hcju/isabelleps huggingface dataset:
finals_def: fixes es :: "expr list" shows "finals es \<equiv> (\<exists>vs. es = map Val vs) \<or> (\<exists>vs r es'. es = map Val vs @ Throw r # es')"
assms_2: fixes y :: "'b" assumes "y \<in> S" shows "orthogonal x y"
simps_1: fixes p :: "'e"   and s :: "Concrete.var \<Rightarrow> nat" shows "cval (Inl p) s = min (1::'e) (max (0::'e) p)"
lookup_higher_when: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and u :: "'a"   and v :: "'a" shows "lookup (punit.higher p u) v = (lookup p v when u \<prec> v)"
disjoint_family_on_mono: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'b set" assumes "A \<subseteq> B"   and "disjoint_family_on f B" shows "disjoint_family_on f A"
has_default_policy: fixes rs :: "'b simple_rule list"   and p :: "('b, 'c) simple_packet_scheme" assumes "has_default_policy rs" shows "simple_fw rs p = Decision FinalAllow \<or> simple_fw rs p = Decision FinalDeny"
updVal_preserves_sWls: fixes SEM :: "('b, 'c, 'd, 'sort, 'e, 'f, 'g) semDom_scheme"   and xs :: "'varSort"   and sX :: "'f"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'f"   and x :: "'var" assumes "sWls SEM (asSort xs) sX"   and "sWlsVal SEM val" shows "sWlsVal SEM (val (x := sX)_xs)"
rep_eq: fixes x :: "proper_base" shows "get_base x = Rep_proper_base x"
e_type_const_list: fixes vs :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list"   and \<C>' :: "t_context" assumes "const_list vs"   and "\<S>\<bullet>\<C> \<turnstile> vs : ts _> ts'" shows "\<exists>tvs. ts' = ts @ tvs \<and> length vs = length tvs \<and> \<S>\<bullet>\<C>' \<turnstile> vs : [] _> tvs"
strict_implies_order: fixes a :: "'b list"   and b :: "'b list" assumes "a <p b" shows "a \<le>p b"
rank_eq_card_iarrays_code: fixes A :: "(('d, 'e) vec, 'f) vec" shows "rank A = (let A' = Gauss_Jordan_iarrays (matrix_to_iarray A) in card {i. \<not> is_zero_iarray (A' !! mod_type_class.to_nat i)})"
label0_def: fixes nil :: "'a"   and atom0 :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "classical nil" shows "classical.label0 atom0 x = (if atom0 x then some else none)"
a_get_child_nodes_locs_def: fixes ptr :: "(_) object_ptr" shows "local.a_get_child_nodes_locs ptr \<equiv> (if is_element_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr then {preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) child_nodes)} else {}) \<union> (if is_document_ptr_kind ptr then {preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) document_element)} else {}) \<union> {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing)}"
nprv_negE01: fixes \<phi> :: "'fmla"   and F :: "'fmla set"   and \<psi> :: "'fmla" assumes "neg \<phi> \<in> F"   and "\<phi> \<in> F"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv F \<psi>"
Pd_shuffle_to_top_simps_2: fixes d :: "'b"   and ds' :: "'b set"   and Y :: "'a set" assumes "d \<in> ds'" shows "Pd_shuffle_to_top ds' Y d = shuffle_to_top Y d"
PSimQ': fixes \<Psi>' :: "'b" shows "\<Psi>' \<rhd> P \<leadsto><Rel> Q"
