hcju/isabelleps huggingface dataset:
atomic_ev_signal_preserves_invariants: fixes s :: "state_t"   and tid :: "thread_id_t"   and partner :: "thread_id_t" assumes "atomic_step_invariant s" shows "atomic_step_invariant (atomic_step_ev_signal tid partner s)"
less_sets_singleton1: fixes a :: "'a"   and A :: "'a set" shows "({a} \<lless> A) = (\<forall>x\<in>A. a < x)"
wn_d: shows "winding_number (c1 +++ reversepath c) z = d"
abs_ok_1: shows "term_ok \<Theta> (Abs_fv x \<tau>' s)"
sym_Un: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "sym r"   and "sym s" shows "sym (r \<union> s)"
insert_subset: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" shows "(x \<triangleright> A \<subseteq> B) = (x \<in> B \<and> A \<subseteq> B)"
axioms_1: fixes H :: "'e set"   and R :: "('f, 'g) ring_scheme"   and M :: "('f, 'e, 'h) module_scheme" assumes "submodule H R M" shows "subgroup H (add_monoid M)"
put_get: fixes \<sigma> :: "'b"   and v :: "'a" shows "get (put \<sigma> v) = v"
simps: fixes t :: "'a" shows "no_abs t = term_cases (\<lambda>_. True) (\<lambda>_. True) (\<lambda>t u. no_abs t \<and> no_abs u) False t"
left_neutral: fixes a :: "'a" shows "bot \<squnion> a = a"
rel_transp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "transp R" shows "transp (rel_pmf R)"
assms: shows "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"
nres_rel_def: fixes R :: "('d \<times> 'e) set" shows "\<langle>R\<rangle>nres_rel \<equiv> {(c, a). c \<le> \<Down> R a}"
flowsto_def: fixes X0 :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and T :: "real set"   and CX :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and X1 :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" shows "flowsto X0 T CX X1 = (\<forall>(x0, d0)\<in>X0. \<exists>h\<in>T. h \<in> existence_ivl0 x0 \<and> (flow0 x0 h, Dflow x0 h o\<^sub>L d0) \<in> X1 \<and> (\<forall>h'\<in>{0<--<h}. (flow0 x0 h', Dflow x0 h' o\<^sub>L d0) \<in> CX))"
H: shows "\<Gamma> \<turnstile> (\<lambda>x:S. s) : U \<rightarrow> U'"
subst_atms_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and AA :: "'b set"   and \<sigma> :: "'c" shows "substitution_ops.subst_atms subst_atm AA \<sigma> = (\<lambda>A. subst_atm A \<sigma>) ` AA"
array_rel_def_internal: fixes R :: "('c \<times> 'd) set" shows "array_rel R \<equiv> {(Array xs, Array ys) |xs ys. (xs, ys) \<in> \<langle>R\<rangle>list_rel}"
target_size: fixes c :: "'a \<Rightarrow> 'b word" shows "target_size c = LENGTH('b)"
case_two_iff_forever_neither: fixes i :: "nat" shows "case_two i = (\<exists>t. \<forall>T\<ge>t. change_on_neither (block i T) (gap i T))"
edge_succ_cyclic: fixes v :: "'a" assumes "v \<in> verts G"   and "out_arcs G v \<noteq> {}" shows "cyclic_on (edge_succ M) (out_arcs G v)"
simps: fixes G :: "prog"   and a1 :: "state"   and a2 :: "state" shows "G\<turnstile>a1 \<midarrow>sxalloc\<rightarrow> a2 = ((\<exists>s. a1 = Norm s \<and> a2 = Norm s) \<or> (\<exists>j s. a1 = (Some (Jump j), s) \<and> a2 = (Some (Jump j), s)) \<or> (\<exists>e s. a1 = (Some (Error e), s) \<and> a2 = (Some (Error e), s)) \<or> (\<exists>a s. a1 = (Some (Xcpt (Loc a)), s) \<and> a2 = (Some (Xcpt (Loc a)), s)) \<or> (\<exists>s0 xn a x s1. a1 = (Some (Xcpt (Std xn)), s0) \<and> a2 = (Some (Xcpt (Loc a)), s1) \<and> G\<turnstile>Norm s0 \<midarrow>halloc CInst (SXcpt xn)\<succ>a\<rightarrow> (x, s1)))"
