hcju/isabelleps huggingface dataset:
distinct_141: fixes x5 :: "Syntax.term"   and x14 :: "Syntax.term" shows "Inj1 x5 \<noteq> Unroll x14"
p_fusion_assoc: fixes ps :: "'a list"   and qs :: "'a list"   and rs :: "'a list" shows "p_fusion ps (p_fusion qs rs) = p_fusion (p_fusion ps qs) rs"
d_IN_ge_point: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and y :: "'k"   and x :: "'k" shows "f (y, x) \<le> d_IN f x"
elims: fixes x :: "'a \<times> 'c \<times> 'b \<times> 'd"   and y :: "'a \<times> 'b \<times> 'c \<times> 'd" assumes "mat_transpose x = y" obtains a :: "'a" and b :: "'c" and c :: "'b" and d :: "'d" where "x = (a, b, c, d)" and "y = (a, c, b, d)"
Comm2: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and Q :: "pi"   and Q'' :: "pi"   and x :: "name"   and Q' :: "pi" assumes "P \<Longrightarrow>\<^sub>l a[b] \<prec> P'"   and "Q \<Longrightarrow>\<^sub>lb in Q''\<rightarrow>a<x> \<prec> Q'" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l \<tau> \<prec> P' \<parallel> Q'"
non_empty_cycle_root_iff: fixes x :: "'a" shows "(\<exists>r. non_empty_cycle_root r x) = (cycle x \<and> x \<noteq> 0)"
triv_if: fixes b :: "bexp"   and c :: "com" shows "IF b THEN c ELSE c \<sim> c"
elims: fixes x :: "('b \<times> nat) option"   and xa :: "'b"   and y :: "nat" assumes "has_locks x xa = y" obtains t :: "'b" where "x = None" and "xa = tb__" and "y = 0"   | t' :: "'b" and n :: "nat" and t :: "'b" where "x = \<lfloor>(t', n)\<rfloor>" and "xa = t" and "y = (if t = t' then Suc n else 0)"
read_only_reads_unowned_def: 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" shows "read_only_reads_unowned ts \<equiv> \<forall>i p\<^sub>i is\<^sub>i \<O>\<^sub>i \<R>\<^sub>i \<D>\<^sub>i \<theta>\<^sub>i sb\<^sub>i j p\<^sub>j is\<^sub>j \<O>\<^sub>j \<R>\<^sub>j \<D>\<^sub>j \<theta>\<^sub>j sb\<^sub>j. i < length ts \<longrightarrow> j < length ts \<longrightarrow> i \<noteq> j \<longrightarrow> ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i) \<longrightarrow> ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j) \<longrightarrow> (\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) \<inter> read_only_reads (acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>i) \<O>\<^sub>i) (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>i) = {}"
D_implies_R: assumes "D \<V> Tr\<^bsub>(induceES SES)\<^esub>" shows "R \<V> Tr\<^bsub>(induceES SES)\<^esub>"
alpha_Tree_eqvt_aux: fixes t :: "('e, 'f, 'g) Tree"   and p :: "perm" assumes "\<And>a b. ((a \<rightleftharpoons> b) \<bullet> t =\<^sub>\<alpha> t) = (p \<bullet> (a \<rightleftharpoons> b) \<bullet> t =\<^sub>\<alpha> p \<bullet> t)" shows "p \<bullet> supp_rel (=\<^sub>\<alpha>) t = supp_rel (=\<^sub>\<alpha>) (p \<bullet> t)"
rtrancl_Int: fixes a :: "'a"   and b :: "'a"   and A :: "('a \<times> 'a) set"   and B :: "('a \<times> 'a) set" assumes "(a, b) \<in> A"   and "(a, b) \<in> B" shows "(a, b) \<in> (A \<inter> B)\<^sup>*"
assoc: shows "M * A * adjoint M *\<^sub>v v = M *\<^sub>v (A *\<^sub>v (adjoint M *\<^sub>v v))"
