hcju/isabelleps huggingface dataset:
sel_4: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "'c"   and x4 :: "'a" shows "tgtOf (Trans x1 x2 x3 x4) = x4"
measurable_in_subalg: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and N :: "'c measure" assumes "subalgebra M F"   and "f \<in> F \<rightarrow>\<^sub>M N" shows "f \<in> restr_to_subalg M F \<rightarrow>\<^sub>M N"
abs_eq_iff: fixes x :: "'a ltln"   and y :: "'a ltln" shows "(abs_ltln\<^sub>L x = abs_ltln\<^sub>L y) = x \<sim>\<^sub>L y"
ctx_cld_ctx_dcl: fixes ctx :: "'a"   and dcl :: "char list"   and cl :: "cl"   and fds :: "fd list"   and mds :: "meth_def list"   and path :: "('a \<times> cld) list" assumes "(ctx, cld_def dcl cl fds mds) \<in> set path" shows "(ctx, dcl) \<in> (\<lambda>(ctx, cld). (ctx, class_name_f cld)) ` set path"
lim: shows "AE x in M. (\<lambda>i. f i x) \<longlonglongrightarrow> u x"
ap_gpv_interchange: fixes u :: "('i \<Rightarrow> 'f, 'g, 'h) gpv"   and x :: "'i" shows "u \<diamondop> Generative_Probabilistic_Value.Done x = Generative_Probabilistic_Value.Done (\<lambda>f. f x) \<diamondop> u"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_val type_wf set_val set_val_locs"   and "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"   and "l_set_val_get_disconnected_nodes_axioms set_val_locs get_disconnected_nodes_locs" shows "l_set_val_get_disconnected_nodes type_wf set_val set_val_locs get_disconnected_nodes get_disconnected_nodes_locs"
contract_conj_overlap: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A" shows "(K \<div> \<phi>) \<inter> (K \<div> \<psi>) \<subseteq> K \<div> (\<phi> .\<and>. \<psi>)"
basis_reduction_swap_def: fixes i :: "nat"   and fs :: "'a list" shows "local.basis_reduction_swap i fs = (False, i - 1, fs[i := fs ! (i - 1), i - 1 := fs ! i])"
v0_notin_S: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and S :: "'c set" assumes "Separation G v0 v1 S" shows "v0 \<notin> S"
starfunNat_real_of_nat: shows "*f* real = hypreal_of_hypnat"
bisimSubstResComm: fixes \<Psi> :: "'b"   and x :: "name"   and y :: "name"   and P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>y\<rparr>P) \<sim>\<^sub>s \<lparr>\<nu>y\<rparr>(\<lparr>\<nu>x\<rparr>P)"
Bell_altdef: fixes A :: "'b set" assumes "finite A" shows "Bell (card A) = card {P. partition_on A P}"
Poly_coeffs: fixes p :: "'b poly" shows "Poly (coeffs p) = p"
all_not_zero: shows "\<forall>n. \<not> is_zero_row_upt_k n k A"
MPLS_s_def: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap" shows "MPLS_s PROB s \<equiv> (\<lambda>(s, as). Inf (PLS s as)) ` {(s, as) |as. as \<in> valid_plans PROB}"
sqrt_aform_err_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real" shows "sqrt_aform_err p X = (let ivl = ivl_of_aform_err p X; l = lower ivl; u = upper ivl in if 0 < l then min_range_mono p floatarith.Sqrt (\<lambda>x. Half (floatarith.Inverse (floatarith.Sqrt x))) l u X else Some (ivl_err (real_interval (sqrt_float_interval p ivl))))"
fps_mult_fps_X_deriv_shift: fixes a :: "'a fps" shows "fps_X * fps_deriv a = Abs_fps (\<lambda>n. of_nat n * fps_nth a n)"
Cons_1: fixes j :: "nat"   and i :: "nat" assumes "j \<le> i"   and "i < length xs" shows "xs ! i \<in> set (drop j xs)"
