hcju/isabelleps huggingface dataset:
subst_fv_unfold: fixes t :: "('b, 'a) Term.term"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "fv (t \<cdot> s) = fv t - subst_domain s \<union> fv\<^sub>s\<^sub>e\<^sub>t (s ` (fv t \<inter> subst_domain s))"
lfp_loop_sound: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "healthy (wp body)"   and "sound P" shows "sound (lfp_exp (\<lambda>Q s. \<guillemotleft> G \<guillemotright> s * wp body Q s + \<guillemotleft> \<N> G \<guillemotright> s * P s))"
assms_1: shows "\<forall>w. prices Mkt x 0 w = cls_val_process Mkt pf2 0 w"
length_greater_0_conv: fixes xs :: "'e list" shows "(0 < length xs) = (xs \<noteq> [])"
\<psi>\<^sub>1\<^sub>0_values: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < dim_row (Matrix.mat (2 ^ n) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ n)))"   and "j < dim_col (Matrix.mat (2 ^ n) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ n)))" shows "Matrix.mat (2 ^ n) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ n)) $$ (i, j) = complex_of_real (1 / sqrt 2 ^ n)"
valid_result_mmsaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "nat \<times> nat \<times> bool list \<times> bool list \<times> (nat \<times> 'a option list set) queue \<times> (nat \<times> 'a option list set) queue \<times> ('a option list, nat) mapping \<times> ('a option list, nat) mapping"   and auxlist :: "(nat \<times> 'a option list set) list" assumes "valid_mmsaux args cur aux auxlist" shows "result_mmsaux args aux = foldr (\<union>) (concat (map (\<lambda>(t, rel). if left (args_ivl args) \<le> cur - t then [rel] else []) auxlist)) {}"
insert_before_list_in_set: fixes x :: "'a"   and v :: "'a"   and ref :: "'a option"   and xs :: "'a list" shows "(x \<in> set (insert_before_list v ref xs)) = (x = v \<or> x \<in> set xs)"
ubound: shows "wmax u1 (wneg l1) \<equiv>\<^sub>U \<bar>r\<bar>"
permutations_of_setD_1: fixes xs :: "'b list"   and A :: "'b set" assumes "xs \<in> permutations_of_set A" shows "set xs = A"
set: fixes x :: "'a" shows "set_node_ptr (node_ptr.Ext x) = {x}"
simps: fixes t1 :: "'a tree"   and t2 :: "'a tree" shows "mrbst_join t1 t2 = (if t1 = \<langle>\<rangle> then return_pmf t2 else if t2 = \<langle>\<rangle> then return_pmf t1 else bernoulli_pmf (real (size t1) / real (size t1 + size t2)) \<bind> (\<lambda>b. if b then case t1 of \<langle>l, x, r\<rangle> \<Rightarrow> map_pmf (Node l x) (mrbst_join r t2) else case t2 of \<langle>l, x, r\<rangle> \<Rightarrow> map_pmf (\<lambda>l'. \<langle>l', x, r\<rangle>) (mrbst_join t1 l)))"
perm_boolI: fixes P :: "bool"   and pi :: "('d \<times> 'd) list" assumes "P" shows "pi \<bullet> P"
configuration_ext_def: fixes states :: "'a \<Rightarrow> 'b"   and msgs :: "nat \<Rightarrow> 'c message list"   and process_snapshot :: "'a \<Rightarrow> 'b option"   and channel_snapshot :: "nat \<Rightarrow> 'c list \<times> recording_state"   and more :: "'d" shows "\<lparr>states = states, msgs = msgs, process_snapshot = process_snapshot, channel_snapshot = channel_snapshot, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons configuration_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple (Record.iso_tuple_cons Record.tuple_iso_tuple states msgs) (Record.iso_tuple_cons Record.tuple_iso_tuple process_snapshot channel_snapshot)) more"
sub_incidence_system_def: fixes \<U> :: "'c set"   and \<A> :: "'c set multiset"   and \<V> :: "'c set"   and \<B> :: "'c set multiset" shows "sub_incidence_system \<U> \<A> \<V> \<B> \<equiv> sub_set_system \<U> \<A> \<V> \<B> \<and> incidence_system \<U> \<A>"
len_gt_eq: shows "dim_col (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets))) \<le> length (rows_to_keep (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets))))"
subspaceI: fixes S :: "'b set" assumes "(0::'b) \<in> S"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x + y \<in> S"   and "\<And>c x. x \<in> S \<Longrightarrow> c *a x \<in> S" shows "m1.subspace S"
