hcju/isabelleps huggingface dataset:
map_simp_ide: fixes a :: "'a" assumes "A.ide a" shows "\<sigma>\<tau>.map a = B (\<tau> a) (\<sigma> a)"
in_lots_1: shows "mix_pmf \<alpha> p r \<in> \<P> outcomes"
c'_def: shows "c' = c * to_fract (lead_coeff p)"
divides_refl: fixes a :: "'a list" assumes "a \<in> carrier (A [X])" shows "a divides\<^bsub>A [X]\<^esub> a"
ifte_6: shows "hom_graph f (insert i S) (ifte (var i) y z) a'"
order_trans: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd" assumes "x \<preceq> y"   and "y \<preceq> z" shows "x \<preceq> z"
disjoint_family_Suc2: fixes A :: "nat \<Rightarrow> 'b set" assumes "\<And>n. A (Suc n) \<subseteq> A n" shows "disjoint_family (\<lambda>i. A i - A (Suc i))"
path_conforms_with_strategy_drop: fixes p :: "Player"   and P :: "'a llist"   and \<sigma> :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "path_conforms_with_strategy p P \<sigma>" shows "path_conforms_with_strategy p (ldropn n P) \<sigma>"
ivl_disj_int_one_2: fixes l :: "'a"   and u :: "'a" shows "{..<l} \<inter> {l..<u} = {}"
chine_assoc_in_hom: shows "\<guillemotleft>doms.chine_assoc : Chn ((local.dom \<mu> \<star> local.dom \<nu>) \<star> local.dom \<pi>) \<rightarrow>\<^sub>C Chn (local.dom \<mu> \<star> local.dom \<nu> \<star> local.dom \<pi>)\<guillemotright>"
ESem_simps_6: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and body :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> Value" shows "\<lbrakk> Terms.Let \<Gamma> body \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> body \<rbrakk>\<^bsub>\<lbrace>\<Gamma>\<rbrace>\<rho>\<^esub>"
length_suc_length: fixes n :: "nat" shows "length (bc (Suc n)) \<le> Suc (length (bc n))"
filter_splitE: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "filter P xs = ys @ zs" obtains us :: "'b list" and vs :: "'b list" where "xs = us @ vs" and "filter P us = ys" and "filter P vs = zs"
emeasure_0_AE: fixes M :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "emeasure M (space M) = 0" shows "almost_everywhere M P"
w_in_hom_1: shows "\<guillemotleft>w : src u \<rightarrow> src f\<guillemotright>"
surjD: fixes f :: "'e \<Rightarrow> 'd"   and y :: "'d" assumes "surj f" shows "\<exists>x. y = f x"
aux_series_summable: shows "summable (\<lambda>n. real_of_int (b n) / real_of_int (prod a {..n}))"
wlsFSw_recAll_termFSwMorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "wlsFSw MOD" shows "termFSwMorph (rec MOD) (recAbs MOD) MOD"
int_of_uint32_mod: fixes x :: "uint32"   and y :: "uint32" shows "int_of_uint32 (x mod y) = int_of_uint32 x mod int_of_uint32 y"
IH: fixes X :: "char list set" shows "L c X \<subseteq> rvars c \<union> X"
subst_exu: fixes \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var"   and "y \<in> var"   and "x \<noteq> y"   and "x \<notin> FvarsT t" shows "subst (exu x \<phi>) t y = exu x (subst \<phi> t y)"
arr_iff_in_hom: fixes f :: "'a2" shows "C2.arr f = \<guillemotleft>f : C2.dom f \<rightarrow>\<^sub>2 C2.cod f\<guillemotright>"
fg: fixes i :: "'a"   and j :: "'b" assumes "i \<in> Basis"   and "j \<in> Basis" shows "f i j = g i j"
