hcju/isabelleps huggingface dataset:
coreflexive_pp_dist_comp: fixes uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.pd_allegory uminus inf less_eq less sup bot top one times conv"   and "less_eq x one"   and "less_eq y one" shows "inf (uminus (inf (uminus (times x y)) one)) one = times (inf (uminus (inf (uminus x) one)) one) (inf (uminus (inf (uminus y) one)) one)"
65791: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - ((top \<squnion> x) \<sqinter> - y) = - - y \<sqinter> - x"
akra_bazzi_term_imp_less: fixes x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat"   and b :: "real"   and t :: "nat \<Rightarrow> nat"   and x :: "nat" assumes "akra_bazzi_term x\<^sub>0 x\<^sub>1 b t"   and "x\<^sub>1 \<le> x" shows "t x < x"
j: shows "atom j \<sharp> (i, s, t, k, del)"
emsapss_decode_help2_def: fixes mHash :: "bit list"   and EM :: "bit list"   and emBits :: "nat" shows "emsapss_decode_help2 mHash EM emBits = (if show_rightmost_bits EM 8 \<noteq> BC then False else emsapss_decode_help3 mHash EM emBits)"
list_ex1_iff: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "list_ex1 P xs = (\<exists>!x. x \<in> set xs \<and> P x)"
inv_unit_factor_eq_0_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "(divide one (unit_factor a) = zero) = (a = zero)"
raa_cor_4: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<^bold>\<not>\<phi> in v]"   and "[\<^bold>\<not>\<psi> \<^bold>\<rightarrow> \<phi> in v]"   and "[\<^bold>\<not>\<phi> in v]" shows "[\<psi> in v]"
real_of_float_dense: fixes x :: "real"   and y :: "real" assumes "x < y" shows "\<exists>q. x < real_of_float q \<and> real_of_float q < y"
PDG_BS_valid_node: fixes n :: "'node"   and S :: "'node set" assumes "n \<in> PDG_BS S" shows "valid_node n"
finsert_is_funion: fixes a :: "'m"   and A :: "'m fset" shows "finsert a A = {|a|} |\<union>| A"
components_are_iso: fixes a :: "'a" assumes "A.ide a" shows "B.iso (\<tau> a)"
counit_simps_2: shows "local.dom \<epsilon>' = f \<star> g'"
two_lists_cases_Cons_Cons: fixes as :: "'c list"   and bs :: "'b list" obtains ys :: "'b list" where "as = []" and "bs = ys__"   | xs :: "'c list" where "as = xs__" and "bs = []"   | x :: "'c" and xs :: "'c list" and y :: "'b" and ys :: "'b list" where "as = x # xs" and "bs = y # ys"
od_fm_fun_inductive: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N" shows "inductive_set (od_fm_fun R M N f H)"
mbuf2t_take_add'_1: fixes f :: "event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and xs :: "event_data option list set list"   and ys :: "event_data option list set list"   and buf :: "event_data option list set list \<times> event_data option list set list"   and nts :: "nat list"   and z' :: "'a"   and buf' :: "event_data option list set list \<times> event_data option list set list"   and nts' :: "nat list"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and j' :: "nat"   and P' :: "String.literal \<Rightarrow> nat option"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula" assumes "mbuf2t_take f z (mbuf2_add xs ys buf) nts = (z', buf', nts')"   and "pred_mapping (\<lambda>x. x \<le> j) P"   and "pred_mapping (\<lambda>x. x \<le> j') P'"   and "rel_mapping (\<le>) P P'"   and "wf_mbuf2' \<sigma> P V j n R \<phi> \<psi> buf"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j)..<j'] nts"   and "list_all2 (\<lambda>i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) [Monitor.progress \<sigma> P \<phi> j..<Monitor.progress \<sigma> P' \<phi> j'] xs"   and "list_all2 (\<lambda>i. qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<psi>)) [Monitor.progress \<sigma> P \<psi> j..<Monitor.progress \<sigma> P' \<psi> j'] ys"   and "j \<le> j'" shows "wf_mbuf2' \<sigma> P' V j' n R \<phi> \<psi> buf'"