map_spmf_bind_spmf: fixes f :: "'d \<Rightarrow> 'b"   and p :: "'e spmf"   and g :: "'e \<Rightarrow> 'd spmf" shows "map_spmf f (p \<bind> g) = p \<bind> map_spmf f \<circ> g"
sum_lessThan_telescope: fixes f :: "nat \<Rightarrow> 'a"   and m :: "nat" shows "(\<Sum>n<m. f (Suc n) - f n) = f m - f 0"
action_of_inverse: fixes y :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "y \<in> {p. is_CSP_process p}" shows "action.relation_of (action_of y) = y"
expand: fixes llist :: "'b llist"   and llist' :: "'b llist" assumes "lnull llist = lnull llist'"   and "\<lbrakk>\<not> lnull llist; \<not> lnull llist'\<rbrakk> \<Longrightarrow> lhd llist = lhd llist' \<and> ltl llist = ltl llist'" shows "llist = llist'"
pdiv_3: fixes K :: "'a set" assumes "K \<in> p" shows "K \<noteq> {}"
smember_def: fixes x :: "'b"   and s :: "'b stream" shows "Stream.smember x s = (x \<in> i.R s)"
NetsCollectedSortQ: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "distinct p"   and "noDenyAll1 p"   and "all_in_list p l"   and "singleCombinators p" shows "NetsCollected (qsort p l)"
sparse_row_vector_def: fixes arr :: "(nat \<times> 'a) list" shows "sparse_row_vector arr = foldl (\<lambda>m x. m + singleton_matrix 0 (fst x) (snd x)) 0 arr"
path_image_def: fixes g :: "real \<Rightarrow> 'a" shows "path_image g = g ` {0..1}"
tsubst_preserves_safe: fixes \<theta> :: "('a, 'b) trm"   and \<sigma> :: "('a, 'c, 'b) subst" assumes "dsafe \<theta>"   and "\<And>i f'. SFunctions \<sigma> i = Some f' \<Longrightarrow> dfree f'" shows "dsafe (Tsubst \<theta> \<sigma>)"
rel_eq: shows "rel_sp\<^sub>\<mu> (=) (=) = (=)"
Markov_Decision_Process_axioms: shows "Markov_Decision_Process K"
simps_5: fixes M :: "('b, 'c, 'd) KripkeStructure"   and w :: "'d"   and as :: "'b list"   and \<phi> :: "('b, 'c) Kform" shows "M, w \<Turnstile> \<^bold>C\<^bsub>as\<^esub> \<phi> = (\<forall>w'\<in>(\<Union> (relations M ` set as))\<^sup>+ `` {w}. M, w' \<Turnstile> \<phi>)"
no_defection_def: fixes s :: "v_state"   and round_votes :: "process \<Rightarrow> val option"   and r :: "nat" shows "no_defection s round_votes r = (\<forall>r'<r. \<forall>a Q v. quorum_for Q v (votes s r') \<and> a \<in> Q \<longrightarrow> round_votes a \<in> {None, Some v})"
dis_rews_7: fixes cfun :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" shows "is_CB\<cdot>(CFn\<cdot>cfun) = FF"
less_finite_2_def: fixes x :: "Enum.finite_2"   and y :: "Enum.finite_2" shows "(x < y) = (x = a\<^sub>1 \<and> y = a\<^sub>2)"
lens_plus_comm: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e" assumes "X \<bowtie> Y" shows "X +\<^sub>L Y \<approx>\<^sub>L Y +\<^sub>L X"
matrix_transpose_mul: fixes A :: "(('a, 'd) vec, 'b) vec"   and B :: "(('a, 'c) vec, 'd) vec" shows "Finite_Cartesian_Product.transpose (A ** B) = Finite_Cartesian_Product.transpose B ** Finite_Cartesian_Product.transpose A"