induct: fixes P :: "(iface \<Rightarrow> ('b word \<times> nat) list option) \<Rightarrow> 'b common_primitive match_expr \<Rightarrow> bool"   and a0 :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and a1 :: "'b common_primitive match_expr" assumes "\<And>uu_. P uu_ MatchAny"   and "\<And>ipassmt ifce. P ipassmt (Match (OIface ifce))"   and "\<And>uv_ v. P uv_ (Match (Src v))"   and "\<And>uv_ v. P uv_ (Match (Dst v))"   and "\<And>uv_ v. P uv_ (Match (IIface v))"   and "\<And>uv_ v. P uv_ (Match (Prot v))"   and "\<And>uv_ v. P uv_ (Match (Src_Ports v))"   and "\<And>uv_ v. P uv_ (Match (Dst_Ports v))"   and "\<And>uv_ v. P uv_ (Match (MultiportPorts v))"   and "\<And>uv_ v. P uv_ (Match (L4_Flags v))"   and "\<And>uv_ v. P uv_ (Match (CT_State v))"   and "\<And>uv_ v. P uv_ (Match (Extra v))"   and "\<And>ipassmt m. P ipassmt m \<Longrightarrow> P ipassmt (MatchNot m)"   and "\<And>ipassmt m1 m2. \<lbrakk>P ipassmt m1; P ipassmt m2\<rbrakk> \<Longrightarrow> P ipassmt (MatchAnd m1 m2)" shows "P a0 a1"
igWlsAllDisj_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'd, 'e, 'i, 'j, 'k, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.igWlsAllDisj Delta MOD \<equiv> igWlsDisj MOD \<and> FixSyn.igWlsAbsDisj Delta MOD"
intro: fixes \<T> :: "'a stateful_policy" assumes "fst ` flows_fix \<T> \<subseteq> hosts \<T>"   and "snd ` flows_fix \<T> \<subseteq> hosts \<T>"   and "flows_state \<T> \<subseteq> flows_fix \<T>"   and "finite (hosts \<T>)" shows "wf_stateful_policy \<T>"
simps_3: fixes C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" shows "preT (C\<^sub>1;; C\<^sub>2) e = preT C\<^sub>1 (preT C\<^sub>2 e)"
pair_loopfree_digraph_axioms: shows "pair_loopfree_digraph (subdivide G e w)"
Act_eq_iff_perm: fixes \<alpha>1 :: "'c"   and x1 :: "('a, 'b, 'c) formula"   and \<alpha>2 :: "'c"   and x2 :: "('a, 'b, 'c) formula" shows "(Act \<alpha>1 x1 = Act \<alpha>2 x2) = (\<exists>p. supp x1 - bn \<alpha>1 = supp x2 - bn \<alpha>2 \<and> (supp x1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> x1 = x2 \<and> supp \<alpha>1 - bn \<alpha>1 = supp \<alpha>2 - bn \<alpha>2 \<and> (supp \<alpha>1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> \<alpha>1 = \<alpha>2)"
noStuck_execD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow>\<notin>{Stuck}"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "t \<noteq> Stuck"
OS_def: fixes A :: "'p"   and B :: "'p"   and P :: "'p"   and Q :: "'p" shows "A B OS P Q \<equiv> \<exists>R. A B TS P R \<and> A B TS Q R"
find_index_inj: fixes xs :: "'a list" shows "inj_on (find_index xs) (set xs)"
gen_new_keys_not_analzd: fixes K :: "nat"   and evs :: "event list"   and KK :: "nat set" assumes "Key K \<notin> used evs"   and "K \<in> symKeys"   and "evs \<in> set_mr" shows "K \<notin> keysFor (analz (Key ` KK \<union> knows Spy evs))"
Gramian_determinant_def: fixes G :: "'c vec list"   and k :: "nat" shows "Gramian_determinant G k = det (Gramian_matrix G k)"
