hcju/isabelleps huggingface dataset:
sheaf_spec_morphisms_are_ring_morphisms: fixes U :: "'a set set"   and V :: "'a set set" assumes "is_zariski_open U"   and "is_zariski_open V"   and "V \<subseteq> U" shows "ring_homomorphism (sheaf_spec_morphisms U V) (\<O> U) (add_sheaf_spec U) (mult_sheaf_spec U) (zero_sheaf_spec U) (one_sheaf_spec U) (\<O> V) (add_sheaf_spec V) (mult_sheaf_spec V) (zero_sheaf_spec V) (one_sheaf_spec V)"
stable_pair_on_Cd: fixes ds :: "'d set"   and XD_XH :: "'x set \<times> 'x set"   and d :: "'d" assumes "stable_pair_on ds XD_XH"   and "d \<in> ds" shows "Cd d (fst XD_XH) = match XD_XH \<inter> Field (Pd d)"
composition_series_length_one: shows "(length \<GG> = 1) = (\<GG> = [{\<one>\<^bsub>G\<^esub>}])"
hyp: shows "(l = PAodv-:4 \<or> l = PAodv-:5 \<or> l = PRreq-:0 \<or> l = PRreq-:2) \<and> sip (\<sigma>' i) \<noteq> oip (\<sigma>' i)"
simps: fixes tp :: "nat"   and tss :: "nat queue"   and len :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and a1_map :: "('a option list, nat) mapping"   and a2_map :: "(nat, ('a option list, nat + nat) mapping) mapping"   and "done" :: "'a option list set list"   and done_length :: "nat" shows "lin_ts_mmuaux (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length) = linearize tss"
sin_double: fixes x :: "'b" shows "sin ((2::'b) * x) = (2::'b) * sin x * cos x"
rep_eq: fixes x :: "'c"   and xa :: "'c cset" shows "(x \<in>\<^sub>c xa) = (x \<in> rcset xa)"
indefinite_integral_continuous_1: fixes f :: "real \<Rightarrow> 'c"   and a :: "real"   and b :: "real" assumes "f integrable_on {a..b}" shows "continuous_on {a..b} (\<lambda>x. integral {a..x} f)"
pair_measure_distr: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and S :: "'b measure"   and g :: "'c \<Rightarrow> 'd"   and N :: "'c measure"   and T :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M S"   and "g \<in> N \<rightarrow>\<^sub>M T"   and "sigma_finite_measure (distr N T g)" shows "distr M S f \<Otimes>\<^sub>M distr N T g = distr (M \<Otimes>\<^sub>M N) (S \<Otimes>\<^sub>M T) (\<lambda>(x, y). (f x, g y))"
F: shows "map (inv p) (filter (\<lambda>x. x \<in> range p) (xs @ [y])) \<in> traces P"
perm_v_tm: fixes t :: "dbtm" assumes "supp t - {atom i} \<subseteq> atom ` Vs" shows "subst i' (Var sn') (ssubst (vquot_dbtm (insert i Vs) t) (insert i Vs) Fi) = ssubst (vquot_dbtm V' (subst_dbtm (DBVar sn) i t)) V' F'"
jg: fixes x :: "'a" shows "(j (g (h x)) = x) = (g (h x) = h x)"
prv_all_gen: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and x :: "'a"   and \<phi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "x \<in> var"   and "\<phi> \<in> fmla"   and "prv \<phi>" shows "prv (all x \<phi>)"
keys_fold_combine_plus: fixes A :: "'c set"   and f :: "'c \<Rightarrow> ('d, real) mapping" assumes "finite A" shows "Mapping.keys (fold_combine_plus f A) = (\<Union>x\<in>A. Mapping.keys (f x))"
cblinfun_of_matrix_works: fixes f :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "cblinfun_of_matrix (\<lambda>i j. i \<bullet>\<^sub>C cblinfun_apply f j) = f"
intersects_clw_def: fixes X :: "'c set"   and sctn :: "'c sctn" shows "intersects_clw X sctn = sets_of_coll (X ::: clw_rel appr_rel) \<bind> (\<lambda>XS. FORWEAK XS (RETURN False) (\<lambda>X. op_intersects X sctn) (\<lambda>a b. RETURN (a \<or> b)))"
dom_map_entry: fixes k :: "'c"   and f :: "'d \<Rightarrow> 'd"   and xs :: "('c \<times> 'd) list" shows "fst ` set (map_entry k f xs) = fst ` set xs"
def_hashmap_size_pState_ext_def: fixes uu :: "'a pState_scheme itself" shows "def_hashmap_size uu \<equiv> 10"