generalized_fw_split: fixes fw :: "('a simple_match \<times> 'b) list"   and p :: "('a, 'c) simple_packet_scheme"   and r :: "'a simple_match \<times> 'b" assumes "generalized_sfw fw p = Some r" shows "\<exists>fw1 fw3. fw = fw1 @ r # fw3 \<and> generalized_sfw fw1 p = None"
axioms_2: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> 'e"   and \<sigma> :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "vertical_composite A B F G H \<sigma> \<tau>" shows "category B"
distinct_map: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "distinct (map f xs) = (distinct xs \<and> inj_on f (set xs))"
simps_4: fixes x :: "'b"   and t :: "'b tree" shows "cost (Delete x) [t] = T_delete x t"
straighten_topdown_def: fixes x :: "wall"   and y :: "wall" shows "straighten_topdown x y \<equiv> x = basic [vert, cup] \<circ> basic [cap, vert] \<and> y = basic [vert] \<circ> basic [vert]"
ltake_fin: fixes r :: "'b llist"   and i :: "nat" shows "r \<down> i \<in> UNIV\<^sup>\<star>"
Digamma_real_ge_three_halves_pos: fixes x :: "real" assumes "3 / 2 \<le> x" shows "0 < Digamma x"
noE: shows "Ifm vs bs (qe (list_conj (snd (partition bound0 (conjuncts p))))) = Ifm vs bs (E (list_conj (snd (partition bound0 (conjuncts p)))))"
assoc_in_hom: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and h :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.ide g"   and "Span.ide h"   and "Span.src f = Span.trg g"   and "Span.src g = Span.trg h" shows "Span.in_hom \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc (\<odot>) Maps.PRJ\<^sub>0 Maps.PRJ\<^sub>1 f g h, Dom = Dom ((f \<circ> g) \<circ> h), Cod = Cod (f \<circ> g \<circ> h)\<rparr> ((f \<circ> g) \<circ> h) (f \<circ> g \<circ> h)"
fold_Nil: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" shows "fold f [] = id"
next_free_release: fixes \<phi> :: "'b pltl"   and \<psi> :: "'b pltl" shows "next_free (\<phi> R\<^sub>p \<psi>) = (next_free \<phi> \<and> next_free \<psi>)"
encode_decode_exp_0: fixes base :: "nat"   and M :: "hmultiset" assumes "2 \<le> base"   and "well_base\<^sub>h base M" shows "encode base 0 (decode base 0 M) = M"
parent_child_rel_parent: fixes h :: "(_) heap"   and child_node :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.get_parent child_node \<rightarrow>\<^sub>r Some parent" shows "(parent, 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 child_node) \<in> Shadow_DOM.parent_child_rel h"
simps: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b list"   and a2 :: "'b list" shows "lexext gt a1 a2 = ((\<exists>y ys. a1 = y # ys \<and> a2 = []) \<or> (\<exists>y x ys xs. a1 = y # ys \<and> a2 = x # xs \<and> gt y x) \<or> (\<exists>ys xs x. a1 = x # ys \<and> a2 = x # xs \<and> lexext gt ys xs))"
space_measure_of_eq: fixes M :: "'b measure"   and \<mu> :: "'b set \<Rightarrow> ennreal" shows "space (measure_of (space M) (sets M) \<mu>) = space M"
mod_pred_Suc_mod: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "Suc ((n - Suc 0) mod m) mod m = n mod m"
F_3: fixes i :: "nat" shows "emeasure M (F i) \<noteq> \<infinity>"
uprod_split_asm: fixes P :: "'a \<Rightarrow> bool"   and f :: "('b, 'a) commute"   and x :: "'b uprod" shows "P (case_uprod f x) = (\<nexists>a b. x = Upair a b \<and> \<not> P (apply_commute f a b))"
mremove_valid: fixes t :: "(int \<times> 'b) Tree"   and i :: "int" assumes "valid_tmap t" shows "valid_tmap (mremove i t)"
read_only_unowned: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length (t # ts)"   and "(t # ts) ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "\<O> \<inter> read_only \<S> = {}"
comm_groupE_1: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "comm_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"
dims_dematricize: fixes rmodes :: "nat set"   and A :: "'m mat"   and ds :: "nat list" shows "dims (dematricize rmodes A ds) = ds"
mod_true: fixes h :: "heap \<times> nat set" shows "(h \<Turnstile> true) = in_range h"
