hcju/isabelleps huggingface dataset:
terminates_Call_body: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and s :: "('d, 'f) xstate" assumes "\<Gamma> p = Some bdy" shows "\<Gamma>\<turnstile>Call p \<down> s = \<Gamma>\<turnstile>the (\<Gamma> p) \<down> s"
fmempty_def: shows "{$$} \<equiv> Abs_fmap Map.empty"
cmp_combine_def: fixes D1 :: "'a set"   and cmp1 :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res"   and D2 :: "'a set"   and cmp2 :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res"   and a :: "'a"   and b :: "'a" shows "cmp_combine D1 cmp1 D2 cmp2 a b \<equiv> if a \<in> D1 \<and> b \<in> D1 then cmp1 a b else if a \<in> D1 \<and> b \<in> D2 then LESS else if a \<in> D2 \<and> b \<in> D1 then GREATER else cmp2 a b"
simps_2: fixes F :: "'n \<Rightarrow> 'o fundef option"   and instr :: "('p, 'q, 'n, 'e, 'f, 'g, 'r, 's) instr"   and p :: "('p, 'q, 'n, 'e, 'f, 'g, 'r, 's) instr list"   and \<Sigma> :: "Unboxed.type option list" shows "sp F (instr # p) \<Sigma> = sp_instr F instr \<Sigma> \<bind> sp F p"
edge_bipartite_webE: fixes \<Gamma> :: "('a, 'b) web_scheme"   and uv :: "'a + 'a"   and uv' :: "'a + 'a" assumes "edge (bipartite_web_of \<Gamma>) uv uv'" obtains u :: "'a" and v :: "'a" where "uv = Inl u__" and "uv' = Inr v" and "edge \<Gamma> u__ v"   | u :: "'a" where "uv = Inl u" and "uv' = Inr u" and "u \<in> \<^bold>V\<^bsub>\<Gamma>\<^esub>" and "u \<notin> A \<Gamma>" and "u \<notin> B \<Gamma>"
proj_poly_syz_eq_zero_iff: fixes n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "(proj_poly_syz n p = 0) = (component_of_term ` keys p \<subseteq> {0..<n})"
lookup_HSem_heap: fixes y :: "Vars.var"   and h :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" assumes "y \<in> domA h" shows "(\<N>\<lbrace>h\<rbrace>\<rho>) y = \<N>\<lbrakk> the (map_of h y) \<rbrakk>\<^bsub>\<N>\<lbrace>h\<rbrace>\<rho>\<^esub>"
idem: fixes a :: "'g" shows "a \<sqinter> a = a"
exhaust: fixes y :: "mode" obtains "y = Nop"   | "y = Fast"   | "y = Slow"
prj1_simps_1: fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "arr \<p>\<^sub>1[f, g]"
quotient_is_domain: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "primeideal I R" shows "domain (R Quot I)"
simps_7: fixes p :: "'b" shows "redex (Call p) = Call p"
of_nat_le_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m \<le> n)"
wf_induct_rule: fixes r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "wf r"   and "\<And>x. (\<And>y. (y, x) \<in> r \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
map_tailrec_rev: fixes f :: "'h \<Rightarrow> 'g"   and as :: "'h list"   and bs :: "'g list" shows "map_tailrec_rev f as bs = rev (map f as) @ bs"
abc_ex_path: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "[[a b c]]" shows "\<exists>Q\<in>\<P>. a \<in> Q \<and> b \<in> Q \<and> c \<in> Q"
subgraph_apath_imp_apath: fixes H :: "('a, 'a \<times> 'a) pre_digraph"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "pre_digraph.apath H u p v"   and "subgraph H (with_proj G)" shows "apath u p v"
backflows_filternew_flows_state: fixes \<T> :: "'a stateful_policy" shows "backflows (filternew_flows_state \<T>) = backflows (flows_state \<T>) - flows_fix \<T>"
simps_5: fixes Ts :: "typ list"   and t :: "Core.term"   and u :: "Core.term" shows "typ_of1 Ts (t $ u) = Option.bind (typ_of1 Ts u) (\<lambda>U. Option.bind (typ_of1 Ts t) (case_typ (\<lambda>fun. case_list None (\<lambda>T1. case_list None (\<lambda>T2. case_list (if fun = STR ''fun'' then if T1 = U then Some T2 else None else None) (\<lambda>ab. Map.empty)))) (\<lambda>variable. Map.empty)))"
qstar_unfoldl: fixes x :: "'a" shows "(1::'a) \<squnion> x \<cdot> qstar x \<le> qstar x"
