hcju/isabelleps huggingface dataset:
fixp_induct: fixes P :: "(('b, 'call, 'ret) gpv \<Rightarrow> 's \<Rightarrow> ('b \<times> 's) spmf) \<Rightarrow> bool" assumes "spmf.admissible (\<lambda>exec_gpv. P (curry exec_gpv))"   and "P (\<lambda>exec_gpv c. return_pmf None)"   and "\<And>s. P s \<Longrightarrow> P (\<lambda>a aa. the_gpv a \<bind> (\<lambda>a. case a of Pure x \<Rightarrow> return_spmf (x, aa) | IO out c \<Rightarrow> callee aa out \<bind> (\<lambda>a. case a of (x, y) \<Rightarrow> s (c x) y)))" shows "P local.exec_gpv"
det_smult: fixes a :: "'a"   and A :: "'a mat" shows "Determinant.det (a \<cdot>\<^sub>m A) = a ^ dim_col A * Determinant.det A"
ttree_rest_substitute: fixes f :: "'b \<Rightarrow> 'b ttree"   and S :: "'b set"   and T :: "'b set"   and t :: "'b ttree" assumes "\<And>x. carrier (f x) \<inter> S = {}" shows "ttree_restr S (substitute f T t) = ttree_restr S t"
lNactGe: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat" assumes "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'"   and "\<parallel>c\<parallel>\<^bsub>t n'\<^esub>" shows "n \<le> n'"
emeasure_measure_pmf_zero_iff: fixes p :: "'c pmf"   and s :: "'c set" shows "(emeasure (measure_pmf p) s = 0) = (set_pmf p \<inter> s = {})"
wf_append_exec: fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and S' :: "('d, 'c) strand_step list" assumes "wf\<^sub>s\<^sub>t V (S @ S')" shows "wf\<^sub>s\<^sub>t (V \<union> wfvarsoccs\<^sub>s\<^sub>t S) S'"
eventually_False_sequentially: shows "\<not> (\<forall>\<^sub>F n in sequentially. False)"
simps_3: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and cfg1 :: "('a, 'c, 'b) configuration"   and v :: "bool"   and cfg2 :: "('a, 'c, 'b) configuration" shows "asynchronousSystem.steps trans sends cfg1 <\<bottom>, outM v> cfg2 = False"
cycleE: fixes xs :: "Vertices list" assumes "cycle xs" shows "path xs \<and> xs \<noteq> [] \<and> 2 < length xs \<and> last xs \<rightarrow>\<^bsub>T2\<^esub> hd xs"
card_Un_Int: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
typed_model_axioms: shows "typed_model arity public Ana \<Gamma>"
no_nonce_NS1_NS2: fixes evs :: "event list"   and C :: "agent"   and NA' :: "msg"   and NA :: "nat"   and D :: "agent"   and B :: "agent"   and A :: "agent" assumes "evs \<in> ns_public"   and "Crypt (pubK C) \<lbrace>NA', Nonce NA, Agent D\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (knows Spy evs)" shows "Nonce NA \<in> analz (knows Spy evs)"
inorder_balR: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and r :: "('b \<times> nat) tree" shows "Tree2.inorder (balR l a r) = Tree2.inorder l @ a # Tree2.inorder r"
lookup_dflt_def: fixes xs :: "('b \<times> 'a) list"   and k :: "'b" shows "lookup_dflt xs k = (case map_of xs k of None \<Rightarrow> 0::'a | Some v \<Rightarrow> v)"
a_a'_rel: shows "\<forall>pp\<in>set (smods p q). 0 \<le> poly pp a * poly pp a'"
conf_f_Throwing: fixes P :: "jvm_prog"   and h :: "heap"   and sh :: "sheap"   and ST :: "ty list"   and LT :: "ty err list"   and "is" :: "bytecode"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and Cs :: "char list list"   and C' :: "char list"   and xcp :: "nat"   and obj :: "obj"   and sfs :: "sfields" assumes "conf_f P h sh (ST, LT) is (stk, loc, C, M, pc, Called Cs)"   and "is_class P C'"   and "h xcp = \<lfloor>obj\<rfloor>"   and "sh C' = \<lfloor>(sfs, Processing)\<rfloor>" shows "conf_f P h sh (ST, LT) is (stk, loc, C, M, pc, Throwing (C' # Cs) xcp)"
