub>)|\<^bsub>C\<cdot>r\<^esub> = (\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>|\<^sup>\<circ>\<^bsub>r\<^esub>\<^esub>)|\<^bsub>C\<cdot>r\<^esub>"
alpha_Tree_eqvt: fixes t1 :: "('a, 'b, 'c, 'd) Tree"   and t2 :: "('a, 'b, 'c, 'd) Tree"   and p :: "perm" assumes "t1 =\<^sub>\<alpha> t2" shows "p \<bullet> t1 =\<^sub>\<alpha> p \<bullet> t2"
Rep_mod_ring_inject: fixes x :: "'a mod_ring"   and y :: "'a mod_ring" shows "(Rep_mod_ring x = Rep_mod_ring y) = (x = y)"
B: assumes "\<forall>\<^sub>F x in at_top. eval_primfuns fs x = 0" shows "False"
l_mult_eqn: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G"   and "a = b" shows "c \<cdot> a = c \<cdot> b"
disjointed_subset: fixes A :: "nat \<Rightarrow> 'b set"   and n :: "nat" shows "disjointed A n \<subseteq> A n"
Abs_assn_eqI_2: fixes P :: "heap \<times> nat set \<Rightarrow> bool"   and Pr :: "assn" assumes "\<And>h. P h = (h \<Turnstile> Pr)" shows "Pr = Abs_assn P"
amin_ge: fixes x :: "ant"   and y :: "ant" shows "x \<le> amin x y \<or> y \<le> amin x y"
rebalance_middle_tree_bal: fixes ls :: "('a btree \<times> 'a) list"   and sub :: "'a btree"   and sep :: "'a"   and rs :: "('a btree \<times> 'a) list"   and t :: "'a btree"   and k :: "nat" assumes "bal (Node (ls @ (sub, sep) # rs) t)" shows "bal (rebalance_middle_tree k ls sub sep rs t)"
sel: fixes callee :: "'e \<Rightarrow> 'h \<Rightarrow> ('i \<times> 'e, 'f, 'g) gpv"   and state :: "('d, 'h, 'i) gpv \<times> 'e + ('d, 'h, 'i) rpv \<times> ('i \<times> 'e, 'f, 'g) gpv" shows "the_gpv (inline_aux callee state) = (case state of Inl (c, s) \<Rightarrow> map_spmf (\<lambda>result. case result of Inl (x, s) \<Rightarrow> Pure (x, s) | Inr (out, oracle, rpv) \<Rightarrow> IO out (\<lambda>input. inline_aux callee (Inr (rpv, oracle input)))) (inline1 callee c s) | Inr (rpv, c) \<Rightarrow> map_spmf (\<lambda>result. case result of Inl (Inl (x, s)) \<Rightarrow> Pure (x, s) | Inl (Inr (out, oracle, rpv)) \<Rightarrow> IO out (\<lambda>input. inline_aux callee (Inr (rpv, oracle input))) | Inr (out, c) \<Rightarrow> IO out (\<lambda>input. inline_aux callee (Inr (rpv, c input)))) (the_gpv c \<bind> (\<lambda>generat. case generat of Pure (x, s') \<Rightarrow> map_spmf Inl (inline1 callee (rpv x) s') | IO out c \<Rightarrow> return_spmf (Inr (out, c)))))"
f'_def: shows "f' = (\<lambda>x. if x \<in> A then the_inv_into {..<card A} ((!) (sorted_list_of_set A)) x else card A)"
vimageI: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d"   and b :: "'c"   and B :: "'c set" assumes "f a = b"   and "b \<in> B" shows "a \<in> f -` B"
bis_coind: fixes Retr :: "('b \<times> 'b) set \<Rightarrow> ('b \<times> 'b) set"   and theta :: "('b \<times> 'b) set" assumes "Bisim.mono Retr"   and "sym theta"   and "theta \<subseteq> Retr (theta \<union> bis Retr)" shows "theta \<subseteq> bis Retr"
t: shows "case_nat t \<Gamma> \<turnstile>\<^sub>c e : REAL"
nos_incD: fixes \<xi> :: "state"   and \<xi>' :: "state" assumes "nos_inc \<xi> \<xi>'" shows "no \<xi> \<le> no \<xi>'"