mask_out_sub_mask: fixes x :: "'a word"   and n :: "nat" shows "x AND NOT (mask n) = x - (x AND mask n)"
ln_Gamma_real_LIMSEQ: fixes x :: "real" assumes "0 < x" shows "ln_Gamma_series x \<longlonglongrightarrow> ln_Gamma x"
generate_img: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g"   and K :: "'e set" assumes "group_hom G H h"   and "K \<subseteq> carrier G" shows "generate H (h ` K) = h ` generate G K"
the_\<nu>_props_1: shows "\<guillemotleft>w.the_\<nu> : local.dom \<sigma> \<Rightarrow> s\<^sub>1 \<star> src s\<^sub>0\<guillemotright>"
char_eq_0_iff: fixes a :: "nat" shows "(\<chi> a = 0) = (a \<notin> carrier G)"
assms_4: fixes i :: "'b" assumes "i \<in> S - T" shows "g i = \<^bold>1"
eventually_weakly_fair: fixes enabled :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and taken :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<diamond>weakly_fair enabled taken = weakly_fair enabled taken"
def_scheme: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0" shows "f self \<equiv> \<lambda>\<tau>. SOME res. let res = \<lambda>_. res in if \<tau> \<Turnstile> \<delta> self then \<tau> \<Turnstile> PRE self \<and> \<tau> \<Turnstile> POST self res else \<tau> \<Turnstile> res \<triangleq> invalid"
var_lookup_out: fixes x :: "'a \<Longrightarrow> 'c"   and A :: "'b"   and A' :: "'c" shows "get\<^bsub>$x\<acute>\<^esub> (A, A') = get\<^bsub>x\<^esub> A'"
fmapU_writerT_def: shows "fmapU \<equiv> writerT_map\<cdot>ID"
strict_trans: fixes a :: "'a Word.word"   and b :: "'a Word.word"   and c :: "'a Word.word" assumes "a <s b"   and "b <s c" shows "a <s c"
assoc_props_1: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and h :: "'a arrow_of_spans_data" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "src \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc (\<cdot>) prj0 prj1 f g h, Dom = Dom ((f \<star> g) \<star> h), Cod = Cod (f \<star> g \<star> h)\<rparr> = src h"
empt_invariant: shows "set [] \<subseteq> reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0"
Neg: fixes n :: "nat"   and p :: "pf" shows "dfa_of_pf n (Neg p) = negate_dfa (dfa_of_pf n p)"
fw_neg_diag_neg_cycle: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat"   and i :: "nat"   and xs :: "nat list" assumes "len (fw m n k) i i xs < (0::'a)"   and "i \<le> n"   and "k \<le> n"   and "set xs \<subseteq> {0..n}" shows "\<exists>i\<le>n. \<exists>ys. set ys \<subseteq> set xs \<union> {0..k} \<and> len m i i ys < (0::'a)"
L1_Controller: fixes s :: "nat \<Rightarrow> nat list"   and l :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and i :: "nat" assumes "Controller_L s (fin_inf_append [Zero] l) l z" shows "fin_make_untimed (inf_truncate z i) \<noteq> []"
fls_residue_def: fixes f :: "'a fls" shows "fls_residue f \<equiv> fls_nth f (- 1)"
inorder_isempty': fixes t :: "'t" assumes "inv t"   and "sorted1 (inorder t)" shows "is_empty t = (inorder t = [])"
hausdorff_distance_infdist_triangle: fixes B :: "'b set"   and C :: "'b set"   and x :: "'b" assumes "B \<noteq> {}"   and "bounded B"   and "bounded C" shows "infdist x C \<le> infdist x B + hausdorff_distance B C"
smooth_compat_le: fixes k :: "enat"   and c1 :: "('c, 'd) chart"   and c2 :: "('c, 'd) chart"   and l :: "enat" assumes "k-smooth_compat c1 c2"   and "l \<le> k" shows "l-smooth_compat c1 c2"
measure_eqI_countable': fixes M :: "'d measure"   and A :: "'d set"   and N :: "'d measure" assumes "space M = A"   and "space N = A"   and "\<And>x. x \<in> A \<Longrightarrow> {x} \<in> sets M"   and "\<And>x. x \<in> A \<Longrightarrow> {x} \<in> sets N"   and "countable A"   and "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}" shows "M = N"