emp_concat_emp: fixes us :: "'b list list"   and S :: "'b list set" assumes "us \<in> lists S\<^sub>+"   and "concat us = \<epsilon>" shows "us = \<epsilon>"
seq_SUP_distl: fixes P :: "('b, 'd) urel"   and Q :: "'e \<Rightarrow> ('d, 'c) urel"   and A :: "'e set" shows "P ;; \<Or>(Q ` A) = \<Or>((\<lambda>i. P ;; Q i) ` A)"
shadow_root_host_dual: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) 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"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and host :: "(_) element_ptr" assumes "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs"   and "h \<turnstile> get_host shadow_root_ptr \<rightarrow>\<^sub>r host" shows "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr"
context_decomp_1: fixes t :: "trm"   and T :: "POPLmarkRecord.type" assumes "[] \<turnstile> t : T" shows "t \<in> value \<or> (\<exists>E t\<^sub>0 t\<^sub>0'. E \<in> ctxt \<and> t = E t\<^sub>0 \<and> t\<^sub>0 \<longmapsto>\<^sub>c t\<^sub>0')"
assms_1: fixes k :: "nat" shows "(f k has_integral y k) S"
closure_of_empty: fixes X :: "'c topology" shows "X closure_of {} = {}"
paths_tl_notnil: fixes xs :: "'a list" assumes "xs \<in> paths" shows "tl xs \<noteq> []"
action_def: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<box>[P]_v \<equiv> %s. \<forall>i. P (s |\<^sub>s i) \<or> (Unchanged v) (s |\<^sub>s i)"
sq_norm_vec_ge_0: fixes v :: "'a vec" shows "(0::'a) \<le> \<parallel>v\<parallel>\<^sup>2"
latestAct_less: fixes n' :: "nat"   and n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and nid :: "'a"   and t :: "nat \<Rightarrow> cnf" assumes "n' < n \<and> active nid (t n')" shows "n' \<le> dynamic_component.latestAct active nid t n"
psimps: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_trd_aux_dom bs (t, p)" shows "sig_trd_aux bs (t, p) = (let p' = case find_sig_reducer (map spp_of bs) (lt p) t 0 of None \<Rightarrow> p | Some i \<Rightarrow> p - monom_mult (lookup (rep_list p) t / punit.lc (rep_list (bs ! i))) (t - punit.lt (rep_list (bs ! i))) (bs ! i); p'' = punit.lower (rep_list p') t in if p'' = 0 then p' else sig_trd_aux bs (punit.lt p'', p'))"
mahler_measure_monic_def: fixes p :: "complex poly" shows "mahler_measure_monic p = (\<Prod>a\<leftarrow>complex_roots_complex p. max 1 (cmod a))"
simps: fixes a :: "'b"   and r :: "(bool \<times> 'b) rexp"   and b :: "'c" shows "delta_b a (r, b) = (let r' = read a r in (follow False r', final r'))"
heron_step: fixes t :: "nat"   and n :: "nat" assumes "0 < t" shows "Discrete.sqrt n \<le> (t + n div t) div 2"
seE_fringe1: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and re :: "('d \<times> nat) edge"   and prb' :: "('d, 'e, 'f) pre_RedBlack"   and c' :: "('e, 'f) conf" assumes "sub_rel_of (red prb) (subs prb)"   and "ui_edge re \<in> edges (black prb) \<and> (src re \<in> red_vertices prb \<and> tgt re \<notin> red_vertices prb \<and> red prb' = add_edge (red prb) re) \<and> src re \<notin> subsumees (subs prb) \<and> se (confs prb (src re)) (labelling (black prb) (ui_edge re)) c' \<and> prb' = \<lparr>red = red prb', black = black prb, subs = subs prb, init_conf = init_conf prb, confs = (confs prb)(tgt re := c'), marked = (marked prb)(tgt re := marked prb (src re)), strengthenings = strengthenings prb\<rparr>"   and "\<not> marked prb (src re)"   and "ui_edge ` out_edges (red prb') (src re) \<subset> out_edges (black prb) (fst (src re))"   and "out_edges (black prb) (fst (tgt re)) \<noteq> {}" shows "fringe prb' = fringe prb \<union> {tgt re}"
llength_ltakeWhile_lt_iff: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "(llength (ltakeWhile P xs) < llength xs) = (\<exists>x\<in>lset xs. \<not> P x)"