zless_nat_conj: fixes w :: "int"   and z :: "int" shows "(nat w < nat z) = (0 < z \<and> w < z)"
irreducible\<^sub>d_multD: fixes a :: "'b poly"   and b :: "'b poly" assumes "irreducible\<^sub>d (a * b)" shows "degree a = 0 \<and> a \<noteq> 0 \<and> irreducible\<^sub>d b \<or> degree b = 0 \<and> b \<noteq> 0 \<and> irreducible\<^sub>d a"
pos_le_divide_eq: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "(0::'a) < c" shows "(a \<le> b / c) = (a * c \<le> b)"
ant_int: fixes n :: "nat" shows "ant (int n) = an n"
in_sccs_vertsI_sccs: fixes S :: "'a set" assumes "S \<in> verts ` sccs" shows "S \<in> sccs_verts"
long_divisionE: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list" assumes "subfield K R"   and "p \<in> carrier (K [X])"   and "q \<in> carrier (K [X])"   and "q \<noteq> []" shows "long_divides p q (p pdiv q, p pmod q)"
connected_space_iff_components_eq: fixes X :: "'a topology" shows "connected_space X = (\<forall>C\<in>connected_components_of X. \<forall>C'\<in>connected_components_of X. C = C')"
Nats_induct: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "class.semiring_1 one times plus zero"   and "x \<in> semiring_1.Nats one plus zero"   and "\<And>n. P (semiring_1.of_nat one plus zero n)" shows "P x"
Aenc: fixes a :: "cmsg"   and b :: "cmsg"   and c :: "cmsg"   and d :: "cmsg" assumes "eq a b"   and "eq c d" shows "eq (cAenc a c) (cAenc b d)"
irreducible_pos_gcd_twice: fixes p :: "int poly"   and q :: "int poly" assumes "irreducible p"   and "0 < lead_coeff p"   and "irreducible q"   and "0 < lead_coeff q" shows "gcd p q = 1 \<or> p = q"
dom_inv: fixes f :: "'a \<times> 'a" assumes "B.VV.iso f" shows "B.VV.dom (B.VV.inv f) = B.VV.cod f"
measurable_gfp_coinduct: fixes P :: "'b measure \<Rightarrow> bool"   and M :: "'b measure"   and F :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c" assumes "P M"   and "inf_continuous F"   and "\<And>M A. \<lbrakk>P M; \<And>N. P N \<Longrightarrow> A \<in> N \<rightarrow>\<^sub>M count_space UNIV\<rbrakk> \<Longrightarrow> F A \<in> M \<rightarrow>\<^sub>M count_space UNIV" shows "gfp F \<in> M \<rightarrow>\<^sub>M count_space UNIV"
d0_def: shows "st3.d0 = prod_list st3.dims0"
l_minus: fixes x :: "int"   and y :: "int" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus> x \<otimes> y = \<ominus> (x \<otimes> y)"
comm_monoid_fun_axioms: shows "comm_monoid_fun (\<^bold>*) \<^bold>1"
Trans: shows "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'"
carrier_A: shows "mat_of_cblinfun A \<in> carrier_mat dB dA"
sets_Collect_neg: fixes M :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "{x \<in> space M. P x} \<in> sets M" shows "{x \<in> space M. \<not> P x} \<in> sets M"
sym_simps_1: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "arr \
[a, b]"
eta_preserves_typ_of1: fixes t :: "Core.term"   and t' :: "Core.term"   and Ts :: "typ list"   and \<tau> :: "typ" assumes "t \<rightarrow>\<^sub>\<eta> t'"   and "typ_of1 Ts t = Some \<tau>" shows "typ_of1 Ts t' = Some \<tau>"