minus_splus: fixes s :: "'a"   and t :: "'a"   and v :: "'t" assumes "s adds t" shows "(t - s) \<oplus> v = t \<oplus> v \<ominus> s"
uminus_Inf: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "class.complete_boolean_algebra Inf Sup inf less_eq less sup bot top minus uminus" shows "uminus (Inf A) = Sup (uminus ` A)"
stopping_time_def: fixes F :: "'c \<Rightarrow> 'd measure"   and T :: "'d \<Rightarrow> 'c" shows "stopping_time F T = (\<forall>t. Measurable.pred (F t) (\<lambda>x. T x \<le> t))"
const_term_def: fixes p :: "'a list list" shows "UP.const_term p = UP.eval p \<zero>\<^bsub>poly_ring R\<^esub>"
h_def: shows "h \<equiv> \<lambda>x. C x f"
forbidden_path_prefix_is_path: fixes v :: "'vertex"   and es :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) stream"   and n :: "nat" assumes "forbidden_path v es" obtains v' :: "'vertex" where "path v' v (rev (stake n es))"
prems_6: shows "InvariantClCurrentLevel (getCl state') (getM state')"
constrains_INT: fixes I :: "'c set"   and F :: "'d program"   and A :: "'c \<Rightarrow> 'd set"   and A' :: "'c \<Rightarrow> 'd set" assumes "\<And>i. i \<in> I \<Longrightarrow> F \<in> A i co A' i" shows "F \<in> \<Inter> (A ` I) co \<Inter> (A' ` I)"
atLeastAtMost_singleton': fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.order less_eq less"   and "a = b" shows "ord.atLeastAtMost less_eq a b = {a}"
arr_cod_iff_arr: fixes f :: "'a" shows "B.arr (B.cod f) = B.arr f"
assms_4: shows "AE x in M. x \<in> \<Omega>"
SPEC_cons_rule: fixes m :: "'b nres"   and \<Phi> :: "'b \<Rightarrow> bool"   and \<Psi> :: "'b \<Rightarrow> bool" assumes "m \<le> SPEC \<Phi>"   and "\<And>x. \<Phi> x \<Longrightarrow> \<Psi> x" shows "m \<le> SPEC \<Psi>"
normalize_ports_generic_normalized_n_primitive: fixes m :: "'b common_primitive match_expr"   and disc :: "'b common_primitive \<Rightarrow> bool"   and sel :: "'b common_primitive \<Rightarrow> ipt_l4_ports"   and C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and rewrite_neg :: "'b common_primitive match_expr \<Rightarrow> 'b common_primitive match_expr"   and normalize_pos :: "'b common_primitive match_expr \<Rightarrow> 'b common_primitive match_expr list" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "\<And>m. normalized_nnf_match m \<Longrightarrow> \<not> has_disc_negated disc False (rewrite_neg m)"   and "\<And>m m'. \<lbrakk>normalized_nnf_match m; \<not> has_disc_negated disc False m; m' \<in> set (normalize_pos m)\<rbrakk> \<Longrightarrow> normalized_nnf_match m'"   and "\<And>m m'. \<lbrakk>normalized_nnf_match m; \<not> has_disc_negated disc False m\<rbrakk> \<Longrightarrow> \<forall>m'\<in>set (normalize_pos m). normalized_n_primitive (disc, sel) (\<lambda>ps. case ps of L4Ports x pts \<Rightarrow> length pts \<le> 1) m'" shows "\<forall>m'\<in>set (normalize_ports_generic normalize_pos rewrite_neg m). normalized_n_primitive (disc, sel) (\<lambda>ps. case ps of L4Ports x pts \<Rightarrow> length pts \<le> 1) m'"
c': shows " t \<longrightarrow>\<^sub>1\<^sup>* t1"