spmf_run_obsf_oracle_obsf_distinguisher: fixes "oracle" :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and gpv :: "('b, 'c) distinguisher"   and s :: "'a" shows "spmf (run_gpv (obsf_oracle oracle) (obsf_distinguisher gpv) (OK s)) True = spmf (run_gpv oracle gpv s) True"
assms_5: shows "a' \<in> \<R>\<^sub>+ \<Psi> v"
word_of_int_2p_len: shows "word_of_int (2 ^ LENGTH('a)) = 0"
simple_match_any_def: shows "simple_match_any \<equiv> \<lparr>iiface = ifaceAny, oiface = ifaceAny, src = (0, 0), dst = (0, 0), proto = ProtoAny, sports = (0, 65535), dports = (0, 65535)\<rparr>"
sum_Suc_diff: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "m \<le> Suc n" shows "(\<Sum>i = m..n. f (Suc i) - f i) = f (Suc n) - f m"
totient_double: fixes n :: "nat" shows "totient (2 * n) = (if even n then 2 * totient n else totient n)"
store_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "store_instr instr \<equiv> let instr_name = fst instr; op_list = snd instr; flagi = get_operand_flag (op_list ! 0); rd = if instr_name \<in> {load_store_type STA, load_store_type STBA, load_store_type STHA, load_store_type STDA} then get_operand_w5 (op_list ! 4) else get_operand_w5 (op_list ! 3) in do psr_val \<leftarrow> gets (cpu_reg_val PSR); s_val \<leftarrow> gets (\<lambda>s. get_S psr_val); if instr_name \<in> {load_store_type STA, load_store_type STDA, load_store_type STHA, load_store_type STBA} \<and> s_val = 0 then do raise_trap privileged_instruction; return () od else if instr_name \<in> {load_store_type STA, load_store_type STDA, load_store_type STHA, load_store_type STBA} \<and> flagi = 1 then do raise_trap illegal_instruction; return () od else store_sub1 instr rd s_val od"
isSPath_no_selfloop: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat"   and u1 :: "nat" assumes "isSimplePath u p v" shows "(u1, u1) \<notin> set p"
subst_eta_expand: fixes n :: "nat"   and e :: "exp"   and x :: "Vars.var"   and y :: "Vars.var" shows "(eta_expand n e)[x::=y] = eta_expand n e[x::=y]"
slice_deterministic: fixes a :: "'edge"   and a' :: "'edge"   and S :: "'node SDG_node set" assumes "valid_edge a"   and "valid_edge a'"   and "intra_kind (kind a)"   and "intra_kind (kind a')"   and "sourcenode a = sourcenode a'"   and "targetnode a \<noteq> targetnode a'" obtains Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool" and Q' :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool" where "slice_kind S a = (Q)\<^sub>\<surd>" and "slice_kind S a' = (Q')\<^sub>\<surd>" and "\<forall>s. (Q s \<longrightarrow> \<not> Q' s) \<and> (Q' s \<longrightarrow> \<not> Q s)"
ZassenhausTr3: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and H1 :: "'e set"   and K :: "'e set"   and K1 :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<guillemotright> H1 "   and "G \<guillemotright> K "   and "G \<guillemotright> K1 "   and "\<natural>\<^bsub>G\<^esub>H \<triangleright> H1"   and "\<natural>\<^bsub>G\<^esub>K \<triangleright> K1" shows "\<natural>\<^bsub>G\<^esub>H1 \<diamondop>\<^bsub>G\<^esub> H \<inter> K \<triangleright> H1 \<diamondop>\<^bsub>G\<^esub> H \<inter> K1"
less_prod_def: fixes p1 :: "'b \<times> 'c"   and p2 :: "'b \<times> 'c" shows "(p1 < p2) = (p1 \<le> p2 \<and> \<not> p2 \<le> p1)"
simps_2: fixes M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V :: "'d list set"   and m :: "nat"   and n :: "nat" shows "RM M2 M1 \<Omega> V m (Suc n) = {xs' \<in> C M2 M1 \<Omega> V m (Suc n). \<not> M1 \<preceq>\<lbrakk>{xs'}\<rbrakk> M2 \<or> (\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}. \<exists>V''\<in>N io M1 V. \<exists>S1 vs xs. io = vs @ xs \<and> mcp (vs @ xs) V'' vs \<and> S1 \<subseteq> nodes M2 \<and> (\<forall>s1\<in>S1. \<forall>s2\<in>S1. s1 \<noteq> s2 \<longrightarrow> (\<forall>io1\<in>RP M2 s1 vs xs V''. \<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and> m < LB M2 M1 vs xs (TS M2 M1 \<Omega> V m n \<union> V) S1 \<Omega> V'')}"