prems: shows "linear (app t\<^sub>1 t\<^sub>2)"
equiv_trans_refl: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "equiv_trans t t"
take_Cons_numeral: fixes v :: "num"   and x :: "'b"   and xs :: "'b list" shows "take (numeral v) (x # xs) = x # take (numeral v - 1) xs"
antisym: fixes b :: "'b list"   and a :: "'b list" assumes "b \<le>s a"   and "a \<le>s b" shows "a = b"
distinct_29: fixes x2 :: "sethi_type"   and x6 :: "arith_type" shows "sethi_type x2 \<noteq> arith_type x6"
zag_pl_def: fixes x :: "int \<times> int"   and y :: "int \<times> int" shows "zag_pl x y \<equiv> if snd x + snd y = 2 then (0, 1) else if snd x + snd y = 1 then (0, 1) else if snd x + snd y = 0 then (fst x + fst y, 0) else if snd x + snd y = - 1 then (0, - 1) else if snd x + snd y = - 2 then (0, - 1) else undefined"
unitaryI2: fixes P :: "'a \<Rightarrow> real" assumes "nneg P"   and "bounded_by 1 P" shows "unitary P"
assms_1: shows "\<exists>M. \<forall>n. sum a {0..n} \<le> M"
IH: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option" assumes "s\<^sub>1 = (h, l(V := None))" shows "P \<turnstile> \<langle>{V:T; e\<^sub>1},(h, l)\<rangle> \<rightarrow>* \<langle>e\<^sub>2,(h\<^sub>2, l\<^sub>2(V := l V))\<rangle>"
a: shows "\<not> fin (M[a\<turnstile>c>b]) y"
order_A: fixes rs :: "nat list"   and y :: "nat"   and shared_weights :: "bool"   and ws :: "nat \<Rightarrow> real" assumes "deep_model_correct_params_y rs y" shows "order (deep_model_correct_params_y.A shared_weights rs y ws) = 2 * deep_model_correct_params.N_half rs"
map_simp: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and b :: "'ah"   and f :: "'ag" assumes "constant_functor A B b"   and "partial_magma.arr A f" shows "constant_functor.map A B b f = b"
coeff_not_in_monomials: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and m :: "'b mpoly" assumes "x \<notin> monomials m" shows "MPoly_Type.coeff m x = (0::'b)"
unitaryI: fixes P :: "'c \<Rightarrow> real" assumes "sound P"   and "bounded_by 1 P" shows "unitary P"
sqn_update_bigger: fixes hops :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and ip :: "nat"   and ip' :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and nhip :: "nat" assumes "1 \<le> hops" shows "sqn rt ip \<le> sqn (update rt ip' (dsn, dsk, flag, hops, nhip)) ip"
Cn_imp: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> p .\<longrightarrow>. q) = (Cn {q} \<subseteq> Cn (A \<union> {p}))"
ex_norm_eq_1: shows "\<exists>x. \<parallel>x\<parallel> = 1"
simps: fixes a :: "'b set"   and A :: "'b set set" shows "(a \<in> \<sigma>_algebra A) = ((\<exists>a. a = a \<and> a \<in> A) \<or> a = UNIV \<or> (\<exists>a. a = - a \<and> a \<in> \<sigma>_algebra A) \<or> (\<exists>a. a = \<Union> (range a) \<and> (\<forall>x. a x \<in> \<sigma>_algebra A)))"
bounded_iff: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "(b \<squnion> c \<sqsubseteq> a) = (b \<sqsubseteq> a \<and> c \<sqsubseteq> a)"
\<phi>_def: fixes y :: "'b"   and h :: "'a" shows "Adj.\<eta>\<epsilon>.\<phi> y h = G h \<cdot>\<^sub>D Adj.\<eta> y"
nprv_clear3_1: 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 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 fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<phi>2 :: "'c"   and \<phi>3 :: "'c"   and \<psi> :: "'c"   and \<phi>1 :: "'c" assumes "Deduct_with_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>2, \<phi>3} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<psi> \<in> fmla" shows "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>2, \<phi>3} \<psi>"
Lim_def: fixes "open" :: "'d set \<Rightarrow> bool"   and A :: "'e filter"   and f :: "'e \<Rightarrow> 'd" assumes "class.t2_space open" shows "t2_space.Lim open A f = (THE l. filterlim f (topological_space.nhds open l) A)"
