hcju/isabelleps huggingface dataset:
Hintikka_def: fixes S :: "'b formula set" shows "Hintikka S \<equiv> \<bottom> \<notin> S \<and> (\<forall>k. Atom k \<in> S \<longrightarrow> \<^bold>\<not> (Atom k) \<in> S \<longrightarrow> False) \<and> (\<forall>F G. F \<^bold>\<and> G \<in> S \<longrightarrow> F \<in> S \<and> G \<in> S) \<and> (\<forall>F G. F \<^bold>\<or> G \<in> S \<longrightarrow> F \<in> S \<or> G \<in> S) \<and> (\<forall>F G. F \<^bold>\<rightarrow> G \<in> S \<longrightarrow> \<^bold>\<not> F \<in> S \<or> G \<in> S) \<and> (\<forall>F. \<^bold>\<not> (\<^bold>\<not> F) \<in> S \<longrightarrow> F \<in> S) \<and> (\<forall>F G. \<^bold>\<not> (F \<^bold>\<and> G) \<in> S \<longrightarrow> \<^bold>\<not> F \<in> S \<or> \<^bold>\<not> G \<in> S) \<and> (\<forall>F G. \<^bold>\<not> (F \<^bold>\<or> G) \<in> S \<longrightarrow> \<^bold>\<not> F \<in> S \<and> \<^bold>\<not> G \<in> S) \<and> (\<forall>F G. \<^bold>\<not> (F \<^bold>\<rightarrow> G) \<in> S \<longrightarrow> F \<in> S \<and> \<^bold>\<not> G \<in> S)"
residual_conjugate_l: fixes x :: "'a"   and y :: "'a" shows "x \<leftarrow> y = - (- x \<lhd> y)"
bind_if2: fixes m :: "'m"   and b :: "bool"   and t :: "'a \<Rightarrow> 'm"   and e :: "'a \<Rightarrow> 'm" shows "bind m (\<lambda>x. if b then t x else e x) = (if b then bind m t else bind m e)"
cbox_borel: fixes a :: "'c"   and b :: "'c" shows "cbox a b \<in> sets borel"
abel_semigroup_axioms: shows "abel_semigroup normalization_euclidean_semiring_class.gcd"
collect_set_map: fixes h1 :: "'l \<Rightarrow> 'o"   and h2 :: "'m \<Rightarrow> 'o"   and h3 :: "'n \<Rightarrow> 'o" shows "Gset \<circ> Gmap h1 h2 h3 = BNF_Def.collect {(`) h1 \<circ> Gset1, (`) h2 \<circ> Gset2, (`) h3 \<circ> Gset3}"
pcompose_eq_0: fixes p :: "'b poly"   and q :: "'b poly" assumes "p \<circ>\<^sub>p q = 0"   and "0 < degree q" shows "p = 0"
dg_Set_is_arrD_1: fixes \<alpha> :: "V"   and A :: "V"   and B :: "V"   and S :: "V" assumes "S : A \<mapsto>\<^bsub>dg_Set \<alpha>\<^esub> B" shows "arr_Set \<alpha> S"
ff_asm: fixes n :: "nat" assumes "m \<le> n" shows "ff (n + 1) < d (n + 1) * Max (ff ` {s..n})"
le_imp_neg_le: 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 a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less_eq a b" shows "less_eq (uminus b) (uminus a)"
R0n0i_def: shows "R0n0i \<equiv> Id"
bounded_linear_def: fixes f :: "'e \<Rightarrow> 'f" shows "bounded_linear f \<equiv> linear f \<and> bounded_linear_axioms f"
rel_transp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "transp R" shows "transp (rel_bset R)"
four_block_carrier_mat: fixes A :: "'b mat"   and nr1 :: "nat"   and nc1 :: "nat"   and D :: "'b mat"   and nr2 :: "nat"   and nc2 :: "nat"   and B :: "'b mat"   and C :: "'b mat" assumes "A \<in> carrier_mat nr1 nc1"   and "D \<in> carrier_mat nr2 nc2" shows "four_block_mat A B C D \<in> carrier_mat (nr1 + nr2) (nc1 + nc2)"
