hcju/isabelleps huggingface dataset:
uf_graph_invar_def: fixes uf_T :: "('a \<times> 'a) set \<times> 'edge set" shows "uf_graph_invar uf_T \<equiv> case uf_T of (uf, T) \<Rightarrow> corresponding_union_find uf T \<and> Domain uf = V"
simps_1: fixes p :: "nat"   and xs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_slp p [] xs = Some xs"
AssocMerge_def: fixes M :: "(('d, 'd, 'd) mrg, 'd) urel" shows "AssocMerge M = (\<langle>[&0-\<Sigma> \<mapsto>\<^sub>s &1-0-\<Sigma>, &1-0-\<Sigma> \<mapsto>\<^sub>s &1-1-\<Sigma>, &1-1-\<Sigma> \<mapsto>\<^sub>s &0-\<Sigma>]\<rangle>\<^sub>a ;; \<^bold>M3(M) = \<^bold>M3(M))"
oseq_bigger_than_next: fixes i :: "nat" shows "optoy i \<Turnstile> (otherwith nos_inc {i} (orecvmsg msg_ok), other nos_inc {i} \<rightarrow>) global (\<lambda>\<sigma>. no (\<sigma> i) \<le> no (\<sigma> (nhid (\<sigma> i))))"
times_iff: fixes a :: "hf"   and b :: "hf"   and A :: "hf"   and B :: "hf" shows "(\<langle>a, b\<rangle> \<^bold>\<in> A * B) = (a \<^bold>\<in> A \<and> b \<^bold>\<in> B)"
safe_privilege: fixes s :: "'b sparc_state"   and s' :: "'b sparc_state" assumes "get_delayed_pool s = [] \<and> get_trap_set s = {} \<and> snd (execute_instruction () s) = False \<and> s' = snd (fst (execute_instruction () s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
subdivision_bidir: fixes G :: "('a, 'b) pre_digraph"   and rev_G :: "'b \<Rightarrow> 'b"   and H :: "('a, 'b) pre_digraph"   and rev_H :: "'b \<Rightarrow> 'b" assumes "subdivision (G, rev_G) (H, rev_H)" shows "bidirected_digraph H rev_H"
Var_def: fixes x1 :: "name" shows "Var x1 \<equiv> Abs_trm (Var_Rep x1)"
a_coset_add_inv2: fixes M :: "'a set"   and x :: "'a"   and y :: "'a" assumes "M +> x = M +> y"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "M \<subseteq> carrier R" shows "M +> x \<oplus> \<ominus> y = M"
intro: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "UP_ring R"
assms_2: shows "stable_translation_length f \<noteq> 0"
f7: shows "Leaf \<notin> set (local.atomics (max (fst w) (Suc (r t))) (snd w)) \<and> list_all2 (\<lambda>t ta. l t = Suc (r ta)) (butlast (local.atomics (max (fst w) (Suc (r t))) (snd w))) (tl (local.atomics (max (fst w) (Suc (r t))) (snd w))) \<and> (local.atomics (max (fst w) (Suc (r t))) (snd w) = [] \<or> l (last (local.atomics (max (fst w) (Suc (r t))) (snd w))) = fst (max (fst w) (Suc (r t)), snd w) \<and> r (hd (local.atomics (max (fst w) (Suc (r t))) (snd w))) = snd (max (fst w) (Suc (r t)), snd w))"
normal_form_zero_iff: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "(normal_form F p = 0) = (p \<in> ideal F)"
same_one_same_zero_2: fixes C :: "'c ring set"   and R :: "'c ring" assumes "\<And>R. R \<in> C \<Longrightarrow> field R"   and "\<And>R S. \<lbrakk>R \<in> C; S \<in> C\<rbrakk> \<Longrightarrow> R \<lesssim> S \<or> S \<lesssim> R"   and "R \<in> C" shows "\<zero>\<^bsub>union_ring C\<^esub> = \<zero>\<^bsub>R\<^esub>"
virt_to_phys_unchanged2_sub1: fixes va :: "word32"   and s2 :: "('b, 'c) sparc_state_scheme"   and v1 :: "'d word"   and v2 :: "'d word"   and addr :: "word36"   and val :: "word8" shows "(case mem_context_val_w32 (word_of_int 9) (ucast ((v1 >> 11 << 11) OR (v2 AND 511 << 2))) (mem s2) of None \<Rightarrow> None | Some lvl1_page_table \<Rightarrow> ptd_lookup va lvl1_page_table (mem s2) 1) = (case mem_context_val_w32 (word_of_int 9) (ucast ((v1 >> 11 << 11) OR (v2 AND 511 << 2))) ((mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))) of None \<Rightarrow> None | Some lvl1_page_table \<Rightarrow> ptd_lookup va lvl1_page_table ((mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))) 1)"
c: shows "abc_steps_l (0, args) A stp = (st, nl)"
eq: fixes x :: "'a" assumes "x \<in> set_spmf p" shows "spmf p x = spmf q (f x)"
