hcju/isabelleps huggingface dataset:
assoc'_is_natural_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and \<nu> :: "'b"   and \<tau> :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>"   and "partial_magma.arr V \<nu>"   and "partial_magma.arr V \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "bicategory.\<a>' V H \<a> src trg \<mu> \<nu> \<tau> = V (H (H \<mu> \<nu>) \<tau>) (bicategory.\<a>' V H \<a> src trg (partial_magma.dom V \<mu>) (partial_magma.dom V \<nu>) (partial_magma.dom V \<tau>))"
branch_existence: fixes \<rho> :: "'c run"   and \<S>\<^sub>1 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list" assumes "\<rho> \<in> \<lbrakk> \<S>\<^sub>1 \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g" shows "\<exists>\<S>\<^sub>2. \<S>\<^sub>1 \<hookrightarrow> \<S>\<^sub>2 \<and> \<rho> \<in> \<lbrakk> \<S>\<^sub>2 \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
insert_def: fixes e :: "'c"   and a :: "'d"   and bq :: "('c, 'd) BinomialTree list" shows "local.insert e a bq = ins (Node e a 0 []) bq"
val_field_1_neq_0: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "1\<^sub>r \<noteq> \<zero>"
inferences_between_def: fixes \<Gamma> :: "'b Inference_System.inference set"   and CC :: "'b literal multiset set"   and C :: "'b literal multiset" shows "inference_system.inferences_between \<Gamma> CC C = {\<gamma> \<in> \<Gamma>. old_infer_from (CC \<union> {C}) \<gamma> \<and> C \<in># old_side_prems_of \<gamma> + {#old_main_prem_of \<gamma>#}}"
af_G_letter_sat_core_lifted: fixes \<G> :: "'b ltl set"   and \<phi> :: "'b ltl_prop_equiv_quotient"   and \<nu> :: "'b set" assumes "Only_G \<G>"   and "\<G> \<Turnstile>\<^sub>P Rep \<phi>" shows "\<G> \<Turnstile>\<^sub>P Rep (\<up>af\<^sub>G \<phi> \<nu>)"
irredh: shows "\<not> is_red (insert q (G - {p})) h"
mset_le_trans: fixes K :: "'b multiset"   and M :: "'b multiset"   and N :: "'b multiset" assumes "K < M"   and "M < N" shows "K < N"
Diagonal_to_Smith_row_i_PQ_eq: fixes P' :: "(('a, 'rows) vec, 'rows) vec"   and B' :: "(('a, 'cols) vec, 'rows) vec"   and Q' :: "(('a, 'cols) vec, 'cols) vec"   and i :: "nat" assumes "(P', B', Q') = Diagonal_to_Smith_row_i_PQ i bezout (P, B, Q)"   and "isDiagonal B"   and "i < min (nrows A) (ncols A)" shows "B' = Diagonal_to_Smith_row_i B i bezout"
weak_tau_modality_def: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and x :: "('a, 'd, 'b) formula" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition" shows "indexed_weak_nominal_ts.weak_tau_modality \<tau> x \<equiv> Disj (map_bset (indexed_weak_nominal_ts.tau_steps \<tau> x) (Abs_bset UNIV))"
measurable_value': fixes M :: "'c measure" shows "value \<in> restrict_space (tree_sigma M) (- {\<langle>\<rangle>}) \<rightarrow>\<^sub>M M"
Unlock1Synchronized: fixes xs :: "'addr val list"   and V :: "nat"   and a' :: "'addr"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and v :: "'addr val"   and h :: "'heap" assumes "xs ! V = Addr a'"   and "V < length xs" shows "uf,P,t \<turnstile>1 \<langle>insync\<^bsub>V\<^esub> (a) Val v,(h, xs)\<rangle> -\<lbrace>(Unlock, a'), SyncUnlock a'\<rbrace>\<rightarrow> \<langle>Val v,(h, xs)\<rangle>"
poly_mult_zero_1: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list" assumes "ring R"   and "set p \<subseteq> carrier R" shows "ring.poly_mult R [] p = []"
one_rel_def: shows "one_rel = \<one> / \<one>"