lr_of_tran_fbs_dropD: fixes rt :: "'b routing_rule list"   and fw :: "'b simple_rule list"   and ifs :: "'d"   and p :: "('b, 'c) simple_packet_scheme"   and r :: "'b simple_match"   and oif :: "char list" assumes "valid_prefixes rt"   and "has_default_route rt"   and "no_oif_match fw"   and "generalized_sfw (lr_of_tran_fbs rt fw ifs) p = Some (r, oif, Drop)" shows "simple_linux_router_nol12 rt fw p = None"
SDG_parent_cdep_edge: fixes n' :: "'node SDG_node"   and m :: "'node"   and n :: "'node SDG_node" assumes "valid_SDG_node n'"   and "m = parent_node n'"   and "n = CFG_node m"   and "n \<noteq> n'" shows "n \<longrightarrow>\<^bsub>cd\<^esub> n'"
assms_4: shows "domA \<Gamma> \<subseteq> S"
update_get: fixes f :: "('v \<Rightarrow> int option) \<Rightarrow> 'v \<Rightarrow> int option"   and g :: "('v \<Rightarrow> int option) \<Rightarrow> 'm" shows "update f (get g) = get (update f \<circ> g \<circ> f)"
assms: fixes x :: "'d" assumes "x \<in> A" shows "f x \<in> L F (\<lambda>_. 1::'b)"
assms_1: shows "\<guillemotleft>f : \<I>\<^sub>D \<rightarrow>\<^sub>D F \<I>\<^sub>C\<guillemotright>"
continuous_map_pullback': fixes T1 :: "'d topology"   and T2 :: "'e topology"   and f :: "'f \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and A :: "'f set" assumes "continuous_map T1 T2 (f \<circ> g)"   and "topspace T1 \<subseteq> g -` A" shows "continuous_map T1 (pullback_topology A f T2) g"
qAbs_preserves_alpha: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and X' :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and xs :: "'varSort"   and x :: "'var" assumes "X #= X'"   and "qGood X \<or> qGood X'" shows "qAbs xs x X $= qAbs xs x X'"
isometryD_1: fixes f :: "'b \<Rightarrow> 'c" assumes "isometry f" shows "isometry_on UNIV f"
simps_3: fixes x2 :: "'b" shows "OK x2 \<noteq> Err"
sets_pair_in_sets: fixes A :: "'c measure"   and B :: "'d measure"   and N :: "('c \<times> 'd) measure" assumes "\<And>a b. \<lbrakk>a \<in> sets A; b \<in> sets B\<rbrakk> \<Longrightarrow> a \<times> b \<in> sets N" shows "sets (A \<Otimes>\<^sub>M B) \<subseteq> sets N"
v_2: shows "v \<noteq> 0\<^sub>v n"
obj_fields_marked_eq_imp: shows "eq_imp (\<lambda>r' s. (gc_field_set s, gc_tmp_ref s, map_option obj_fields (sys_heap s r'), map_option obj_mark (sys_heap s r'), sys_fM s, filter (\<lambda>s. is_mw_Mutate s \<or> is_mw_Mutate_Payload s) (sys_mem_store_buffers gc s))) obj_fields_marked"
inverse_unique: fixes a :: "'b"   and b :: "'b" assumes "a + b = (0::'b)" shows "- a = b"
perm_rem_def: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'c" shows "perm_rem x f \<equiv> if f x \<noteq> x then x \<rightleftharpoons>\<^sub>F f x \<circ> f else f"
assigns_SNOCs_zero: fixes k :: "'k"   and m :: "nat"   and \<AA> :: "'i"   and idx :: "'n"   and i :: "nat" assumes "LESS k m (#\<^sub>V \<AA>)"   and "#\<^sub>V \<AA> = idx" shows "m\<^bsup>(SNOC (zero idx) ^^ i) \<AA>\<^esup>k = m\<^bsup>\<AA>\<^esup>k"