degree_smult_eq: fixes a :: "'b"   and p :: "'b poly" shows "degree (Polynomial.smult a p) = (if a = (0::'b) then 0 else degree p)"
M_M_pow_is_M: fixes n :: "nat"   and p :: "int"   and f :: "int" assumes "n \<noteq> 0"   and "1 < p" shows "poly_mod.M p (poly_mod.M (p ^ n) f) = poly_mod.M p f"
T_Abs: fixes x :: "vrs"   and T\<^sub>1 :: "ty"   and \<Gamma> :: "binding list"   and t\<^sub>2 :: "trm"   and T\<^sub>2 :: "ty" assumes "VarB x T\<^sub>1 # \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2" shows "\<Gamma> \<turnstile> (\<lambda>x:T\<^sub>1. t\<^sub>2) : T\<^sub>1 \<rightarrow> T\<^sub>2"
simps: fixes p :: "('a, 'b) form" shows "dest_Exists (Exists p) = p"
simps_1: fixes k :: "nat"   and T :: "POPLmark.type" shows "\<down>\<^sub>\<tau> 0 k T = T"
target: fixes T1 :: "'a"   and T2 :: "'a"   and TRel :: "('a \<times> 'a) set"   and Enc :: "'b \<Rightarrow> 'a" assumes "(T1, T2) \<in> TRel" shows "(TargetTerm T1, TargetTerm T2) \<in> encoding.indRelT Enc TRel"
unat_of_nat: fixes x :: "nat" shows "unat (word_of_nat x) = x mod 2 ^ LENGTH('a)"
iTrigger_def: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "iTrigger I P Q \<equiv> (\<box> t I. Q t) \<or> (\<diamond> t I. P t \<and> (\<box> t' I \<down>\<ge> t. Q t'))"
correctCompKS_Keys: fixes C :: "specID" assumes "correctCompositionKS C" shows "correctCompositionKeys C"
simps_2: fixes x2 :: "'b" shows "SOURCE \<noteq> Inner x2"
one_relcycle_group: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "\<one>\<^bsub>relcycle_group p X S\<^esub> = 0"
has_integral_le: fixes f :: "'b \<Rightarrow> real"   and i :: "real"   and S :: "'b set"   and g :: "'b \<Rightarrow> real"   and j :: "real" assumes "(f has_integral i) S"   and "(g has_integral j) S"   and "\<And>x. x \<in> S \<Longrightarrow> f x \<le> g x" shows "i \<le> j"
tObsC_ii_abs_jview_eq: fixes a :: "'a"   and t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace" assumes "spr_jview a t' = spr_jview a t" shows "tObsC_ii_abs t = tObsC_ii_abs t'"
msort_map: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "distinct xs"   and "sorted xs"   and "inj_on f (set xs)"   and "order_class.mono_on f (set xs)" shows "msort (map f xs) = map f xs"
rel_converter_Grp: fixes f :: "'l \<Rightarrow> 'h"   and B :: "'i set"   and g :: "'i \<Rightarrow> 'm"   and C :: "'j set"   and h :: "'j \<Rightarrow> 'n"   and k :: "'o \<Rightarrow> 'k" shows "rel_converter (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (BNF_Def.Grp B g) (BNF_Def.Grp C h) (BNF_Def.Grp UNIV k)\<inverse>\<inverse> = BNF_Def.Grp {conv. results_converter (\<I>_uniform (range f) UNIV) (\<I>_uniform UNIV (range k)) conv \<subseteq> B \<and> outs_converter (\<I>_uniform (range f) UNIV) (\<I>_uniform UNIV (range k)) conv \<subseteq> C} (map_converter f g h k)"
wrap_wrap_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and x :: "'f" assumes "digraph_isomorphism hom"   and "f \<in> arcs G \<rightarrow> arcs G"   and "g \<in> arcs G \<rightarrow> arcs G" shows "wrap_iso_arcs hom f (wrap_iso_arcs hom g x) = wrap_iso_arcs hom (f \<circ> g) x"
walk_takeWhile: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and f :: "'c \<Rightarrow> bool" assumes "Graph G"   and "Graph.walk G xs" shows "Graph.walk G (takeWhile f xs)"
get_update_eq: fixes a :: "'a array"   and i :: "nat"   and v :: "'a"   and h :: "heap" shows "Array.get (Array.update a i v h) a = (Array.get h a)[i := v]"
between_front: fixes v :: "'a"   and us :: "'a list"   and u :: "'a"   and vs :: "'a list" assumes "v \<notin> set us" shows "between (u # us @ v # vs) u v = us"
prognosis_not_called: fixes prognosis :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<Rightarrow> Arity list \<Rightarrow> Arity \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and x :: "Vars.var"   and as :: "Arity list"   and a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "CardinalityPrognosisShape prognosis"   and "ae x = \<bottom>" shows "prognosis ae as a (\<Gamma>, e, S) \<sqsubseteq> prognosis ae as a (delete x \<Gamma>, e, S)"
