hcju/isabelleps huggingface dataset:
with_def: fixes zero :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" shows "span.with zero plus \<equiv> \<lambda>scale b. {sum_with plus zero (\<lambda>a. scale (r a) a) t |t r. finite t \<and> t \<subseteq> b}"
ORD_OF_EQ_diff: fixes x :: "nat"   and y :: "nat" assumes "x \<noteq> y" shows "{ORD_OF x EQ ORD_OF y} \<turnstile> Fls"
cong_def: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b c a = (modulo b a = modulo c a)"
"1_3": shows "length slist_1 = Suc (length as_1)"
rn_price_borel_adapt: fixes cash_flow :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and N :: "bool stream measure"   and q :: "real"   and n :: "nat" assumes "cash_flow \<in> borel_measurable (G matur)"   and "N = bernoulli_stream q"   and "0 < q"   and "q < 1"   and "n \<le> matur" shows "rn_price N cash_flow matur n \<in> borel_measurable (G n)"
pos_divide_less_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and c :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less (divide b c) a = less b (times a c)"
inject_1: fixes x1 :: "\<omega>"   and y1 :: "\<omega>" shows "(\<omega>\<upsilon> x1 = \<omega>\<upsilon> y1) = (x1 = y1)"
nr_bounds: shows "\<forall>\<^sup>\<infinity>n. 2 \<le> nat \<lceil>r n\<rceil> \<and> nat \<lceil>r n\<rceil> \<le> n"
insert_commute: fixes x :: "'c"   and y :: "'c"   and A :: "'c set" shows "Set.insert x (Set.insert y A) = Set.insert y (Set.insert x A)"
open_vec_def: fixes S :: "('a, 'b) vec set" shows "open S = (\<forall>x\<in>S. \<exists>A. (\<forall>i. open (A i) \<and> x $ i \<in> A i) \<and> (\<forall>y. (\<forall>i. y $ i \<in> A i) \<longrightarrow> y \<in> S))"
simps_5: shows "J.Cod J.BT = J.TT"
a1: shows "s' = snd (fst (call_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
a: shows "\<forall>x y. (x, y) \<in> r\<^sup>* \<and> x \<in> r\<^sup>* `` A \<and> y \<in> r\<^sup>* `` A \<longrightarrow> (x, y) \<in> (restrict r (r\<^sup>* `` A))\<^sup>*"
strips_operator_inverse_is: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op' :: "('c \<times> 'd) strips_operator" assumes "is_valid_problem_sas_plus \<Psi>"   and "op' \<in> set ((\<phi> \<Psi> )\<^sub>\<O>)" shows "\<phi>\<^sub>O \<Psi> \<phi>\<^sub>O\<inverse> \<Psi> op' = op'"
C_disj_le_gz: fixes i :: "nat"   and j :: "nat"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat" assumes "i \<le> j"   and "0 < i" shows "C M2 M1 \<Omega> V m i \<inter> C M2 M1 \<Omega> V m (Suc j) = {}"
distinct_28: fixes x151 :: "floatarith"   and x152 :: "nat"   and x11 :: "floatarith"   and x12 :: "floatarith" shows "x151 ^\<^sub>e x152 \<noteq> Add x11 x12"
intros_3: fixes P :: "complex mat"   and Q :: "complex mat"   and R :: "complex mat"   and S1 :: "com"   and S2 :: "com" assumes "is_quantum_predicate P"   and "is_quantum_predicate Q"   and "is_quantum_predicate R"   and "\<turnstile>\<^sub>p {P} S1 {Q}"   and "\<turnstile>\<^sub>p {Q} S2 {R}" shows "\<turnstile>\<^sub>p {P} S1;; S2 {R}"
nterm_to_term_closed_wellformed: fixes t :: "nterm"   and \<Gamma> :: "name list" assumes "closed t" shows "wellformed (nterm_to_term \<Gamma> t)"
monotone_ennreal_add2: fixes x :: "ennreal" shows "monotone (\<le>) (\<le>) ((+) x)"
check_ipresIGFresh: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "ipresIGFresh check (errMOD MOD) MOD"
d_OUT_cmult: fixes c :: "ennreal"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" shows "d_OUT (\<lambda>e. c * f e) x = c * d_OUT f x"
assms: fixes s :: "'a" assumes "s \<in> S" shows "0 \<in> U s \<and> is_interval (U s) \<and> U s \<subseteq> T"