Q1: shows "Coplanar A B C X \<and> Bet P X S"
countable_Diff_eq: fixes A :: "'a set"   and x :: "'a" shows "countable (A - {x}) = countable A"
a1: shows "\<forall>A\<in>W. \<forall>a1\<in>A. \<forall>a2\<in>A. same_fw_behaviour_one a1 a2 c rs"
spr_simRels_def: shows "spr_simRels \<equiv> \<lambda>a. rel_ext (\<lambda>uu_. \<exists>s s'. uu_ = (s, s') \<and> envObs a (sprFst s) = envObs a (sprFst s') \<and> envObs a (sprLst s) = envObs a (sprLst s') \<and> sprCRel s = sprCRel s')"
to_polynomial_def: fixes R :: "('c, 'd) ring_scheme" shows "to_polynomial R = (\<lambda>a. monom (UP R) a 0)"
Segment_A: fixes C :: "Class"   and m :: "Method"   and l :: "int"   and l1 :: "int"   and code :: "(int \<times> Instr) list"   and ll :: "int" assumes "Segment C m l l1 code"   and "l \<le> ll"   and "ll < l1" shows "Segment C m ll l1 code"
lincomb_restrict: fixes U :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and b :: "'c \<Rightarrow> 'a" assumes "U \<subseteq> carrier V"   and "a \<in> U \<rightarrow> carrier K"   and "restrict a U = restrict b U" shows "lincomb a U = lincomb b U"
poly_roots_finite: fixes p :: "'c poly" assumes "p \<noteq> 0" shows "finite {x. poly p x = (0::'c)}"
obj_simps_3: fixes a :: "'b" assumes "B.obj a" shows "trg\<^sub>B a = a"
lincomb_list_carrier: fixes vs :: "'a vec list"   and c :: "nat \<Rightarrow> 'a" assumes "set vs \<subseteq> carrier_vec n" shows "lincomb_list c vs \<in> carrier_vec n"
ennreal_mult_strict_left_mono: fixes a :: "ennreal"   and c :: "ennreal"   and b :: "ennreal" assumes "a < c"   and "0 < b"   and "b < top" shows "b * a < b * c"
cases: fixes x :: "char list" obtains "x = []"   | c :: "char" and cs :: "char list" where "x = c # cs"
Inverse_relE: fixes f :: "'b"   and g :: "'b" assumes "cinv f g" obtains "f \<approx>> g" and "f ;; g = id (dom f)" and "g ;; f = id (cod f)"
simps_2: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and uu :: "'a"   and uv :: "'a list" shows "lex_ext_unbounded f (uu # uv) [] = (True, True)"
leq_trans: fixes x :: "'g +\<^sub>\<bottom> 'h"   and y :: "'g +\<^sub>\<bottom> 'h"   and z :: "'g +\<^sub>\<bottom> 'h" assumes "sum_bot_ord x y"   and "sum_bot_ord y z" shows "sum_bot_ord x z"
inject: fixes x1 :: "'g"   and x2 :: "'h"   and y1 :: "'g"   and y2 :: "'h" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
tabulation_data_axioms: shows "tabulation_data (\<cdot>) (\<star>) \<a> \<i> src trg r ((r \<star> \<phi>) \<cdot> \<rho>) f' g"
magQ_qtimes: fixes x :: "'a['b, 'd]"   and y :: "'a['c, 'd]" shows "\<lbrakk>x \<^bold>\<cdot> y\<rbrakk>\<^sub>Q = \<lbrakk>x\<rbrakk>\<^sub>Q \<cdot> \<lbrakk>y\<rbrakk>\<^sub>Q"
nrows_eq_card_rows: fixes A :: "(('a, 'b) vec, 'c) vec" shows "nrows_iarray (matrix_to_iarray A) = CARD('c)"
is_extensional: fixes f :: "'d" assumes "\<not> D.arr f" shows "\<I>\<^sub>1 \<otimes>\<^sub>D f = D.null"
map_ds_map_SSup_iff: fixes f :: "'a downset \<Rightarrow> 'b downset"   and g :: "'a \<Rightarrow> 'b" shows "(f = map_ds g) = (map_SSup f = g)"