P_Cons_4: fixes \<Gamma>\<^sub>2 :: "binding list"   and fs :: "(char list \<times> trm) list"   and \<Gamma>\<^sub>1 :: "binding list"   and t\<^sub>2 :: "trm"   and T\<^sub>2 :: "POPLmarkRecord.type"   and ts :: "trm list" assumes "\<Gamma>\<^sub>2 \<turnstile> fs [:] fTs"   and "\<Gamma>\<^sub>1 @ \<Delta>\<^sub>2 @ \<Gamma>\<^sub>2 \<turnstile> t\<^sub>2 : T\<^sub>2"   and "\<turnstile> fps [\<rhd>] fs \<Rightarrow> ts" shows "\<down>\<^sub>e \<parallel>\<Delta>\<^sub>2\<parallel> 0 \<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2 \<turnstile> t\<^sub>2[\<parallel>\<Gamma>\<^sub>1\<parallel> \<mapsto>\<^sub>s ts] : \<down>\<^sub>\<tau> \<parallel>\<Delta>\<^sub>2\<parallel> \<parallel>\<Gamma>\<^sub>1\<parallel> T\<^sub>2"
is_aligned_0': fixes n :: "nat" shows "is_aligned 0 n"
Integer_arith_strict_4: fixes x :: "Integer" shows "x * \<bottom> = \<bottom>"
eq: shows "local.add a p\<^sub>1 p\<^sub>2 = p\<^sub>3"
numeral_Bit1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit1 n) = plus (plus (numeral.numeral one plus n) (numeral.numeral one plus n)) one"
axioms_1: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "ideal I R" shows "additive_subgroup I R"
eval_fps_diff: fixes z :: "'a"   and f :: "'a fps"   and g :: "'a fps" assumes "ereal (norm z) < fps_conv_radius f"   and "ereal (norm z) < fps_conv_radius g" shows "eval_fps (f - g) z = eval_fps f z - eval_fps g z"
DEFL_eq_errorT: shows "DEFL('e\<cdot>('f, 'g) errorT) = DEFL('e\<cdot>'g error\<cdot>'f)"
da_e2_BinOp: fixes G :: "prog"   and accC :: "qtname"   and L :: "lenv"   and s0 :: "state"   and binop :: "binop"   and e1 :: "expr"   and e2 :: "expr"   and A :: "assigned"   and e1T :: "ty"   and e2T :: "ty"   and s1 :: "state"   and v1 :: "val" assumes "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>In1l (BinOp binop e1 e2)\<guillemotright> A"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e1\<Colon>-e1T"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e2\<Colon>-e2T"   and "wt_binop G binop e1T e2T"   and "s0\<Colon>\<preceq>(G, L)"   and "normal s1"   and "G\<turnstile>s0 \<midarrow>e1-\<succ>v1\<rightarrow> s1"   and "G,snd s1\<turnstile>v1\<Colon>\<preceq>e1T"   and "wf_prog G" shows "\<exists>E2. \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s1)) \<guillemotright>(if need_second_arg binop v1 then In1l e2 else In1r Skip)\<guillemotright> E2"
assms_1: shows "disjoint ({M} \<union> {N} \<union> V)"
a: shows "abc_steps_l (0, lm) B n = (sb, lmb)"
order_iff: fixes a :: "'e multiset"   and b :: "'e multiset" shows "(a \<subseteq># b) = (a = a \<inter># b)"
collect_locks_def: fixes las :: "'c lock_actions" shows "collect_locks las = {l. Lock \<in> set (las $ l)}"
induct: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> (('c \<times> 'd) \<times> nat) tree \<Rightarrow> bool"   and a0 :: "'c"   and a1 :: "'d"   and a2 :: "(('c \<times> 'd) \<times> nat) tree" assumes "\<And>x y. P x y \<langle>\<rangle>"   and "\<And>x y t1 a b lv t2. \<lbrakk>cmp x a = LT \<Longrightarrow> P x y t1; cmp x a = GT \<Longrightarrow> P x y t2\<rbrakk> \<Longrightarrow> P x y \<langle>t1, ((a, b), lv), t2\<rangle>" shows "P a0 a1 a2"
redT_updIs_insert_Interrupt: fixes t :: "'b"   and "is" :: "'b set"   and ias :: "'b interrupt_action list" assumes "t \<in> redT_updIs is ias"   and "t \<notin> is" shows "Interrupt t \<in> set ias"
Lower_empty: fixes L :: "('a, 'b) gorder_scheme" shows "Lower L {} = carrier L"
cf_sol_def: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme"   and X :: "'e"   and p :: "'e"   and c :: "nat \<times> (nat \<Rightarrow> 'e)" shows "cf_sol R S X p c = (pol_coeff S c \<and> p = polyn_expr R X (fst c) c)"
mult_of_def: fixes R :: "('c, 'd) ring_scheme" shows "Ring_Divisibility.mult_of R \<equiv> \<lparr>carrier = carrier R - {\<zero>\<^bsub>R\<^esub>}, monoid.mult = (\<otimes>\<^bsub>R\<^esub>), one = \<one>\<^bsub>R\<^esub>\<rparr>"
power_Suc0_right: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.monoid_mult one times" shows "power.power one times a (Suc 0) = a"
ac_operator_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "ac_operator f \<equiv> (\<forall>a b c. f (f a b) c = f a (f b c)) \<and> (\<forall>a b. f a b = f b a)"