amin_le_r: fixes x :: "ant"   and y :: "ant" shows "amin x y \<le> y"
share_all_until_volatile_write_unowned: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and a :: "nat" assumes "ownership_distinct ts"   and "sharing_consis \<S> ts"   and "\<forall>i p is \<theta> sb \<D> \<O> \<R>. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> a \<notin> \<O> \<union> all_acquired sb" shows "share_all_until_volatile_write ts \<S> a = \<S> a"
mult_le_cancel1_hmset: fixes k :: "hmultiset"   and m :: "hmultiset"   and n :: "hmultiset" shows "(k * m \<le> k * n) = (0 < k \<longrightarrow> m \<le> n)"
rem_implicit_pres_ops_astI: fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.astI (rem_implicit_pres_ops prob) = ast_problem.astI prob"
star_fix: fixes x :: "'a" shows "x ^ * = x * x ^ * \<sqinter> (1::'a)"
card_of_Un_diff_infinite: fixes A :: "'d set"   and B :: "'d set" assumes "infinite A"   and "|B| <o |A|" shows "|A - B| =o |A|"
neq_LNil_conv: fixes xs :: "'b llist" shows "(xs \<noteq> LNil) = (\<exists>x xs'. xs = LCons x xs')"
cases: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c \<times> 'd Filters.filter" assumes "triple phi" obtains xa :: "'c" and y :: "'d Filters.filter" where "x = (xa, y)"
SUP_upper2: fixes i :: "'b"   and A :: "'b set"   and u :: "'c"   and f :: "'b \<Rightarrow> 'c" assumes "i \<in> A"   and "u \<le> f i" shows "u \<le> \<Squnion>(f ` A)"
min_elt_mem: fixes B :: "'f set" assumes "B \<subseteq> UNIV"   and "B \<noteq> {}" shows "min_elt B \<in> B"
assms_4: shows "(- h \<sqinter> - - g)\<^sup>\<star> \<le> forest_components f"
intf: fixes k :: "nat" shows "f k integrable_on cbox a b"
InvariantEquivalent_def: fixes F0 :: "Literal list list"   and F :: "Literal list list" shows "InvariantEquivalent F0 F \<equiv> equivalentFormulae F0 F"
trmc_conj_inclusion: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A"   and "\<phi> \<notin> K \<div>\<^bsub>\<gamma>\<^sub>T\<^sub>R\<^esub> (\<phi> .\<and>. \<psi>)" shows "K \<div>\<^bsub>\<gamma>\<^sub>T\<^sub>R\<^esub> (\<phi> .\<and>. \<psi>) \<subseteq> K \<div>\<^bsub>\<gamma>\<^sub>T\<^sub>R\<^esub> \<phi>"
intro: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>a b c. times (times a b) c = times a (times b c)" shows "class.semigroup_mult times"
REC_def: fixes B :: "('f \<Rightarrow> 'e) \<Rightarrow> 'f \<Rightarrow> 'e"   and x :: "'f" shows "REC B x \<equiv> if trimono B then lfp B x else top"
sel_3: fixes x11 :: "('a, 'b) graph_node"   and x12 :: "'a"   and x13 :: "nat"   and x14 :: "'c" shows "iAnnot' (I x11 x12 x13 x14) = x13"
ccINF_sup: fixes B :: "'d set"   and f :: "'d \<Rightarrow> 'a"   and a :: "'a" assumes "countable B" shows "sup (Inf (f ` B)) a = (INF b\<in>B. sup (f b) a)"
inject: fixes x2 :: "t"   and y2 :: "t" shows "(TSome x2 = TSome y2) = (x2 = y2)"
that: fixes xs :: "'a list" assumes "v \<leadsto>xs\<leadsto> w" shows "thesis"
cmult': fixes c :: "'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "L F (\<lambda>x. f x * c) = L F f"
box_ne_empty_2: fixes a :: "'a"   and b :: "'a" shows "(box a b \<noteq> {}) = (\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i)"
cblinfun_image_sup: fixes U :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and A :: "'b ccsubspace"   and B :: "'b ccsubspace" shows "U *\<^sub>S (A \<squnion> B) = U *\<^sub>S A \<squnion> U *\<^sub>S B"
union_ts_iff: fixes c :: "'a"   and A :: "'b \<Rightarrow> 'a set"   and B :: "'b \<Rightarrow> 'a set"   and k :: "'b" shows "(c \<in> (A \<Rightarrow>\<union> B) k) = (c \<in> A k \<or> c \<in> B k)"