SLC: fixes s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s\<^sub>1 \<leadsto> s\<^sub>2"   and "s\<^sub>1 \<leadsto> s\<^sub>2'"   and "reachable s\<^sub>1"   and "infinite UNIV"   and "infinite UNIV" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (s\<^sub>2, s\<^sub>3) \<in> [\<leadsto>]\<^sup>= \<and> (s\<^sub>2', s\<^sub>3') \<in> [\<leadsto>]\<^sup>="
trace_rest_eq_def: fixes rest1 :: "('j, 'k, 'l, 'm, 'n, 'o, 'p) rest_scheme"   and rest2 :: "('q, 'k, 'l, 'm, 'n, 'o, 'r) rest_scheme"   and IA :: "'l set"   and IU :: "'m set"   and p :: "'j spmf"   and q :: "'q spmf" shows "trace_rest_eq rest1 rest2 IA IU p q = (\<forall>tr. set tr \<subseteq> IA \<times> UNIV <+> IU \<times> UNIV \<longrightarrow> rel_prod (rel_fun (eq_onp (\<lambda>ia. ia \<in> IA)) (=)) (rel_fun (eq_onp (\<lambda>iu. iu \<in> IU)) (=)) (trace_rest' rest1 p tr) (trace_rest' rest2 q tr))"
first_edge_hd_tl: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "v \<leadsto>xs\<leadsto>\<^bsub>H\<^esub> w"   and "tl xs \<noteq> []" shows "H.first_edge_of_walk xs = (v, hd (tl xs))"
prems_3: shows "fst cf1 \<approx>01 fst cf2"
dghm_comp_ObjMap_vsv: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "vsv ((\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>ObjMap\<rparr>)"
node_sos_dest: fixes i :: "nat"   and p :: "'c"   and R :: "nat set"   and a :: "'d node_action"   and s' :: "'c net_state"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(NodeS i p R, a, s') \<in> node_sos S" shows "\<exists>P' R'. s' = NodeS i P' R'"
max_path_2: shows "length max_path = (GREATEST x. exist_path_length n x)"
geometric_sum: fixes x :: "'a"   and n :: "nat" assumes "x \<noteq> (1::'a)" shows "sum ((^) x) {..<n} = (x ^ n - (1::'a)) / (x - (1::'a))"
r_def: shows "r \<equiv> run\<^sub>t \<delta> q\<^sub>0 w"
simps: fixes x :: "'a"   and xs :: "'a list"   and n :: "nat" shows "push_array_basic_fun x (xs, n) = (xs[n := x], n + 1)"
at_neg_neg1: fixes c :: "cars"   and \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool" shows "\<forall>ts v. (\<forall>v'. v = c > v' \<longrightarrow> \<phi> ts v') \<longrightarrow> \<not> (\<forall>v'. v = c > v' \<longrightarrow> \<not> \<phi> ts v')"
strong_exhaust: fixes y :: "fm"   and c :: "'a" obtains x11 :: "tm" and x12 :: "tm" where "y = x11 IN x12"   | x21 :: "tm" and x22 :: "tm" where "y = x21 EQ x22"   | x31 :: "fm" and x32 :: "fm" where "y = x31 OR x32"   | x4 :: "fm" where "y = Neg x4"   | x51 :: "name" and x52 :: "fm" where "{atom x51} \<sharp>* c" and "y = SyntaxN.Ex x51 x52"
ntrs_valid_preserve_s: fixes s :: "'n list"   and c :: "'n list multiset"   and w :: "('p, 'ba) label list"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), w, s', c') \<in> ntrs fg"   and "valid fg ({#s#} + c)" shows "valid fg ({#s'#} + c')"
with_proj_simps_4: fixes G :: "'b pair_pre_digraph" shows "tail (with_proj G) = fst"
finite: fixes A :: "'k set" assumes "class.finite TYPE('k)" shows "finite A"
code_status_assn_def: shows "code_status_assn = hr_comp (status_assn raw_string_assn) code_status_status_rel"