unit_char_2: fixes a :: "'b" assumes "B.obj a" shows "C.iso (F.unit a)"
simps_1: fixes uu :: "'b dres" shows "(dSUCCEEDi \<le> uu) = True"
polyhedron_imp_closed: fixes S :: "'b set" assumes "polyhedron S" shows "closed S"
adj_sub_prop_fact: fixes i :: "'d"   and x2 :: "'f \<Rightarrow> ('d, 'e) trm"   and k :: "'f"   and j :: "'g + 'g"   and \<sigma> :: "('d, 'h, 'g, 'i) subst_scheme"   and x1 :: "'g" assumes "i \<in> SIGT (x2 k)"   and "j \<in> (case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)" shows "Inl i \<in> ({Inl x |x. x \<in> dom (SFunctions \<sigma>)} \<union> {Inr (Inl x) |x. x \<in> dom (SContexts \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPredicates \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPrograms \<sigma>)}) \<inter> insert (Inr (Inr x1)) {Inl x |x. \<exists>xa. x \<in> SIGT (x2 xa)}"
simps_3: shows "eval_ternary_Imp TernaryFalse TernaryTrue = TernaryTrue"
v'_def: shows "v' = \<lparr>ext = Abs_real_int (left (ext v4), right (ext v2)), lan = lan v, own = own v\<rparr>"
epm_11: fixes v :: "'b"   and w :: "'b"   and e :: "'b" assumes "bounded_distrib_lattice_class.comp_inf.vector v" shows "w \<sqinter> - stone_algebra_class.comp_inf.prim_EP w v e \<sqinter> stone_algebra_class.comp_inf.prim_P w v e = bot"
less_iff_diff_less_0: 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" shows "less a b = less (minus a b) zero"
reachable_induce_ss: fixes S :: "'a set"   and u :: "'a"   and v :: "'a"   and T :: "'a set" assumes "u \<rightarrow>\<^sup>*\<^bsub>(G \<restriction> {v. u \<rightarrow>\<^sup>* v}) \<restriction> S\<^esub> v"   and "S \<subseteq> T" shows "u \<rightarrow>\<^sup>*\<^bsub>(G \<restriction> {v. u \<rightarrow>\<^sup>* v}) \<restriction> T\<^esub> v"
theNFTBoth_length_finished_1: fixes n :: "nat"   and cl :: "('d, 'e, 'f) cmd list"   and dl :: "('g, 'h, 'i) cmd list" assumes "n \<in> theNFTBoth cl dl" shows "n < length cl"
ccloses_extend: fixes \<theta>_c :: "(coname \<times> name \<times> trm) list"   and \<Delta> :: "(coname \<times> ty) list"   and a :: "coname"   and x :: "name"   and P :: "trm"   and B :: "ty" assumes "\<theta>_c ccloses \<Delta>"   and "a \<sharp> \<Delta>"   and "a \<sharp> \<theta>_c"   and "x:P \<in> \<parallel>(B)\<parallel>" shows "(a, x, P) # \<theta>_c ccloses (a, B) # \<Delta>"
sup_Sup: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "sup_prod x y = Sup_prod {x, y}"
VUnion_vempty: shows "\<Union>\<^sub>\<circ> 0 = 0"
rel_def: fixes x :: "'a set set \<times> ('a set \<Rightarrow> ('a \<times> 'a) set)"   and y :: "'a set set \<times> ('a set \<Rightarrow> ('a \<times> 'a) set)" shows "st.rel x y \<equiv> (fst x \<in> {U. is_zariski_open U \<and> \<pp> \<in> U} \<and> fst y \<in> {U. is_zariski_open U \<and> \<pp> \<in> U}) \<and> (snd x \<in> \<O> fst x \<and> snd y \<in> \<O> fst y) \<and> (\<exists>W. W \<in> {U. is_zariski_open U \<and> \<pp> \<in> U} \<and> W \<subseteq> fst x \<inter> fst y \<and> sheaf_spec_morphisms (fst x) W (snd x) = sheaf_spec_morphisms (fst y) W (snd y))"
assms_2: shows "\<forall>j<i. xs ! j \<noteq> x"
assms_2: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<noteq> b"   and "c \<noteq> d"   and "a = c \<and> b = d \<or> a = c \<and> b \<noteq> d \<or> a \<noteq> c \<and> b = d \<or> a \<noteq> c \<and> a \<noteq> d \<and> b \<noteq> c \<and> b \<noteq> d" shows "P a b c d"
strict_implies_not_eq: fixes b :: "'x \<Rightarrow>\<^sub>0 nat"   and a :: "'x \<Rightarrow>\<^sub>0 nat" assumes "b \<prec> a" shows "a \<noteq> b"
unwrap_strict: shows "unwrap\<cdot>\<bottom> = \<bottom>"
Exit_does_not_control: fixes n' :: "'node" assumes "(_Exit_) controls n'" shows "False"