assms_3: fixes \<rho> :: "complex mat" assumes "\<rho> \<in> carrier_mat n n"   and "partial_density_operator \<rho>" shows "trace (A * \<rho>) \<le> trace (B * \<rho>)"
cases: fixes x :: "'a + 'b + 'c" obtains a :: "'a" where "x = Inl a"   | b :: "'b" where "x = Inr (Inl b)"   | c :: "'c" where "x = Inr (Inr c)"
get_component_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok get_component ptr"
Un_least: fixes A :: "'d set"   and C :: "'d set"   and B :: "'d set" assumes "A \<subseteq> C"   and "B \<subseteq> C" shows "A \<union> B \<subseteq> C"
norm1_def: fixes p :: "'a poly" shows "norm1 p = sum_list (map abs (coeffs p))"
eq_\<I>_converterD: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and conv :: "('h, 'i, 'j, 'k) converter"   and conv' :: "('h, 'i, 'j, 'k) converter"   and q :: "'h" assumes "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'"   and "q \<in> outs_\<I> \<I>" shows "eq_\<I>_gpv (rel_prod (eq_onp (\<lambda>r. r \<in> responses_\<I> \<I> q)) (eq_\<I>_converter \<I> \<I>')) \<I>' (run_converter conv q) (run_converter conv' q)"
pathstart_linepath: fixes a :: "'c"   and b :: "'c" shows "pathstart (linepath a b) = a"
r: fixes i :: "nat" assumes "i < length xs" shows "r i *\<^sub>R z = xs ! i"
measure_eqI_countable_AE': fixes M :: "'b measure"   and B :: "'b set"   and N :: "'b measure"   and \<Omega> :: "'b set" assumes "sets M = Pow B"   and "sets N = Pow B"   and "\<Omega> \<subseteq> B"   and "AE x in M. x \<in> \<Omega>"   and "AE x in N. x \<in> \<Omega>"   and "countable \<Omega>"   and "\<And>x. x \<in> \<Omega> \<Longrightarrow> emeasure M {x} = emeasure N {x}" shows "M = N"
map_topology_def: fixes p :: "'c \<Rightarrow> 'b"   and X :: "'c topology" shows "map_topology p X = final_topology (p ` topspace X) (\<lambda>_. X) (\<lambda>_. p)"
discrete_diagram_axioms: shows "discrete_diagram J.comp C (\<lambda>j. null)"
strict_trans: fixes b :: "'g"   and a :: "'g"   and c :: "'g" assumes "b < a"   and "c < b" shows "c < a"
h6: shows "\<forall>i<n. msg (Suc 0) (nReturn i)"
add_point_blocks_blocks_alt: fixes p :: "'a"   and bs :: "'a set set" shows "add_point_to_blocks p bs = image_mset (insert p) {#b \<in># \<B>. b \<in> bs#} + {#b \<in># \<B>. b \<notin> bs#}"
int_pow_mult_distrib: fixes x :: "'a"   and y :: "'a"   and i :: "int" assumes "x \<otimes> y = y \<otimes> x"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "(x \<otimes> y) [^] i = x [^] i \<otimes> y [^] i"
le_inf_concat_start: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w :: "'a list"   and w\<^sub>1 :: "'a stream"   and w\<^sub>2 :: "'a stream" assumes "traces ind" shows "traces.le_inf ind (w @- w\<^sub>1) (w @- w\<^sub>2) = traces.le_inf ind w\<^sub>1 w\<^sub>2"
rell_F_compp_Grp: fixes L1 :: "'n \<Rightarrow> 'x \<Rightarrow> bool"   and L2 :: "'o \<Rightarrow> 'y \<Rightarrow> bool"   and L3 :: "'p \<Rightarrow> 'z \<Rightarrow> bool" shows "rell_F L1 L2 L3 = (Grp {x. set1_F x \<subseteq> {(x, y). L1 x y} \<and> set2_F x \<subseteq> {(x, y). L2 x y} \<and> set3_F x \<subseteq> {(x, y). L3 x y}} (mapl_F fst fst fst))\<inverse>\<inverse> OO Grp {x. set1_F x \<subseteq> {(x, y). L1 x y} \<and> set2_F x \<subseteq> {(x, y). L2 x y} \<and> set3_F x \<subseteq> {(x, y). L3 x y}} (mapl_F snd snd snd)"
fun_sum_commute: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "f (0::'d) = (0::'c)"   and "\<And>x y. f (x + y) = f x + f y" shows "f (sum g A) = (\<Sum>a\<in>A. f (g a))"