Diagonal_to_Smith_row_i_PQ': fixes P' :: "(('a, 'rows) vec, 'rows) vec"   and B' :: "(('a, 'cols) vec, 'rows) vec"   and Q' :: "(('a, 'cols) vec, 'cols) vec"   and i :: "nat" assumes "(P', B', Q') = Diagonal_to_Smith_row_i_PQ i bezout (P, B, Q)"   and "isDiagonal B"   and "i < min (nrows A) (ncols A)" shows "B' = P' ** A ** Q' \<and> invertible P' \<and> invertible Q'"
right_minus: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.group_add minus plus zero uminus" shows "plus a (uminus a) = zero"
treewidth_def: shows "treewidth \<equiv> Min treewidth_cards"
UN_bool_eq: fixes A :: "bool \<Rightarrow> 'b set" shows "\<Union> (range A) = A True \<union> A False"
closed_hyperplane: fixes a :: "'c"   and b :: "real" shows "closed {x. a \<bullet> x = b}"
union_open: fixes M :: "'a set set" assumes "\<forall>m\<in>M. m open\<^bsub>T\<^esub>" shows "\<Union> M open\<^bsub>T\<^esub>"
s_r: shows "s r = Some (\<sigma>, \<tau>, \<E> [Ite (VE (CV F)) e1 e2])"
target: fixes T1 :: "'procT"   and T2 :: "'procT"   and TRel :: "('procT \<times> 'procT) set" assumes "(T1, T2) \<in> TRel" shows "TargetTerm T1 \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> TargetTerm T2"
tst1: shows "ts1 t = \<lfloor>(x1, ln1)\<rfloor>"
equivP_pick_preserves: fixes P :: "'d \<Rightarrow> bool"   and \<phi> :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and X :: "'d set" assumes "equivP P \<phi>"   and "(P /// \<phi>) X" shows "P (pick X)"
factor_gauss_int_prime_nat_correct: fixes p :: "nat" assumes "prime p" shows "set (factor_gauss_int_prime_nat p) = prime_factors (of_nat p)"
simps: fixes xs :: "'b list"   and l :: "nat"   and r :: "nat" shows "quicksort xs l r = (if r \<le> l then xs else if length xs \<le> r then xs else let (p, xs1) = Quicksort.partition xs l r; xs2 = quicksort xs1 l (p - 1) in quicksort xs2 (p + 1) r)"
one_uexpr_def: shows "1 = \<guillemotleft>1::'c\<guillemotright>"
nodes_target: fixes p :: "'state"   and r :: "'action list" assumes "p \<in> nodes"   and "path r p" shows "target r p \<in> nodes"
rel_refl_strong: fixes x :: "'c list"   and Ra :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>z. z \<in> set x \<Longrightarrow> Ra z z" shows "list_all2 Ra x x"
small: shows "small {[a, b]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>}"
assms_5: shows "n \<equiv> df - J + (dg - J)"
delayedR_correct_2: fixes u :: "'c \<Rightarrow> real"   and R :: "('c \<Rightarrow> real) set"   and I :: "'c \<Rightarrow> Regions.intv"   and r :: "('c \<times> 'c) set" assumes "u \<in> R"   and "R \<in> \<R>"   and "Regions.valid_region \<X> k I r"   and "\<forall>c\<in>\<X>. \<not> Regions.isConst (I c)"   and "Regions.region \<X> I r \<in> Succ \<R> R" shows "\<exists>t\<ge>0. delayedR (Regions.region \<X> I r) u = u \<oplus> t \<and> (1 - Max ({frac (u c) |c. c \<in> \<X> \<and> Regions.isIntv (I c)} \<union> {0})) / 2 \<le> t"
eq_iff: fixes a :: "'a"   and b :: "'a" shows "(a = b) = (b \<succeq> a \<and> a \<succeq> b)"
cosh_real_nonneg_le_iff: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "0 \<le> y" shows "(cosh x \<le> cosh y) = (x \<le> y)"
silent_tl: fixes s :: "'x2 \<times> 'm2"   and ta :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and s' :: "'x2 \<times> 'm2" assumes "\<tau>move2 s ta s'" shows "ta = \<lbrace>\<rbrace>"
ennreal_top_mult: fixes a :: "ennreal" shows "top * a = (if a = 0 then 0 else top)"
mem_cball: fixes y :: "'a"   and x :: "'a"   and e :: "real" shows "(y \<in> cball x e) = (dist x y \<le> e)"
soundness: fixes A :: "'c set"   and n :: "nat"   and branch :: "(('d, 'c) fm list \<times> 'c) list"   and M :: "('e, 'd) model"   and g :: "'c \<Rightarrow> 'e"   and w :: "'e" assumes "A, n \<turnstile> branch" shows "\<exists>block\<in>set branch. \<exists>p. p on block \<and> \<not> M, g, w \<Turnstile> p"