is_ring_morphism: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and U :: "'d set"   and V :: "'d set" assumes "presheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"   and "is_open U"   and "is_open V"   and "V \<subseteq> U" shows "ring_homomorphism (\<rho> U V) (\<FF> U) (add_str U) (mult_str U) (zero_str U) (one_str U) (\<FF> V) (add_str V) (mult_str V) (zero_str V) (one_str V)"
subtermeq_set_if_vars: fixes x :: "'c"   and M :: "('d, 'c) Term.term set" assumes "x \<in> fv\<^sub>s\<^sub>e\<^sub>t M" shows "Var x \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M"
assms_2: fixes A :: "'a set" assumes "A \<in> sets M"   and "\<forall>n>0. (T ^^ n) -` A \<inter> A = {}" shows "A \<in> null_sets M"
elims_2: fixes x :: "('form \<times> nat list) \<times> ('form, 'var) antecedent" assumes "valid_in_port x" obtains v :: "'form \<times> nat list" and p :: "('form, 'var) antecedent" where "x = (v, p)" and "v |\<in>| vertices \<and> p |\<in>| inPorts (nodeOf v)"
assms_4: shows "2 ^ card (fmdom' s) - 1 < length as"
assms: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1 @ [Rule m Log] @ rs\<^sub>2, s\<rangle> \<Rightarrow> t"
COUPLING': shows "coupling ((relax_outgoing' u du (M_lookup V) (Q_lookup Q))(u := None)) (M_lookup V(u \<mapsto> du)) (relax_outgoing u D) (Set.insert u S)"
unique_root_lr: fixes plr :: "int poly \<times> rat \<times> rat" assumes "\<exists>!x. root_cond plr x" shows "rai_lb plr \<le> rai_ub plr"
assms: assumes "0 < degree p" shows "\<exists>q r. degree q < degree p \<and> degree r < degree p \<and> p = q * r"
cod_MkArr: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'a"   and B :: "'a"   and F :: "'b" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) (concrete_category.MkArr A B F)" shows "partial_magma.cod (concrete_category.COMP Obj Hom Comp) (concrete_category.MkArr A B F) = concrete_category.MkArr B B (Id B)"
disjoint_correct: fixes s1 :: "'a dlist"   and s2 :: "('a, unit) HashMap.hashmap" assumes "ls.invar s1"   and "hs.invar s2" shows "lh_set_xx.g_disjoint s1 s2 = (ls.\<alpha> s1 \<inter> hs.\<alpha> s2 = {})"
order_refl: fixes less_eq :: "'m \<Rightarrow> 'm \<Rightarrow> bool"   and less :: "'m \<Rightarrow> 'm \<Rightarrow> bool"   and x :: "'m" assumes "class.preorder less_eq less" shows "less_eq x x"
evaldjf_def: fixes f :: "'b \<Rightarrow> fm"   and ps :: "'b list" shows "evaldjf f ps \<equiv> foldr (djf f) ps F"
bind_commute_spmf: fixes p :: "'d spmf"   and q :: "'e spmf"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c spmf" shows "p \<bind> (\<lambda>x. q \<bind> f x) = q \<bind> (\<lambda>y. p \<bind> (\<lambda>x. f x y))"
run_snth: fixes A :: "('c, 'd) nba"   and r :: "('c \<times> 'd) stream"   and p :: "'d"   and k :: "nat" assumes "run A r p" shows "fst (r !! k) \<in> alphabet A \<and> snd (r !! k) \<in> transition A (fst (r !! k)) (target (stake k r) p)"
decompose_fragments: fixes c :: "('g, 'h, 'i, 'j) com"   and ictxt :: "('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com"   and fctxt :: "('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com list"   and c0 :: "('g, 'h, 'i, 'j) com"   and cs :: "('g, 'h, 'i, 'j) com list" assumes "(c, ictxt, fctxt) \<in> decompose_com c0" shows "(c, loc_compC c (fctxt c @ cs)) \<in> fragments c0 (atCs cs)"