canI_associator_0: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f"   and "partial_magma.ide V g"   and "partial_magma.ide V h"   and "src f = trg g"   and "src g = trg h" shows "\<a> f g h = bicategory.can V H \<a> \<i> src trg (bicategorical_language.Hcomp (bicategorical_language.Prim f) (bicategorical_language.Hcomp (bicategorical_language.Prim g) (bicategorical_language.Prim h))) (bicategorical_language.Hcomp (bicategorical_language.Hcomp (bicategorical_language.Prim f) (bicategorical_language.Prim g)) (bicategorical_language.Prim h))"
N: shows "f absolutely_integrable_on {y \<in> g ` S. f y < 0}"
stable_on_CH: fixes ds :: "D2 set"   and X :: "X3 set" assumes "Theorem_9.stable_on ds X" shows "Theorem_9.CH X = X"
IH: shows "list_of (\<Union> (f ` (I - {Min I}))) = concat (map (list_of \<circ> f) (list_of (I - {Min I})))"
a_Lam_inv_P: fixes v :: "Syntax.term"   and x :: "Syntax.var"   and vP' :: "Syntax.term"   and vV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "{$$} \<turnstile> v, Lam x vP', vV : Fun \<tau>\<^sub>1 \<tau>\<^sub>2" obtains v' :: "Syntax.term" and vV' :: "Syntax.term" where "v = Lam x v'" and "vV = Lam x vV'" and "{$$}(x $$:= \<tau>\<^sub>1) \<turnstile> v', vP', vV' : \<tau>\<^sub>2"
mult_mono': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq c d"   and "less_eq zero a"   and "less_eq zero c" shows "less_eq (times a c) (times b d)"
lens_member_get_override: fixes x :: "'a \<Longrightarrow> 'b"   and a :: "'b scene"   and b :: "'b"   and b' :: "'b" assumes "x \<in>\<^sub>S a" shows "get\<^bsub>x\<^esub> (b \<oplus>\<^sub>S b' on a) = get\<^bsub>x\<^esub> b'"
isidx_def: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and xs :: "nat list" assumes "Abstract_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions" shows "isidx xs = to_nat (Some xs)"
ring_iso_imp_img_domain: fixes h :: "'c \<Rightarrow> 'e"   and S :: "('e, 'f) ring_scheme" assumes "h \<in> ring_iso (S\<lparr>zero := h \<zero>\<rparr>) S" shows "domain (S\<lparr>zero := h \<zero>\<^bsub>S\<lparr>zero := h \<zero>\<rparr>\<^esub>\<rparr>)"
m_inv_consistent: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set"   and x :: "'a" assumes "Group.group G"   and "subgroup H G"   and "x \<in> H" shows "inv\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> x = inv\<^bsub>G\<^esub> x"
E0: shows "T\<^sub>p init qs Strat = (\<Sum>i<length qs. t\<^sub>p (steps' init qs Strat i) (qs ! i) (Strat ! i))"
a: shows "\<forall>xa. a xa \<in> \<rat> \<and> a xa < x \<bullet> xa \<and> x \<bullet> xa - a xa < e'"
good2: shows "is_interp (adjointFO I \<sigma> \<omega>)"
mask_inj_hlp1: shows "inj_on mask {0..16}"
lb_root: shows "\<forall>p\<in>set ps. \<forall>x. poly p x = 0 \<longrightarrow> lb < x"
map_simp_ide: fixes a :: "'a" assumes "\<tau>.A.ide a" shows "\<tau>\<tau>'.map a = B (\<tau>'.map a) (\<tau> a)"
complement_closure: fixes x :: "i" assumes "CMC"   and "\<exists>w. \<not> O w x" shows "\<exists>z. \<forall>w. P w z = (\<not> O w x)"
mem_context_val_def: fixes asi :: "word8"   and add :: "word36"   and m :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option" shows "mem_context_val asi add m \<equiv> let asi8 = word_of_int 8; r1 = m asi add in if r1 = None then m asi8 add else r1"