neg_cycle_imp_inf_\<mu>: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and w :: "'a"   and c :: "'b list"   and f :: "'b \<Rightarrow> real" assumes "awalk u p v"   and "awalk w c w"   and "w \<in> set (awalk_verts u p)"   and "awalk_cost f c < 0" shows "\<mu> f u v = - \<infinity>"
simps: fixes a1 :: "nat"   and a2 :: "Lholed"   and a3 :: "e list"   and a4 :: "e list" shows "Lfilled_exact a1 a2 a3 a4 = ((\<exists>lholed es. a1 = 0 \<and> a2 = lholed \<and> a3 = es \<and> a4 = es \<and> lholed = LBase [] []) \<or> (\<exists>vs lholed n es' l es'' k es lfilledk. a1 = k + 1 \<and> a2 = lholed \<and> a3 = es \<and> a4 = vs @ [Label n es' lfilledk] @ es'' \<and> const_list vs \<and> lholed = LRec vs n es' l es'' \<and> Lfilled_exact k l es lfilledk))"
slpa_number_Calls_eq_number_Returns: fixes cs :: "'edge list"   and as :: "'edge list" assumes "same_level_path_aux cs as"   and "upd_cs cs as = []"   and "\<forall>a\<in>set as. valid_edge a"   and "\<forall>c\<in>set cs. valid_edge c" shows "length (filter (\<lambda>a. \<exists>Q r p fs. kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs) (as @ cs)) = length (filter (\<lambda>a. \<exists>Q p f. kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f) as)"
vrat_inverse_def: shows "vrat_inverse = (\<lambda>x\<in>\<^sub>\<circ>\<rat>\<^sub>\<circ>. inverse (rat_of_vrat x)\<^sub>\<rat>)"
insert_seq_after_start: fixes ops :: "('a \<times> 'a option) list"   and ref :: "'a"   and xs :: "('a \<times> 'a option) list" assumes "insert_ops ops"   and "insert_seq (Some ref) xs"   and "insert_ops xs"   and "set xs \<subseteq> set ops"   and "ref \<in> set (interp_ins ops)" shows "\<forall>i\<in>set (map fst xs). list_order ops ref i"
Ks_2: shows "combine Ks (map (([^]) x) (rev [0..<Suc n])) = \<zero>"
compact_imp_seq_compact: fixes U :: "'c set" assumes "compact U" shows "seq_compact U"
subst_lit_comp_subst: fixes L :: "'a literal"   and \<tau> :: "'s"   and \<sigma> :: "'s" shows "L \<cdot>l (\<tau> \<odot> \<sigma>) = L \<cdot>l \<tau> \<cdot>l \<sigma>"
0: shows "rels pointsTo \<subseteq> r1 \<union> r2"
scc_verts_da: shows "H.sccs_verts = G.sccs_verts - {G.scc_of (tail G a)} \<union> {H.scc_of (tail G a), H.scc_of (head G a)}"
non_z: shows "\<forall>x\<in>s - {z}. f x \<noteq> 0"
assms_1: shows "a \<in>\<^sub>\<circ> \<Inter>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)"
Dp_Mp_eq: fixes f :: "int poly" shows "f = pn.Mp f + Polynomial.smult (p ^ n) (pn.Dp f)"
prems_2: shows "set (\<phi>1 # \<phi>1s) \<subseteq> fmla"
simps_2: fixes v :: "'v"   and v1 :: "'v"   and v2 :: "'v"   and p :: "('v \<times> 'v) list"   and v' :: "'v" shows "is_path v ((v1, v2) # p) v' = (v = v1 \<and> (v1, v2) \<in> E \<and> is_path v2 p v')"
R1_def: shows "R1 = {(I1, I2). \<exists>c1 c1' c2 c2' b b'. I1 = [if b then c1 else c2 fi] \<and> I2 = [if b' then c1' else c2' fi] \<and> [c1] \<approx>\<^bsub>d\<^esub> [c1'] \<and> [c2] \<approx>\<^bsub>d\<^esub> [c2'] \<and> b \<equiv>\<^bsub>d\<^esub> b'}"
intro: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map"   and "design \<V> \<B>"   and "design \<V>' \<B>'" shows "design_isomorphism \<V> \<B> \<V>' \<B>' bij_map"
type_update_select_length0: fixes cts :: "ct list"   and tm :: "checker_type" assumes "type_update_select (TopType cts) = tm"   and "length cts = 0"   and "tm \<noteq> Bot" shows "tm = TopType [TAny]"
prime_exp: fixes p :: "nat"   and n :: "nat" shows "prime_elem (p ^ n) = (prime_elem p \<and> n = 1)"