resultant_impl_idom_divide_def: shows "resultant_impl_idom_divide = resultant_impl_generic basic_div_exp"
empty_path_nodes: fixes n :: "'node"   and n' :: "'node" assumes "n -[]\<rightarrow>* n'" shows "n = n'"
ListSum_le: fixes V :: "'a list"   and f :: "'a \<Rightarrow> nat"   and g :: "'a \<Rightarrow> nat" assumes "\<And>v. v \<in> set V \<Longrightarrow> f v \<le> g v" shows "ListSum V f \<le> ListSum V g"
sorted_spvec_cons3: fixes a :: "nat \<times> 'a"   and b :: "nat \<times> 'a"   and t :: "(nat \<times> 'a) list" assumes "sorted_spvec (a # b # t)" shows "fst a < fst b"
log_powr_cancel: fixes a :: "real"   and y :: "real" assumes "0 < a"   and "a \<noteq> 1" shows "log a (a powr y) = y"
pth_eq_iff_index_eq: fixes m :: "('a, 'b) pointermap_scheme"   and p1 :: "nat"   and p2 :: "nat" assumes "pointermap_sane m"   and "pointermap_p_valid p1 m"   and "pointermap_p_valid p2 m" shows "(pm_pth m p1 = pm_pth m p2) = (p1 = p2)"
l_adopt_node_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_adopt_node_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf ShadowRootClass.known_ptr DocumentClass.type_wf DocumentClass.known_ptr Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs Core_DOM_Functions.set_child_nodes Core_DOM_Functions.set_child_nodes_locs get_shadow_root get_shadow_root_locs set_disconnected_nodes set_disconnected_nodes_locs Shadow_DOM.get_owner_document Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.remove_child Shadow_DOM.remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs get_ancestors_di get_ancestors_di_locs Shadow_DOM.adopt_node Shadow_DOM.adopt_node_locs adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M adopt_node_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs ShadowRootClass.known_ptrs get_host get_host_locs get_disconnected_document get_disconnected_document_locs Shadow_DOM.remove"
that: fixes k :: "'a" assumes "k \<in> fst ` set (fst xs) \<union> fst ` set (fst ys)"   and "aux = f k (lookup_raw xs k) (lookup_raw ys k)"   and "\<And>k'. \<lbrakk>k' \<in> fst ` set (fst xs) \<union> fst ` set (fst ys); lt ko k' k\<rbrakk> \<Longrightarrow> f k' (lookup_raw xs k') (lookup_raw ys k') = Some Eq" shows "thesis"
wf_update: fixes i :: "'agent"   and Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "i \<in> agents"   and "total_preorder_on alts Ri'" shows "is_pref_profile (R1(i := Ri'))"
poss3: shows "squash (rpoly q x2) = 0 \<longrightarrow> (\<exists>r>x1. r \<le> x2 \<and> (\<exists>q\<in>set qs. rpoly q r = 0))"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "(a1 >\<^sub>t a2) = ((\<exists>t s. a1 = t \<and> a2 = s \<and> vars_mset s \<subseteq># vars_mset t \<and> wt s < wt t) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> wt t = wt s \<and> \<not> head t \<le>\<ge>\<^sub>h\<^sub>d head s \<and> num_args t = 1 \<and> (\<exists>f\<in>local.ground_heads (head t). arity_sym f = 1 \<and> wt_sym f = 0) \<and> arg t \<ge>\<^sub>t s) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> vars_mset s \<subseteq># vars_mset t \<and> wt t = wt s \<and> head t >\<^sub>h\<^sub>d head s) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> vars_mset s \<subseteq># vars_mset t \<and> wt t = wt s \<and> head t = head s \<and> (\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s))))"
H_simp: shows "N <#> N1 = H1 <#> H \<inter> K"
not_eq_order_implies_strict: fixes a :: "('a, 'b) Term.term"   and b :: "('a, 'b) Term.term" assumes "a \<noteq> b"   and "b \<unrhd> a" shows "b \<rhd> a"
expectation_gpv_Done: fixes fail :: "ennreal"   and \<I> :: "('a, 'b) \<I>"   and f :: "'c \<Rightarrow> ennreal"   and x :: "'c" shows "expectation_gpv fail \<I> f (Generative_Probabilistic_Value.Done x) = f x"