set_cases_1: fixes e :: "'a"   and a :: "('a, 'b) Term.term" assumes "e \<in> funs_term a" obtains z2 :: "('a, 'b) Term.term list" where "a = Fun e z2__"   | z1 :: "'a" and z2 :: "('a, 'b) Term.term list" and x :: "('a, 'b) Term.term" where "a = Fun z1 z2" and "x \<in> set z2" and "e \<in> funs_term x"
mod1: fixes r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and R' :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M :: "'b"   and N :: "'b"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and seq :: "('a, 'b) sequent" assumes "r \<in> p_e R' M N"   and "r \<in> R" shows "extendConc seq r \<in> ext R R' M N"
less_\<I>_def: shows "(<) = mk_less (\<le>)"
cod_simp: fixes f :: "(bool, unit) concrete_category.arr" assumes "partial_magma.arr (concrete_category.COMP UNIV (\<lambda>a b. if a = b then {()} else {}) (\<lambda>_ _ _ _ _. ())) f" shows "partial_magma.cod (concrete_category.COMP UNIV (\<lambda>a b. if a = b then {()} else {}) (\<lambda>_ _ _ _ _. ())) f = f"
step_3_main_dims_main: fixes n :: "nat"   and k :: "nat"   and A :: "'b mat" shows "dim_row (step_3_main n k A) = dim_row A \<and> dim_col (step_3_main n k A) = dim_col A"
Gromov_extension_homeomorphism: fixes lambda :: "real"   and C :: "real"   and f :: "'c \<Rightarrow> 'd"   and X :: "'c set" assumes "lambda C -quasi_isometry f"   and "homeomorphism_on X f" shows "homeomorphism_on (to_Gromov_completion ` X \<union> Gromov_boundary) (Gromov_extension f)"
select_convs_2: fixes variables_of :: "'c list"   and operators_of :: "'c strips_operator list"   and initial_of :: "'c \<Rightarrow> bool option"   and goal_of :: "'c \<Rightarrow> bool option"   and more :: "'d" shows "\<lparr>strips_problem.variables_of = variables_of, operators_of = operators_of, initial_of = initial_of, goal_of = goal_of, \<dots> = more\<rparr>\<^sub>\<O> = operators_of"
6: shows "r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> * (v * v\<^sup>T * e \<sqinter> g) \<le> r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> * e"
assms: shows "A \<approx>\<^sub>\<circ> B"
conc_trans_additional_6: fixes A :: "'b nres"   and B :: "'b nres"   and C :: "'b nres" assumes "A \<le> B"   and "B \<le> \<Down> Id C" shows "A \<le> C"
iT_Div_mod_inext_nth: fixes I :: "nat set"   and k :: "nat"   and n :: "nat" assumes "I \<noteq> {}"   and "\<forall>x\<in>I. \<forall>y\<in>I. x mod k = y mod k" shows "(I \<oslash> k) \<rightarrow> n = (I \<rightarrow> n) div k"
hyps: shows "Q \<parallel> T \<longmapsto> \<alpha> \<prec> QT'"
set_disconnected_nodes_get_parent: fixes ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "\<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>Shadow_DOM.get_parent_locs. r h h')"
AB: shows "\<mu>.dom.leg0 \<cdot> doms.Prj\<^sub>1\<^sub>1 = \<nu>.dom.leg1 \<cdot> doms.Prj\<^sub>0\<^sub>1"
comp_eq_dest_lhs: fixes a :: "'e \<Rightarrow> 'd"   and b :: "'b \<Rightarrow> 'e"   and c :: "'b \<Rightarrow> 'd"   and v :: "'b" assumes "a \<circ> b = c" shows "a (b v) = c v"
bounded_iff: fixes a :: "'t"   and b :: "'t"   and c :: "'t" shows "(a \<preceq>\<^sub>t ord_term_lin.min b c) = (a \<preceq>\<^sub>t b \<and> a \<preceq>\<^sub>t c)"