trace_measurement2: fixes n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and A :: "complex mat" assumes "measurement n 2 M"   and "A \<in> carrier_mat n n" shows "trace (M 0 * A * adjoint (M 0)) + trace (M 1 * A * adjoint (M 1)) = trace A"
card_A_eq: shows "card A = card A1 + card A2 + card A3"
disj_of_clause_simps_1: shows "disj_of_clause [] = \<bottom>"
eucl_le: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and abs :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "less_eq x y = (\<forall>i\<in>Basis. inner x i \<le> inner y i)"
not_eq_extremum: fixes a :: "'a list" shows "(a \<noteq> []) = strict_border [] a"
PTrans: shows "P \<Longrightarrow>\<^sup>^rev perm \<bullet> \<alpha> \<prec> P'"
set_rev: fixes xs :: "'d list" shows "set (rev xs) = set xs"
assms_10: shows "(x1, y1) \<noteq> (x2, y2)"
cases: fixes x :: "pheap" obtains h :: "heap" and as :: "nat set" where "x = pHeap h as"
limit_ordinal_def: fixes x :: "ordinal" shows "limit_ordinal x = (x \<noteq> 0 \<and> (\<forall>y. x \<noteq> oSuc y))"
big_step_t_induct: fixes x1a :: "com"   and x1b :: "char list \<Rightarrow> int"   and x2a :: "nat"   and x3a :: "char list \<Rightarrow> int"   and P :: "com \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "(x1a, x1b) \<Rightarrow> x2a \<Down> x3a"   and "\<And>s. P SKIP s (Suc 0) s"   and "\<And>x a s. P (x ::= a) s (Suc 0) (s(x := aval a s))"   and "\<And>c1 s1 x s2 c2 y s3 z. \<lbrakk>(c1, s1) \<Rightarrow> x \<Down> s2; P c1 s1 x s2; (c2, s2) \<Rightarrow> y \<Down> s3; P c2 s2 y s3; z = x + y\<rbrakk> \<Longrightarrow> P (c1;; c2) s1 z s3"   and "\<And>b s c1 x t y c2. \<lbrakk>bval b s; (c1, s) \<Rightarrow> x \<Down> t; P c1 s x t; y = x + 1\<rbrakk> \<Longrightarrow> P (IF b THEN c1 ELSE c2) s y t"   and "\<And>b s c2 x t y c1. \<lbrakk>\<not> bval b s; (c2, s) \<Rightarrow> x \<Down> t; P c2 s x t; y = x + 1\<rbrakk> \<Longrightarrow> P (IF b THEN c1 ELSE c2) s y t"   and "\<And>b s c. \<not> bval b s \<Longrightarrow> P (WHILE b DO c) s (Suc 0) s"   and "\<And>b s1 c x s2 y s3 z. \<lbrakk>bval b s1; (c, s1) \<Rightarrow> x \<Down> s2; P c s1 x s2; (WHILE b DO c, s2) \<Rightarrow> y \<Down> s3; P (WHILE b DO c) s2 y s3; 1 + x + y = z\<rbrakk> \<Longrightarrow> P (WHILE b DO c) s1 z s3" shows "P x1a x1b x2a x3a"
DomainI: fixes a :: "'l"   and b :: "'m"   and r :: "('l \<times> 'm) set" assumes "(a, b) \<in> r" shows "a \<in> Domain r"
strand_subst_hom_2: fixes x :: "('c, 'd) strand_step"   and S :: "('c, 'd) strand_step list"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "x # S \<cdot>\<^sub>s\<^sub>t \<theta> = (x \<cdot>\<^sub>s\<^sub>t\<^sub>p \<theta>) # (S \<cdot>\<^sub>s\<^sub>t \<theta>)"
assms: assumes "PROP P"   and "CONSTRAINT_SLOT PROP S" shows "PROP Q"
fps_cutoff_left_mult_nth: fixes k :: "nat"   and n :: "nat"   and f :: "'a fps"   and g :: "'a fps" assumes "k < n" shows "fps_nth (fps_cutoff n f * g) k = fps_nth (f * g) k"
vfsequence_iff: fixes xs :: "V" shows "vfsequence xs = (vsv xs \<and> \<D>\<^sub>\<circ> xs \<in>\<^sub>\<circ> \<omega>)"
le_infI1: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and x :: "'b"   and b :: "'b" assumes "class.semilattice_inf inf less_eq less"   and "less_eq a x" shows "less_eq (inf a b) x"