mult_strict_left_mono: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less zero c" shows "less (times c a) (times c b)"
assms_1: shows "case_nat t1 \<Gamma> \<turnstile>\<^sub>c g : t2"
1: shows "spoly g1 g2 = 0 + (\<Sum>g\<in>G. q g \<odot> g)"
less_list_def: fixes xs :: "'a list"   and ys :: "'a list" shows "xs < ys \<equiv> xs \<le> ys \<and> \<not> ys \<le> xs"
gcd_1_int: fixes m :: "int" shows "gcd m 1 = 1"
preserves_assoc_1: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "F \<a>\<^sub>C[f, g, h] = \<Phi> (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F h] \<cdot>\<^sub>D (D.inv (\<Phi> (f, g)) \<star>\<^sub>D F h) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, h))"
Fin: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and c1 :: "stmt"   and c2 :: "stmt" assumes "E,dt\<Turnstile>c1\<Colon>\<surd>"   and "E,dt\<Turnstile>c2\<Colon>\<surd>" shows "E,dt\<Turnstile>c1 Finally c2\<Colon>\<surd>"
OCC1: shows "\<forall>t\<in>FP. \<forall>f\<in>funs_term t. is_Abs f \<longrightarrow> f \<in> Abs ` OCC"
trms_subsumes_def: fixes C :: "'b eclause"   and D :: "'b eclause"   and \<sigma> :: "('b \<times> 'b trm) list" shows "trms_subsumes C D \<sigma> = (subst_cl (cl_ecl C) \<sigma> = cl_ecl D \<and> subst_set (trms_ecl C) \<sigma> \<subseteq> trms_ecl D)"
hd_def: fixes list :: "'e list" shows "hd list = (case list of x21 # x22 \<Rightarrow> x21)"
dmus': shows "iarray_update dmus i (iarray_append (dmus !! i) (dmus !! j !! j * (fi \<bullet> fs !! Suc j) - sig)) = dmus'"
final_2: shows "\<forall>p. ps final p \<noteq> None"
zmult_zless_mono2: fixes i :: "int"   and j :: "int"   and k :: "int" assumes "i < j"   and "0 < k" shows "k * i < k * j"
fixp_above_the_while_option: shows "fixp_above a = the (while_option (\<lambda>A. f A \<noteq> A) f a)"
hcomp_in_vhomE: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data" assumes "\<guillemotleft>\<nu> \<star> \<mu> : f \<Rightarrow> g\<guillemotright>" obtains "arr \<mu>" and "arr \<nu>" and "src \<nu> = trg \<mu>" and "src \<mu> = src f" and "src \<mu> = src g" and "trg \<nu> = trg f" and "trg \<nu> = trg g"
elims: fixes x :: "color"   and xa :: "('b \<times> color) tree"   and y :: "('b \<times> color) tree" assumes "paint x xa = y" obtains c :: "color" where "x = c__" and "xa = \<langle>\<rangle>" and "y = \<langle>\<rangle>"   | c :: "color" and l :: "('b \<times> color) tree" and a :: "'b" and uu_ :: "color" and r :: "('b \<times> color) tree" where "x = c" and "xa = \<langle>l, (a, uu_), r\<rangle>" and "y = \<langle>l, (a, c), r\<rangle>"
order_refl: fixes x :: "'b list" shows "x \<le>s x"
inject_7: fixes x10 :: "'c rexp"   and y10 :: "'c rexp" shows "(Pr x10 = Pr y10) = (x10 = y10)"
lm014: fixes X :: "'a set"   and x :: "'b"   and Y :: "'a set"   and y :: "'b" shows "runiq (X \<times> {x} +< Y \<times> {y})"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vreal_mult"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> vreal_mult" shows "b = c"
get: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and l :: "'b" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Read (VE (Loc l))])"   and "l \<in> dom (\<sigma>;;\<tau>)" shows "substitution.revision_step subst r s (s(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (the ((\<sigma>;;\<tau>) l))])))"
substC_congC: fixes c :: "('c, 'd) lit list"   and \<pi>1 :: "Preliminaries.var \<Rightarrow> 'c trm"   and \<pi>2 :: "Preliminaries.var \<Rightarrow> 'c trm" assumes "\<And>x. x \<in> varsC c \<Longrightarrow> \<pi>1 x = \<pi>2 x" shows "substC \<pi>1 c = substC \<pi>2 c"
order_trans_rules_22: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x < y"   and "y \<le> z" shows "x < z"