timpl_closure_Var_inv: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and x :: "'d"   and TI :: "('c set \<times> 'c set) set" assumes "t \<in> timpl_closure (Var x) TI" shows "t = Var x"
mojmir_accept_token_set_def1: assumes "accept\<^sub>M" shows "\<exists>i<max_rank. finite fail \<and> finite (local.merge i) \<and> infinite (succeed i) \<and> (\<forall>j<i. finite (succeed j))"
det: fixes n :: "nat"   and fs :: "'a vec list" assumes "gram_schmidt_fs_Rn n fs"   and "length fs = n" shows "det (mat_of_rows n fs) = det (mat_of_rows n (map (gram_schmidt_fs.gso n fs) [0..<length fs]))"
pa_det_algo_def: fixes H1 :: "('a, 'c) hashedTa"   and H2 :: "('b, 'c) hashedTa" shows "pa_det_algo H1 H2 \<equiv> \<lparr>dwa_cond = pa_cond, dwa_step = pa_step H1 H2, dwa_initial = pa_initial H1 H2, dwa_invar = pa_invar H1 H2\<rparr>"
game_sem_mono: fixes I :: "interp"   and \<alpha> :: "game" shows "mono (game_sem I \<alpha>)"
f1: fixes gpv :: "('a, 'g, 'e) gpv"   and gpv' :: "('b, 'h, 'f) gpv"   and x :: "('a, 'g, ('a, 'g, 'e) rpv) generat"   and y :: "('b, 'h, ('b, 'h, 'f) rpv) generat"   and a :: "('a, 'g, 'e) rpv"   and b :: "('b, 'h, 'f) rpv"   and a' :: "'e"   and \<alpha> :: "'f"   and \<beta> :: "'e"   and b' :: "'f" assumes "local.foo results_gpv results_gpv gpv gpv' A A'"   and "x \<in> set_spmf (the_gpv gpv)"   and "y \<in> set_spmf (the_gpv gpv')"   and "a \<in> generat_conts x"   and "b \<in> generat_conts y"   and "R' a' \<alpha>"   and "R' \<beta> b'" shows "local.foo results_gpv results_gpv (a a') (b b') A A'"
set_takeWhileD: fixes x :: "'f"   and P :: "'f \<Rightarrow> bool"   and xs :: "'f list" assumes "x \<in> set (takeWhile P xs)" shows "x \<in> set xs \<and> P x"
abs_power_nonneg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.ring_1_abs_real_line plus zero minus less_eq less times abs uminus one" shows "less_eq zero (power.power one times (abs a) n)"
assms_5: shows "R (xs ! (hi - 1)) (xs ! hi)"
arr_char: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.VV.arr f = (Span.arr (fst f) \<and> Span.arr (snd f) \<and> Span.src (fst f) = Span.trg (snd f))"
SucFbd_Cinfinite: fixes r2a :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set" shows "Cinfinite (cardSuc ((bd_F1 +c r2a) +c r2))"
indRelRTPO_relates_source_target: fixes S :: "'procS"   and T :: "'procT"   and TRel :: "('procT \<times> 'procT) set" assumes "SourceTerm S \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> TargetTerm T" shows "(TargetTerm (\<lbrakk>S\<rbrakk>), TargetTerm T) \<in> (indRelRTPO TRel)\<^sup>="
inter_guards_execn_noFault: fixes c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and s :: "('d, 'f) xstate"   and n :: "nat"   and t :: "('d, 'f) xstate" assumes "(c1 \<inter>\<^sub>g c2) = Some c"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "\<not> isFault t" shows "\<Gamma>\<turnstile> \<langle>c1,s\<rangle> =n\<Rightarrow> t \<and> \<Gamma>\<turnstile> \<langle>c2,s\<rangle> =n\<Rightarrow> t"
non_comm: shows "x \<cdot> y \<noteq> y \<cdot> x"
assms_2: shows "\<exists>\<^sub>F x in F. f x = c"
monoid_add_hom_axioms: shows "monoid_add_hom poly_y_x"