quasi_inversesI: fixes f :: "'a"   and g :: "'a" assumes "B.ide f"   and "B.ide g"   and "src f \<cong>\<^sub>B g \<star> f"   and "f \<star> g \<cong>\<^sub>B trg f" shows "B.quasi_inverses f g"
inorder_delete: fixes t :: "('c \<times> 'd) tree"   and x :: "'c" assumes "sorted1 (inorder t)" shows "inorder (Tree_Map.delete x t) = AList_Upd_Del.del_list x (inorder t)"
simps_2: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and lt :: "dag"   and a :: "ref"   and rt :: "dag" shows "Dag p l r (Node lt a rt) = (p = a \<and> p \<noteq> Null \<and> Dag (l p) l r lt \<and> Dag (r p) l r rt)"
fun_top: shows "fun \<top> = (\<lambda>x. \<top>)"
gt_compat_fun: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   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 t' :: "('b, 'a) tm"   and t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "epo ground_heads_var gt_sym arity_sym arity_var extf"   and "epo.gt ground_heads_var gt_sym extf t' t" shows "epo.gt ground_heads_var gt_sym extf (App s t') (App s t)"
intros_4: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set" assumes "\<^bold>\<not> F \<triangleright> \<Gamma> \<turnstile> \<bottom>" shows "\<Gamma> \<turnstile> F"
circline_type: fixes H :: "circline" shows "circline_type H = - 1 \<or> circline_type H = 0 \<or> circline_type H = 1"
extend_unless: fixes F :: "'a program"   and C :: "'c set"   and A :: "'a set"   and B :: "'a set" assumes "extend h F \<in> stable C"   and "F \<in> A unless B" shows "extend h F \<in> C \<inter> extend_set h A unless extend_set h B"
rel_flip: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "('e, 'c) fmap"   and b :: "('e, 'd) fmap" shows "fmrel R\<inverse>\<inverse> a b = fmrel R b a"
diff_add_cancel: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "plus (minus a b) b = a"
assms_1: fixes x :: "nat" shows "eval r_phi [i, x] = eval f [x]"
BB_def: shows "BB \<equiv> \<lambda>j0 j K. list_of (a j0 \<union> (\<Union>i<j. b (enum K i) (j0, i)))"
simps_2: fixes G :: "prog"   and I :: "qtname"   and dynC :: "qtname"   and static_membr :: "bool" shows "G,IfaceT I \<turnstile> dynC valid_lookup_cls_for static_membr = (if static_membr then dynC = Object else G\<turnstile>Class dynC\<preceq>Iface I)"
eval_in_hom: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide b"   and "ide c" shows "\<guillemotleft>eval b c : HF'.prod (exp b c) b \<rightarrow> c\<guillemotright>"
galois_bijection_def: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" shows "galois_bijection G \<equiv> connection G \<and> galois_bijection_axioms G"
orthonormal_partial_sum_diff: fixes S :: "real set"   and w :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and I :: "nat set"   and a :: "nat \<Rightarrow> real" assumes "orthonormal_system S w"   and "\<And>i. w i square_integrable S"   and "f square_integrable S"   and "finite I" shows "(l2norm S (\<lambda>x. f x - (\<Sum>i\<in>I. a i * w i x)))\<^sup>2 = (l2norm S f)\<^sup>2 + (\<Sum>i\<in>I. (a i)\<^sup>2) - 2 * (\<Sum>i\<in>I. a i * orthonormal_coeff S w f i)"
cyclicI: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" assumes "ccw' p q r" shows "ccw' q r p"