ffold_finsert: fixes x :: "val \<times> val"   and A :: "(val \<times> val) fset"   and z :: "bool" assumes "x |\<notin>| A" shows "ffold (good_entry xa__ e \<rho> \<circ> good_prod) z (finsert x A) = (good_entry xa__ e \<rho> \<circ> good_prod) x (ffold (good_entry xa__ e \<rho> \<circ> good_prod) z A)"
Some_lcl_casts_values: fixes wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and V :: "char list"   and v :: "val"   and T :: "ty"   and v'' :: "val"   and v' :: "val" assumes "wf_prog wf_md P"   and "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "l V = \<lfloor>v\<rfloor>"   and "E V = \<lfloor>T\<rfloor>"   and "P \<turnstile> T casts v'' to v "   and "l' V = \<lfloor>v'\<rfloor>" shows "P \<turnstile> T casts v' to v' "
redT_updT_new_thread_ts: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and t :: "'f"   and x :: "'g"   and m :: "'i" assumes "thread_ok ts (NewThread t x m)" shows "redT_updT ts (NewThread t x m) t = \<lfloor>(x, no_wait_locks)\<rfloor>"
exec_move_InstanceOf: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and T :: "ty"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (e instanceof T) h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
rel_interior_nonempty_interior: fixes S :: "'b set" assumes "interior S \<noteq> {}" shows "rel_interior S = interior S"
inter: shows "I\<^sub>1 \<inter> I\<^sub>2 \<noteq> {}"
simps_2: fixes nodes :: "'a set"   and edges :: "('a \<times> 'b \<times> 'a) set"   and more :: "'c" shows "edges \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = edges"
convex_less_ri_domain: fixes f :: "'b \<Rightarrow> ereal"   and a :: "ereal" assumes "Lower_Semicontinuous.convex_on UNIV f"   and "\<exists>x. f x < a" shows "\<exists>x\<in>rel_interior (domain f). f x < a"
coprime_commute: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one b a = algebraic_semidom.coprime times one a b"
cdSp_Nil: fixes n :: "'node SDG_node" assumes "valid_SDG_node n" shows "n cd-[]\<rightarrow>\<^sub>d* n"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and index :: "int"   and sizes :: "int set" assumes "pairwise_balance point_set block_collection index"   and "K_block_design point_set block_collection sizes"   and "\<Lambda>_PBD_axioms sizes" shows "\<Lambda>_PBD point_set block_collection index sizes"
set_tries_of_list: fixes key :: "'a \<Rightarrow> 'b list"   and xs :: "'a list" shows "set_tries (tries_of_list key xs) = set xs"
right_idem: fixes a :: "'c option"   and b :: "'c option" shows "least (least a b) b = least a b"
starlike_convex_tweak_boundary_points: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "S \<noteq> {}"   and "rel_interior S \<subseteq> T"   and "T \<subseteq> closure S" shows "starlike T"
node_sos_dest_is_net_state: fixes s :: "'c net_state"   and a :: "'d node_action"   and s' :: "'c net_state"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(s, a, s') \<in> node_sos S" shows "\<exists>i' P' R'. s' = NodeS i' P' R'"
is_acc_run_def: fixes m :: "'b"   and r :: "nat \<Rightarrow> 'Q" shows "gb_graph.is_acc_run (to_gbg_ext m) r \<equiv> graph_defs.is_run (to_gbg_ext m) r \<and> gb_graph.is_acc (to_gbg_ext m) r"
sparse_row_matrix_cons: fixes a :: "nat \<times> (nat \<times> 'a) list"   and arr :: "(nat \<times> (nat \<times> 'a) list) list" shows "sparse_row_matrix (a # arr) = move_matrix (sparse_row_vector (snd a)) (int (fst a)) 0 + sparse_row_matrix arr"