language_def: shows "D.language \<equiv> {xs. D.nextl (dfa.init \<lparr>dfa.states = {0}, init = 0, final = {}, nxt = \<lambda>q x. q\<rparr>) xs \<in> dfa.final \<lparr>dfa.states = {0}, init = 0, final = {}, nxt = \<lambda>q x. q\<rparr>}"
simps_5: fixes b :: "bexp"   and c :: "com" shows "kill (WHILE b DO c) = {}"
pr_algo_maintains_hb_labeling: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and a :: "pr_operation"   and f' :: "nat \<times> nat \<Rightarrow> 'a"   and l' :: "nat \<Rightarrow> nat" assumes "Height_Bounded_Labeling c s t f l"   and "((f, l), a, f', l') \<in> Network.pr_algo_lts c t" shows "Height_Bounded_Labeling c s t f' l'"
inject_1: fixes x11 :: "subject"   and x12 :: "name \<Rightarrow> pi noption"   and y11 :: "subject"   and y12 :: "name \<Rightarrow> pi noption" shows "(BoundR_Rep x11 x12 = BoundR_Rep y11 y12) = (x11 = y11 \<and> x12 = y12)"
assms_2: fixes a :: "'a"   and b :: "'b"   and list :: "('a \<times> 'b) list" assumes "xs = (a, b) # list" shows "P"
clop_iso: fixes f :: "'b \<Rightarrow> 'b" assumes "clop f" shows "mono f"
L10_Controller: fixes s :: "nat \<Rightarrow> nat list"   and l :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and i :: "nat"   and r :: "nat" assumes "Controller_L s (fin_inf_append [Zero] l) l z"   and "fin_make_untimed (inf_truncate z i) ! (length (fin_make_untimed (inf_truncate z i)) - Suc 0) \<noteq> Zero"   and "last (fin_make_untimed (inf_truncate z i)) = l i"   and "hd (s (Suc i)) = hd (s i) + r"   and "fin_make_untimed (inf_truncate z i) \<noteq> []"   and "r \<le> 10" shows "hd (s (Suc i)) \<le> 800"
offs_pred_ub: fixes ns :: "nat list"   and ub :: "nat"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and i :: "nat" assumes "offs_pred ns ub xs index key mi ma"   and "i < length ns"   and "0 < offs_num (length ns) xs index key mi ma i" shows "ns ! i + offs_num (length ns) xs index key mi ma i \<le> ub"
dim_zero_eq: fixes A :: "('a, 'b) vec set" assumes "vec.dim A = 0" shows "A = {} \<or> A = {0}"
collision_react_subset: fixes s :: "real set"   and t :: "real set" assumes "collision_react s"   and "s \<subseteq> t" shows "collision_react t"
inter_restrict: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and B :: "'b set" assumes "finite A" shows "sum g (A \<inter> B) = (\<Sum>x\<in>A. if x \<in> B then g x else (0::'c))"
equal_neg_zero: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "ell_field R"   and "x \<in> carrier R" shows "(x = \<ominus>\<^bsub>R\<^esub> x) = (x = \<zero>\<^bsub>R\<^esub>)"
equalityD1: fixes A :: "'e set"   and B :: "'e set" assumes "A = B" shows "A \<subseteq> B"
ih: fixes y :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "size y < size t"   and "wary (App s y)" shows "App s y >\<^sub>t y"
i: shows "((\<lambda>x. if g x = l then f l else f (g x)) \<longlongrightarrow> f l) F"
simps_2: fixes r :: "'b routing_rule"   and rs :: "'b routing_rule list"   and p :: "'b word" shows "routing_table_semantics (r # rs) p = (if prefix_match_semantics (routing_match r) p then routing_action r else routing_table_semantics rs p)"