EqOnGL_def: fixes p :: "pvar"   and p' :: "pvar" shows "EqOnGL p p' \<equiv> Scon {EqOnUOut p p' u ou |u ou. (u, ou) \<in> GL \<times> UNIV}"
ld_asi_def: fixes instr :: "sparc_operation \<times> inst_operand list"   and s_val :: "word1" shows "ld_asi instr s_val \<equiv> if fst instr \<in> {load_store_type LDD, load_store_type LD, load_store_type LDUH, load_store_type LDSB, load_store_type LDUB, load_store_type LDSH} then if s_val = 0 then word_of_int 10 else word_of_int 11 else get_operand_asi (snd instr ! 3)"
string_list_trans: fixes xa :: "char list list"   and ya :: "char list list"   and z :: "char list list" assumes "(xa, ya) \<in> lexord (lexord {(x, y). x < y})"   and "(ya, z) \<in> lexord (lexord {(x, y). x < y})" shows "(xa, z) \<in> lexord (lexord {(x, y). x < y})"
simple_function_If_set: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "simple_function M f"   and "simple_function M g"   and "A \<inter> space M \<in> sets M" shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)"
UM_Struct_axioms: shows "UM_Struct wtFsym wtPsym arOf parOf D intF intP"
tm_max_range: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and t1 :: "taylor_model"   and y :: "real"   and t2 :: "taylor_model"   and I :: "float interval list"   and a :: "float interval list"   and prec :: "nat" assumes "x \<in>\<^sub>i range_tm e t1"   and "y \<in>\<^sub>i range_tm e t2"   and "num_params (tm_poly t1) \<le> length I"   and "num_params (tm_poly t2) \<le> length I"   and "develops_at_within e a I" shows "max x y \<in>\<^sub>i range_tm e (tm_max prec I a t1 t2)"
lens_comp_assoc: fixes X :: "'c \<Longrightarrow> 'e"   and Y :: "'e \<Longrightarrow> 'f"   and Z :: "'f \<Longrightarrow> 'd" shows "X ;\<^sub>L (Y ;\<^sub>L Z) = X ;\<^sub>L Y ;\<^sub>L Z"
psubstAll_compose: fixes P :: "('d, 'e, 'varSort, 'var, 'f) param"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" assumes "goodPar P" shows "(good X \<longrightarrow> (\<forall>rho rho'. {rho, rho'} \<subseteq> envsOf P \<longrightarrow> X #[rho] #[rho'] = X #[rho &[rho']])) \<and> (goodAbs A \<longrightarrow> (\<forall>rho rho'. {rho, rho'} \<subseteq> envsOf P \<longrightarrow> A $[rho] $[rho'] = A $[rho &[rho']]))"
step_2_correct: fixes p :: "prog" assumes "spec\<^sub>2 p" shows "spec\<^sub>1 p"
fls_X_inv_power_base_factor_to_fps: fixes n :: "nat" shows "fls_base_factor_to_fps (fls_X_inv ^ n) = 1"
similar_mat_witD2_3: fixes A :: "'a mat"   and n :: "nat"   and m :: "nat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat" assumes "A \<in> carrier_mat n m"   and "similar_mat_wit A B P Q" shows "A = P * B * Q"
subset: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set"   and B :: "'e set" assumes "semilattice_set f"   and "finite A"   and "B \<noteq> {}"   and "B \<subseteq> A" shows "f (semilattice_set.F f B) (semilattice_set.F f A) = semilattice_set.F f A"
fps_right_inverse_idempotent_ring1: fixes f :: "'a fps"   and x :: "'a"   and y :: "'a" assumes "fps_nth f 0 * x = (1::'a)"   and "x * y = (1::'a)" shows "fps_right_inverse (fps_right_inverse f x) y = f"
card_Diff_singleton: fixes A :: "'d set"   and x :: "'d" assumes "finite A"   and "x \<in> A" shows "card (A - {x}) = card A - 1"