s_Tr: fixes \<pi>\<^sub>1 :: "Syntax.term list"   and e\<^sub>1 :: "Syntax.term"   and m :: "mode"   and i :: "nat"   and \<pi>\<^sub>2 :: "Syntax.term list"   and e\<^sub>2 :: "Syntax.term"   and \<pi>\<^sub>3 :: "Syntax.term list"   and e\<^sub>3 :: "Syntax.term" assumes "\<lless>\<pi>\<^sub>1, e\<^sub>1\<ggreater> m\<rightarrow>i \<lless>\<pi>\<^sub>2, e\<^sub>2\<ggreater>"   and "\<lless>\<pi>\<^sub>2, e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>\<^sub>3, e\<^sub>3\<ggreater>" shows "\<lless>\<pi>\<^sub>1, e\<^sub>1\<ggreater> m\<rightarrow>i + 1 \<lless>\<pi>\<^sub>3, e\<^sub>3\<ggreater>"
conforms_Err: fixes e :: "error"   and s :: "st"   and G :: "prog"   and L :: "lenv" shows "(Some (Error e), s)\<Colon>\<preceq>(G, L) = Norm s\<Colon>\<preceq>(G, L)"
distinct_concat: fixes xs :: "'c list list" assumes "distinct xs"   and "\<And>ys. ys \<in> set xs \<Longrightarrow> distinct ys"   and "\<And>ys zs. \<lbrakk>ys \<in> set xs; zs \<in> set xs; ys \<noteq> zs\<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}" shows "distinct (concat xs)"
range_to_nat: shows "range Bij_Nat.to_nat = {0..<CARD('a)}"
fixp_mono: fixes Sup :: "'c set \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'c" assumes "class.ccpo Sup less_eq less"   and "monotone less_eq less_eq f"   and "monotone less_eq less_eq g"   and "\<And>Z. less_eq (f Z) (g Z)" shows "less_eq (ccpo.fixp Sup less_eq f) (ccpo.fixp Sup less_eq g)"
mult_mat_vec_def: fixes A :: "'a mat"   and v :: "'a vec" shows "A *\<^sub>v v \<equiv> Matrix.vec (dim_row A) (\<lambda>i. row A i \<bullet> v)"
induct: fixes x :: "'val"   and g :: "'g"   and \<phi> :: "'val"   and P :: "'val \<Rightarrow> bool" assumes "x \<in> reachable g \<phi>"   and "unnecessaryPhi g \<phi> \<Longrightarrow> P \<phi>"   and "\<And>\<phi>'' \<phi>'''. \<lbrakk>\<phi>'' \<in> reachable g \<phi>; P \<phi>''; phiArg g \<phi>'' \<phi>'''; unnecessaryPhi g \<phi>'''\<rbrakk> \<Longrightarrow> P \<phi>'''" shows "P x"
iso_verts_iso_inv: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism"   and u :: "'e" assumes "H.digraph_isomorphism hom"   and "u \<in> verts (app_iso hom (app_iso hom G))" shows "iso_verts hom (iso_verts (H.inv_iso hom) u) = u"
in_BC: shows "U\<^sub>0 \<union> V\<^sub>0 \<in> BC"
of_nat_less_iff: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m < n)"
eq_upred_sym: fixes x :: "('b, 'a) uexpr"   and y :: "('b, 'a) uexpr" shows "(x =\<^sub>u y) = (y =\<^sub>u x)"
finite_pdevs_domain: fixes x :: "'a pdevs" shows "finite (pdevs_domain x)"
bad_init_spec: shows "test_owner \<notin> bad_init \<and> test_partner \<notin> bad_init"
s_type_unfold_3: fixes \<S> :: "s_context"   and rs :: "t list option"   and i :: "nat"   and vs :: "v list"   and es :: "e list"   and ts :: "t list" assumes "\<S>\<bullet>rs \<tturnstile>_ i vs;es : ts" shows "\<S>\<bullet>(s_inst \<S> ! i)\<lparr>local := local (s_inst \<S> ! i) @ map typeof vs, return := rs\<rparr> \<turnstile> es : [] _> ts"
assms: shows "(x, y) \<in> (Restr r A)\<^sup>*"
return_pmf_bind_option: fixes x :: "'d option"   and f :: "'d \<Rightarrow> 'c option" shows "return_pmf (x \<bind> f) = return_pmf x \<bind> return_pmf \<circ> f"