not_neutral_contains_not_neutral: fixes g :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "sum g A \<noteq> (0::'d)" obtains a :: "'e" where "a \<in> A" and "g a \<noteq> (0::'d)"
fps_to_fls_def: fixes f :: "'b fps" shows "fps_to_fls f \<equiv> Abs_fls (\<lambda>k. if k < 0 then 0::'b else f $ nat k)"
bounded_cbox: fixes a :: "'b"   and b :: "'b" shows "bounded (cbox a b)"
act_conj_act_1: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<A>(\<^bold>\<A>\<phi> \<^bold>\<rightarrow> \<phi>) in v]"
discI: fixes list :: "'a list"   and x21 :: "'a"   and x22 :: "'a list" assumes "list = x21 # x22" shows "list \<noteq> []"
hom_compl: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b formula" shows "hom f (- x) = - hom f x"
inst: shows "SecurityInvariant sinvar_spec default_node_properties receiver_violation"
nth_take: fixes i :: "nat"   and n :: "nat"   and xs :: "'a list" assumes "i < n" shows "take n xs ! i = xs ! i"
p_def: fixes x :: "nat" shows "p x = c_pair x 0"
good_item_simps_29: fixes Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xsa :: "'varSort"   and x1 :: "'var"   and xa :: "'var" assumes "good Y" shows "good (Y #[x1 // xa]_xsa)"
assms_2: shows "2 \<cdot> g \<cdot> x - 2 \<cdot> g \<cdot> h = v \<cdot> v"
dg_prod_vunion_Obj_in_Obj: fixes J :: "V"   and K :: "V"   and b :: "V"   and \<AA> :: "V \<Rightarrow> V"   and c :: "V" assumes "J \<inter>\<^sub>\<circ> K = []\<^sub>\<circ>"   and "b \<in>\<^sub>\<circ> dg_prod J \<AA>\<lparr>Obj\<rparr>"   and "c \<in>\<^sub>\<circ> dg_prod K \<AA>\<lparr>Obj\<rparr>" shows "b \<union>\<^sub>\<circ> c \<in>\<^sub>\<circ> dg_prod (J \<union>\<^sub>\<circ> K) \<AA>\<lparr>Obj\<rparr>"
IH: fixes Ks :: "('a, 'b) Term.term literal list"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term option" assumes "sorted_wrt leq_lit Ls" shows "subsumes_list Ls Ks \<sigma> = subsumes_list_filter Ls Ks \<sigma>"
case_cong: fixes seq :: "'b Predicate.seq"   and seq' :: "'b Predicate.seq"   and f1 :: "'c"   and g1 :: "'c"   and f2 :: "'b \<Rightarrow> 'b Predicate.pred \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'b Predicate.pred \<Rightarrow> 'c"   and f3 :: "'b Predicate.pred \<Rightarrow> 'b Predicate.seq \<Rightarrow> 'c"   and g3 :: "'b Predicate.pred \<Rightarrow> 'b Predicate.seq \<Rightarrow> 'c" assumes "seq = seq'"   and "seq' = seq.Empty \<Longrightarrow> f1 = g1"   and "\<And>x21 x22. seq' = seq.Insert x21 x22 \<Longrightarrow> f2 x21 x22 = g2 x21 x22"   and "\<And>x31 x32. seq' = seq.Join x31 x32 \<Longrightarrow> f3 x31 x32 = g3 x31 x32" shows "(case seq of seq.Empty \<Rightarrow> f1 | seq.Insert x21 x22 \<Rightarrow> f2 x21 x22 | seq.Join x31 x32 \<Rightarrow> f3 x31 x32) = (case seq' of seq.Empty \<Rightarrow> g1 | seq.Insert x21 x22 \<Rightarrow> g2 x21 x22 | seq.Join x31 x32 \<Rightarrow> g3 x31 x32)"
Collect_cong: fixes P :: "'x \<Rightarrow> bool"   and Q :: "'x \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "{x. P x} = {x. Q x}"
nat_pow_eone: fixes x :: "'a" assumes "x \<in> carrier \<G>" shows "x [^] 1 = x"
inverts_components: fixes a :: "'c" assumes "Cop.ide a" shows "S.inverse_arrows (\<Phi> a) (\<Psi>.map a)"
exec_move_TryI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e' :: "'addr expr1"   and C :: "String.literal"   and V :: "nat" assumes "exec_move ci P t e h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_move ci P t (try e' catch(C V) e) h (stk, loc, Suc (Suc (length (compE2 e') + pc)), xcp) ta h' (stk', loc', Suc (Suc (length (compE2 e') + pc')), xcp')"