joinable_Cons_l_iff: fixes p :: "'a list"   and v :: "'a"   and q :: "'a list" assumes "p \<noteq> []" shows "joinable (v # p) q = joinable p q"
t: shows "0 < the (eval r_result1 [t, i, f \<triangleright> n])"
wf_fixp_uinduct_pure_ueq_gen: fixes fp :: "(('d, 'e) urel \<Rightarrow> ('d, 'e) urel) \<Rightarrow> ('d, 'e) urel"   and B :: "('d, 'e) urel \<Rightarrow> ('d, 'e) urel"   and R :: "('f \<times> 'f) set"   and Pre :: "('d, 'e) urel"   and e :: "('f, 'd) uexpr"   and Post :: "('d, 'e) urel" assumes "fp B = B (fp B)"   and "wf R"   and "\<And>f st. \<lbrakk>\<And>st'. (st', st) \<in> R \<Longrightarrow> (Pre \<and> \<lceil>e\<rceil>\<^sub>< =\<^sub>u \<guillemotleft>st'\<guillemotright> \<Rightarrow> Post) \<sqsubseteq> f; fp B = f\<rbrakk> \<Longrightarrow> (Pre \<and> \<lceil>e\<rceil>\<^sub>< =\<^sub>u \<guillemotleft>st\<guillemotright> \<Rightarrow> Post) \<sqsubseteq> B f" shows "(Pre \<Rightarrow> Post) \<sqsubseteq> fp B"
f: fixes s :: "complex" assumes "a < Re s" shows "(f has_laplace F) s"
bound0_I: fixes p :: "fm"   and vs :: "'b list"   and b :: "'b"   and bs :: "'b list"   and b' :: "'b" assumes "bound0 p" shows "Ifm vs (b # bs) p = Ifm vs (b' # bs) p"
simps_1: fixes I :: "'at set"   and A :: "'at" shows "I \<Turnstile> Pos A = (A \<in> I)"
semiring_norm_64: fixes a :: "'d" shows "a * (0::'d) = (0::'d)"
inj_on_funpow_dist1: fixes y :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "y \<in> orbit f x" shows "inj_on (\<lambda>n. (f ^^ n) x) {0..<funpow_dist1 f x y}"
Abs_bset_inverse': fixes A :: "'a set" assumes "|A| <o natLeq +c |UNIV|" shows "set_bset (Abs_bset A) = A"
assms: fixes x :: "Preliminaries.var" assumes "x \<in> vars T" shows "\<pi>1 x = \<pi>2 x"
sgn_1_def: fixes x :: "int poly \<times> rat \<times> rat" shows "sgn_1 x = sgn (rai_ub x)"
valid_cubes: shows "\<forall>(k1, \<gamma>1)\<in>one_chain1. valid_path (snd (f (k1, \<gamma>1)))"
simps_2: fixes T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty" shows "V (T\<^sub>1\<rightarrow>T\<^sub>2) = {Lam [x].e |x e. \<forall>v\<in>V T\<^sub>1. \<exists>v'. e[x::=v] \<Down> v' \<and> v' \<in> V T\<^sub>2}"
wtC_faxCD: fixes f :: "'fsym" assumes "wtFsym f"   and "protCl (resOf f)" shows "GE.wtC (faxCD f)"
replication_numbers_def: fixes p :: "'a" shows "add_point_sys.replication_numbers p \<equiv> {\<B> rep x |x. x \<in> add_point p}"
one_right_assertion: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "times x one = x"
smult_mat_mat_one_element: fixes A :: "'a mat"   and B :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat 1 1"   and "B \<in> carrier_mat 1 n" shows "A * B = A $$ (0, 0) \<cdot>\<^sub>m B"
to_int': fixes ff_ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c mod_ring \<Rightarrow> bool"   and p :: "int"   and y :: "'b" assumes "mod_ring_gen ff_ops R p"   and "0 \<le> arith_ops_record.to_int ff_ops y"   and "arith_ops_record.to_int ff_ops y < p" shows "R y (of_int (arith_ops_record.to_int ff_ops y))"
Dom_comp: fixes f :: "('o, 'a) arr"   and g :: "('o, 'a) arr" assumes "arr f"   and "arr g"   and "Dom g = Cod f" shows "Dom (COMP g f) = Dom f"