essential_RF: fixes G :: "('a, 'b) graph_scheme"   and B :: "'a set"   and S :: "'a set" shows "essential G B (roofed_gen G B S) = essential G B S"
assms_2: shows "\<sigma>\<^bsup>l\<^esup> \<restriction> local.reads (path \<sigma> l) = \<sigma>'\<^bsup>l'\<^esup> \<restriction> local.reads (path \<sigma> l)"
fundantipermutation1: fixes v :: "'a" assumes "v \<in> C0" shows "the_inv_into (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) fundantivertex v \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
induct: fixes P :: "'b tree \<Rightarrow> 'b tree \<Rightarrow> bool"   and a0 :: "'b tree"   and a1 :: "'b tree" assumes "\<And>r. P \<langle>\<rangle> r"   and "\<And>l a r rr. P l r \<Longrightarrow> P \<langle>l, a, r\<rangle> rr" shows "P a0 a1"
Diag_Diagonalize_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Arr C t" shows "monoidal_language.Diag C (monoidal_language.Diagonalize C t)"
USdB_comp_head_tail: fixes c :: "('exp, 'id) MWLfCom"   and c' :: "('exp, 'id) MWLfCom"   and d :: "'d"   and V :: "('exp, 'id) MWLfCom list"   and V' :: "('exp, 'id) MWLfCom list" assumes "[c] \<approx>\<^bsub>d\<^esub> [c']"   and "V \<approx>\<^bsub>d\<^esub> V'" shows "(c # V) \<approx>\<^bsub>d\<^esub> c' # V'"
enum_nxt_ge: fixes N :: "nat set"   and a :: "nat"   and n :: "nat" assumes "infinite N" shows "a \<le> enum (nxt N a) n"
not_zero_j: shows "\<not> is_zero_row (mod_type_class.from_nat j) A"
simps_3: fixes x :: "'a"   and xs :: "[:'a:]" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "prefix\<cdot>(x :# xs)\<cdot>[::] = FF"
squash_prod_list: fixes ls :: "real list" shows "squash (prod_list ls) = prod_list (map squash ls)"
div_div_div_same: fixes d :: "'a"   and b :: "'a"   and a :: "'a" assumes "d dvd b"   and "b dvd a" shows "a div d div (b div d) = a div b"
hyps_3: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val"   and cmn' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "n < length cms"   and "lcn'' = (cms ! n, mem)"   and "lcn' = (cmn', mem')" shows "(cms, mem) \<rightarrow>\<^bsub>replicate k n\<^esub> (cms[n := cmn'], mem')"
Post_def: shows "Post = (\<lambda>k. tensor_P post (1\<^sub>m K))"
\<R>_def: shows "\<R> \<equiv> {region X I J r |I J r. valid_region X k I J r}"
char_poly_0_block: fixes A :: "'b mat"   and B :: "'b mat"   and C :: "'b mat"   and m :: "nat"   and n :: "nat"   and D :: "'b mat" assumes "A = four_block_mat B C (0\<^sub>m m n) D"   and "\<exists>es. char_poly B = (\<Prod>a\<leftarrow>es. [:- a, 1::'b:])"   and "\<exists>es. char_poly D = (\<Prod>a\<leftarrow>es. [:- a, 1::'b:])"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n m"   and "D \<in> carrier_mat m m" shows "char_poly A = char_poly B * char_poly D"
differentiable: fixes f :: "'e \<Rightarrow> 'a"   and x :: "'e"   and S :: "'e set"   and g :: "'e \<Rightarrow> 'b" assumes "f differentiable at x within S"   and "g differentiable at x within S" shows "(\<lambda>x. f x ** g x) differentiable at x within S"
mu_F1_i: shows "\<bar>gs1.\<mu> i (i - 1)\<bar> \<le> 1 / 2"
landau_function_family_pair_trans: fixes Ftr :: "'a filter"   and F :: "('a \<Rightarrow> real) set"   and G :: "('a \<Rightarrow> real) set"   and f :: "'a \<Rightarrow> real"   and H :: "('a \<Rightarrow> real) set"   and g :: "'a \<Rightarrow> real" assumes "landau_function_family_pair Ftr F G f"   and "landau_function_family_pair Ftr G H g" shows "landau_function_family_pair Ftr F (G * H) f"