brec_step_invar: fixes rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap"   and \<delta> :: "('a, 'b) ta_rule set"   and Qi :: "('a, unit) HashMap.hashmap"   and \<Sigma> :: "('a, 'b tree) HashMap.hashmap \<times> ('a list \<times> 'a list) \<times> (('a, 'b) ta_rule, nat) HashMap.hashmap \<times> 'a option" assumes "rqrm_invar rqrm"   and "rqrm_prop \<delta> rqrm"   and "hs.invar Qi"   and "\<Sigma> \<in> brec_invar_add (hs.\<alpha> Qi)"   and "brec_\<alpha> \<Sigma> \<in> brw_invar \<delta>"   and "brec_cond \<Sigma>" shows "brec_step rqrm Qi \<Sigma> \<in> brec_invar_add (hs.\<alpha> Qi)"
markedElementIsMarkedTrue: fixes m :: "'b"   and M :: "('b \<times> bool) list" shows "(m \<in> set (markedElements M)) = ((m, True) \<in> set M)"
inject_1: fixes x11 :: "name"   and x12 :: "name \<Rightarrow> pi noption"   and y11 :: "name"   and y12 :: "name \<Rightarrow> pi noption" shows "(BoundOutputR_Rep x11 x12 = BoundOutputR_Rep y11 y12) = (x11 = y11 \<and> x12 = y12)"
totient_def: fixes n :: "nat" shows "totient n = card (totatives n)"
holf1: shows "f holomorphic_on {z \<in> S. d \<bullet> z < k}"
add_strict_increasing: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "{#} \<subset># a"   and "b \<subseteq># c" shows "b \<subset># a + c"
simps_2: fixes l :: "int tree"   and k :: "int"   and r :: "int tree" shows "nodes \<langle>l, k, r\<rangle> = {(k, 1)} \<union> incr2 ` (nodes l \<union> nodes r)"
always_def: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and at :: "'a \<Rightarrow> nat \<Rightarrow> 'a"   and p :: "'a" assumes "class.temporal Inf Sup inf less_eq less sup bot top minus uminus at" shows "temporal.always Inf at p = Inf (range (at p))"
Sup_eqI: fixes A :: "'a set"   and x :: "'a" assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"   and "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y" shows "\<Squnion>A = x"
simplex_like_map: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "OrderingSetIso less_eq less less_eq' less' P f"   and "ordering.simplex_like less_eq less P" shows "ordering.simplex_like less_eq' less' (f ` P)"
N: shows "set n_as \<inter> UNIV \<times> {ev} = set n_as \<inter> UNIV \<times> {ev} - {0} \<times> UNIV"
weakStepSimDef_1: fixes P :: "pi"   and Rs :: "residual"   and C :: "'a"   and Rel :: "(pi \<times> pi) set" shows "weakStepSimAct P Rs C Rel \<equiv> (\<forall>Q' a x. Rs = a<\<nu>x> \<prec> Q' \<longrightarrow> x \<sharp> C \<longrightarrow> (\<exists>P'. P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel)) \<and> (\<forall>Q' a x. Rs = a<x> \<prec> Q' \<longrightarrow> x \<sharp> C \<longrightarrow> (\<exists>P''. \<forall>u. \<exists>P'. P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P' \<and> (P', Q'[x::=u]) \<in> Rel)) \<and> (\<forall>Q' \<alpha>. Rs = \<alpha> \<prec> Q' \<longrightarrow> (\<exists>P'. P \<Longrightarrow>\<^sub>l \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel))"
prime_factorization_unique'': fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "\<forall>p\<in>#M. normalization_semidom.prime zero times one normalize p"   and "\<forall>p\<in>#N. normalization_semidom.prime zero times one normalize p"   and "normalize (comm_monoid_mult.prod_mset times one {#i. i \<in># M#}) = normalize (comm_monoid_mult.prod_mset times one {#i. i \<in># N#})" shows "M = N"
freshAlphaSwap: fixes p :: "(name \<times> name) list"   and Xs :: "name set"   and Ys :: "name set"   and x :: "name" assumes "set p \<subseteq> Xs \<times> Ys"   and "Xs \<sharp>* x"   and "Ys \<sharp>* x" shows "x \<sharp> p"
lift_un_eq: fixes x :: "float interval option"   and f :: "float interval \<Rightarrow> float interval option" shows "lift_un x f = Option.bind x f"
ide_is_iso: fixes a :: "'a" assumes "B.ide a" shows "B.iso a"