rep_eq: fixes x :: "('a, 'b) pfun" shows "pdom x = dom (Rep_pfun x)"
AnalBinds_not_there: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and exp :: "exp \<Rightarrow> 'd \<rightarrow> 'c"   and ae :: "Vars.var \<Rightarrow> 'd\<^sub>\<bottom>" assumes "x \<notin> domA \<Gamma>" shows "(ExpAnalysis.AnalBinds exp \<Gamma>\<cdot>ae) x = \<bottom>"
dist_fst_abs: fixes p :: "int \<times> int"   and l :: "int" shows "dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case (l, snd p) of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) = real_of_int \<bar>fst p - l\<bar>"
old_redundant_infer_iff: fixes N :: "'a literal multiset set"   and \<gamma> :: "'a Inference_System.inference" shows "src.redundant_infer N \<gamma> = (\<exists>DD\<subseteq>N. DD \<union> set_mset (old_side_prems_of \<gamma>) \<TTurnstile>e {old_concl_of \<gamma>} \<and> (\<forall>D\<in>DD. D < old_main_prem_of \<gamma>))"
Gp_inherited: fixes K :: "'a set"   and L :: "'a set" assumes "G \<guillemotright> K "   and "G \<guillemotright> L "   and "K \<subseteq> L" shows "\<natural>\<^bsub>\<natural>L\<^esub>K = \<natural>K"
left_right_inverse_eq: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'a" assumes "f \<circ> g = id"   and "g \<circ> h = id" shows "f = h"
case_prod_conv: fixes f :: "'aq \<Rightarrow> 'ar \<Rightarrow> 'ap"   and a :: "'aq"   and b :: "'ar" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
refl_onPD: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "refl_onP A r"   and "a \<in> A" shows "r a a"
IH: shows "d_OUT (h_plus i) x \<le> d_IN (h_plus i) x"
noops_valid: fixes \<Psi> :: "('a, 'b) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "is_valid_problem_sas_plus (prob_with_noop \<Psi>)"
fin: shows "finite {X \<in> \<F>. \<forall>Y\<in>\<F>. \<not> X \<subset> Y}"
dg_prod_Cod_vsv: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V" shows "vsv (dg_prod I \<AA>\<lparr>Cod\<rparr>)"
simps_6: fixes \<Sigma> :: "'b list"   and w :: "'b list"   and I :: "(nat + nat set) list"   and \<phi>\<^sub>1 :: "'b formula"   and \<phi>\<^sub>2 :: "'b formula" assumes "formula \<Sigma>" shows "formula.satisfies \<Sigma> (w, I) (FAnd \<phi>\<^sub>1 \<phi>\<^sub>2) = (formula.satisfies \<Sigma> (w, I) \<phi>\<^sub>1 \<and> formula.satisfies \<Sigma> (w, I) \<phi>\<^sub>2)"
IP: fixes \<sigma> :: "'a"   and k :: "'x"   and it :: "'x set" assumes "c \<sigma>"   and "k \<in> it"   and "\<forall>j\<in>it. k \<le> j"   and "\<forall>j\<in>\<alpha> s - it. j \<le> k"   and "it \<subseteq> \<alpha> s"   and "I it \<sigma>" shows "I (it - {k}) (f k \<sigma>)"
widens_trans: fixes P :: "'a prog"   and Ss :: "ty list"   and Ts :: "ty list"   and Us :: "ty list" assumes "P \<turnstile> Ss [\<le>] Ts"   and "P \<turnstile> Ts [\<le>] Us" shows "P \<turnstile> Ss [\<le>] Us"
step_invar: fixes s :: "'a"   and s' :: "'a" assumes "s \<in> wa_invar (det_wa_wa DWA)"   and "s \<in> wa_cond (det_wa_wa DWA)"   and "(s, s') \<in> wa_step (det_wa_wa DWA)" shows "s' \<in> wa_invar (det_wa_wa DWA)"