UN_extend_simps_1: fixes a :: "'a"   and B :: "'b \<Rightarrow> 'a set"   and C :: "'b set" shows "insert a (\<Union> (B ` C)) = (if C = {} then {a} else \<Union>x\<in>C. insert a (B x))"
simps_2: fixes n :: "nat"   and p :: "nat"   and ps :: "nat list" shows "check_prime_factors_subset n (p # ps) = (if n = 0 then False else if 1 < p \<and> p dvd n then check_prime_factors_subset (n div p) (p # ps) else check_prime_factors_subset n ps)"
delete_first_eq_none: fixes P :: "'a \<Rightarrow> bool"   and l :: "'a list" shows "(delete_first P l = None) = (\<forall>x\<in>set l. \<not> P x)"
unity_def: shows "\<I>\<^sub>D \<equiv> D.cod \<iota>\<^sub>D"
cases: fixes x :: "nat \<times> nat" obtains y :: "nat" and xa :: "nat" where "x = (y, xa)"
extend_act_def: fixes h :: "'d \<times> 'f \<Rightarrow> 'e" shows "extend_act h = (\<lambda>act. \<Union>(s, s')\<in>act. \<Union>y. {(h (s, y), h (s', y))})"
transfer_lookup: shows "rel_fun (=) (local.rel_state (=)) lookup' lookup"
functor_axioms: shows "functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) (\<lambda>f. f \<otimes>\<^sub>C C.cod \<iota>\<^sub>C)"
simps: fixes \<delta> :: "('d \<times> 'e \<times> 'd) set"   and q\<^sub>0 :: "'d"   and \<alpha> :: "(('d \<times> 'e \<times> 'd) set \<times> ('d \<times> 'e \<times> 'd) set set) set"   and w :: "nat \<Rightarrow> 'e" shows "accept\<^sub>G\<^sub>R_LTS (\<delta>, q\<^sub>0, \<alpha>) w = (\<exists>(Fin, Inf)\<in>\<alpha>. accepting_pair\<^sub>G\<^sub>R_LTS \<delta> q\<^sub>0 (Fin, Inf) w)"
ssubst_Pair_rhs: fixes r :: "'d"   and s :: "'e"   and R :: "('d \<times> 'e) set"   and s' :: "'e" assumes "(r, s) \<in> R"   and "s' = s" shows "(r, s') \<in> R"
leg1_simps_4: shows "trg t\<^sub>0u\<^sub>1.p\<^sub>1 = trg (t\<^sub>0\<^sup>* \<star> u\<^sub>1)"
case_rews_2: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case LT of LT \<Rightarrow> f1 | EQ \<Rightarrow> f2 | GT \<Rightarrow> f3) = f1"
dsr2: fixes x :: "'a"   and y :: "'a" shows "d (x \<cdot> d y) = d (x \<cdot> y)"
sub_inter_left1: fixes A :: "funty"   and C :: "funty"   and B :: "funty" assumes "A <:: C" shows "A \<sqinter> B <:: C"
list_upd_conv: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref"   and y :: "ref"   and q :: "ref" assumes "islist p h"   and "y \<notin> set (list p h)" shows "list p (h(y := q)) = list p h"
normal_Suc: fixes m :: "nat"   and ms :: "nat list" shows "normal (m # ms) = (0 < m)"
indecomposable_succ: fixes \<alpha> :: "V" shows "indecomposable (ZFC_in_HOL.succ \<alpha>) = (\<alpha> = []\<^sub>\<circ>)"
is_distinguishing_formula_def: fixes x :: "('idx, unit, ('act, 'pred) S_action) formula"   and P :: "'state"   and Q :: "'state" shows "S_transform.is_distinguishing_formula x P Q \<equiv> P \<Turnstile>\<^sub>S x \<and> \<not> Q \<Turnstile>\<^sub>S x"
2344: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> y) \<squnion> - - y = top"
io_last: shows "inj_on last {p. \<exists>u. gen_iapath V u p v}"
essential_classD3: fixes K :: "'a \<Rightarrow> 'a pmf"   and C :: "'a set"   and x :: "'a"   and y :: "'a" assumes "MC_syntax.essential_class K C"   and "x \<in> C"   and "y \<in> C" shows "(x, y) \<in> MC_syntax.communicating K"
WeakControlDependencePDG_axioms: shows "WeakControlDependencePDG sourcenode targetnode kind valid_edge (_Entry_) Def Use state_val (_Exit_)"