simps_1: fixes b :: "nat" shows "evalC b (C []) = 0"
Seq_decomp_relpowp_Fault: fixes n :: "nat"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p\<^sub>1 :: "('d, 'e, 'f) com"   and p\<^sub>2 :: "('d, 'e, 'f) com"   and s :: "'d"   and f :: "'f" assumes "\<Gamma>\<turnstile> (Seq p\<^sub>1 p\<^sub>2, Normal s) \<rightarrow>\<^sup>nn (Skip, Fault f)" shows "(\<exists>n1. \<Gamma>\<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>nn1 (Skip, Fault f)) \<or> (\<exists>t n1 n2. \<Gamma>\<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>nn1 (Skip, Normal t) \<and> n1 < n \<and> n2 < n \<and> \<Gamma>\<turnstile> (p\<^sub>2, Normal t) \<rightarrow>\<^sup>nn2 (Skip, Fault f))"
l7: shows "r ^ 3 = Fract (p ^ 3) (q ^ 3)"
distinct_11: fixes x21 :: "('a, 'b) com"   and x22 :: "('a, 'b) com"   and x41 :: "'a"   and x42 :: "('a, 'b) com" shows "x21 ;; x22 \<noteq> while x41 do x42"
l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_shadow_root get_shadow_root_locs get_host get_host_locs \<equiv> get_host = l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_host get_shadow_root \<and> get_host_locs = l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_host_locs get_shadow_root_locs"
ordering2: shows "\<forall>x\<in>{a, b, c, d}. \<exists>n. (finite {a, b, c, d} \<longrightarrow> n < card {a, b, c, d}) \<and> (if n = 0 then a else if n = 1 then b else if n = 2 then c else d) = x"
make_affine_proj_eq_iff: fixes p :: "'a \<times> 'a \<times> 'a"   and p' :: "'a \<times> 'a \<times> 'a" assumes "in_carrierp p"   and "in_carrierp p'" shows "proj_eq p p' = (make_affine p = make_affine p')"
Inf_if_Inf_from: fixes \<iota> :: "'b inference"   and Inf :: "'b inference set"   and N :: "'b set" assumes "\<iota> \<in> inference_system.Inf_from Inf N" shows "\<iota> \<in> Inf"
assms_1: shows "(v', n', def) = lookupDef g n v"
simps_7: fixes var :: "nat"   and a :: "atom fm"   and b :: "atom fm" shows "freeIn var (And a b) = (freeIn var a \<and> freeIn var b)"
lv_ivl_def: fixes xrs :: "'a list"   and yrs :: "'a list" shows "lv_ivl xrs yrs = {zrs. list_all2 (\<le>) xrs zrs \<and> list_all2 (\<le>) zrs yrs}"
shiftr_mask_eq: fixes x :: "'a word"   and n :: "nat" shows "(x >> n) && mask (size x - n) = x >> n"
prod_rel_def: fixes R1 :: "('a \<times> 'd) set"   and R2 :: "('c \<times> 'e) set" shows "R1 \<times>\<^sub>r R2 \<equiv> {((a, b), a', b'). (a, a') \<in> R1 \<and> (b, b') \<in> R2}"
components_nonoverlap: fixes c :: "'a set"   and s :: "'a set"   and c' :: "'a set" assumes "c \<in> components s"   and "c' \<in> components s" shows "(c \<inter> c' = {}) = (c \<noteq> c')"
preorder_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
strategyproofness: shows "pmf (sds R14) c + pmf (sds R14) d \<le> pmf (sds R16) d + pmf (sds R16) c"
dom_restrict_shared: fixes m :: "nat \<Rightarrow> bool option"   and A :: "nat set"   and L :: "nat set" shows "dom (m \<ominus>\<^bsub>A\<^esub> L) = dom m - L"
fstzero_add_zerosnd: shows "fstzero + zerosnd = id_blinfun"