\<psi>_mapsto: fixes b :: "'d"   and a :: "'d" assumes "D.ide b"   and "D.ide a" shows "HomD.\<psi> (b, a) \<in> HomD.set (b, a) \<rightarrow> D.hom b a"
ab_group_add_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "ab_group_add_hom hom \<equiv> group_add_hom hom"
size_queue_sum_list: fixes bq :: "('c, 'd) SkewBinomialTree list" shows "size (queue_to_multiset bq) = sum_list (map (size \<circ> tree_to_multiset) bq)"
circ_decompose_4: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>\<star> = x\<^sup>\<star> * (y\<^sup>\<star> * x\<^sup>\<star>)\<^sup>\<star>"
sym_group_carrier: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" shows "(p \<in> carrier (sym_group n)) = (p permutes {1..n})"
uminus_Integer_def: fixes a :: "Integer" shows "- a = (case a of MkI\<cdot>x \<Rightarrow> MkI\<cdot>(- x))"
shadow_root_ptr_kinds_commutes: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap" shows "(cast shadow_root_ptr |\<in>| document_ptr_kinds h) = (shadow_root_ptr |\<in>| shadow_root_ptr_kinds h)"
permutation_swap_id: fixes a :: "'d"   and b :: "'d" shows "permutation (Fun.swap a b id)"
induct: fixes P :: "int poly \<Rightarrow> int poly list \<Rightarrow> bool"   and a0 :: "int poly"   and a1 :: "int poly list" assumes "\<And>f us. \<lbrakk>\<And>x xa xb xaa xab xc xd xe y. \<lbrakk>x = degree f; xa = choose_u us; xb = LLL_short_polynomial pl x xa; xaa = gcd f xb; xab = degree xaa; \<not> (xab = 0 \<or> x \<le> xab); xc = f div xaa; xd = List.partition (\<lambda>gi. p.dvdm gi xc) us; (xe, y) = xd\<rbrakk> \<Longrightarrow> P xc xe; \<And>x xa xb xaa xab xc xd xe y. \<lbrakk>x = degree f; xa = choose_u us; xb = LLL_short_polynomial pl x xa; xaa = gcd f xb; xab = degree xaa; \<not> (xab = 0 \<or> x \<le> xab); xc = f div xaa; xd = List.partition (\<lambda>gi. p.dvdm gi xc) us; (xe, y) = xd\<rbrakk> \<Longrightarrow> P xaa y\<rbrakk> \<Longrightarrow> P f us" shows "P a0 a1"
return_time_n: fixes A :: "'a set"   and n :: "nat" assumes "A \<in> sets M" shows "return_time_function A -` {Suc n} = return_partition A (Suc n)"
simps_2: fixes uv :: "'b" shows "|Inl uv|\<^sub>r = undefined"
all_f_in: shows "\<forall>i<length (pivot_positions A). (i, f i) \<in> set (pivot_positions A)"
bij_betw_ball: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and phi :: "'f \<Rightarrow> bool" assumes "bij_betw f A B" shows "(\<forall>b\<in>B. phi b) = (\<forall>a\<in>A. phi (f a))"
ftv_Ctxt_eqvt: fixes pi :: "(W.tvar \<times> W.tvar) list"   and \<Gamma> :: "(W.var \<times> tyS) list" shows "pi \<bullet> ftv \<Gamma> = ftv (pi \<bullet> \<Gamma>)"
subdiv_step_def: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and u :: "'c"   and v :: "'c"   and w :: "'c"   and uv :: "'d"   and uw :: "'d"   and vw :: "'d" shows "subdiv_step G rev_G H rev_H u v w uv uw vw \<equiv> subdivision_step G rev_G H rev_H (u, v, w) (uv, uw, vw)"
satPB_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi> :: "('fsym, 'psym) lit list set" shows "I.satPB \<xi> \<Phi> \<equiv> \<forall>c\<in>\<Phi>. I.satC \<xi> c"