other_bothE: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "'b"   and \<sigma> :: "'b \<Rightarrow> 'a"   and \<sigma>' :: "'b \<Rightarrow> 'a" assumes "other P {i} \<sigma> \<sigma>'" obtains "\<sigma>' i = \<sigma> i" and "\<forall>j. j \<noteq> i \<longrightarrow> P (\<sigma> j) (\<sigma>' j)"
intro: fixes hom :: "'d \<Rightarrow> 'c" assumes "hom (1::'d) = (1::'c)" shows "one_hom hom"
subst_scls_mset_empty: fixes \<sigma> :: "'s" shows "{#} \<cdot>cm \<sigma> = {#}"
p\<^sub>1_def: shows "p\<^sub>1 = find_closest p\<^sub>0 (dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (p\<^sub>2 # ps)"
simps_13: fixes V :: "'c"   and T :: "ty"   and e :: "'c exp" shows "ss_exp {V:T; e} = ss_exp e"
conjugate_square_greater_0_vec: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "((0::'a) < inner_prod v v) = (v \<noteq> 0\<^sub>v n)"
cr_cblinfun_def: shows "cr_cblinfun \<equiv> \<lambda>x y. x = (*\<^sub>V) y"
prems_1: shows "charslength (p @ s # ts) = charslength p"
fps_of_ratfps_ratfps_of_poly: fixes p :: "'b poly" shows "fps_of_ratfps (ratfps_of_poly p) = fps_of_poly p"
simps_2: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c stream"   and a :: "'b" shows "stl (sscan f xs a) = sscan f (stl xs) (f (shd xs) a)"
induct: fixes P :: "freemsg \<Rightarrow> bool"   and freemsg :: "freemsg" assumes "\<And>x. P (NONCE x)"   and "\<And>x1a x2. \<lbrakk>P x1a; P x2\<rbrakk> \<Longrightarrow> P (MPAIR x1a x2)"   and "\<And>x1a x2. P x2 \<Longrightarrow> P (CRYPT x1a x2)"   and "\<And>x1a x2. P x2 \<Longrightarrow> P (DECRYPT x1a x2)" shows "P freemsg"
reciprocal_add: fixes P :: "'b poly"   and p :: "nat"   and Q :: "'b poly" assumes "degree P \<le> p"   and "degree Q \<le> p" shows "reciprocal_poly p (P + Q) = reciprocal_poly p P + reciprocal_poly p Q"
multiplicity_times_same: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<not> dvd.dvd times p one"   and "p \<noteq> zero" shows "factorial_semiring.multiplicity times one p (times p x) = Suc (factorial_semiring.multiplicity times one p x)"
cases: fixes a1 :: "(bool list option \<times> unit \<times> bool list cstate) spmf"   and a2 :: "(bool list option \<times> bool list cstate) spmf" assumes "local.S a1 a2" obtains "a1 = return_spmf (None, (), Void)" and "a2 = return_spmf (None, Void)"   | plain :: "bool list" where "a1 = return_spmf (None, (), Store plain__)" and "a2 = map_spmf (\<lambda>key. (Some key, Store (key [\<oplus>] plain__))) (spmf_of_set (nlists UNIV \<eta>))" and "length plain__ = id' \<eta>"   | plain :: "bool list" where "a1 = return_spmf (None, (), cstate.Collect plaina__)" and "a2 = map_spmf (\<lambda>key. (Some key, cstate.Collect (key [\<oplus>] plaina__))) (spmf_of_set (nlists UNIV \<eta>))" and "length plaina__ = id' \<eta>"   | key :: "bool list" and plain :: "bool list" where "a1 = return_spmf (Some (key__ [\<oplus>] plainb__), (), Store plainb__)" and "a2 = return_spmf (Some key__, Store (key__ [\<oplus>] plainb__))" and "length plainb__ = id' \<eta>" and "length key__ = id' \<eta>"   | key :: "bool list" and plain :: "bool list" where "a1 = return_spmf (Some (keya__ [\<oplus>] plainc__), (), cstate.Collect plainc__)" and "a2 = return_spmf (Some keya__, cstate.Collect (keya__ [\<oplus>] plainc__))" and "length plainc__ = id' \<eta>" and "length keya__ = id' \<eta>"   | "a1 = return_spmf (None, (), cstate.Fail)" and "a2 = map_spmf (\<lambda>x. (Some x, cstate.Fail)) (spmf_of_set (nlists UNIV \<eta>))"   | key :: "bool list" and plain :: "bool list" where "a1 = return_spmf (Some (key [\<oplus>] plain), (), cstate.Fail)" and "a2 = return_spmf (Some key, cstate.Fail)" and "length plain = id' \<eta>" and "length key = id' \<eta>"