in_arcs_def: fixes G :: "('f, 'e) pre_digraph"   and v :: "'f" shows "in_arcs G v \<equiv> {e \<in> arcs G. head G e = v}"
elims_3: fixes x :: "lock_action list" assumes "\<not> must_acquire_lock x" obtains "x = []"   | las :: "lock_action list" where "x = Unlock # lasa__"   | las :: "lock_action list" where "x = UnlockFail # lasb__" and "\<not> must_acquire_lock lasb__"   | las :: "lock_action list" where "x = ReleaseAcquire # las" and "\<not> must_acquire_lock las"
iface_subset: fixes i1 :: "iface"   and i2 :: "iface" shows "iface_subset i1 i2 = ({i. match_iface i1 i} \<subseteq> {i. match_iface i2 i})"
GPV_bind: fixes f :: "('g, 'e, ('g, 'e, 'f) rpv) generat spmf"   and g :: "'g \<Rightarrow> ('d, 'e, 'f) gpv" shows "GPV f \<bind> g = GPV (f \<bind> (\<lambda>generat. case generat of Pure x \<Rightarrow> the_gpv (g x) | IO out c \<Rightarrow> return_spmf (IO out (\<lambda>input. c input \<bind> g))))"
hd_F_in_F: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "hd (F k) \<in> set (F n)"
inner_nonneg_nonneg: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" 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"   and "less_eq zero a"   and "less_eq zero b" shows "0 \<le> inner a b"
hom_1_iff: fixes x :: "'b" shows "([:x:] = 1) = (x = (1::'b))"
rep_eq: fixes x :: "'e \<Longrightarrow> 'f"   and xa :: "('g, 'f) uexpr" shows "(x \<sharp> xa) = (\<forall>b v. \<lbrakk>xa\<rbrakk>\<^sub>e (put\<^bsub>x\<^esub> b v) = \<lbrakk>xa\<rbrakk>\<^sub>e b)"
assms: shows "bidirected_digraph (with_proj G) rev_G"
that: assumes "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> a \<in>\<^sub>\<circ> f i" shows "thesis"
that: shows "\<forall>i\<in>S. length xs \<le> i"
class_leq_trans: fixes cs :: "(String.literal \<times> String.literal) set"   and x :: "String.literal"   and y :: "String.literal"   and z :: "String.literal" assumes "wf_subclass_loc cs"   and "class_leq cs x y"   and "class_leq cs y z" shows "class_leq cs x z"
refine: fixes succi :: "'b \<Rightarrow> 'b list"   and E :: "('c \<times> 'c) set"   and node_rel :: "('b \<times> 'c) set"   and Pimpl :: "'b \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and node_eq_impl :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and U0' :: "'b list"   and U0 :: "'c set"   and node_hash_impl :: "nat \<Rightarrow> 'b \<Rightarrow> nat"   and node_def_hash_size :: "nat" assumes "(succi, E) \<in> \<langle>node_rel\<rangle>slg_rel"   and "(Pimpl, P) \<in> node_rel \<rightarrow> bool_rel"   and "(node_eq_impl, (=)) \<in> node_rel \<rightarrow> node_rel \<rightarrow> bool_rel"   and "(U0', U0) \<in> \<langle>node_rel\<rangle>list_set_rel"   and "is_bounded_hashcode node_rel node_eq_impl node_hash_impl"   and "is_valid_def_hm_size TYPE('b) node_def_hash_size" shows "(wset_find_path_impl node_hash_impl node_def_hash_size node_eq_impl succi U0' Pimpl, wset_find_path' E U0 P) \<in> \<langle>\<langle>\<langle>node_rel\<rangle>list_rel \<times>\<^sub>r node_rel\<rangle>option_rel\<rangle>nres_rel"
diff_right_commute: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and c :: "'d"   and b :: "'d" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (minus a c) b = minus (minus a b) c"
f_merge: fixes x :: "'a"   and y :: "'a" assumes "f x = f y" shows "f (merge x y) = f x"
incseq_nn_integral: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "incseq f" shows "incseq (\<lambda>i. integral\<^sup>N M (f i))"
colossless_gpv_lossless_spmfD: fixes \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv" assumes "colossless_gpv \<I> gpv" shows "lossless_spmf (the_gpv gpv)"
substPB_def: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'c trm"   and \<Phi> :: "('c, 'd) lit list set" shows "substPB \<pi> \<Phi> = {substC \<pi> c |c. c \<in> \<Phi>}"