dghm_comp_ArrMap_vrange: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<R>\<^sub>\<circ> ((\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>ArrMap\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
ContractsWithUnilateralSubstitutesAndIRC_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set" shows "ContractsWithUnilateralSubstitutesAndIRC Xd Xh Pd Ch \<equiv> ContractsWithUnilateralSubstitutes Xd Xh Pd Ch \<and> ContractsWithIRC Xd Xh Pd Ch"
simps_1: fixes uu :: "(real, 'b) vec" shows "KD_Tree.size_kdt (Leaf uu) = 1"
ap_spmf_C: fixes f :: "('e \<Rightarrow> 'f \<Rightarrow> 'd) spmf"   and x :: "'f spmf"   and y :: "'e spmf" shows "return_spmf (\<lambda>f x y. f y x) \<diamondop> f \<diamondop> x \<diamondop> y = f \<diamondop> y \<diamondop> x"
preserves_cod: fixes f :: "'b" assumes "arr f" shows "cod (\<ll> f) = local.map (cod f)"
g': fixes tx :: "real \<times> 'a" assumes "tx \<in> T \<times> X" shows "g' tx = ($) (f' tx)"
anonymous_profile_permute: fixes \<sigma> :: "'alt \<Rightarrow> 'alt" assumes "\<sigma> permutes alts"   and "finite agents" shows "anonymous_profile (permute_profile \<sigma> R2) = image_mset (map ((`) \<sigma>)) (anonymous_profile R2)"
free_generator_sub: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set" assumes "R module M"   and "free_generator R M H" shows "H \<subseteq> carrier M"
comp_null_1: fixes f :: "'c" shows "C.null \<cdot> f = C.null"
simps_1: fixes x1 :: "'f"   and x2 :: "'g"   and x3 :: "('d, 'e, 'f, 'g, 'h) qTerm"   and y1 :: "'f"   and y2 :: "'g"   and y3 :: "('d, 'e, 'f, 'g, 'h) qTerm" shows "(qAbs x1 x2 x3 = qAbs y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
int_rdvd_iff: fixes i :: "int"   and t :: "int" shows "(real_of_int i rdvd real_of_int t) = (i dvd t)"
deleteAround_nextVertex: fixes f :: "face"   and g :: "graph"   and a :: "nat"   and b :: "nat"   and ps :: "(nat \<times> nat) list" assumes "f \<in> set (facesAt g a)" shows "(f \<bullet> a, b) \<notin> set (deleteAround g a ps)"
except_empty: fixes p :: "'b \<Rightarrow>\<^sub>0 'c" shows "except p {} = p"
c_union_def: shows "c_union = (\<lambda>u v. set_to_nat (nat_to_set u \<union> nat_to_set v))"
proportionality_def: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "vector_space scale" shows "vector_space.proportionality scale \<equiv> {(x, y). x \<noteq> (0::'b) \<and> y \<noteq> (0::'b) \<and> (\<exists>k. x = scale k y)}"
simps: fixes a1 :: "s_context"   and a2 :: "t list option"   and a3 :: "nat"   and a4 :: "v list"   and a5 :: "e list"   and a6 :: "t list" shows "a1\<bullet>a2 \<tturnstile>_ a3 a4;a5 : a6 = (\<exists>i \<S> tvs vs \<C> rs es ts. a1 = \<S> \<and> a2 = rs \<and> a3 = i \<and> a4 = vs \<and> a5 = es \<and> a6 = ts \<and> i < length (s_inst \<S>) \<and> tvs = map typeof vs \<and> \<C> = (s_inst \<S> ! i)\<lparr>local := local (s_inst \<S> ! i) @ tvs, return := rs\<rparr> \<and> \<S>\<bullet>\<C> \<turnstile> es : [] _> ts \<and> (rs = Some ts \<or> rs = None))"
cgenideal_is_principalideal: fixes R :: "('c, 'd) ring_scheme"   and i :: "'c" assumes "cring R"   and "i \<in> carrier R" shows "principalideal (PIdl\<^bsub>R\<^esub> i) R"