subprob_space_density_not_empty: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "subprob_space (density M f)" shows "space M \<noteq> {}"
inv_max_hi_delete: fixes t :: "('b ivl \<times> 'b) tree"   and x :: "'b ivl" assumes "inv_max_hi t" shows "inv_max_hi (delete x t)"
right_neutral: fixes a :: "'d upred" shows "(a \<or> false) = a"
subcls_ObjectI1: fixes C :: "qtname"   and G :: "prog" assumes "C \<noteq> Object"   and "is_class G C"   and "ws_prog G" shows "G\<turnstile>C\<prec>\<^sub>C Object"
intro: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'f set"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "vector_space_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1"   and "vector_space_on U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2" shows "vector_space_pair_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2"
unit_simps_1: fixes a :: "'c" assumes "C.obj a" shows "C.arr \<i>\<^sub>C[a]"
ex_prop_equiv: fixes X :: "'b program set" shows "ex_prop X = (\<forall>G. (G \<in> X) = (\<forall>H. G component_of H \<longrightarrow> H \<in> X))"
emb_step_equiv: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" shows "(t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s) = ((\<exists>p d. emb_step_at p d t = s) \<and> t \<noteq> s)"
eval_fds_truncate: fixes n :: "nat"   and f :: "'a fds"   and s :: "'a" shows "eval_fds (fds_truncate n f) s = (\<Sum>k = 1..n. fds_nth f k / nat_power k s)"
set_iterator_weaken_R: fixes iti :: "('e \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e"   and S0 :: "'f set"   and R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and R' :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "set_iterator_genord iti S0 R"   and "\<And>x y. \<lbrakk>x \<in> S0; y \<in> S0; R x y\<rbrakk> \<Longrightarrow> R' x y" shows "set_iterator_genord iti S0 R'"
output_map_generat: fixes x :: "('g, 'h, 'i) generat"   and f :: "'g \<Rightarrow> 'k"   and g :: "'h \<Rightarrow> 'j"   and h :: "'i \<Rightarrow> 'l" assumes "\<not> is_Pure x" shows "generat.output (map_generat f g h x) = g (generat.output x)"
nat_add_left_cancel_less: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "(k + m < k + n) = (m < n)"
mult_2: fixes z :: "'a" shows "(2::'a) * z = z + z"
iMODb_set_UN_set_eq: shows "iMODb_set = iMODb_UN_set"
max_bag_card_upper_bound_V: fixes G :: "('c, 'd) Graph_scheme"   and T :: "('e, 'f) Graph_scheme"   and bag :: "'e \<Rightarrow> 'c set" assumes "TreeDecomposition G T bag"   and "V\<^bsub>G\<^esub> \<noteq> {}" shows "TreeDecomposition.max_bag_card T bag \<le> card V\<^bsub>G\<^esub>"
hom_dvd: fixes p :: "'c poly"   and q :: "'c poly" assumes "p dvd q" shows "map_poly to_fract p dvd map_poly to_fract q"
\<theta>_2: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"
1: shows "ord_class.filter F \<and> ord_class.filter G"
prems: shows "\<forall>a\<in>insert a A'. finite (B a)"
lselect_to_lproject: fixes w :: "'b llist"   and s :: "nat set" shows "lselect s w = lmap fst (lproject (UNIV \<times> s) (w \<bar>\<bar> iterates Suc 0))"
vp_prime: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "prime_ideal (Vr K v) (vp K v)"
C\<^sub>4s: shows "\<^bold>\<forall>x. ((\<^bold>\<forall>xa. cod x \<cdot> xa \<^bold>= xa \<^bold>\<leftarrow> E (cod x \<cdot> xa)) \<^bold>\<and> (\<^bold>\<forall>xa. xa \<cdot> cod x \<^bold>= xa \<^bold>\<leftarrow> E (xa \<cdot> cod x))) \<^bold>\<and> E (cod x \<cdot> x)"
iRelease_not_iUntil_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t'. t' \<R> t I. Q t) = (\<not> (\<not> P t'. t' \<U> t I. \<not> Q t))"
ordinal_dvd_mod: fixes y :: "ordinal"   and x :: "ordinal" shows "(y dvd x) = (x mod y = 0)"