valid_Act: fixes P :: "'state"   and \<alpha> :: "'act"   and x :: "('a, 'pred, 'act) formula" shows "P \<Turnstile> Act \<alpha> x = (\<exists>\<alpha>' x' P'. Act \<alpha> x = Act \<alpha>' x' \<and> P \<rightarrow> \<langle>\<alpha>',P'\<rangle> \<and> P' \<Turnstile> x')"
hsize_args: fixes t :: "('c, 'd) tm" shows "1 + sum_list (map hsize (args t)) = hsize t"
cases: fixes a :: "msg"   and H :: "msg set" assumes "a \<in> synth H" obtains X :: "msg" where "a = Xa__" and "Xa__ \<in> H"   | agt :: "agent" where "a = Agent agt"   | n :: "nat" where "a = Number n"   | X :: "msg" where "a = Hash Xb__" and "Xb__ \<in> synth H"   | X :: "msg" and Y :: "msg" where "a = \<lbrace>Xc__, Y\<rbrace>" and "Xc__ \<in> synth H" and "Y \<in> synth H"   | X :: "msg" and K :: "nat" where "a = Crypt K X" and "X \<in> synth H" and "Key K \<in> H"
assms_2: shows "\<forall>t\<in>C. open t"
vsv_axioms: shows "vsv (\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>)"
DDDC: fixes f :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans" assumes "NatTrans f" shows "CatDom (NTDom f) = CatDom (NTCod f)"
left_mult_zero_mat: fixes A :: "'b mat"   and n :: "nat"   and nc :: "nat"   and nr :: "nat" assumes "A \<in> carrier_mat n nc" shows "0\<^sub>m nr n * A = 0\<^sub>m nr nc"
rpr_in_set: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "rpr A r"   and "x \<^bsub>r\<^esub>\<preceq> y" shows "{x, y} \<subseteq> A"
call_result: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and t :: "state" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some rs"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m (Call chain)], Undecided\<rangle> \<Rightarrow> t"
wf_matchF_aux_Cons: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex"   and entry :: "nat \<times> event_data option list set list \<times> event_data option list set"   and aux :: "(nat \<times> event_data option list set list \<times> event_data option list set) list"   and ne :: "nat"   and i :: "nat" assumes "wf_matchF_aux \<sigma> V n R I r (entry # aux) ne i" shows "wf_matchF_aux \<sigma> V n R I r aux (Suc ne) i"
choose_u_Cons: fixes x :: "int poly"   and xs :: "int poly list" shows "choose_u (x # xs) = (if xs = [] then x else min_degree_poly x (choose_u xs))"
lm062: fixes P :: "('c \<times> 'd) set" shows "inj_on fst P = inj_on snd (P\<inverse>)"
eq: shows "(\<i> * z * csqrt (1 - z\<^sup>2))\<^sup>2 = z\<^sup>2 * (z\<^sup>2 - 1)"
rtrancl_into_rtrancl: fixes a :: "'g"   and b :: "'g"   and r :: "('g \<times> 'g) set"   and c :: "'g" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>*"
vector_smult_lid: fixes x :: "('b, 'c) vec" shows "(1::'b) *s x = x"
delete_block_subset: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and b :: "'b set" assumes "incidence_system point_set block_collection" shows "incidence_system.del_block block_collection b \<subseteq># block_collection"
compose_eq: fixes x :: "'d"   and A :: "'d set"   and g :: "'f \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'f" assumes "x \<in> A" shows "compose A g f x = g (f x)"
assms_3: fixes x :: "'a" assumes "x \<in> T" shows "f x = g x"