chop_leq1: fixes r :: "real_int"   and s :: "real_int"   and t :: "real_int" shows "R_Chop(r,s,t) \<longrightarrow> s \<le> r"
twoCisTypeI_2: shows "\<forall>x\<in>{a..b}. g2 x \<le> g1 x"
f': fixes tx :: "real \<times> 'a" assumes "tx \<in> T \<times> X" shows "(f has_derivative ($) (f' tx)) (at tx)"
hom_1_iff: fixes x :: "'b mod_ring" shows "(to_int_mod_ring x = 1) = (x = 1)"
less_1: fixes s :: "(accent \<times> 'a) list"   and t :: "(accent \<times> 'a) list" assumes "length s + length t < length sa__ + length ta__"   and "(s, t) \<in> greek_less r" shows "(inv_greek s, inv_greek t) \<in> greek_less r"
sel_1: fixes x1 :: "'f"   and x2 :: "'g" shows "foo.field_1 (make_foo x1 x2) = x1"
homeomorphism_imp_closed_map: fixes S :: "'c set"   and T :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c"   and U :: "'c set" assumes "homeomorphism S T f g"   and "closedin (top_of_set S) U" shows "closedin (top_of_set T) (f ` U)"
LIMSEQ_iff: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a" shows "X \<longlonglongrightarrow> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. \<parallel>X n - L\<parallel> < r)"
divides_fcount: fixes a :: "'a"   and b :: "'a" assumes "a divides b"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "factorcount G a \<le> factorcount G b"
neg_finite: shows "AE M' in M. \<integral>\<^sup>+ x. ennreal (- f x) \<partial>M' \<noteq> \<infinity>"
vimage_comp: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'e"   and x :: "'e set" shows "f -` g -` x = (g \<circ> f) -` x"
elims_2: fixes x :: "('b, 'c) Combinators list" assumes "singleCombinators x" obtains "x = []"   | xs :: "('b, 'c) Combinators list" where "x = DenyAll # xs__" and "singleCombinators xs__"   | v :: "'b" and va :: "'b" and xs :: "('b, 'c) Combinators list" where "x = DenyAllFromTo v__ va__ # xsa__" and "singleCombinators xsa__"   | v :: "'b" and va :: "'b" and vb :: "'c" and xs :: "('b, 'c) Combinators list" where "x = AllowPortFromTo v va vb # xs" and "singleCombinators xs"
span_zero: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'c set" assumes "module scale" shows "(0::'c) \<in> module.span scale S"
nozer: shows "\<not> (\<exists>x>- r. poly [:c, b, a:] x = 0)"
mem_restr_def: fixes A :: "'b list set"   and x :: "'b option list" shows "mem_restr A x = (\<exists>y\<in>A. list_all2 (\<lambda>a b. a \<noteq> None \<longrightarrow> a = Some b) x y)"
plus_times_le_one_lemma: fixes i :: "real"   and u :: "real" assumes "i \<le> 1"   and "u \<le> 1" shows "i + u - i * u \<le> 1"
lprefix_lnull: fixes ys :: "'f llist"   and xs :: "'f llist" assumes "lnull ys" shows "lprefix xs ys = lnull xs"
collapse: fixes prod :: "'f \<times> 'g" shows "(fst prod, snd prod) = prod"
map_replicate_trivial: fixes x :: "'b"   and i :: "nat" shows "map (\<lambda>i. x) [0..<i] = replicate i x"
mat_eqI: fixes nr :: "nat"   and nc :: "nat"   and m1 :: "'e list list"   and m2 :: "'e list list" assumes "mat nr nc m1"   and "mat nr nc m2"   and "\<And>i j. \<lbrakk>i < nc; j < nr\<rbrakk> \<Longrightarrow> m1 ! i ! j = m2 ! i ! j" shows "m1 = m2"
disc_5: fixes x21 :: "(Term_Class.term \<times> sterm) list"   and x22 :: "(name, value) fmap" shows "is_Vabs (Vabs x21 x22)"
zeroes_ni_non_special_solutions: shows "(zeroes m, zeroes n) \<notin> set (non_special_solutions a b)"
alpha_qFresh_qSwap_id: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "qGood X"   and "qFresh zs z1 X"   and "qFresh zs z2 X" shows "X #[[z1 \<and> z2]]_zs #= X"