steps_eq_in: fixes ly :: "nat list"   and ap :: "abc_inst list"   and tp :: "(action \<times> nat) list"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and ins :: "abc_inst"   and stp :: "nat"   and s' :: "nat"   and l' :: "cell list"   and r' :: "cell list" assumes "ly = layout_of ap"   and "tp = tm_of ap"   and "crsp ly (as, lm) (s, l, r) ires"   and "abc_fetch as ap = Some ins"   and "steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp = (s', l', r')"   and "s' \<noteq> 0" shows "steps0 (s, l, r) tp stp = (s', l', r')"
Abs_eq: fixes x :: "'a"   and y :: "'a" shows "(Abs x = Abs y) = eq x y"
comp2_simp: fixes h :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'c"   and x :: "'d"   and y :: "'e" shows "(h o2 (f,g)) x y = h (f x) (g y)"
measure_cond_spmf: fixes p :: "'c spmf"   and A :: "'c set"   and B :: "'c set" shows "Sigma_Algebra.measure (measure_spmf (cond_spmf p A)) B = Sigma_Algebra.measure (measure_spmf p) (A \<inter> B) / Sigma_Algebra.measure (measure_spmf p) A"
ipurge_ref_subset_insert: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list"   and x :: "'c"   and X :: "'c set" shows "ipurge_ref I D u xs (insert x X) \<subseteq> insert x (ipurge_ref I D u xs X)"
cmp_rfn3: fixes r :: "nat"   and s :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set" assumes "0 < r"   and "0 < s"   and "compseries G r f"   and "compseries G s g" shows "cmp_rfn G r f s g 0 = carrier G \<and> cmp_rfn G r f s g (s * r) = {\<one>}"
minimize_wrt_ni: fixes x :: "'a"   and xs :: "'a list"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "x \<in> set xs"   and "x \<notin> set (minimize_wrt Q xs)" shows "\<exists>y\<in>set xs. \<not> Q y x \<and> x \<noteq> y"
marg_dens_integral: fixes x :: "nat"   and X :: "val set"   and \<rho> :: "nat \<Rightarrow> val" assumes "x \<in> V"   and "X \<in> sets (stock_measure (\<Gamma> x))"   and "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "set_nn_integral (stock_measure (\<Gamma> x)) X (marg_dens \<Y> x \<rho>) = \<integral>\<^sup>+\<sigma>\<in>(\<lambda>\<sigma>. \<sigma> x) -` X \<inter> space (state_measure V \<Gamma>). \<delta> (merge V V' (\<sigma>, \<rho>))\<partial>state_measure V \<Gamma>"
assms: shows "prv (neg (PP \<langle>fls\<rangle>))"
pcr_cr_eq: shows "pcr_ttree (=) = cr_ttree"
m_3: shows "\<forall>t\<in>S_Max_Le. m \<le> t"
hyps_3: assumes "P \<noteq> P''" shows "P \<oplus> Q \<Longrightarrow>\<^sub>\<tau> P''"
fi_scalar_prod_gso: fixes i :: "nat"   and j :: "nat" assumes "i < length fs"   and "j < length fs" shows "fs ! i \<bullet> gso j = \<mu> i j * \<parallel>gso j\<parallel>\<^sup>2"
awalk_verts_arc2: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and e :: "'a \<times> 'a" assumes "S.awalk u p v"   and "e \<in> set p" shows "snd e \<in> set (pawalk_verts u p)"
merge_restrict_1: fixes I :: "'b set"   and J :: "'b set"   and x :: "'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'c" shows "merge I J (restrict x I, y) = merge I J (x, y)"
Limit_Vfrom_eq: fixes i :: "V"   and a :: "V" assumes "Limit i" shows "Vfrom a i = \<Squnion> (Vfrom a ` elts i)"