norm_uk_eq: fixes A :: "((real, 'a) vec, 'b) vec"   and k :: "'a" assumes "rank A = ncols A" shows "norm (column k (Gram_Schmidt_matrix A)) = column k (fst (QR_decomposition A)) \<bullet> column k A"
NUM_E_2: fixes U :: "(nat \<Rightarrow> nat option) set" assumes "U \<in> NUM" shows "\<exists>\<psi>\<in>\<R>\<^sup>2. \<forall>f\<in>U. \<exists>i. \<psi> i = f"
pid_init_gc_def: shows "pid_init_gc \<equiv> \<lparr>pid_global_config.processes = map local.init_pc (program.processes prog), state = \<lparr>global_state.variables = init_valuation (global_vars prog)\<rparr>\<rparr>"
N_1: fixes i :: "'i" shows "N i \<in> null_sets M"
not_in_Gromov_boundary: fixes x :: "'c Gromov_completion" assumes "x \<notin> Gromov_boundary" shows "\<exists>a. x = to_Gromov_completion a"
h_def: shows "h \<equiv> \<lambda>a. a ^ (n - 1) mod n"
hyps_3: fixes Ts :: "ty list" assumes "P,E,hp s'' \<turnstile> es'' [:] Ts"   and "P,E \<turnstile> s'' \<surd>" shows "P,E \<turnstile> s' \<surd>"
Nats_def: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.semiring_1 one times plus zero" shows "semiring_1.Nats one plus zero = range (semiring_1.of_nat one plus zero)"
rtranclD: fixes a :: "'c"   and b :: "'c"   and R :: "('c \<times> 'c) set" assumes "(a, b) \<in> R\<^sup>*" shows "a = b \<or> a \<noteq> b \<and> (a, b) \<in> R\<^sup>+"
rep_Tree\<^sub>\<alpha>_def: shows "rep_Tree\<^sub>\<alpha> \<equiv> quot_type.rep Rep_Tree\<^sub>\<alpha>"
mset_sub_decompose: fixes ds :: "'b list"   and bs :: "'b list"   and as :: "'b multiset" assumes "mset ds \<subseteq># mset bs + as"   and "length ds < length bs" shows "\<exists>b1 b b2. bs = b1 @ b # b2 \<and> mset ds \<subseteq># mset (b1 @ b2) + as"
tautology_deletion: fixes A :: "'a"   and C :: "'a literal multiset"   and N :: "'a literal multiset set"   and P :: "'a literal multiset set"   and Q :: "'a literal multiset set" assumes "Neg A \<in># C"   and "Pos A \<in># C" shows "(N \<union> {C}, P, Q) \<leadsto> (N, P, Q)"
normf: fixes x :: "'a" assumes "x \<in> cbox a b" shows "norm (f x) \<le> g x"
map_transfer: fixes Rb :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and Sd :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "((Rb ===> Sd) ===> rel_mset Rb ===> rel_mset Sd) image_mset image_mset"
simps_3: fixes \<phi> :: "'a dfmla" shows "wff (Neg \<phi>) = wff \<phi>"
iSp_Append_cdep: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "n i-ns\<rightarrow>\<^sub>d* n''"   and "n'' \<longrightarrow>\<^bsub>cd\<^esub> n'" shows "n i-ns @ [n'']\<rightarrow>\<^sub>d* n'"
residue_mult_comm: fixes p :: "int"   and k :: "nat"   and x :: "int"   and y :: "int" assumes "padic_integers p" shows "x \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> y = y \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> x"
peirce_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x ; y \<cdot> z\<^sup>\<smile> = 0" shows "y ; z \<cdot> x\<^sup>\<smile> = 0"
word_plus_mcs_3: fixes v :: "'b word"   and w :: "'b word"   and x :: "'b word" assumes "v \<le> w"   and "x \<le> w + x" shows "v + x \<le> w + x"
sum_nonneg: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> {#} \<subseteq># f x" shows "{#} \<subseteq># comm_monoid_add.sum (+) {#} f A"
bypass: fixes z :: "'a" assumes "z \<in> set p" shows "z \<notin> S"