Rep_inverse: fixes x :: "'a word" shows "of_bl (to_bl x) = x"
R0_def: shows "R0 = {(F1, F2). \<exists>c1 c1' W W'. F1 = [fork c1 W] \<and> F2 = [fork c1' W'] \<and> [c1] \<approx>\<^bsub>d\<^esub> [c1'] \<and> W \<approx>\<^bsub>d\<^esub> W'}"
tvalyz_nonzero: shows "tval z - tval y \<noteq> (0::'a)"
phull_subset_row_space: fixes ps :: "('c \<Rightarrow>\<^sub>0 'd) list"   and ts :: "'c list" assumes "Keys (set ps) \<subseteq> set ts" shows "phull (set ps) \<subseteq> row_to_poly ts ` row_space (polys_to_mat ts ps)"
index_uminus_vec_1: fixes i :: "nat"   and v :: "'a vec" assumes "i < dim_vec v" shows "(- v) $ i = - v $ i"
that: fixes a' :: "'a"   and b' :: "real" assumes "a' \<noteq> (0::'a)"   and "S \<inter> {x. a' \<bullet> x \<le> b'} = S \<inter> {x. a \<bullet> x \<le> b}"   and "S \<inter> {x. a' \<bullet> x = b'} = S \<inter> {x. a \<bullet> x = b}"   and "\<And>w. w \<in> S \<Longrightarrow> w + a' \<in> S" shows "thesis"
Abs_lifted_boolean_algebra_inverse: fixes y :: "('c, 'd) phi \<Rightarrow> 'c" assumes "y \<in> {xf. True}" shows "Rep_lifted_boolean_algebra (Abs_lifted_boolean_algebra y) = y"
AE_commute: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "{x \<in> space (lborel \<Otimes>\<^sub>M lborel). P (fst x) (snd x)} \<in> sets (lborel \<Otimes>\<^sub>M lborel)" shows "(AE x in lborel. almost_everywhere lborel (P x)) = (AE y in lborel. AE x in lborel. P x y)"
neqE: fixes x :: "'a"   and y :: "'a"   and ko :: "'a key_order" assumes "x \<noteq> y" obtains "lt_of_key_order ko x y"   | "lt_of_key_order ko y x"
wiring: shows "wiring \<I>1 \<I>2 conv w"
MP_subset_SMP_2: fixes S :: "('fun, 'var) strand_step list" shows "trms\<^sub>s\<^sub>t S \<subseteq> SMP (trms\<^sub>s\<^sub>t S)"
setops\<^sub>s\<^sub>s\<^sub>t_subst: fixes S :: "('b, 'a) stateful_strand_step list"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "bvars\<^sub>s\<^sub>s\<^sub>t S \<inter> subst_domain \<theta> = {}" shows "setops\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>) = setops\<^sub>s\<^sub>s\<^sub>t S \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<theta>"
facet_map: fixes C :: "'a set"   and z :: "'a set" assumes "folding.chamber C"   and "z \<lhd> C" shows "f ` z \<lhd> f ` C"
ar_coset_same4: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and a :: "'c" assumes "Ring R"   and "ideal R I"   and "a \<in> I" shows "a \<uplus>\<^bsub>R\<^esub> I = I"
fdemodalisation2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(\<langle>x| y \<le> ar ar z) = (ar ar y \<cdot> x \<le> x \<cdot> ar ar z)"
scale_sum_right: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'d"   and f :: "'f \<Rightarrow> 'e"   and A :: "'f set" assumes "module scale" shows "scale a (sum f A) = (\<Sum>x\<in>A. scale a (f x))"
encode_problem_def: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "\<Phi> \<Pi> t \<equiv> \<Phi>\<^sub>I \<Pi> \<^bold>\<and> (encode_operators \<Pi> t \<^bold>\<and> (encode_all_frame_axioms \<Pi> t \<^bold>\<and> (\<Phi>\<^sub>G \<Pi>) t))"
outgoing'_edges: fixes U :: "nat set" shows "outgoing' U \<subseteq> E"