has_limits_of_shape_def: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "J.has_limits_of_shape J \<equiv> \<forall>D. diagram J (\<cdot>\<^sub>J) D \<longrightarrow> (\<exists>a \<chi>. limit_cone J (\<cdot>\<^sub>J) D a \<chi>)"
coinduction: fixes R :: "'s \<Rightarrow> 's \<Rightarrow> bool"   and s1 :: "'s"   and s2 :: "'s" assumes "bisimulation R"   and "wf_state s1"   and "wf_state s2"   and "R s1 s2" shows "L s1 = L s2"
pequiv_pr_trans: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and c :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "a \<simeq> b"   and "b \<sqsubseteq> c" shows "a \<sqsubseteq> c"
assms: shows "\<forall>i j k. f (i, j) = f (i, k)"
nonpart_implies_proper_product: fixes x :: "i"   and y :: "i" assumes "\<not> P x y \<and> O x y" shows "PP (x \<otimes> y) x"
is_lb_def: fixes s :: "'a"   and s1 :: "'a"   and s2 :: "'a" shows "is_lb s s1 s2 \<equiv> s \<preceq> s2 \<and> s \<preceq> s1"
that: fixes as :: "'a" assumes "R as cs"   and "A.reachable as" shows "thesis"
prems_3: shows "InvariantEquivalentZL (getF state') (getM state') Phi"
closed_seqs_memE: fixes s :: "nat \<Rightarrow> 'c"   and R :: "('c, 'd) ring_scheme"   and k :: "nat" assumes "s \<in> carrier R\<^bsup>\<omega>\<^esup>" shows "s k \<in> carrier R"
assms: shows "C\<^sub>1 \<subseteq> C\<^sub>2"
cols_transpose: fixes A :: "'a mat" shows "cols A\<^sup>T = rows A"
cfcomp1: fixes f :: "'t \<rightarrow> 's"   and g :: "'r \<rightarrow> 't" shows "f oo g = (\<Lambda> x. f\<cdot>(g\<cdot>x))"
encode_Suc_exp: fixes base :: "nat"   and e :: "nat"   and n :: "nat" assumes "2 \<le> base" shows "encode base (Suc e) n = encode base e (base * n)"
s0_secrecy: shows "reach s0 \<subseteq> s0_secrecy"
monad_prob_stateT: fixes return :: "'a \<times> 'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'c"   and sample :: "'d pmf \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_prob return bind sample" shows "monad_prob (return_state return) (bind_state bind) (sample_state sample)"
assms_2: fixes u :: "'a" assumes "u \<in> S" shows "gauge (f u)"
arrow_from_functor_axioms: shows "arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) \<Delta>.map (G (J_C.MkIde D)) (J_C.MkIde D) (Adj.\<epsilon> (J_C.MkIde D))"
simps_16: shows "extN.Top + extN.Top = extN.N 2"
hd_cong: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and t :: "pdf_type"   and f' :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal" assumes "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f"   and "density_context V V' \<Gamma> \<delta>"   and "\<Gamma> \<turnstile> e : t"   and "free_vars e \<subseteq> V \<union> V'"   and "\<And>\<rho> x. \<lbrakk>\<rho> \<in> space (state_measure V' \<Gamma>); x \<in> space (stock_measure t)\<rbrakk> \<Longrightarrow> f \<rho> x = f' \<rho> x" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f'"
Cons_eq_filterD: fixes x :: "'c"   and xs :: "'c list"   and P :: "'c \<Rightarrow> bool"   and ys :: "'c list" assumes "x # xs = filter P ys" shows "\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