filter_mset_neq: fixes y :: "'b"   and M :: "'b multiset" shows "{#x \<in># M. x \<noteq> y#} = removeAll_mset y M"
section_retraction_compose: fixes e :: "'s"   and m :: "'s"   and e' :: "'s"   and m' :: "'s" assumes "S.ide (e \<cdot>\<^sub>S m)"   and "S.ide (e' \<cdot>\<^sub>S m')"   and "S.seq m' m" shows "S.ide ((e \<cdot>\<^sub>S e') \<cdot>\<^sub>S m' \<cdot>\<^sub>S m)"
invertible: fixes u :: "'a" assumes "u \<in> I" shows "additive.sub.invertible u"
symI: fixes r :: "('a \<times> 'a) set" assumes "\<And>a b. (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r" shows "sym r"
subsumeseq_term_iff: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term" shows "(t \<le>\<cdot> s) = (\<exists>\<sigma>. s = t \<cdot> \<sigma>)"
add_diff_assoc_enat: fixes z :: "enat"   and y :: "enat"   and x :: "enat" assumes "z \<le> y" shows "x + (y - z) = x + y - z"
N: shows "\<forall>n\<ge>N. dist (X n) (X N) < 1"
effectI: fixes c :: "'b Heap"   and h :: "heap"   and r :: "'b"   and h' :: "heap" assumes "execute c h = Some (r, h')" shows "effect c h h' r"
total_cons_def: fixes s :: "nat \<Rightarrow> nat option" shows "total_cons s \<equiv> s \<in> \<R>"
mk_until_def: fixes x :: "'a ltln"   and y :: "'a ltln" shows "mk_until x y \<equiv> case x of true\<^sub>n \<Rightarrow> mk_finally y | false\<^sub>n \<Rightarrow> y | _ \<Rightarrow> case y of true\<^sub>n \<Rightarrow> true\<^sub>n | false\<^sub>n \<Rightarrow> false\<^sub>n | _ \<Rightarrow> x U\<^sub>n y"
column_eq_Gram_Schmidt_matrix: fixes A :: "((real, 'a) vec, 'b) vec"   and i :: "'a"   and ia :: "'a" assumes "rank A = ncols A"   and "column i (Gram_Schmidt_matrix A) = column ia (Gram_Schmidt_matrix A)" shows "i = ia"
iso_in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<cong> b\<guillemotright>" obtains "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" and "local.iso f"
simps: fixes \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'd"   and x :: "nat"   and n :: "nat" shows "semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n = run \<delta> q\<^sub>0 (suffix x w) (n - x)"
Gateway_def: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list" shows "Gateway req dt a stop lose d ack i vc \<equiv> \<exists>i1 i2 x y. Sample req dt x stop lose ack i1 vc \<and> Delay y i1 d x i2 \<and> Loss lose a i2 y i"
c: shows "\<forall>s t. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (R s t) c s t Q,A"
IH: fixes i :: "'c" assumes "Vagree \<nu> \<nu>' (FVT (args i))" shows "dterm_sem I (args i) \<nu> = dterm_sem I (args i) \<nu>'"
asAbs_equal_imp_alphaAbs: fixes qA :: "('e, 'f, 'varSort, 'var, 'g) qAbs"   and qB :: "('e, 'f, 'varSort, 'var, 'g) qAbs" assumes "qGoodAbs qA"   and "asAbs qA = asAbs qB" shows "qA $= qB"
select_convs_3: fixes orderA :: "('a, 'c) gorder_scheme"   and orderB :: "('b, 'd) gorder_scheme"   and lower :: "'a \<Rightarrow> 'b"   and upper :: "'b \<Rightarrow> 'a"   and more :: "'e" shows "\<pi>\<^sup>*\<^bsub>\<lparr>orderA = orderA, orderB = orderB, lower = lower, upper = upper, \<dots> = more\<rparr>\<^esub> = lower"