is_pr_Iso_rec: fixes Fs\<^sub>1 :: "'a list set"   and Fs\<^sub>2 :: "'b list set"   and F\<^sub>1 :: "'a list"   and \<phi> :: "'a \<Rightarrow> 'b" assumes "inj_on (\<lambda>xs. {xs} // {\<cong>}) Fs\<^sub>1"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) Fs\<^sub>2"   and "F\<^sub>1 \<in> Fs\<^sub>1" shows "is_pr_Iso \<phi> Fs\<^sub>1 Fs\<^sub>2 = (\<exists>F\<^sub>2\<in>Fs\<^sub>2. length F\<^sub>1 = length F\<^sub>2 \<and> is_pr_Iso \<phi> (Fs\<^sub>1 - {F\<^sub>1}) (Fs\<^sub>2 - {F\<^sub>2}) \<and> (\<exists>n. map \<phi> F\<^sub>1 = rotate n F\<^sub>2) \<and> inj_on \<phi> (\<Union> (set ` Fs\<^sub>1)))"
prod_mset_pos: fixes A :: "'a multiset" assumes "\<And>x. x \<in># A \<Longrightarrow> (0::'a) < x" shows "(0::'a) < \<Prod>\<^sub># A"
f_map: fixes r :: "'f"   and m :: "'v" assumes "r \<in> UNIV"   and "m \<in> V" shows "T (r \<cdot> m) = (r \<cdot>\<cdot> T) m"
vfun_le: fixes t1 :: "(val \<times> val) list"   and t2 :: "(val \<times> val) list" assumes "t1 \<lesssim> t2" shows "VFun t1 \<sqsubseteq> VFun t2"
ipurge_ref_aux_single_dom: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and u :: "'f"   and xs :: "'e list"   and X :: "'e set" shows "ipurge_ref_aux I D {u} xs X = ipurge_ref I D u xs X"
simps_1: fixes \<Gamma> :: "char list set"   and c :: "nat" shows "wfe \<Gamma> (Const c) = True"
vsubst_psubst_idEnv: fixes ys :: "'varSort"   and z :: "'var"   and y :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" shows "X #[z // y]_ys = X #[idEnv [y \<leftarrow> Var ys z]_ys]"
snd_eq_Range: fixes R :: "('b \<times> 'a) set" shows "snd ` R = Range R"
index_unit_vec_2: fixes i :: "nat"   and n :: "nat" assumes "i < n" shows "unit_vec n i $ i = (1::'b)"
emeasure_lborel_cball_finite: fixes c :: "'a"   and r :: "real" shows "emeasure lborel (cball c r) < \<infinity>"
simps_9: fixes m :: "nat"   and p :: "('a, 'b) form" shows "closed m (Exists p) = closed (Suc m) p"
MkArr_in_hom: fixes A :: "'o"   and B :: "'o"   and f :: "'a" assumes "A \<in> Obj"   and "B \<in> Obj"   and "f \<in> Hom A B" shows "in_hom (MkArr A B f) (MkIde A) (MkIde B)"
ord_strict_p_irreflexive: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "\<not> ordered_term.ord_strict_p ord_term_strict p p"
x_grid: shows "x \<in> grid (base (ds' \<union> ds) b) ds"
cmp_pol_coeff_e: fixes A :: "('e, 'f) Ring_scheme"   and B :: "('e, 'g) Ring_scheme"   and Y :: "'e"   and f :: "'a \<Rightarrow> 'e"   and n :: "nat"   and c :: "nat \<Rightarrow> 'a" assumes "PolynRg A B Y"   and "f \<in> pHom R S X, A B Y"   and "pol_coeff S (n, c)" shows "pol_coeff B (n, cmp f c)"
comp_assoc_assoc'_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "\<a>[f, g, h] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, h] = f \<star> g \<star> h"
generate_idem': fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> carrier G"   and "B \<subseteq> carrier G" shows "generate G (generate G A \<union> B) = generate G (A \<union> B)"
sig_red_mono: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and F' :: "('t \<Rightarrow>\<^sub>0 'b) set" assumes "sig_red sing_reg top_tail F p q"   and "F \<subseteq> F'" shows "sig_red sing_reg top_tail F' p q"
surj_Poly_list_of_vec: assumes "0 < degree u" shows "(Poly \<circ> list_of_vec) ` carrier local.V = carrier (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u})"