real_powr_at_top_neg: fixes a :: "real" assumes "0 < a"   and "a < 1" shows "((powr) a \<longlongrightarrow> 0) at_top"
is_rewrite_ord_def: fixes rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool" shows "is_rewrite_ord rword = (reflp rword \<and> transp rword \<and> (\<forall>a b. rword a b \<or> rword b a) \<and> (\<forall>a b. rword a b \<longrightarrow> rword b a \<longrightarrow> fst a = fst b) \<and> (\<forall>d G a b. dickson_grading d \<longrightarrow> is_sig_GB_upt d G (lt b) \<longrightarrow> a \<in> G \<longrightarrow> b \<in> G \<longrightarrow> a \<noteq> 0 \<longrightarrow> b \<noteq> 0 \<longrightarrow> lt a adds\<^sub>t lt b \<longrightarrow> \<not> is_sig_red (\<prec>\<^sub>t) (=) G b \<longrightarrow> rword (spp_of a) (spp_of b)))"
pirreducibleE_2: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "pirreducible\<^bsub>K [X]\<^esub> K p" shows "p \<notin> Units (K [X]\<^bsub>K [X]\<^esub>)"
antisym_def: fixes r :: "('d \<times> 'd) set" shows "antisym r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"
h': shows "h2 \<turnstile> set_child_nodes parent (remove1 child ptr_children) \<rightarrow>\<^sub>h h'"
zless_imp_add1_zle: fixes w :: "int"   and z :: "int" assumes "w < z" shows "w + 1 \<le> z"
always_induct: fixes P :: "(nat \<Rightarrow> 'g) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'g" assumes "P \<sigma>"   and "(\<box>(\<lambda>s. P s \<longrightarrow> (\<circle>P) s)) \<sigma>" shows "(\<box>P) \<sigma>"
cases: fixes a1 :: "'b nmultiset"   and a2 :: "'b nmultiset" assumes "(a1, a2) \<in> sub_nmset" obtains X :: "'b nmultiset" and M :: "'b nmultiset multiset" where "a1 = X" and "a2 = MSet M" and "X \<in># M"
chamber_in_image: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and y :: "'c set" assumes "ChamberComplexMorphism X Y f"   and "y \<in> f \<turnstile> X"   and "SimplicialComplex.maxsimp Y y" shows "SimplicialComplex.maxsimp (f \<turnstile> X) y"
subtopologyE: fixes s :: "'a set" assumes "s open" obtains t :: "'a set" where "t open\<^bsub>T\<^esub>" and "s = t \<inter> carrier"
sorted_insort_key: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (map f (linorder.insort_key less_eq f x xs)) = linorder.sorted less_eq (map f xs)"
post_cond_def: fixes a :: "nat list"   and b :: "nat list" shows "post_cond a b = (\<lambda>(x, y). static_bounds a b x y \<and> a \<bullet> x = b \<bullet> y \<and> boundr_impl a b x y)"
is_expansion_zero: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and basis :: "(real \<Rightarrow> real) list" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "basis_wf basis"   and "length basis = expansion_level TYPE('a)" shows "is_expansion zero_expansion basis"
execute_parallel_operator_effect_3: fixes s :: "'a \<Rightarrow> bool option"   and ops :: "'a strips_operator list"   and v :: "'a" assumes "are_all_operators_applicable s ops"   and "are_all_operator_effects_consistent ops" shows "(\<forall>op\<in>set ops. v \<notin> set (add_effects_of op) \<and> v \<notin> set (delete_effects_of op)) \<longrightarrow> execute_parallel_operator s ops v = s v"