A8: shows "(\<sigma>', q) \<in> oreachable A S U"
cl_UN: fixes L :: "'b set set"   and r :: "'c \<Rightarrow> 'b set"   and I :: "'c set" assumes "lattice L" shows "cl L (\<Union> (r ` I)) = (\<Union>i\<in>I. cl L (r i))"
left_total_vimage2p: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and g :: "'e \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> 'c" assumes "left_total P"   and "surj g" shows "left_total (BNF_Def.vimage2p f g P)"
csl\<rho>': shows "cs\<^bsup>\<pi>'\<^esup> l' = cs\<^bsup>\<rho>'\<^esup> l'"
i_Exec_Stream_Acc_Output_eq_last_message_hold: fixes k :: "nat"   and output_fun :: "'b \<Rightarrow> 'a message_af"   and trans_fun :: "'c message_af \<Rightarrow> 'b \<Rightarrow> 'b"   and input :: "nat \<Rightarrow> 'c message_af"   and c :: "'b" assumes "0 < k" shows "i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c = (output_fun \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c) \<longmapsto> k \<div>\<^sub>l k"
fls_of_int_nonneg: fixes n :: "nat" shows "of_int (int n) = fls_const (of_int (int n))"
path_split_2: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and a :: "'a"   and as' :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n (as @ a # as') n'" shows "valid_edge a"
assms_2: shows "f \<in> M \<rightarrow>\<^sub>M P"
realpow_square_minus_le: fixes u :: "real"   and x :: "real" shows "- u\<^sup>2 \<le> x\<^sup>2"
F_def: fixes P :: "'a process" shows "F P = FAILURES (Rep_process P)"
As_s: shows "\<forall>i\<in>Field r. As (succ r i) \<noteq> As i"
pderiv_eval_deriv_monom: fixes a :: "'a"   and b :: "'a"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "to_fun (pderiv (monom P a n)) b = deriv (monom P a n) b"
MClkReplyNotRetry: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState"   and p :: "PrIds" shows "\<turnstile> MClkReply send rcv cst p \<longrightarrow> \<not> MClkRetry send rcv cst p"
set_map_\<alpha>_cong: fixes \<alpha>' :: "'a \<Rightarrow> 'a set" assumes "\<And>x. x \<in> R \<Longrightarrow> \<alpha>' x = additive.kernel.Class x"   and "\<alpha>' \<in> extensional R" shows "Set_Theory.map \<alpha>' R additive.kernel.Partition"
xor_compl_right: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "boolean_algebra.xor conj disj compl x (compl y) = compl (boolean_algebra.xor conj disj compl x y)"
empty_ran: fixes f :: "'b \<Rightarrow> 'a option" shows "(ran f = {}) = (\<forall>x. f x = None)"
lemma_termdiff5: fixes k :: "real"   and f :: "nat \<Rightarrow> real"   and g :: "'a \<Rightarrow> nat \<Rightarrow> 'b" assumes "0 < k"   and "summable f"   and "\<And>h n. \<lbrakk>h \<noteq> (0::'a); norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h" shows "(\<lambda>h. suminf (g h)) \<midarrow>0::'a\<rightarrow> (0::'b)"
l2_inv3D_HashNonce2: fixes s :: "l2_state"   and X :: "msg"   and R :: "fid_t"   and N :: "nat" assumes "s \<in> l2_inv3"   and "hmac X (NonceF (R $ N)) \<in> synth (analz (extr (bad s) (ik s) (chan s)))" shows "R \<in> bad_runs s"
arc_distinct_ends: fixes g :: "real \<Rightarrow> 'b" assumes "arc g" shows "pathfinish g \<noteq> pathstart g"
multiple_1_same: shows "sys2.multiple_blocks 1 = \<B>'"
N_1: shows "N \<in> sets (M1 \<Otimes>\<^sub>M M2)"
bit_minus_int_iff: fixes k :: "int"   and n :: "nat" shows "bit (- k) n = (\<not> bit (k - 1) n)"
finter_left_commute: fixes A :: "'c fset"   and B :: "'c fset"   and C :: "'c fset" shows "A |\<inter>| (B |\<inter>| C) = B |\<inter>| (A |\<inter>| C)"
ltake_lappend1: fixes n :: "enat"   and xs :: "'a llist"   and ys :: "'a llist" assumes "n \<le> llength xs" shows "ltake n (xs @\<^sub>l ys) = ltake n xs"
finite_UN: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "finite A" shows "finite (\<Union> (B ` A)) = (\<forall>x\<in>A. finite (B x))"