list_gcd_simps_2: fixes x :: "'a"   and xs :: "'a list" shows "gcd_list (x # xs) = gcd x (gcd_list xs)"
no_ofail_oreturn: fixes x :: "'d" shows "no_ofail (\<lambda>_. True) (oreturn x)"
combine_options_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a option"   and y :: "'a option" shows "combine_options f x y = (case x of None \<Rightarrow> y | Some x \<Rightarrow> case y of None \<Rightarrow> Some x | Some y \<Rightarrow> Some (f x y))"
Says_Server_message_form: fixes B :: "agent"   and A :: "agent"   and NA :: "msg"   and K :: "nat"   and NB :: "msg"   and evs :: "event list" assumes "Says Server B \<lbrace>Crypt (shrK A) \<lbrace>NA, Agent A, Agent B, Key K\<rbrace>, Crypt (shrK B) \<lbrace>NB, Agent A, Agent B, Key K\<rbrace>\<rbrace> \<in> set evs"   and "evs \<in> otway" shows "K \<notin> range shrK \<and> (\<exists>i. NA = Nonce i) \<and> (\<exists>j. NB = Nonce j)"
Invariant_func_is_AE_constant: fixes f :: "'a \<Rightarrow> 'c" assumes "f \<in> borel_measurable Invariants" shows "\<exists>y. AE x in M. f x = y"
C: shows "Max (offs_set_prev ns xs index key mi ma j) = i"
a_ensure_pre_insertion_validity_def: fixes node :: "(_) node_ptr"   and parent :: "(_) object_ptr"   and child_opt :: "(_) node_ptr option" shows "local.a_ensure_pre_insertion_validity node parent child_opt = Heap_Error_Monad.bind (if is_character_data_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r parent then error HierarchyRequestError else return ()) (\<lambda>_. Heap_Error_Monad.bind (get_ancestors parent) (\<lambda>ancestors. Heap_Error_Monad.bind (if cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node \<in> set ancestors then error HierarchyRequestError else return ()) (\<lambda>_. Heap_Error_Monad.bind (case child_opt of None \<Rightarrow> return () | Some child \<Rightarrow> Heap_Error_Monad.bind (get_parent child) (\<lambda>child_parent. if child_parent \<noteq> Some parent then error NotFoundError else return ())) (\<lambda>_. Heap_Error_Monad.bind (get_child_nodes parent) (\<lambda>children. Heap_Error_Monad.bind (if children \<noteq> [] \<and> is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r parent then error HierarchyRequestError else return ()) (\<lambda>_. if is_character_data_ptr\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node \<and> is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r parent then error HierarchyRequestError else return ()))))))"
smap_less_def: fixes m1 :: "'a \<Rightarrow> 'b set"   and m2 :: "'a \<Rightarrow> 'b set" shows "(m1 \<subseteq>. m2) = (\<forall>k. m1 k \<subseteq> m2 k)"
in_mono: fixes A :: "'g set"   and B :: "'g set"   and x :: "'g" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
tauStepChainInduct: fixes x1 :: "('a, 'b, 'c) psi"   and x2 :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> x1 \<Longrightarrow>\<^sub>\<tau> x2"   and "\<And>a b. \<Psi> \<rhd> a \<longmapsto> \<tau> \<prec> b \<Longrightarrow> P a b"   and "\<And>a b c. \<lbrakk>\<Psi> \<rhd> a \<Longrightarrow>\<^sub>\<tau> b; P a b; \<Psi> \<rhd> b \<longmapsto> \<tau> \<prec> c\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
absorb_iff2: fixes b :: "'a"   and a :: "'a" shows "(b \<le> a) = (a \<cdot> b = b)"
intros: fixes s :: "'a + 'b"   and x :: "'b" assumes "s = Inr x" shows "setrp s x"