Inf_aform_def: fixes X :: "'b \<times> 'b pdevs" shows "Inf_aform X = fst X - Radius X"
vimage_singleton_eq: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b"   and b :: "'b" shows "(a \<in> f -` {b}) = (f a = b)"
HOinitConfig_def: fixes A :: "('a, 'b, 'c, 'd) CHOAlgorithm_scheme"   and cfg :: "'a \<Rightarrow> 'b" shows "HOinitConfig A cfg \<equiv> CHOinitConfig A cfg (\<lambda>q. undefined)"
cases_scheme: fixes r :: "'a s_context_scheme" obtains s_inst :: "t_context list" and s_funcs :: "tf list" and s_tab :: "nat list" and s_mem :: "nat list" and s_globs :: "tg list" and more :: "'a" where "r = \<lparr>s_inst = s_inst, s_funcs = s_funcs, s_tab = s_tab, s_mem = s_mem, s_globs = s_globs, \<dots> = more\<rparr>"
fv_eq_FunI: fixes T :: "('a, 'b) Term.term list"   and S :: "('c, 'b) Term.term list"   and f :: "'a"   and g :: "'c" assumes "length T = length S"   and "\<And>i. i < length T \<Longrightarrow> fv (T ! i) = fv (S ! i)" shows "fv (Fun f T) = fv (Fun g S)"
SCnotConc: fixes a :: "('a, 'b) Combinators"   and b :: "('a, 'b) Combinators"   and p :: "('a, 'b) Combinators list" assumes "a \<oplus> b \<in> set p"   and "singleCombinators p" shows "False"
ln_gt_zero: fixes x :: "real" assumes "1 < x" shows "0 < ln x"
proj_psi_is_proj_psi_l0: shows "proj_psi = proj_psi_l 0"
poly_add_coeff: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "R.coeff (R.poly_add p1 p2) = (\<lambda>i. R.coeff p1 i \<oplus> R.coeff p2 i)"
split_list: fixes A :: "'b list"   and x :: "'b"   and X :: "'b set" assumes "set A \<subseteq> {x} \<union> X"   and "x \<in>. A" shows "\<exists>B. set (x # B) = set A \<and> x \<notin> set B"
ts_lose_ack_st2vc_com: fixes lose :: "'a \<Rightarrow> bool list"   and t :: "'a"   and ack :: "'a \<Rightarrow> 'b list"   and x :: "'b"   and i1 :: "'a \<Rightarrow> 'c list"   and vc :: "'a \<Rightarrow> vcType list"   and st_out :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "lose t = [True] \<or> lose t = [False]"   and "lose t = [True] \<longrightarrow> ack t = [x] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = x"   and "lose t = [False] \<longrightarrow> ack t = [y] \<and> i1 t = [] \<and> vc t = [vc_com] \<and> st_out t = y" shows "ack t = [st_out t]"
in_quotient_imp_closed: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and X :: "'c set"   and x :: "'c"   and y :: "'c" assumes "equiv A r"   and "X \<in> A // r"   and "x \<in> X"   and "(x, y) \<in> r" shows "y \<in> X"
induct: fixes x1 :: "state"   and x2 :: "com"   and x3 :: "nat"   and x4 :: "state"   and P :: "state \<Rightarrow> com \<Rightarrow> nat \<Rightarrow> state \<Rightarrow> bool" assumes "x1 -x2-x3\<rightarrow> x4"   and "\<And>t f s n. t \<in> f s \<Longrightarrow> P s (Do f) n t"   and "\<And>s0 c1 n s1 c2 s2. \<lbrakk>s0 -c1-n\<rightarrow> s1; P s0 c1 n s1; s1 -c2-n\<rightarrow> s2; P s1 c2 n s2\<rbrakk> \<Longrightarrow> P s0 (c1; c2) n s2"   and "\<And>b s c1 n t c2. \<lbrakk>b s; s -c1-n\<rightarrow> t; P s c1 n t\<rbrakk> \<Longrightarrow> P s (IF b THEN c1 ELSE