insert_dom: fixes f :: "'c \<Rightarrow> 'b option"   and x :: "'c"   and y :: "'b" assumes "f x = Some y" shows "insert x (dom f) = dom f"
splitFace_f12_f21_neq_norm: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and vs :: "nat list"   and f12 :: "face"   and f21 :: "face"   and newGraph :: "graph" assumes "pre_splitFace g ram1 ram2 oldF vs"   and "minGraphProps g"   and "(f12, f21, newGraph) = splitFace g ram1 ram2 oldF vs" shows "normFace f12 \<noteq> normFace f21"
cmax_com: fixes r :: "('c \<times> 'c) set"   and s :: "('d \<times> 'd) set" shows "cmax r s =o cmax s r"
valid_cert_appendI: fixes r :: "pratt list"   and s :: "pratt list" assumes "valid_cert r"   and "valid_cert s" shows "valid_cert (r @ s)"
msed_rel_invR: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'a multiset"   and b :: "'b"   and N :: "'b multiset" assumes "rel_mset R M (add_mset b N)" shows "\<exists>M1 a. M = add_mset a M1 \<and> R a b \<and> rel_mset R M1 N"
arr_def: fixes f :: "(bool, 'arr list) concrete_category.arr" shows "arr f \<equiv> domains f \<noteq> {} \<or> codomains f \<noteq> {}"
wf_subclass_def: fixes rel :: "('a \<times> 'a) set" shows "wf_subclass rel = (trans rel \<and> antisym rel \<and> Refl rel)"
card_of_Times_singl2: fixes A :: "'a set"   and b :: "'b" shows "|A| =o |{b} \<times> A|"
lNactLe_nact: fixes n' :: "nat"   and n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "n' = n \<or> n' < n \<and> \<not> (\<exists>k\<ge>n'. k < n \<and> \<parallel>c\<parallel>\<^bsub>t k\<^esub>)" shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'"
dist_execution_def: fixes events :: "'a \<Rightarrow> ('a, 'b) event list" shows "dist_execution events \<equiv> dist_execution_preliminary TYPE('a) \<and> dist_execution_axioms events"
metric_Cauchy_iff2: fixes X :: "nat \<Rightarrow> 'a" shows "Cauchy X = (\<forall>j. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < inverse (real (Suc j)))"
assms: shows "n' = n \<or> n' < n \<and> \<not> (\<exists>k\<ge>n'. k < n \<and> \<parallel>c\<parallel>\<^bsub>t k\<^esub>)"
assms_2: shows "set ops' \<subseteq> set ((\<phi> \<Psi> )\<^sub>\<O>)"
closed_f: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "Semilat A r f"   and "x \<in> A"   and "y \<in> A" shows "x \<squnion>\<^bsub>f\<^esub> y \<in> A"
rel_compp: fixes R :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and S :: "'j \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_option (R OO S) = rel_option R OO rel_option S"
brD: fixes c :: "'e"   and a :: "'f"   and \<alpha> :: "'e \<Rightarrow> 'f"   and I :: "'e \<Rightarrow> bool" assumes "(c, a) \<in> br \<alpha> I" shows "a = \<alpha> c \<and> I c"
FG_advice_af_letter: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and x :: "'b set"   and Y :: "'b ltln set" assumes "w \<Turnstile>\<^sub>n af_letter \<phi> x[Y]\<^sub>\<mu>" shows "(x ## w) \<Turnstile>\<^sub>n \<phi>[Y]\<^sub>\<mu>"
tdghm_op_tdghm_op_tdghm: shows "op_tdghm (op_tdghm \<NN>') = \<NN>'"
RedWhile: fixes b :: "char list exp"   and c :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>while (b) c,s,b'\<rangle> \<rightarrow> \<langle>if (b) (c;; while (b) c) else unit,s,b'\<rangle>"
hd_tl: fixes xs :: "'x" assumes "fst (list_of_oalist xs) \<noteq> []" shows "list_of_oalist xs = (local.hd xs # fst (list_of_oalist (local.tl xs)), snd (list_of_oalist (local.tl xs)))"