plus_sq_mtx_def: shows "(+) \<equiv> map_fun to_vec (map_fun to_vec to_mtx) (+)"
l_get_disconnected_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   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" shows "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs \<equiv> ((\<forall>document_ptr h h'. reads (get_disconnected_nodes_locs document_ptr) (get_disconnected_nodes document_ptr) h h') \<and> (\<forall>h document_ptr. type_wf h \<longrightarrow> document_ptr |\<in>| document_ptr_kinds h \<longrightarrow> h \<turnstile> ok get_disconnected_nodes document_ptr)) \<and> (\<forall>h document_ptr. h \<turnstile> ok get_disconnected_nodes document_ptr \<longrightarrow> document_ptr |\<in>| document_ptr_kinds h) \<and> (\<forall>document_ptr h. pure (get_disconnected_nodes document_ptr) h)"
Obtuse_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p" shows "Obtuse A B C \<equiv> \<exists>A' B' C'. Per A' B' C' \<and> A' B' C' LtA A B C"
l2_inv3_init: shows "init l2 \<subseteq> l2_inv3"
simps: fixes M :: "('b, 'c, 'a) FSM"   and p :: "'a" shows "d_reachable M p = {q. \<exists>xs. d_reaches M p xs q}"
ct_1: fixes s :: "'s" assumes "s \<in> S" shows "ct s \<in> K s"
assms_3: shows "(l, m, r) = partition_by_median k ps"
prod_not_prime: fixes x :: "nat"   and y :: "nat" assumes "prime x"   and "prime y"   and "2 < x"   and "2 < y" shows "\<not> prime ((x - 1) * (y - 1))"
sigma_algebra_trivial: fixes \<Omega> :: "'b set" shows "sigma_algebra \<Omega> {{}, \<Omega>}"
sds_anonymous_neutral: fixes \<sigma> :: "'alt \<Rightarrow> 'alt"   and R1 :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and R2 :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "\<sigma> permutes alts"   and "is_pref_profile R1"   and "is_pref_profile R2"   and "anonymous_profile R1 = image_mset (map ((`) \<sigma>)) (anonymous_profile R2)" shows "sds R1 = map_pmf \<sigma> (sds R2)"
size_index_conv: fixes xs :: "'b list"   and n :: "nat"   and x :: "'b" assumes "length xs = n" shows "(n = index xs x) = (x \<notin> set xs)"
comp_termFSbImorph: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) model_scheme"   and h' :: "'c \<Rightarrow> 'f"   and hA' :: "'d \<Rightarrow> 'g"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'f, 'g, 'h) model_scheme" assumes "termFSbImorph h hA MOD"   and "FSbImorph h' hA' MOD MOD'" shows "termFSbImorph (h' \<circ> h) (hA' \<circ> hA) MOD'"
iso_locally_ringed_spaces_def: fixes X :: "'d set"   and is_open\<^sub>X :: "'d set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'d set \<Rightarrow> 'e set"   and \<rho>\<^sub>X :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and one_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and Y :: "'f set"   and is_open\<^sub>Y :: "'f set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'f set \<Rightarrow> 'g set"   and \<rho>\<^sub>Y :: "'f set \<Rightarrow> 'f set \<Rightarrow> 'g \<Rightarrow> 'g"   and d :: "'g"   and add_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and mult_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and one_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and f :: "'d \<Rightarrow> 'f"   and \<phi>\<^sub>f :: "'f set \<Rightarrow> 'g \<Rightarrow> 'e" shows "iso_locally_ringed_spaces X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f \<equiv> morphism_locally_ringed_spaces X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f \<and> iso_locally_ringed_spaces_axioms X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f"
exec_Normal_elim_cases_9: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Cond b c1 c2,Normal s\<rangle> \<Rightarrow> t" obtains "s \<in> b" and "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> t"   | "s \<notin> b" and "\<Gamma>\<turnstile> \<langle>c2,Normal s\<rangle> \<Rightarrow> t"
assms: shows "HT'set \<pi> (insert (f, P, c, Q) \<Theta>)"