rad_of_mono: fixes x :: "real"   and y :: "real" shows "(rad_of x \<le> rad_of y) = (x \<le> y)"
arr_Rel_converse_Rel: shows "arr_Rel \<alpha> (T\<inverse>\<^sub>R\<^sub>e\<^sub>l)"
simps: fixes xs :: "('c \<times> 'd) list" shows "to_list (T xs) = xs"
lad_onD: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and C :: "'a set"   and B :: "'a set" assumes "lad_on A f"   and "C \<subseteq> B"   and "B \<subseteq> A" shows "card (f C) \<le> card (f B)"
strand_sem_split_left_1: fixes M :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and S' :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<lbrakk>M; S @ S'\<rbrakk>\<^sub>c \<theta>" shows "\<lbrakk>M; S\<rbrakk>\<^sub>c \<theta>"
and_mask_wi: fixes i :: "int"   and n :: "nat" shows "word_of_int i AND mask n = word_of_int (take_bit n i)"
in_domain: fixes s :: "'a" assumes "s \<in> S" shows "(\<lambda>t. \<phi> t s) \<in> T \<rightarrow> S"
produces_imp_Max_literal: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset"   and A :: "'b" assumes "ground_resolution_with_selection S"   and "ground_resolution_with_selection.production S N C = {A}" shows "A = atm_of (Max_mset C)"
can_coherence: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f \<and> partial_magma.arr (free_monoidal_category.comp C) g \<and> partial_magma.dom (free_monoidal_category.comp C) f = partial_magma.dom (free_monoidal_category.comp C) g \<and> partial_magma.cod (free_monoidal_category.comp C) f = partial_magma.cod (free_monoidal_category.comp C) g"   and "free_monoidal_category.can C f"   and "free_monoidal_category.can C g" shows "f = g"
cmult': fixes c :: "'a"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "\<Theta>[F](\<lambda>x. f x * c) = \<Theta>[F](f)"
image_gen_preserves_col: fixes A :: "'p"   and A' :: "'p"   and X :: "'p"   and Y :: "'p"   and B :: "'p"   and B' :: "'p"   and C :: "'p"   and C' :: "'p" assumes "A A' Reflect X Y"   and "B B' Reflect X Y"   and "C C' Reflect X Y"   and "Col A B C" shows "Col A' B' C'"
freshen_def: fixes n :: "nat" shows "predicate.freshen n = map_lc (curry to_nat n)"
fold_delta_b_init_b_mrexps: fixes w :: "'b list"   and s :: "'b rexp" shows "fold delta_b w (init_b s) \<in> mrexps s \<times> UNIV"
that: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "distinct [a, b, c, d]"   and "{a, b, c, d} \<subseteq> A" shows "thesis"
Restrp_well_founded: shows "well_founded UNIV ((\<sqsubset>) \<restriction> A)"
tail_in_verts: fixes e :: "'b" assumes "e \<in> arcs G" shows "tail G e \<in> verts G"
1: shows "(\<lambda>z. (f z - x) / (z - g x)) \<midarrow>g x\<rightarrow> D"
le: fixes z :: "'a" shows "cmod (f z powr g z) \<le> h z"
matrix_to_iarray_mult_row: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and q :: "'a" shows "matrix_to_iarray (mult_row A i q) = mult_row_iarray (matrix_to_iarray A) (mod_type_class.to_nat i) q"
bn_fresh_Act: fixes \<alpha> :: "'c"   and x :: "('a, 'b, 'c) formula" shows "bn \<alpha> \<sharp>* Formula.Act \<alpha> x"
finfun_upd_triv: fixes f :: "'a \<Rightarrow>f 'b"   and x :: "'a" shows "f(x $:= f $ x) = f"
Sup_iter_unfold: fixes x :: "'a \<times> 'b" shows "sup_prod (sd_power x 0) (Sup_prod (range (\<lambda>n. sd_power x (Suc n)))) = Sup_prod (range (sd_power x))"
preserves_cod: fixes f :: "'a" assumes "arr f" shows "cod (trg f) = trg (cod f)"
FreeGroupD: fixes x :: "'b freeword"   and S :: "'b set" assumes "x \<in> FreeGroup S" shows "order.greater_eq S (fst ` set (freeword x))"
bangRelSubWeakBisim: shows "bangRel weakBisim \<subseteq> weakBisim"
semiattractive_subset: fixes B :: "'a set set" assumes "B \<subseteq> Q" shows "semiattractive B (\<lambda>x y. y \<sqsubseteq>\<^sup>s x)"