lunit_simps_4: fixes f :: "'a" assumes "C.ide f" shows "C.dom \<l>\<^sub>C[f] = trg\<^sub>C f \<star>\<^sub>C f"
decomps\<^sub>e\<^sub>s\<^sub>t_exist_subst: fixes \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and A :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term" assumes "ik\<^sub>e\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t \<cdot> \<I>"   and "sem\<^sub>e\<^sub>s\<^sub>t_c {} \<I> A"   and "wf\<^sub>e\<^sub>s\<^sub>t {} A"   and "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)"   and "Ana_invar_subst (ik\<^sub>e\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t A)"   and "well_analyzed A" shows "\<exists>D\<in>decomps\<^sub>e\<^sub>s\<^sub>t (ik\<^sub>e\<^sub>s\<^sub>t A) (assignment_rhs\<^sub>e\<^sub>s\<^sub>t A) \<I>. ik\<^sub>e\<^sub>s\<^sub>t (A @ D) \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c t \<cdot> \<I>"
coset_mult_inv2: fixes M :: "'a set"   and x :: "'a"   and y :: "'a" assumes "M #>\<^bsub>H'\<^esub> x = M #>\<^bsub>H'\<^esub> y"   and "x \<in> carrier H'"   and "y \<in> carrier H'"   and "M \<subseteq> carrier H'" shows "M #>\<^bsub>H'\<^esub> x \<otimes>\<^bsub>H'\<^esub> inv\<^bsub>H'\<^esub> y = M"
cinner_minus_left: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b"   and y :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner (uminus x) y = - cinner x y"
S: fixes i :: "'a"   and e :: "real" assumes "i \<in> Basis"   and "0 < e" shows "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R i \<in> S"
trace_callee_resource_of_oracle: fixes callee :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and p :: "'g spmf" shows "trace_callee run_resource (map_spmf (RES callee) p) = trace_callee callee p"
lset_conv_fixp: shows "lset \<equiv> lfp.fixp_fun (\<lambda>lset xs. case xs of LNil \<Rightarrow> {} | LCons x xs \<Rightarrow> insert x (lset xs))"
assms_4: shows "D \<notin> f \<turnstile> X"
rel_cases: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "('a, 'c) tllist"   and b :: "('b, 'd) tllist" assumes "tllist_all2 R1 R2 a b" obtains x :: "'c" and y :: "'d" where "a = TNil x" and "b = TNil y" and "R2 x y"   | x1a :: "'a" and x2 :: "('a, 'c) tllist" and y1a :: "'b" and y2 :: "('b, 'd) tllist" where "a = TCons x1a x2" and "b = TCons y1a y2" and "R1 x1a y1a" and "tllist_all2 R1 R2 x2 y2"
simps_2: fixes p :: "fm"   and q :: "fm" shows "minusinf (Or p q) = MIR.disj (minusinf p) (minusinf q)"
set_literal_atm_of: fixes L :: "'a literal" shows "set_literal L = {atm_of L}"
mult: fixes A :: "((real, 'n) vec, 'n) vec"   and B :: "((real, 'n) vec, 'n) vec" assumes "P A"   and "P B" shows "P (A ** B)"
nodes_to_list_def: fixes g :: "('a, 'b, 'c) graph_scheme" shows "nodes_to_list g \<equiv> it_to_sorted_list (\<lambda>_ _. True) (nodes g)"
weakly_future_consistent_def: fixes P :: "'c process"   and I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> ('c list \<times> 'c list) set" shows "weakly_future_consistent P I D R \<equiv> \<forall>u\<in>range D \<inter> (- I) `` range D. \<forall>xs ys. (xs, ys) \<in> R u \<longrightarrow> next_dom_events P D u xs = next_dom_events P D u ys \<and> ref_dom_events P D u xs = ref_dom_events P D u ys"
AssertAllState''_tableau_id: fixes t :: "(nat \<times> linear_poly) list"   and ats :: "('i \<times> 'a atom) list" assumes "\<triangle> t" shows "\<T> (assert_bound_loop ats (init t)) = \<T> (init t)"