uw\<theta>\<omega>\<nu>_def: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c"   and \<omega> :: "'c"   and \<nu> :: "'c" shows "uw\<theta>\<omega>\<nu> V H \<a> \<i> src trg r \<rho> f g u w \<theta> \<omega> \<nu> \<equiv> uw\<theta> V H \<a> \<i> src trg r \<rho> f g u w \<theta> \<and> uw\<theta>\<omega>\<nu>_axioms V H \<a> r \<rho> f g u w \<theta> \<omega> \<nu>"
take_first: fixes xs :: "'d list" assumes "xs \<noteq> []" shows "xs \<down> Suc 0 = [xs ! 0]"
finite_supp_implies_hereditarily_fs_Conj\<^sub>\<alpha>: fixes xset :: "('a, 'b, 'c, 'd) formula set['a]" assumes "finite (supp xset)" shows "hereditarily_fs (Conj\<^sub>\<alpha> (map_bset Rep_formula xset))"
da_elim_cases_30: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and T :: "ref_ty"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1l (e InstOf T)\<guillemotright> A" obtains "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A"
length_trace_minus_one_lt_length_trace: fixes I :: "'a \<Rightarrow> bool option"   and \<pi> :: "'a strips_operator list list" shows "length (trace_parallel_plan_strips I \<pi>) - 1 < length (trace_parallel_plan_strips I \<pi>)"
mem_list1: shows "\<forall>r\<in>set (characterize_root_list_p p). consistent_sign_vec_copr qs1 r \<in> set (map (consistent_sign_vec_copr qs1) (characterize_root_list_p p))"
ht_hash_in_dom_in_dom_bounded_hashcode_nat_eq: fixes l :: "('a \<times> 'b) list list"   and k :: "'a" assumes "ht_hash l"   and "1 < length l" shows "(k \<in> dom (map_of (concat l))) = (k \<in> dom (map_of (l ! bounded_hashcode_nat (length l) k)))"
ValueCarAudioSystem: shows "HAInitValue CarAudioSystem = LiftInitData [V0 0, V1 0]"
tup_to_sum_app: fixes p :: "('g discr \<rightarrow> 'i) \<times> ('h discr \<rightarrow> 'i)" shows "tup_to_sum\<cdot>p = (\<Lambda> s. case p of (f, g) \<Rightarrow> case undiscr s of Inl x \<Rightarrow> f\<cdot>(Discr x) | Inr x \<Rightarrow> g\<cdot>(Discr x))"
hyps: shows "jordan_block n a ^\<^sub>m r = mat n n (\<lambda>b. case b of (i, j) \<Rightarrow> if i \<le> j then of_nat (r choose (j - i)) * a ^ (r + i - j) else (0::'a))"
at_most_one_in_maximal: fixes C :: "'b fm set set"   and V :: "'b fm set"   and p :: "'b fm" assumes "consistency C"   and "V \<in> C"   and "maximal V C" shows "\<not> (p \<in> V \<and> (\<^bold>\<not> p) \<in> V)"
triangle_left: shows "(\<xi> \<cdot> (f \<star> local.inv \<eta> \<star> g) \<cdot> (local.inv \<xi> \<star> f \<star> g) \<star> f) \<cdot> (f \<star> \<eta>) = f"
i_def: shows "i = (LEAST i. w i \<in> A)"
approx_mem_monad_zero: fixes x :: "'a star"   and y :: "'a star" assumes "x \<approx> y"   and "x \<in> monad 0" shows "y \<in> monad 0"
joinable_Nil_l_iff: fixes p :: "'a list" shows "joinable [] p = False"
pass\<^sub>2_struct: fixes hs1 :: "'b tree"   and x :: "'b"   and hs :: "'b tree" shows "\<exists>l a. pass\<^sub>2 \<langle>hs1, x, hs\<rangle> = \<langle>l, a, \<langle>\<rangle>\<rangle>"