CFG_Construct_wf_def: fixes \<alpha>e :: "'f \<Rightarrow> ('g \<times> 'h \<times> 'g) set"   and \<alpha>n :: "'f \<Rightarrow> 'g list"   and invar :: "'f \<Rightarrow> bool"   and inEdges' :: "'f \<Rightarrow> 'g \<Rightarrow> ('g \<times> 'h) list"   and Entry :: "'f \<Rightarrow> 'g"   and defs :: "'f \<Rightarrow> 'g \<Rightarrow> 'i set"   and "uses" :: "'f \<Rightarrow> 'g \<Rightarrow> 'i set" shows "CFG_Construct_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<equiv> CFG_Construct \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<and> CFG_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses"
seq2: fixes c1 :: "('exp, 'id) MWLsCom"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and c1' :: "('exp, 'id) MWLsCom"   and m' :: "'id \<Rightarrow> 'val"   and c2 :: "('exp, 'id) MWLsCom" assumes "\<langle>c1,m\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>Some c1',m'\<rangle>" shows "\<langle>c1;c2,m\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>Some (c1';c2),m'\<rangle>"
alpha_par: fixes x :: "'a"   and y :: "'a" shows "\<nu> (x \<parallel> y) = \<nu> x \<parallel> y + x \<parallel> \<nu> y"
fun_cong: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "f = g" shows "f x = g x"
wf_inv_image: fixes r :: "('b \<times> 'b) set"   and f :: "'c \<Rightarrow> 'b" assumes "wf r" shows "wf (inv_image r f)"
interior: fixes ccw :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and p :: "'a"   and q :: "'a"   and r :: "'a"   and t :: "'a" assumes "ccw_system4 ccw S"   and "distinct4 p q r t"   and "in4 S p q r t"   and "ccw t q r"   and "ccw p t r"   and "ccw p q t" shows "ccw p q r"
ivl_disj_int_two_3: fixes l :: "'b"   and m :: "'b"   and u :: "'b" shows "{l..<m} \<inter> {m..<u} = {}"
valid_I_def: shows "valid_I = {I. card I = n \<and> lin_indpt (a ` I) \<and> I \<subseteq> {0..<m}}"
ennreal_minus_pos_iff: fixes a :: "ennreal"   and b :: "ennreal" assumes "a < \<top> \<or> b < \<top>"   and "0 < a - b" shows "b < a"
comp_cod_arr: fixes f :: "'j"   and b :: "'j" assumes "J.arr f"   and "J.cod f = b" shows "b \<cdot>\<^sub>J f = f"
N: shows "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < r / sqrt 2"
0: shows "map (int \<xi>) (GE.getTvars (arOf f)) = al"
mult_adjugate_det: fixes A :: "('b, 'c) sq_matrix" shows "A * Square_Matrix.adjugate A = diag (Square_Matrix.det A)"
simps_4: fixes x :: "nat"   and e :: "cexpr"   and oper :: "pdf_operator"   and e' :: "cexpr" shows "cexpr_comp_aux x e (oper $$\<^sub>c e') = oper $$\<^sub>c (cexpr_comp_aux x e e')"
is_path_undir_subgraph: fixes H :: "('v, 'w) graph"   and x :: "'v"   and p :: "('v \<times> 'w \<times> 'v) list"   and y :: "'v" assumes "is_path_undir H x p y"   and "subgraph H G" shows "is_path_undir G x p y"
mk_ps_chain_snoc: fixes \<Pi> :: "(node \<rightharpoonup>\<^sub>f topbot, node + node fset \<times> command_gadget \<times> node fset) chain"   and \<pi> :: "(node + node fset \<times> command_gadget \<times> node fset) list"   and x :: "node + node fset \<times> command_gadget \<times> node fset" shows "mk_ps_chain \<Pi> (\<pi> @ [x]) = cSnoc (mk_ps_chain \<Pi> \<pi>) x (next_ps (post (mk_ps_chain \<Pi> \<pi>)) x)"