P4: shows "\<lfloor>\<lambda>w. \<forall>x. x actualizedAt w \<longrightarrow> Dependent x w \<longrightarrow> (\<exists>xa. ((actualizedAt) xa \<^bold>\<and> Independent xa \<^bold>\<and> x dependsOn xa) w)\<rfloor>"
on_triple_def: fixes R :: "('f \<times> 'g) set" shows "on_triple R \<equiv> {((l, s, t), l', s', t'). l = l' \<and> (s, s') \<in> R \<and> (t, t') \<in> R}"
IH: assumes "\<And>vs. liveVal' g (v # vs) \<Longrightarrow> thesis" shows "thesis"
moebius_mu_def: fixes n :: "nat" shows "moebius_mu n = (if squarefree n then (- (1::'a)) ^ card (prime_factors n) else (0::'a))"
heap_axioms: shows "heap addr2thread_id thread_id2addr allocate (\<lambda>_. typeof_addr) heap_write P"
foreach_impl_def: fixes Qi :: "'q"   and \<pi>i :: "'m"   and u :: "'v"   and adjs :: "('v \<times> nat) list" shows "foreach_impl Qi \<pi>i u adjs = foldr (foreach_impl_body u) adjs (Qi, \<pi>i)"
seqE: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" assumes "A1xA2.seq g f" obtains "A1.seq (fst g) (fst f)" and "A2.seq (snd g) (snd f)"
IH: assumes "(A, B) \<in> set (pairs As)" shows "A \<in> set As \<and> B \<in> set As"
distinct_9: fixes x21 :: "nat"   and x22 :: "v list"   and x4 :: "e list" shows "RSBreak x21 x22 \<noteq> RSNormal x4"
K_def: shows "K = alpha * ln 2 / (5 * (4 + (L + 2 * delta) / D) * delta * lambda)"
eqvtI: fixes f :: "'b" assumes "\<And>p. p \<bullet> f \<equiv> f" shows "eqvt f"
invertible_mult_row': fixes q :: "'a"   and a :: "'b" assumes "q \<noteq> (0::'a)" shows "invertible (mult_row (mat (1::'a)) a q)"
sat\<^sub>2\<sigma>: shows "eval\<^sub>c F G (C\<^sub>2 \<cdot>\<^sub>l\<^sub>s \<sigma>)"
set_less_aux_insert_same: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "(x \<in> A) = (x \<in> B)" shows "(insert x A \<sqsubset>' insert x B) = (A \<sqsubset>' B)"
induct: fixes x1 :: "nat"   and x2 :: "'b tensor"   and P :: "nat \<Rightarrow> 'b tensor \<Rightarrow> bool" assumes "cprank_max x1 x2"   and "\<And>ds. P 0 (tensor0 ds)"   and "\<And>A B j. \<lbrakk>dims A = dims B; cprank_max1 A; cprank_max j B; P j B\<rbrakk> \<Longrightarrow> P (Suc j) (A + B)" shows "P x1 x2"
change: fixes ts :: "traffic"   and ts' :: "traffic"   and ts'' :: "traffic" assumes "\<exists>c f.  ts \<^bold>\<midarrow> dyn(c,f) \<^bold>\<rightarrow> ts'"   and "ts' \<^bold>\<leadsto> ts''" shows "ts \<^bold>\<leadsto> ts''"
tab_in_hom_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b"   and \<rho> :: "'b"   and f :: "'b"   and g :: "'b" assumes "tabulation_data V H \<a> \<i> src trg r \<rho> f g" shows "partial_magma.in_hom V \<rho> g (H r f)"
scale_cancel_left: fixes a :: "'a"   and x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" shows "(a *s x = a *s y) = (x = y \<or> a = (0::'a))"
simps: fixes queue :: "'queue"   and n :: "nat"   and T :: "'t set" shows "local.round_robin_invar (queue, n) T = (queue_invar queue \<and> Round_Robin.round_robin_invar (queue_\<alpha> queue, n) T)"
induct: fixes P :: "exp \<Rightarrow> bool"   and a0 :: "exp" assumes "\<And>x. P (Var x)"   and "\<And>x e. P (Lam [x]. e)"   and "\<And>e x. P (App e x)"   and "\<And>as e. P (Terms.Let as e)"   and "\<And>b. P (Bool b)"   and "\<And>scrut e1 e2. P (scrut ? e1 : e2)" shows "P a0"