univalent_linorder_matrix_1: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and i :: "'c"   and j :: "'c"   and k :: "'c" assumes "matrix_stone_relation_algebra.univalent f"   and "f (i, j) \<noteq> bot"   and "f (i, k) \<noteq> bot" shows "j = k"
Nil_eq_concat_conv: fixes xss :: "'a list list" shows "([] = concat xss) = (\<forall>xs\<in>set xss. xs = [])"
emeasure_pair_measure: fixes X :: "('c \<times> 'b) set"   and N :: "'c measure" assumes "X \<in> sets (N \<Otimes>\<^sub>M density M2 g)" shows "emeasure (N \<Otimes>\<^sub>M density M2 g) X = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator X (x, y) \<partial>density M2 g \<partial>N"
permutes_others: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and x :: "'a" assumes "p permutes S"   and "x \<notin> S" shows "p x = x"
memo_def: fixes f :: "'k \<Rightarrow> 'm"   and x :: "'k" shows "memo f x = get (\<lambda>table. case table x of None \<Rightarrow> bind (f x) (\<lambda>y. update (\<lambda>m. m(x \<mapsto> y)) (return y)) | Some y \<Rightarrow> return y)"
F_D_FD: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<sqsubseteq>\<^sub>F Q"   and "P \<sqsubseteq>\<^sub>D Q" shows "P \<sqsubseteq>\<^sub>F\<^sub>D Q"
well_analyzedD: fixes A :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term" assumes "well_analyzed A"   and "Decomp t \<in> set A" shows "\<exists>f T. t = Fun f T"
sig_red_single_sing_reg_cases: fixes top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" shows "sig_red_single (\<preceq>\<^sub>t) top_tail p q f t = (sig_red_single (=) top_tail p q f t \<or> sig_red_single (\<prec>\<^sub>t) top_tail p q f t)"
antisym_conv: fixes y :: "'t"   and x :: "'t" assumes "y \<preceq>\<^sub>t x" shows "(x \<preceq>\<^sub>t y) = (x = y)"
div_diff_mult_self2: fixes n :: "nat"   and m :: "nat"   and k :: "nat" shows "(n - m * k) div m = n div m - k"
ESPEC_def: fixes \<Phi> :: "'d \<Rightarrow> bool"   and \<Psi> :: "'e \<Rightarrow> bool" shows "ESPEC \<Phi> \<Psi> \<equiv> SPEC (\<lambda>x. case x of Inl e \<Rightarrow> \<Phi> e | Inr r \<Rightarrow> \<Psi> r)"
False: shows "\<nexists>W. W \<in> \<C> \<and> S \<subseteq> W"
topology_cases: fixes x :: "'c topology" obtains y :: "'c set \<Rightarrow> bool" where "x = topology y" and "y \<in> {L. istopology L}"
ltl_prop_equiv_def: fixes \<phi> :: "'b ltl"   and \<psi> :: "'b ltl" shows "\<phi> \<equiv>\<^sub>P \<psi> \<equiv> \<forall>\<A>. \<A> \<Turnstile>\<^sub>P \<phi> = \<A> \<Turnstile>\<^sub>P \<psi>"
antisymmetric_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.antisymmetric f = (\<forall>i j. i \<noteq> j \<longrightarrow> f (i, j) = bot \<or> f (j, i) = bot)"
trivial_group_subgroup_generated: fixes G :: "('a, 'b) monoid_scheme"   and S :: "'a set" assumes "Group.group G"   and "S \<subseteq> {\<one>\<^bsub>G\<^esub>}" shows "trivial_group (subgroup_generated G S)"
swapped_preserves_wls: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "wls s X"   and "(X, Y) \<in> swapped" shows "wls s Y"
rewrite_iter_fast_def: fixes \<phi> :: "'a ltln" shows "rewrite_iter_fast \<phi> \<equiv> iterate (rewrite_modal \<circ> rewrite_X) \<phi> (size \<phi>)"