mono_Inf: fixes A :: "('b \<Rightarrow> 'c) set" assumes "A \<subseteq> {f. mono f}" shows "mono (\<Sqinter>A)"
child_parent_dual: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "l_get_parent_wf type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_parent"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "child \<in> set children" shows "h \<turnstile> get_parent child \<rightarrow>\<^sub>r Some ptr"
hh1: shows "(A1, B1, C1, D1) = H1"
cxt_inv_assign: fixes E :: "('Var, 'AExp, 'BExp) Stmt list"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and x :: "'Var"   and e :: "'AExp" assumes "cxt_to_stmt E c = x \<leftarrow> e" shows "c = x \<leftarrow> e \<and> E = []"
exists_basis: shows "\<exists>X. is_basis (set_of_vector X)"
elims_3: fixes outPorts :: "'a \<Rightarrow> 'b fset"   and vertices :: "'c fset"   and nodeOf :: "'c \<Rightarrow> 'a"   and x :: "'c" assumes "\<not> Vertex_Graph.terminal_vertex outPorts vertices nodeOf x" obtains v :: "'c" where "x = v" and "\<not> (v |\<in>| vertices \<and> Vertex_Graph.terminal_node outPorts (nodeOf v))"
fps_is_unit_iff: fixes f :: "'b fps" shows "is_unit f = (f $ 0 \<noteq> (0::'b))"
axioms_3: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and 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_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs" shows "l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs get_disconnected_document get_disconnected_document_locs type_wf"
comp_char: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" shows "C.VV.comp g f = (if C.VV.arr f \<and> C.VV.arr g \<and> C.VxV.seq g f then C.VxV.comp g f else C.VxV.null)"
less_2_cases_iff: fixes n :: "nat" shows "(n < 2) = (n = 0 \<or> n = Suc 0)"
intro: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'f set"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "finite_dimensional_vector_space_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 basis1"   and "vector_space_on U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2" shows "finite_dimensional_vector_space_pair_1_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2 basis1"
equalityCE: fixes A :: "'c set"   and B :: "'c set"   and c :: "'c" assumes "A = B" obtains "s2p A c" and "s2p B c"   | "c \<notin> A" and "c \<notin> B"
assms_2: shows "e2 \<sqsubset>\<^sup>i e1"
assms_4: shows "\<forall>v1 v2. v1 \<noteq> v2 \<longrightarrow> {n. R v1 n} \<inter> {n. R v2 n} = {}"
inner_commute: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c"   and y :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x y = inner y x"
assms: shows "P \<rightarrow> \<langle>\<alpha>,P'\<rangle>"
success_p: fixes h :: "heap" assumes "I h" shows "success p h"
insert_Collect: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "insert a (Collect P) = {u. u \<noteq> a \<longrightarrow> P u}"
square_subtract: fixes p :: "int poly fract"   and q :: "int poly fract" shows "(p - q)\<^sup>2 = rat_poly_plus (p\<^sup>2 - rat_poly_times (rat_poly_times 2 p) q) (q\<^sup>2)"
src_dom: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "src\<^sub>C (C.dom \<mu>) = src\<^sub>C \<mu>"
fminus_fempty: fixes A :: "'b fset" shows "A |-| {||} = A"
int: shows "integrable (lebesgue_on {0..pi}) (Dirichlet_kernel n)"
new: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and v :: "('a, 'b, 'c) val"   and l :: "'b" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Ref (VE v)])"   and "l \<notin> LID\<^sub>G s" shows "substitution.revision_step subst r s (s(r \<mapsto> (\<sigma>, \<tau>(l \<mapsto> v), \<E> [VE (Loc l)])))"
unlabel_subseqsD: fixes A :: "'a list"   and B :: "('b \<times> 'a) list" assumes "A \<in> set (subseqs (unlabel B))" shows "\<exists>C\<in>set (subseqs B). unlabel C = A"
convergent_subseq_is_subseq: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "is_subseq_of s (convergent_subseq s)"