left_right_impl_times: fixes a :: "'a"   and b :: "'a" shows "(a l\<rightarrow> b) * a = a * (a r\<rightarrow> b)"
fin_cosets: shows "finite (rcosets\<^bsub>G\<^esub> kernel G H f)"
distinct_305: fixes x17 :: "nat"   and x18 :: "float" shows "Var x17 \<noteq> Num x18"
Lim_emeasure_incseq: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range A \<subseteq> sets M"   and "incseq A" shows "(\<lambda>i. emeasure M (A i)) \<longlonglongrightarrow> emeasure M (\<Union> (range A))"
CSP_Inf: fixes A :: "('c, 'd) action set" assumes "A \<noteq> {}" shows "is_CSP_process (\<lambda>A. A \<in> \<Union> {{p. P p} |P. P \<in> action.relation_of ` A})"
Conj_def: fixes xset :: "('e, 'f, 'g, 'h) formula set['e]" shows "Conj xset = Abs_formula (Conj\<^sub>\<alpha> (map_bset Rep_formula xset))"
CompDefinedE: fixes C :: "('g, 'h, 'i) Category_scheme"   and f :: "'h"   and g :: "'h" assumes "f \<approx>>\<^bsub>C\<^esub> g" obtains "f \<in> mor\<^bsub>C\<^esub>" and "g \<in> mor\<^bsub>C\<^esub>" and "cod\<^bsub>C\<^esub> f = dom\<^bsub>C\<^esub> g"
is_kth_obs_def: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and i :: "nat" shows "is_kth_obs \<pi> k i = (card (obs_ids \<pi> \<inter> {..<i}) = k \<and> local.att (\<pi> i) \<noteq> None)"
simps: fixes A :: "('b \<Rightarrow> 'c) \<times> ('b \<times> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<times> 'c)" shows "On_Off.embed A = (\<lambda>s. return_pmf (fst A s), \<lambda>s r. return_pmf (snd A s r))"
simps_1: fixes x :: "bool" shows "fromC (C x) = Some x"
Quotient3_def: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e" shows "Quotient3 R Abs Rep = ((\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and> (\<forall>r s. R r s = (R r r \<and> R s s \<and> Abs r = Abs s)))"
smerge_def: fixes ss :: "'d stream stream" shows "smerge ss = flat (smap (\<lambda>n. map (\<lambda>s. s !! n) (stake (Suc n) ss) @ stake n (ss !! n)) nats)"
nth_Cons: fixes x :: "'b"   and xs :: "['b]"   and n :: "Integer" shows "(x : xs) !! n = If eq\<cdot>n\<cdot>0 then x else xs !! (n - 1)"
2: shows "((\<lambda>x. \<Sum>b\<in>basis. f (g2 x) \<bullet> b * (vector_derivative g2 (at x within {0..1}) \<bullet> b)) has_integral i2) {0..1}"
negative_imp_inverse_negative: fixes a :: "'a" assumes "a < (0::'a)" shows "inverse a < (0::'a)"
compact_interval: fixes a :: "'c"   and b :: "'c" shows "compact {a..b}"
simps: fixes a :: "nat \<Rightarrow> nat" shows "PrimRec1'p a = (a = (\<lambda>x. 0) \<or> a = Suc \<or> a = c_fst \<or> a = c_snd \<or> (\<exists>f g. a = (\<lambda>x. f (g x)) \<and> PrimRec1'p f \<and> PrimRec1'p g) \<or> (\<exists>f g. a = c_f_pair f g \<and> PrimRec1'p f \<and> PrimRec1'p g) \<or> (\<exists>f g. a = UnaryRecOp f g \<and> PrimRec1'p f \<and> PrimRec1'p g))"
recurse_Node: fixes f :: "'e \<Rightarrow> ('c, 'd) state"   and l :: "'e Tree_Relabelling.tree"   and r :: "'e Tree_Relabelling.tree" shows "recurse f (Tree_Relabelling.tree.Node l r) = pure (\<lambda>r l. Tree_Relabelling.tree.Node l r) \<diamondop> recurse f r \<diamondop> recurse f l"