jw_Prod_inv: fixes e :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "{$$} \<turnstile>\<^sub>W e : Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2" obtains e :: "Syntax.term" and \<tau>\<^sub>1' :: "ty" and e' :: "Syntax.term" where "ea__ = App eb__ e'" and "{$$} \<turnstile>\<^sub>W eb__ : Fun \<tau>\<^sub>1'__ (Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2)" and "{$$} \<turnstile>\<^sub>W e' : \<tau>\<^sub>1'__"   | x :: "Syntax.var" and e\<^sub>1 :: "Syntax.term" and \<tau>\<^sub>1' :: "ty" and e\<^sub>2 :: "Syntax.term" where "ea__ = Syntax.Let e\<^sub>1__ x e\<^sub>2__" and "atom x \<sharp> ({$$}, e\<^sub>1__)" and "{$$} \<turnstile>\<^sub>W e\<^sub>1__ : \<tau>\<^sub>1''__" and "{$$}(x $$:= \<tau>\<^sub>1''__) \<turnstile>\<^sub>W e\<^sub>2__ : Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2"   | e :: "Syntax.term" and \<tau>\<^sub>1' :: "ty" and \<tau>\<^sub>2' :: "ty" and e\<^sub>1 :: "Syntax.term" and e\<^sub>2 :: "Syntax.term" where "ea__ = Case ec__ e\<^sub>1'__ e\<^sub>2'__" and "{$$} \<turnstile>\<^sub>W ec__ : Syntax.Sum \<tau>\<^sub>1'''__ \<tau>\<^sub>2'__" and "{$$} \<turnstile>\<^sub>W e\<^sub>1'__ : Fun \<tau>\<^sub>1'''__ (Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2)" and "{$$} \<turnstile>\<^sub>W e\<^sub>2'__ : Fun \<tau>\<^sub>2'__ (Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2)"   | e\<^sub>1 :: "Syntax.term" and e\<^sub>2 :: "Syntax.term" where "ea__ = Syntax.Pair e\<^sub>1 e\<^sub>2" and "{$$} \<turnstile>\<^sub>W e\<^sub>1 : \<tau>\<^sub>1" and "{$$} \<turnstile>\<^sub>W e\<^sub>2 : \<tau>\<^sub>2"   | e :: "Syntax.term" and \<tau>\<^sub>2' :: "ty" where "ea__ = Prj1 ed__" and "{$$} \<turnstile>\<^sub>W ed__ : Syntax.Prod (Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2) \<tau>\<^sub>2'"   | e :: "Syntax.term" and \<tau>\<^sub>1' :: "ty" where "ea__ = Prj2 ee__" and "{$$} \<turnstile>\<^sub>W ee__ : Syntax.Prod \<tau>\<^sub>1' (Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2)"   | \<alpha> :: "Syntax.tvar" and e :: "Syntax.term" and \<tau> :: "ty" where "ea__ = Unroll ef__" and "Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2 = subst_type \<tau> (Mu \<alpha> \<tau>) \<alpha>" and "{$$} \<turnstile>\<^sub>W ef__ : Mu \<alpha> \<tau>"   | e :: "Syntax.term" where "ea__ = Auth eg__" and "{$$} \<turnstile>\<^sub>W eg__ : Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2"   | e :: "Syntax.term" where "ea__ = Unauth e" and "{$$} \<turnstile>\<^sub>W e : Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2"
disc_22: fixes x21 :: "variable"   and x22 :: "typ" shows "\<not> is_App (Fv x21 x22)"
assms_1: fixes i :: "nat" shows "poly.coeff p i \<in> A"
length_greater_0_conv: fixes xs :: "'a list" shows "(0 < |xs|) = (xs \<noteq> [])"
subset'_code_2: fixes ys :: "'a list"   and B :: "'a set" shows "Cardinality.subset' (set ys) B = (\<forall>y\<in>set ys. y \<in> B)"
diamond_cube_def: shows "diamond_cube = (\<lambda>(x, y). (diamond_x x, (2 * y - 1) * diamond_y (diamond_x x)))"
rA': shows "local.sub2 k l (r, A) = (r', A')"
Act_Basic_2: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<A>(\<phi> \<^bold>& \<psi>) \<^bold>\<equiv> \<^bold>\<A>\<phi> \<^bold>& \<^bold>\<A>\<psi> in v]"