RPC_action_defs_6: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState"   and p :: "PrIds" shows "RPCNext send rcv rst p \<equiv> RPCFwd send rcv rst p \<or> RPCReject send rcv rst p \<or> RPCFail send rcv rst p \<or> RPCReply send rcv rst p"
aux_logic: fixes a :: "bool"   and b :: "bool"   and c :: "bool" assumes "a \<Longrightarrow> b"   and "b \<Longrightarrow> a = c" shows "a = (b \<and> c)"
l2_run_ended_trans: fixes s :: "l2_state"   and R :: "fid_t"   and s' :: "l2_state" assumes "run_ended (progress s R)"   and "(s, s') \<in> TS.trans l2" shows "run_ended (progress s' R)"
add2_None: fixes x :: "'b option" shows "total_partial.add2 x None = None"
assms_1: fixes x :: "real" assumes "x \<in> set (map snd xs)" shows "0 \<le> x"
subsetD: fixes A :: "'m set"   and B :: "'m set"   and c :: "'m" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
intros_1: fixes \<tau> :: "'a OCL_Types.type" shows "element_type (Collection \<tau>) \<tau>"
trancl_Image_advance_ss: fixes u :: "'a"   and v :: "'a"   and E :: "('a \<times> 'a) set" assumes "(u, v) \<in> E" shows "E\<^sup>+ `` {v} \<subseteq> E\<^sup>+ `` {u}"
r_def: fixes x :: "(real, 'n) Finite_Cartesian_Product.vec" shows "r x = Min {(A *v x) $h j / x $h j |j. x $h j \<noteq> 0}"
aHom_0_0: fixes F :: "('e, 'f) aGroup_scheme"   and G :: "('g, 'h) aGroup_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "aGroup F"   and "aGroup G"   and "f \<in> aHom F G" shows "f \<zero>\<^bsub>F\<^esub> = \<zero>\<^bsub>G\<^esub>"
image_uminus_atLeast: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "uminus ` ord.atLeast less_eq x = ord.atMost less_eq (uminus x)"
simps: fixes n :: "lname" shows "the_LVar_name (LVar n) = n"
empty_suffix_is_empty: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "emptyseq s" shows "emptyseq (s |\<^sub>s n)"
WHILET_refine: fixes x :: "'c"   and x' :: "'d"   and R :: "('c \<times> 'd) set"   and b :: "'c \<Rightarrow> bool"   and b' :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and f' :: "'d \<Rightarrow> 'd nres" assumes "(x, x') \<in> R"   and "\<And>x x'. (x, x') \<in> R \<Longrightarrow> b x = b' x'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; b x; b' x'\<rbrakk> \<Longrightarrow> f x \<le> \<Down> R (f' x')" shows "WHILE\<^sub>T b f x \<le> \<Down> R (WHILE\<^sub>T b' f' x')"
set_insert: fixes x :: "'b"   and xs :: "'b list" shows "set (List.insert x xs) = insert x (set xs)"
sup_loc_widens_conv: fixes P :: "'a prog"   and Ts :: "ty list"   and Ts' :: "ty list" shows "P \<turnstile> map OK Ts [\<le>\<^sub>\<top>] map OK Ts' = P \<turnstile> Ts [\<le>] Ts'"
distinct_1: fixes x1 :: "'b"   and x2 :: "'a" shows "Var x1 \<noteq> Sym x2"
build_succ_aux1: fixes nds :: "('a, 'b) node_scheme set" assumes "finite nds"   and "\<And>q. q \<in> nds \<Longrightarrow> finite (incoming q)" shows "build_succ nds \<le> SPEC (\<lambda>r. r = (\<lambda>qn. dflt_None_set {qn'. \<exists>q'. q' \<in> nds \<and> qn' = name q' \<and> qn \<in> incoming q' \<and> qn \<noteq> expand_init}))"
less_sort_dir_def: shows "(<) \<equiv> rec_sort_dir (case_sort_dir False True) (case_sort_dir False False)"
tranclp_rtranclp_tranclp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "r\<^sup>+\<^sup>+ a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>+\<^sup>+ a c"