HStartBallot_HInv4d_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HStartBallot s s' p"   and "HInv4d s p" shows "HInv4d s' p"
WT_gpv_try_gpvI: fixes \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and gpv' :: "('j, 'h, 'i) gpv" assumes "\<I> \<turnstile>g gpv \<surd>"   and "\<not> colossless_gpv \<I> gpv \<Longrightarrow> \<I> \<turnstile>g gpv' \<surd>" shows "\<I> \<turnstile>g TRY gpv ELSE gpv' \<surd>"
c_0: fixes x :: "'a" shows "x \<sqinter> 1\<^sub>\<pi> = x \<cdot> (0::'a)"
rSim: fixes R :: "ccs"   and S :: "ccs" assumes "(R, S) \<in> X" shows "R \<leadsto>\<^sup>^<(weakBisimulation O (X \<union> weakBisimulation) O bisim)> S"
B: shows "y * \<top> \<sqinter> y ^ o = y"
open_Bonk_Schramm_extension_def: fixes U :: "'a Bonk_Schramm_extension set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
not_Cons_self2: fixes x :: "'b"   and xs :: "'b list" shows "x # xs \<noteq> xs"
eval_fmla_Iff: fixes e :: "(name, nat) finfun"   and A :: "fmla"   and B :: "fmla" shows "eval_fmla e (A IFF B) = (eval_fmla e A = eval_fmla e B)"
square_free_smult_iff: fixes c :: "'a"   and f :: "'a poly" assumes "c \<noteq> (0::'a)" shows "square_free (Polynomial.smult c f) = square_free f"
cl_ivl_Suc: fixes g :: "nat \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat" shows "sum g {m..Suc n} = (if Suc n < m then 0::'c else sum g {m..n} + g (Suc n))"
int_and_extra_simps_1: fixes w :: "int" shows "w AND 0 = 0"
UnitsI_1: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "a \<otimes> b = \<one>" shows "a \<in> Units Zp"
assms: shows "I \<circ> K \<circ> I = I"
snd_sym_preproc_complete: fixes v :: "'a"   and gs :: "('a \<Rightarrow>\<^sub>0 'd) list"   and fs :: "('a \<Rightarrow>\<^sub>0 'd) list"   and g :: "'a \<Rightarrow>\<^sub>0 'd" assumes "v \<in> Keys (set (snd (punit.sym_preproc gs fs)))"   and "g \<in> set gs"   and "local.punit.adds_term (punit.lt g) v" shows "local.punit.monom_mult (1::'d) (local.punit.pp_of_term v - punit.lp g) g \<in> set (snd (punit.sym_preproc gs fs))"
hom_graph_inf: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a formula"   and a :: "'b"   and y :: "'a formula"   and b :: "'b" assumes "hom_graph f S x a"   and "hom_graph f S y b" shows "hom_graph f S (x \<sqinter> y) (a \<sqinter> b)"
rel_reflp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "reflp R" shows "reflp (rel_pmf R)"
sequence_number_increases: fixes i :: "nat" shows "paodv i \<TTurnstile>\<^sub>A onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>((\<xi>, uu_), uu_, \<xi>', uu_). sn \<xi> \<le> sn \<xi>')"
jkbpC_jkbpCn_jAction_eq: fixes t :: "'s Trace"   and n :: "nat" assumes "t \<in> Clock.jkbpCn n" shows "jAction Clock.MC t = jAction (Clock.MCn n) t"
wt_App_ge_arg: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and s :: "('b, 'a) tm"   and t :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym (App s t)) (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym t)"
assms: fixes x :: "'a" assumes "x \<in> set xs" shows "f x \<le> SPEC (I x)"
sup_finfun_def: fixes f :: "'a \<Rightarrow>f 'b"   and g :: "'a \<Rightarrow>f 'b" shows "sup f g = (\<lambda>(x, y). sup x y) \<circ>$ ($f, g$)"