prems: shows "cs\<^bsup>\<pi>\<^esup> i = xs @ [x, x'] @ []"
frac_add_le_preservation: fixes a :: "real"   and b :: "nat"   and d :: "real" assumes "a < real b"   and "d < 1 - frac a" shows "a + d < real b"
insert_closed: fixes g :: "'c"   and gens :: "'c set" assumes "g \<in> gens" shows "\<iota> g \<in> carrier \<F>\<^bsub>gens\<^esub>"
eNorm_Norm': fixes x :: "'b"   and N :: "'b quasinorm" assumes "x \<notin> space\<^sub>N N" shows "Norm N x = 0"
cnj_sum: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "cnj_class.cnj (sum f A) = (\<Sum>xa\<in>A. cnj_class.cnj (f xa))"
flatf_fp_induct_pointwise: fixes pre :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and post :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d"   and B :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd"   and a :: "'b"   and x :: "'c" assumes "\<And>a x. pre a x \<Longrightarrow> post a x b"   and "flatf_mono b B"   and "pre a x"   and "\<And>f a x. \<lbrakk>\<And>a' x'. pre a' x' \<Longrightarrow> post a' x' (f x'); pre a x; flatf_ord b f (flatf_fp b B)\<rbrakk> \<Longrightarrow> post a x (B f x)" shows "post a x (flatf_fp b B x)"
atomic_step_invariant_def: fixes s :: "state_t" shows "atomic_step_invariant s \<equiv> sp_subset s"
assms_1: shows "p\<^sub>0\<^sub>L \<noteq> p\<^sub>1\<^sub>L"
mset_exists_imply: fixes x :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b multiset" assumes "x \<in># image_mset f A" shows "\<exists>y\<in>#A. x = f y"
writes_singleton: fixes f :: "'aa \<Rightarrow> ('x, 'y, 'z) prog"   and a :: "'aa"   and h :: "'x"   and h' :: "'x" shows "writes (all_args f) (f a) h h'"
some_gcd_ff_list: fixes xs :: "'b fract list" shows "gcd_ff_list xs (some_gcd_ff_list xs)"
sgn_div_norm: fixes sgn :: "'b \<Rightarrow> 'b"   and norm :: "'b \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.sgn_div_norm sgn norm scaleR" shows "sgn x = scaleR (inverse (norm x)) x"
homeomorphic_space_sym: fixes X :: "'a topology"   and Y :: "'b topology" shows "(X homeomorphic_space Y) = (Y homeomorphic_space X)"
list_of_oalist_tl: fixes xs :: "'x" shows "list_of_oalist (local.tl xs) = tl_raw (list_of_oalist xs)"
exhaust: fixes y :: "ereal" obtains x1 :: "real" where "y = ereal x1"   | "y = PInfty"   | "y = MInfty"
leq: shows "Q2 \<le>\<^sub>L matrix_sum d (\<lambda>k. tensor_P (1\<^sub>m N) (proj_k k)) R"
map\<^sub>0_simps_2: fixes a :: "'d" assumes "D.obj a" shows "src\<^sub>D (\<eta>\<^sub>0 a) = I\<^sub>D.map\<^sub>0 a"
simps: fixes xs :: "'a"   and x :: "'b"   and ys :: "'a"   and y :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm" shows "qFreshAbs xs x (qAbs ys y X) = (xs = ys \<and> x = y \<or> qFresh xs x X)"
ENF_offending_subseteq_lhs: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar_all_edges_normal_form P" shows "set_offending_flows G nP \<subseteq> {{(e1, e2). (e1, e2) \<in> edges G \<and> \<not> P (nP e1) (nP e2)}}"
u_ts_i: shows "u_ts ! i = (u_p, u_is, u_tmps, u_sb, u_dirty, u_owns, u_rels)"
verit_minus_simplify_4: fixes b :: "'b" shows "- (- b) = b"