simps_4: fixes b :: "bexp"   and C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom" shows "strip (IF b THEN C\<^sub>1 ELSE C\<^sub>2) = IF b THEN strip C\<^sub>1 ELSE strip C\<^sub>2"
sum_list_partition: fixes p :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "sum_list (filter p xs) + sum_list (filter (Not \<circ> p) xs) = sum_list xs"
on_omit: fixes p :: "('b, 'c) fm"   and xs :: "nat set"   and ps :: "('b, 'c) fm list"   and i :: "'c" assumes "p on (omit xs ps, i)" shows "p on (ps, i)"
not_Some_eq: fixes x :: "\<langle>'a\<rangle>\<^sub>\<bottom>" shows "(\<forall>y. x \<noteq> \<lfloor>y\<rfloor>) = (x = \<bottom>)"
comparator_sum_simps_3: fixes comp\<^sub>'\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and comp\<^sub>'\<^sub>b :: "'d \<Rightarrow> 'd \<Rightarrow> order"   and x :: "'d"   and y :: "'c" shows "comparator_sum comp\<^sub>'\<^sub>a comp\<^sub>'\<^sub>b (Inr x) (Inl y) = Gt"
collect_cond_actions_def: fixes cts :: "'a conditional_action list" shows "collect_cond_actions cts = {t. Join t \<in> set cts}"
F_link: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "f \<in> Field leq" shows "linkage \<Gamma> f"
agree_weakening_env: fixes e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau> :: "ty"   and \<Gamma> :: "(Syntax.var, ty) fmap" assumes "{$$} \<turnstile> e, eP, eV : \<tau>" shows "\<Gamma> \<turnstile> e, eP, eV : \<tau>"
integrate_var_def: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and v :: "nat"   and e :: "cexpr" shows "integrate_var \<Gamma> v e = \<integral>\<^sub>c map_vars (\<lambda>w. if v = w then 0 else Suc w) e \<partial>\<Gamma> v"
less_M: fixes n :: "nat"   and x :: "'a" assumes "root DIM('a) (1 + real n) *\<^sub>R x \<in> S" shows "norm x < M"
flush_all_until_volatile_write_buffered_val_conv: fixes sb :: "'b memref list"   and i :: "nat"   and ls :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and a :: "nat"   and m :: "nat \<Rightarrow> nat" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}"   and "i < length ls"   and "ls ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)"   and "\<forall>j<length ls. i \<noteq> j \<longrightarrow> (let (uu_, uu_, uu_, sb\<^sub>j, uu_, uu_, uu_) = ls ! j in a \<notin> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j))" shows "flush_all_until_volatile_write ls m a = (case buffered_val sb a of None \<Rightarrow> m a | Some v \<Rightarrow> v)"
hyps_2: shows "\<exists>n. (nxt ^^ n) \<phi> (stl xs)"
open_def: fixes Q :: "'a \<Rightarrow> bool" shows "open Q = (\<forall>C. chain_on (\<sqsubset>) C A \<and> C \<noteq> {} \<and> (\<exists>x\<in>A. glb (\<sqsubset>) C x \<and> Q x) \<longrightarrow> (\<exists>y\<in>C. Q y))"
app_vrrestriction_def: fixes r :: "V"   and A :: "V" shows "r \<restriction>\<^sup>r\<^sub>\<circ> A = set {\<langle>a, b\<rangle> |a b. b \<in>\<^sub>\<circ> A \<and> \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r}"
R'_simps_1: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "x \<le> n"   and "y \<le> n" shows "((x, y) \<in> R' n) = ((x, y) \<in> R)"
IH_1: assumes "sorted (map fst kvs)"   and "distinct (map fst kvs)" shows "rbt_sorted (fst (rbtreeify_f n kvs))"