simps_15: fixes v :: "atom fm" shows "push_forall (AllQ (ExQ v)) = (if freeIn 0 (ExQ v) then lowerFm 0 1 (ExQ v) else AllQ (ExQ v))"
project_set_def: fixes h :: "'d \<times> 'e \<Rightarrow> 'f"   and C :: "'f set" shows "project_set h C = {x. \<exists>y. h (x, y) \<in> C}"
of_int_nonneg: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "0 \<le> z" shows "less_eq zero (ring_1.of_int one plus zero minus z)"
PiE_eq_iff_not_empty: fixes I :: "'c set"   and F :: "'c \<Rightarrow> 'd set"   and F' :: "'c \<Rightarrow> 'd set" assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}" shows "(Pi\<^sub>E I F = Pi\<^sub>E I F') = (\<forall>i\<in>I. F i = F' i)"
3: shows "bprv (imp (PP \<langle>\<phi>G\<rangle>) (PP \<langle>PP \<langle>\<phi>G\<rangle>\<rangle>))"
Cd_simps_1: fixes d :: "'b" shows "Bossy''.Cd d {} = {}"
distinct_filter: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "distinct xs" shows "distinct (List.filter P xs)"
sorted1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq [x] = True"
thm_2_3_7: fixes X :: "'a option set"   and fs :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) list"   and b :: "nat" assumes "finite X"   and "set fs \<subseteq> P[X]"   and "extended_ord.is_GB_cofactor_bound (set fs) b" shows "(1 \<in> ideal (set fs)) = (1 \<in> set (extended_ord.punit.Macaulay_list (extended_ord.deg_shifts X b fs)))"
antisym: fixes x :: "'k multiset"   and y :: "'k multiset" assumes "x \<subseteq># y"   and "y \<subseteq># x" shows "x = y"
left_inverse: fixes a :: "'e" shows "- a + a = (0::'e)"
single_add: fixes k :: "'a"   and a :: "'b"   and b :: "'b" shows "Poly_Mapping.single k (a + b) = Poly_Mapping.single k a + Poly_Mapping.single k b"
mat_zero_def: shows "mat_zero = (0, 0, 0, 0)"
simps_1: fixes s :: "('a, 'b) Term.term" shows "\<box>\<langle>s\<rangle> = s"
same_level_path_aux_valid_path_aux: fixes cs :: "'edge list"   and as :: "'edge list" assumes "same_level_path_aux cs as" shows "valid_path_aux cs as"
subclseq_superD: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog"   and c :: "class" assumes "G\<turnstile>C\<preceq>\<^sub>C D"   and "C \<noteq> D"   and "class G C = Some c" shows "G\<turnstile>super c\<preceq>\<^sub>C D"
right_stoch_mat_st_mat: fixes A :: "'a st_mat" shows "right_stoch_mat (st_mat A)"
weak_pareto_def: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set"   and Pcond :: "'d set \<Rightarrow> 'c set \<Rightarrow> ('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> bool" shows "weak_pareto scf A Is Pcond \<equiv> \<forall>P x y. Pcond A Is P \<and> x \<in> A \<and> y \<in> A \<and> (\<forall>i\<in>Is. x \<^bsub>(P i)\<^esub>\<prec> y) \<longrightarrow> x \<^bsub>(scf P)\<^esub>\<prec> y"
nextVertex_nth: fixes f :: "face"   and i :: "nat" assumes "distinct (vertices f)"   and "i < |vertices f|" shows "f \<bullet> (vertices f ! i) = vertices f ! ((i + 1) mod |vertices f|)"
test_preserve1: fixes p :: "'a"   and x :: "'a"   and y :: "'a" assumes "test p"   and "p \<cdot> x \<le> p \<cdot> x \<cdot> p" shows "p \<cdot> (p \<cdot> x + !p \<cdot> y)\<^sup>\<dagger> = (p \<cdot> x)\<^sup>\<dagger> \<cdot> p"
pathSubs: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and gamma :: "'b"   and n :: "nat" assumes "inherited subs P"   and "fans subs"   and "\<not> P (tree subs gamma)"   and "\<not> terminal subs (path subs gamma P n)" shows "path subs gamma P (Suc n) \<in> subs (path subs gamma P n)"