c2) n t"   and "\<And>b s c2 n t c1. \<lbrakk>\<not> b s; s -c2-n\<rightarrow> t; P s c2 n t\<rbrakk> \<Longrightarrow> P s (IF b THEN c1 ELSE c2) n t"   and "\<And>b s c n. \<not> b s \<Longrightarrow> P s (WHILE b DO c) n s"   and "\<And>b s c n t u. \<lbrakk>b s; s -c-n\<rightarrow> t; P s c n t; t -WHILE b DO c-n\<rightarrow> u; P t (WHILE b DO c) n u\<rbrakk> \<Longrightarrow> P s (WHILE b DO c) n u"   and "\<And>s n t. \<lbrakk>s -body-n\<rightarrow> t; P s body n t\<rbrakk> \<Longrightarrow> P s CALL (Suc n) t"   and "\<And>f s c n t g. \<lbrakk>f s -c-n\<rightarrow> t; P (f s) c n t\<rbrakk> \<Longrightarrow> P s (LOCAL f; c; g) n (g s t)" shows "P x1 x2 x3 x4"
bintrunc_mod2p: fixes n :: "nat"   and w :: "int" shows "take_bit n w = w mod 2 ^ n"
map_simp: fixes f :: "'j" assumes "\<chi>.J.arr f" shows "Fa.map f = F a"
mset_del_min: fixes insert :: "'b \<Rightarrow> 'b tree \<Rightarrow> 'b tree"   and del_min :: "'b tree \<Rightarrow> 'b tree"   and q :: "'b tree" assumes "Heap insert del_min"   and "heap q"   and "q \<noteq> \<langle>\<rangle>" shows "mset_tree (del_min q) = mset_tree q - {#value q#}"
incl_Rel_def: fixes A :: "V"   and B :: "V" shows "incl_Rel A B = [vid_on A, A, B]\<^sub>\<circ>"
prv_prv_neg_imp_neg: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "bprv \<phi>"   and "bprv \<chi>" shows "bprv (neg (imp \<phi> (neg \<chi>)))"
design_isomorphism_def: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd" shows "design_isomorphism \<V> \<B> \<V>' \<B>' bij_map \<equiv> incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map \<and> design \<V> \<B> \<and> design \<V>' \<B>'"
distinct_29: fixes x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'a" shows "x31\<not>x32:arrive(x33) \<noteq> \<tau>\<^sub>n"
induct: fixes valid_edge :: "'g \<Rightarrow> bool"   and src :: "'g \<Rightarrow> 'h"   and trg :: "'g \<Rightarrow> 'h"   and knd :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and E :: "'h"   and X :: "'h"   and x :: "'h LDCFG_node \<times> ('i, 'j, 'k, 'l) edge_kind \<times> 'h LDCFG_node"   and P :: "'h LDCFG_node \<times> ('i, 'j, 'k, 'l) edge_kind \<times> 'h LDCFG_node \<Rightarrow> bool" assumes "lift_valid_edge valid_edge src trg knd E X x"   and "\<And>a e. \<lbrakk>valid_edge a; src a \<noteq> E \<or> trg a \<noteq> X; e = (Node (src a), knd a, Node (trg a))\<rbrakk> \<Longrightarrow> P e"   and "\<And>e. e = (NewEntry, (\<lambda>s. True)\<^sub>\<surd>, Node E) \<Longrightarrow> P e"   and "\<And>e. e = (Node X, (\<lambda>s. True)\<^sub>\<surd>, NewExit) \<Longrightarrow> P e"   and "\<And>e. e = (NewEntry, (\<lambda>s. False)\<^sub>\<surd>, NewExit) \<Longrightarrow> P e" shows "P x"
strong_rev_children: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.tree_invar t" shows "SkewBinomialHeapStruc.invar (rev (filter (\<lambda>t. 0 < SkewBinomialHeapStruc.rank t) (SkewBinomialHeapStruc.children t)))"
prj_chine_2: shows "THfgh.p\<^sub>1 \<star> TTfgh_THfgh.chine \<cong> Tfg_Hfg.chine \<star> TTfgh.p\<^sub>1"