sel_1: fixes x1 :: "bool"   and x2 :: "'a \<Rightarrow> 'a language" shows "\<oo> (Lang x1 x2) = x1"
single_stateE: assumes "state_not_singleton" shows "\<forall>t. (\<forall>s. s = t) \<longrightarrow> False"
path_strategies_strategy: fixes G :: "('c, 'd) ParityGame_scheme"   and S :: "'c set"   and p :: "Player"   and good :: "'c \<Rightarrow> ('c \<Rightarrow> 'c) set"   and r :: "(('c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> 'c)) set"   and P :: "'c llist"   and n :: "nat" assumes "WellOrderedStrategies G S p good r"   and "lset P \<subseteq> S"   and "enat n < llength P" shows "ParityGame.strategy G p (WellOrderedStrategies.path_strategies good r P $ n)"
t_A_A'_leq: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and n :: "nat" assumes "BIT_Off acts qs init"   and "n < length (BIT_Off.paid_A' acts)" shows "BIT_Off.t_A acts qs init n \<le> BIT_Off.t_A' acts qs init n"
jozsa_transform_coeff_is_zero: fixes i :: "nat"   and j :: "nat" assumes "i < dim_row U\<^sub>f \<and> j < dim_col U\<^sub>f" shows "i \<noteq> j \<and> \<not> (i = j + 1 \<and> odd i) \<and> \<not> (i = j - 1 \<and> even i \<and> 1 \<le> j) \<longrightarrow> U\<^sub>f $$ (i, j) = 0"
sets_measure_pmf_count_space: fixes M :: "'a pmf" shows "measure_pmf.events M = sets (count_space UNIV)"
inject: fixes x :: "nat"   and ya :: "nat" shows "(Var x = Var ya) = (x = ya)"
powr_less_one: fixes x :: "real"   and y :: "real" assumes "1 < x"   and "y < 0" shows "x powr y < 1"
M0_mult_M0: shows "M0 * M0 = M0"
map_gpv_conv_map_gpv': fixes f :: "'f \<Rightarrow> 'i"   and g :: "'g \<Rightarrow> 'j" shows "map_gpv f g = map_gpv' f g id"
spair_sigs_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" shows "spair_sigs p q = (let t1 = punit.lt (rep_list p); t2 = punit.lt (rep_list q); l = lcs t1 t2 in ((l - t1) \<oplus> lt p, (l - t2) \<oplus> lt q))"
det_identical_cols: fixes i :: "'c"   and i' :: "'c"   and A :: "('d, 'c) sq_matrix" assumes "i \<noteq> i'"   and "col A i = col A i'" shows "Square_Matrix.det A = (0::'d)"
prems_3: shows "InvariantWatchListsUniq (getWatchList state)"
mult_matrix_n_zero_right: fixes fadd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and fmul :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and n :: "nat"   and A :: "'c matrix" assumes "fadd (0::'b) (0::'b) = (0::'b)"   and "\<forall>a. fmul a (0::'d) = (0::'b)" shows "mult_matrix_n n fmul fadd A 0 = 0"
multivariate_Taylor_has_integral: fixes n :: "nat"   and Df :: "'b \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'd"   and H :: "'b"   and f :: "'b \<Rightarrow> 'd"   and X :: "'b"   and G :: "'b set" assumes "0 < n"   and "\<And>a x. Df a 0 H H = f a"   and "\<And>a i d. \<lbrakk>a \<in> {X--X + H}; i < n\<rbrakk> \<Longrightarrow> ((\<lambda>a. Df a i H H) has_derivative Df a (Suc i) H) (at a within G)"   and "{X--X + H} \<subseteq> G" shows "((\<lambda>x. ((1 - x) ^ (n - 1) / fact (n - 1)) *\<^sub>R Df (X + x *\<^sub>R H) n H H) has_integral f (X + H) - (\<Sum>i<n. (1 / fact i) *\<^sub>R Df X i H H)) {0..1}"