ran_sorted_def: fixes a :: "int \<Rightarrow> int"   and l :: "int"   and h :: "int" shows "ran_sorted a l h \<equiv> \<forall>i\<in>{l..<h}. \<forall>j\<in>{l..<h}. i \<le> j \<longrightarrow> a i \<le> a j"
stopping_timeI: fixes T :: "'c \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'c measure" assumes "\<And>t. Measurable.pred (F t) (\<lambda>x. T x \<le> t)" shows "stopping_time F T"
Cd_Xd: fixes x :: "'a"   and d :: "'b"   and X :: "'a set" assumes "x \<in> Bossy.Cd d X" shows "Xd x = d"
infiter_induct: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice"   and x :: "'b::refinement_lattice" assumes "infinite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)"   and "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c::'b::refinement_lattice) (x::'b::refinement_lattice) \<sqsubseteq> x" shows "infinite_iteration.infiter (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c::'b::refinement_lattice) \<sqsubseteq> (x::'b::refinement_lattice)"
trans_reflclI: fixes r :: "('a \<times> 'a) set" assumes "trans r" shows "trans (r\<^sup>=)"
analz_image_freshK_simps_64: fixes K :: "nat"   and H :: "msg set" shows "insert (Key K) H = Key ` {K} \<union> H"
set_2: fixes x21 :: "'k"   and x22 :: "'k list" shows "set (x21 # x22) = insert x21 (set x22)"
lb_sin_cos_aux_zero_le_one: fixes prec :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" shows "lb_sin_cos_aux prec n i j 0 \<le> 1"
assms_4: shows "h \<turnstile> insert_before ptr node child \<rightarrow>\<^sub>h h'"
approx_minus_zero_cancel_iff: fixes x :: "'a star" shows "(- x \<approx> 0) = (x \<approx> 0)"
transpos_id: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and x :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i \<noteq> j"   and "x \<in> {k. k \<le> n} - {i, j}" shows "transpos i j x = x"
IS_RIGHT_UNIQUED: fixes r :: "('d \<times> 'e) set"   and x :: "'d"   and y :: "'e"   and z :: "'e" assumes "single_valued r"   and "(x, y) \<in> r"   and "(x, z) \<in> r" shows "y = z"
nths_vars1': shows "nths a.dims0 a.vars1' = d.dims1"
Exe1S: fixes v :: "i"   and F :: "\<Pi>\<^sub>1"   and x :: "\<kappa>" shows "[\<lparr>F,x\<rparr> in v] = (\<exists>r o\<^sub>1. Some r = d\<^sub>1 F \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 \<in> ex1 r v)"
vector_to_cblinfun_add: fixes x :: "'b"   and y :: "'b" shows "vector_to_cblinfun (x + y) = vector_to_cblinfun x + vector_to_cblinfun y"
zcount_of_mset: fixes M :: "'a multiset"   and x :: "'a" shows "zcount (zmset_of M) x = int (count M x)"
less_le_trans: fixes D :: "('d, 'e) Order_scheme"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "c \<in> carrier D"   and "a \<prec>\<^bsub>D\<^esub> b"   and "b \<preceq>\<^bsub>D\<^esub> c" shows "a \<prec>\<^bsub>D\<^esub> c"
G_def: shows "G \<equiv> \<lambda>t. C + K * integral {0..t} g"
is_Guard_simps_9: fixes F :: "'f"   and g :: "'d set"   and c :: "('d, 'e, 'f) com" shows "is_Guard (Guard F g c) = True"
fact4: assumes "\<exists>v1 v2. fetch_instruction s'' = Inr v1 \<and> decode_instruction v1 = Inr v2 \<and> annul_val s'' = False \<and> supported_instruction (fst v2) = False" shows "False"
assms: fixes j :: "'a" assumes "j \<in> CBasis" shows "((\<lambda>n. cblinfun_apply (b n) j) \<longlongrightarrow> cblinfun_apply a j) F"
getFresh_def: fixes V :: "'var set" shows "getFresh V = (SOME x. x \<in> var \<and> x \<notin> V)"
to_tree_order_wf_is_l_to_tree_order_wf: shows "l_to_tree_order_wf Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.to_tree_order Shadow_DOM.get_parent Shadow_DOM.get_child_nodes"
f2f2': shows "1 / (f2 - f2') \<le> real l0"