ordertype_UNIV_\<omega>2: shows "ordertype UNIV pair_less = \<omega> \<up> ord_of_nat 2"
new\<^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_get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "new\<^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 h = (new_character_data_ptr, h')"   and "ptr \<noteq> cast (cast new_character_data_ptr)" shows "get ptr h = get ptr h'"
f_shrink_0: fixes xs :: "'d message_af list" shows "xs \<div> 0 = []"
reachable_If: fixes b :: "bexp"   and c1 :: "com"   and c2 :: "com" shows "reachable (IF b THEN c1 ELSE c2) \<subseteq> {IF b THEN c1 ELSE c2} \<union> reachable c1 \<union> reachable c2"
IdI: fixes a :: "'b" shows "(a, a) \<in> Id"
foreach_refine: fixes adjs :: "('v \<times> nat) list"   and u :: "'v"   and Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'v option" assumes "set adjs = {(v, d). (u, v) \<in> edges (G_\<alpha>g g) \<and> G_\<alpha>w g {u, v} = d}" shows "foreach u adjs (Q, \<pi>) = (Qinter Q \<pi> u, \<pi>' Q \<pi> u)"
is_sentence_concat: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and x :: "symbol list"   and y :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.is_sentence \<NN> \<TT> (x @ y) = (CFG.is_sentence \<NN> \<TT> x \<and> CFG.is_sentence \<NN> \<TT> y)"
lim_imp_Gromov_converging_at_boundary: fixes u :: "nat \<Rightarrow> 'c"   and x :: "'c Gromov_completion" assumes "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> x"   and "x \<in> Gromov_boundary" shows "Gromov_converging_at_boundary u"
closure_of_memI: fixes S :: "('a, 'b) eq_object_scheme"   and a :: "'a"   and A :: "'a set" assumes "a .\<in>\<^bsub>S\<^esub> A"   and "a \<in> carrier S" shows "a \<in> closure_of\<^bsub>S\<^esub> A"
IH: shows "\<forall>y. (y, x) \<in> V \<longrightarrow> WHILE\<^sub>T\<^bsup>I\<^esup> b f y \<le> WHILE\<^bsup>I\<^esup> b f y"
commutative_monoid_axioms_def: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "commutative_monoid_axioms M composition \<equiv> \<forall>x y. x \<in> M \<longrightarrow> y \<in> M \<longrightarrow> composition x y = composition y x"
rbt_comp_2: assumes "ID ccompare \<noteq> None" shows "rbt_comp_lookup ccomp = ord.rbt_lookup cless"
llist_all2_lnthD: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist"   and n :: "nat" assumes "llist_all2 P xs ys"   and "enat n < llength xs" shows "P (lnth xs n) (lnth ys n)"
get_mode_reads: fixes element_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "reads (get_mode_locs element_ptr) (get_mode element_ptr) h h'"
endofunctor_axioms: shows "endofunctor (\<cdot>\<^sub>D) trg\<^sub>D"
total1_def: fixes f :: "nat \<Rightarrow> nat option" shows "total1 f \<equiv> \<forall>x. f x \<down>"
emsapss_decode_help4_def: fixes mHash :: "bit list"   and maskedDB :: "bit list"   and H :: "bit list"   and emBits :: "nat" shows "emsapss_decode_help4 mHash maskedDB H emBits = (if take (roundup emBits 8 * 8 - emBits) maskedDB \<noteq> bv_prepend (roundup emBits 8 * 8 - emBits) \<zero> [] then False else emsapss_decode_help5 mHash maskedDB (MGF H (roundup emBits 8 * 8 - length mHash - 8)) H emBits)"
hd_map: fixes xs :: "'d list"   and f :: "'d \<Rightarrow> 'e" assumes "xs \<noteq> []" shows "hd (map f xs) = f (hd xs)"