in_set_subseqs: fixes xs :: "'a list"   and ys :: "'a list" shows "(xs \<in> set (subseqs ys)) = subseq xs ys"
bf_def: shows "bf \<equiv> \<lambda>M q. wfrec pair_less (\<lambda>f (j, i). let R = case prev j i of None \<Rightarrow> M | Some u \<Rightarrow> snd (f u) in grab R (q j i))"
remove_shadow_root_get_child_nodes_is_l_remove_shadow_root_get_child_nodes: shows "l_remove_shadow_root_get_child_nodes Shadow_DOM.get_child_nodes_locs remove_shadow_root_locs"
invar_pass2: fixes hs :: "'b hp list" assumes "\<forall>h\<in>set hs. php h" shows "invar (pass\<^sub>2 hs)"
uint_ge_0: fixes x :: "'b word" shows "0 \<le> uint x"
SPEC_refine: fixes S :: "'c nres"   and R :: "('c \<times> 'd) set"   and \<Phi> :: "'d \<Rightarrow> bool" assumes "S \<le> SPEC (\<lambda>x. \<exists>x'. (x, x') \<in> R \<and> \<Phi> x')" shows "S \<le> \<Down> R (SPEC \<Phi>)"
rep_eq: fixes x :: "('c, 'd) lifted_boolean_algebra"   and xa :: "('c, 'd) lifted_boolean_algebra" shows "Rep_lifted_boolean_algebra (x \<sqinter> xa) = (\<lambda>f. Rep_lifted_boolean_algebra x f \<sqinter> Rep_lifted_boolean_algebra xa f)"
get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ok: fixes h :: "(_) heap"   and ptr :: "(_) character_data_ptr"   and getter :: "'h option RCharacterData_scheme \<Rightarrow> 'l" assumes "DocumentClass.type_wf h"   and "ptr |\<in>| character_data_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr getter"
Int_insert_left: fixes a :: "'a"   and B :: "'a set"   and C :: "'a set" shows "insert a B \<inter> C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
semiring_normalization_rules_23: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.comm_semiring_1 times one plus zero" shows "plus (plus a b) c = plus (plus a c) b"
matchers_map_upd: fixes \<sigma> :: "'f \<Rightarrow> ('d, 'e) Term.term option"   and x :: "'f"   and t :: "('d, 'e) Term.term" assumes "\<sigma> x = None \<or> \<sigma> x = Some t" shows "matchers_map (\<lambda>y. if y = x then Some t else \<sigma> y) = matchers_map \<sigma> \<inter> {\<tau>. \<tau> x = t}"
UNION_empty_conv_1: fixes B :: "'b \<Rightarrow> 'a set"   and A :: "'b set" shows "({} = \<Union> (B ` A)) = (\<forall>x\<in>A. B x = {})"
pigeon_cols_2: shows "r2 \<in> {..<k + 1}"
seq_stop_eval\<^sub>w: fixes c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" shows "\<langle>Stop ;; c', mds, mem\<rangle> \<leadsto> \<langle>c', mds, mem\<rangle>"
in_quotient_imp_closed: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and X :: "'b set"   and x :: "'b"   and y :: "'b" assumes "equiv A r"   and "X \<in> A // r"   and "x \<in> X"   and "(x, y) \<in> r" shows "y \<in> X"
oseq_no_leq_num: fixes i :: "nat"   and ANY :: "state \<Rightarrow> state \<Rightarrow> bool" shows "optoy i \<Turnstile> (act TT, other ANY {i} \<rightarrow>) onl \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (seql i (\<lambda>(\<xi>, l). l \<in> {PToy-:7..PToy-:8} \<longrightarrow> no \<xi> \<le> num \<xi>))"
TC_add: fixes x :: "V"   and y :: "V" shows "TC (x + y) = TC x \<squnion> lift x (TC y)"
nz0: shows "1 + \<i> * z \<noteq> 0"