subcategory_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool" shows "subcategory C Arr \<equiv> category C \<and> subcategory_axioms C Arr"
non_terminating_F: fixes P :: "'d process" shows "non_terminating P = CHAOS UNIV \<sqsubseteq>\<^sub>F P"
sum_subset_iff: fixes A :: "hf"   and B :: "hf"   and C :: "hf"   and D :: "hf" shows "(A + B \<le> C + D) = (A \<le> C \<and> B \<le> D)"
lookup_of_nat: fixes n :: "nat"   and k :: "'r" shows "lookup (of_nat n) k = (of_nat n when k = (0::'r))"
hyps_4: shows "sh''_ = sh'(sh'' \<mapsto> (C, Error))"
two_in_Vset: fixes x :: "V"   and y :: "V" obtains \<alpha> :: "V" where "x \<in> elts (Vset \<alpha>)" and "y \<in> elts (Vset \<alpha>)"
Atm_transC_simp: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and atm :: "'c"   and s :: "'b"   and cf :: "('a, 'c) com \<times> 'b" shows "\<not> PL.transC tval aval (Atm atm, s) cf"
ort: shows "orthogonal \<equiv> \<lambda>A B. fst A * fst B + snd A * snd B = 0"
conforms_lupd: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lenv"   and vn :: "lname"   and T :: "ty"   and v :: "val" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "L vn = Some T"   and "G,s\<turnstile>v\<Colon>\<preceq>T" shows "(x, lupd(vn\<mapsto>v) s)\<Colon>\<preceq>(G, L)"
qmax: fixes k :: "nat" shows "\<bar>poly.coeff q k\<bar> \<le> qmax"
increasing4: fixes as :: "'a list"   and bs :: "'a list" assumes "increasing (as @ bs)" shows "increasing as"
induct: fixes x :: "'a pfm"   and P :: "'a pfm \<Rightarrow> bool" assumes "\<turnstile>\<^sub>! x"   and "\<And>p. ptautology p \<Longrightarrow> P p"   and "\<And>i p q. P (K\<^sub>! i p \<^bold>\<and>\<^sub>! K\<^sub>! i (p \<^bold>\<longrightarrow>\<^sub>! q) \<^bold>\<longrightarrow>\<^sub>! K\<^sub>! i q)"   and "\<And>p q. \<lbrakk>\<turnstile>\<^sub>! p; P p; \<turnstile>\<^sub>! (p \<^bold>\<longrightarrow>\<^sub>! q); P (p \<^bold>\<longrightarrow>\<^sub>! q)\<rbrakk> \<Longrightarrow> P q"   and "\<And>p i. \<lbrakk>\<turnstile>\<^sub>! p; P p\<rbrakk> \<Longrightarrow> P (K\<^sub>! i p)"   and "\<And>p r. \<lbrakk>\<turnstile>\<^sub>! p; P p\<rbrakk> \<Longrightarrow> P ([r]\<^sub>! p)"   and "\<And>r. P ([r]\<^sub>! \<^bold>\<bottom>\<^sub>! \<^bold>\<longleftrightarrow>\<^sub>! r \<^bold>\<longrightarrow>\<^sub>! \<^bold>\<bottom>\<^sub>!)"   and "\<And>r x. P ([r]\<^sub>! Pro\<^sub>! x \<^bold>\<longleftrightarrow>\<^sub>! r \<^bold>\<longrightarrow>\<^sub>! Pro\<^sub>! x)"   and "\<And>r p q. P ([r]\<^sub>! (p \<^bold>\<or>\<^sub>! q) \<^bold>\<longleftrightarrow>\<^sub>! [r]\<^sub>! p \<^bold>\<or>\<^sub>! [r]\<^sub>! q)"   and "\<And>r p q. P ([r]\<^sub>! (p \<^bold>\<and>\<^sub>! q) \<^bold>\<longleftrightarrow>\<^sub>! [r]\<^sub>! p \<^bold>\<and>\<^sub>! [r]\<^sub>! q)"   and "\<And>r p q. P ([r]\<^sub>! (p \<^bold>\<longrightarrow>\<^sub>! q) \<^bold>\<longleftrightarrow>\<^sub>! [r]\<^sub>! p \<^bold>\<longrightarrow>\<^sub>! [r]\<^sub>! q)"   and "\<And>r i p. P ([r]\<^sub>! K\<^sub>! i p \<^bold>\<longleftrightarrow>\<^sub>! r \<^bold>\<longrightarrow>\<^sub>! K\<^sub>! i ([r]\<^sub>! p))" shows "P x"