expectation_gpv_fixp_induct: fixes P :: "(('a, 'b, 'c) gpv \<Rightarrow> ennreal) \<Rightarrow> bool"   and fail :: "ennreal"   and \<I> :: "('b, 'c) \<I>"   and f :: "'a \<Rightarrow> ennreal" assumes "lfp.admissible P"   and "P (\<lambda>_. 0)"   and "\<And>expectation_gpv'. \<lbrakk>\<And>gpv. expectation_gpv' gpv \<le> expectation_gpv fail \<I> f gpv; P expectation_gpv'\<rbrakk> \<Longrightarrow> P (\<lambda>gpv. \<integral>\<^sup>+ generat. (case generat of Pure x \<Rightarrow> f x | IO out c \<Rightarrow> \<Sqinter>r\<in>responses_\<I> \<I> out. expectation_gpv' (c r)) \<partial>measure_spmf (the_gpv gpv) + fail * ennreal (pmf (the_gpv gpv) None))" shows "P (expectation_gpv fail \<I> f)"
cinner_ket_adjointI: fixes F :: "'c ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd ell2"   and G :: "'d ell2 \<Rightarrow>\<^sub>C\<^sub>L 'c ell2" assumes "\<And>i j. (F *\<^sub>V ket i) \<bullet>\<^sub>C ket j = ket i \<bullet>\<^sub>C (G *\<^sub>V ket j)" shows "F = G*"
bind_DONE: fixes x :: "('f, 'g, 'h) Resumption.resumption" shows "x \<bind> DONE = x"
reflpE: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g" assumes "reflp r" obtains "r x x"
bisimParPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim> Q" shows "\<Psi> \<rhd> P \<parallel> R \<sim> Q \<parallel> R"
enum_ex_latin_type_def: fixes P :: "latin_type \<Rightarrow> bool" shows "enum_class.enum_ex P = (\<exists>x. P x)"
prod_prime_factors: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "multiplicative_function f"   and "0 < n" shows "f n = (\<Prod>p\<in>prime_factors n. f (p ^ multiplicity p n))"
Trg_Cod: fixes t :: "'a local.term" assumes "Arr t" shows "Trg (Cod t) = Trg t"
cyclic_on_def: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set" shows "cyclic_on f S = (\<exists>s\<in>S. S = orbit f s)"
list_comb: fixes f :: "'a"   and xs :: "'a list" shows "(consts (list_comb f xs) |\<subseteq>| all_consts) = (consts f |\<subseteq>| all_consts \<and> list_all (\<lambda>t. consts t |\<subseteq>| all_consts) xs)"
norm2_norm1_main_inequality: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "(\<Sum>i = 0..<n. f i * f i) \<le> (\<Sum>i = 0..<n. \<bar>f i\<bar>)\<^sup>2"
thmD6: fixes x :: "item"   and \<delta> :: "symbol list"   and \<omega> :: "symbol list"   and k :: "nat" assumes "wellformed_item x"   and "item_\<beta> x = \<delta> @ \<omega>"   and "item_end x = k"   and "derives \<delta> []" shows "inc_dot (length \<delta>) x \<in> \<pi> k {} {x}"
cqt_1: fixes \<phi> :: "'b \<Rightarrow> \<o>"   and \<alpha> :: "'b" shows "[[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<rightarrow> \<phi> \<alpha>]]"
no_collision_union: fixes S :: "real set"   and T :: "real set" assumes "no_collision_react S"   and "no_collision_react T" shows "no_collision_react (S \<union> T)"
pp_lub: fixes z :: "'a"   and y :: "'a"   and xs :: "'a list" assumes "z \<in> A"   and "y \<in> A"   and "set xs \<subseteq> A"   and "\<forall>x\<in>set xs. x \<sqsubseteq>\<^bsub>r\<^esub> z"   and "y \<sqsubseteq>\<^bsub>r\<^esub> z" shows "xs \<Squnion>\<^bsub>f\<^esub> y \<sqsubseteq>\<^bsub>r\<^esub> z"
coboundedI1: fixes c :: "'a word"   and a :: "'a word"   and b :: "'a word" assumes "c \<le>s a" shows "c \<le>s signed.max a b"