congs_def: fixes F\<^sub>1 :: "'a list"   and F\<^sub>2 :: "'a list" shows "F\<^sub>1 \<cong> F\<^sub>2 \<equiv> \<exists>n. F\<^sub>2 = rotate n F\<^sub>1"
sup_eq_bot_iff: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and bot :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "(sup x y = bot) = (x = bot \<and> y = bot)"
obj_simps_5: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.obj a" shows "Span.cod a = a"
ln_one: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and ln :: "'b \<Rightarrow> 'b" assumes "class.ln dist uniformity open minus norm plus zero uminus scaleR sgn times one ln" shows "ln one = zero"
that: fixes x :: "'a" assumes "x \<in> S"   and "f x = x" shows "thesis"
isint_iff: fixes n :: "MIR.num"   and bs :: "real list" shows "isint n bs = (real_of_int \<lfloor>Inum bs n\<rfloor> = Inum bs n)"
subgroup_self: shows "subgroup (carrier (G\<lparr>carrier := S\<rparr>)) (G\<lparr>carrier := S\<rparr>)"
f': fixes T :: "real set"   and X :: "'e set"   and f :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and f' :: "real \<times> 'e \<Rightarrow> real \<times> 'e \<Rightarrow> 'e"   and tx :: "real \<times> 'e" assumes "derivative_on_prod T X f f'"   and "tx \<in> T \<times> X" shows "((\<lambda>(t, x). f t x) has_derivative f' tx) (at tx within T \<times> X)"
finite_nominals_extend: fixes S :: "(('c, 'b) fm list \<times> 'b) set"   and A :: "'b set"   and f :: "nat \<Rightarrow> ('c, 'b) fm list \<times> 'b"   and n :: "nat" assumes "finite (\<Union> (block_nominals ` S))" shows "finite (\<Union> (block_nominals ` extend A S f n))"
Eq_Eats_Iff: fixes i :: "name"   and z :: "tm"   and x :: "tm"   and y :: "tm" assumes "atom i \<sharp> (z, x, y)" shows "{} \<turnstile> z EQ Eats x y IFF All2 i z (Var i IN x OR Var i EQ y) AND x SUBS z AND y IN z"
changes_alt_poly_at_def: fixes ps :: "'a poly list"   and a :: "'a" shows "changes_alt_poly_at ps a = changes_alt (map (\<lambda>p. poly p a) ps)"
find_nu_measure_def: shows "find_nu_measure \<equiv> measure ((-) nu_witness)"
cltn2_transpose_abs: fixes M :: "((real, 3) vec, 3) vec" assumes "invertible M" shows "cltn2_transpose (cltn2_abs M) = cltn2_abs (Finite_Cartesian_Product.transpose M)"
idem: fixes a :: "'a Word.word" shows "signed.min a a = a"
derived_set_of_def: fixes X :: "'b topology"   and S :: "'b set" shows "X derived_set_of S \<equiv> {x \<in> topspace X. \<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y. y \<noteq> x \<and> y \<in> S \<and> y \<in> T)}"
im_is_submodule: shows "LinearCombinations.submodule R l.im M"
PO_s0g_dom_init: shows "init s0g \<subseteq> s0g_dom"
get_root_node_pure: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "pure (get_root_node ptr) h"
induct: fixes P :: "base \<Rightarrow> bool"   and base :: "base" assumes "\<And>x. P (Repeats x)"   and "\<And>x. P (Shares x)" shows "P base"
io_targets_observable_singleton_ob: fixes M :: "('d, 'e, 'f) FSM"   and io :: "('d \<times> 'e) list"   and q1 :: "'f" assumes "observable M"   and "io \<in> LS M q1" obtains q2 :: "'f" where "io_targets M q1 io = {q2}"
set_pmf_eq: fixes M :: "'c pmf" shows "set_pmf M = {x. pmf M x \<noteq> 0}"
derivation_ge_cons: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and d :: "nat \<times> symbol \<times> symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and i :: "nat" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "derivation_ge (d # D) i = (i \<le> fst d \<and> derivation_ge D i)"
gfp_fixpoint: fixes f :: "'a \<Rightarrow> 'a" assumes "mono f" shows "f (gfp f) = gfp f"
prems_2: shows "\<forall>y. ListMem y (a # l2) \<longrightarrow> R1 x1 y = R2 x2 y"