hcju/isabelleps huggingface dataset:
map_proj_id: fixes \<sigma>l :: "'tp list"   and al :: "univ list" assumes "list_all2 intT \<sigma>l al" shows "map2 proj \<sigma>l al = al"
simps_4: fixes P :: "'state"   and \<alpha> :: "'act"   and t :: "('b, 'pred, 'act) Tree" shows "valid_Tree P (tAct \<alpha> t) = (\<exists>\<alpha>' t' P'. tAct \<alpha> t =\<^sub>\<alpha> tAct \<alpha>' t' \<and> P \<rightarrow> \<langle>\<alpha>',P'\<rangle> \<and> valid_Tree P' t')"
l11_10: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and A' :: "'a"   and C' :: "'a"   and D' :: "'a"   and F' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F"   and "Tarski_neutral_dimensionless.Out Bet B A' A"   and "Tarski_neutral_dimensionless.Out Bet B C' C"   and "Tarski_neutral_dimensionless.Out Bet E D' D"   and "Tarski_neutral_dimensionless.Out Bet E F' F" shows "Tarski_neutral_dimensionless.CongA Bet Cong A' B C' D' E F'"
list_all_simps_1: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "list_all P (x # xs) = (P x \<and> list_all P xs)"
homeomorphism_on_compact: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on S f"   and "compact S"   and "inj_on f S" shows "homeomorphism_on S f"
last_messageI2_aux: fixes i :: "nat"   and xs :: "'b message_af list" assumes "i < length xs"   and "xs ! i \<noteq> NoMsg"   and "\<forall>j. i < j \<and> j < length xs \<longrightarrow> xs ! j = NoMsg" shows "last_message xs = xs ! i"
list_all2_map2: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and as :: "'b list"   and f :: "'d \<Rightarrow> 'c"   and bs :: "'d list" shows "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
matrix_eq: fixes A :: "(('a, 'b) vec, 'c) vec"   and B :: "(('a, 'b) vec, 'c) vec" shows "(A = B) = (\<forall>x. A *v x = B *v x)"
finite_set_mset: fixes M :: "'e multiset" shows "finite (set_mset M)"
simpnum_nz: fixes t :: "MIR.num" shows "nozerocoeff (simpnum t)"
simps_8: fixes e\<^sub>1 :: "'a exp"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "'a exp"   and A :: "'a set option" shows "\<D> (e\<^sub>1\<bullet>F{D} := e\<^sub>2) A = (\<D> e\<^sub>1 A \<and> \<D> e\<^sub>2 (A \<squnion> \<A> e\<^sub>1))"
nonneg: shows "AE x in M. 0 \<le> f x"
pell_solution_exists: shows "\<exists>x y. y \<noteq> 0 \<and> x\<^sup>2 = 1 + D * y\<^sup>2"
smult_assoc: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and r :: "'a"   and s :: "'a"   and m :: "'b" assumes "RModule R smult M"   and "r \<in> R"   and "s \<in> R"   and "m \<in> M" shows "smult r (smult s m) = smult (r * s) m"
simps_4: fixes x5 :: "'a OCL_Types.type"   and y5 :: "'a OCL_Types.type" shows "(Set x5 = Set y5) = (x5 = y5)"
Union_eq: fixes A :: "'c set set" shows "\<mu> A = {x. \<exists>B\<in>A. s2p B x}"
Trg_Cod: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Arr V src trg t" shows "bicategorical_language.Trg trg (bicategorical_language.Cod V src trg t) = bicategorical_language.Trg trg t"
DList_set_empty: shows "DList_set DList_Set.empty = {}"
divide_less_eq_numeral1_1: fixes b :: "'c"   and w :: "num"   and a :: "'c" shows "(b / numeral w < a) = (b < a * numeral w)"
subcls1D: fixes C :: "cname"   and D :: "cname" assumes "C \<prec>C1 D" shows "C \<noteq> Object \<and> (\<exists>c. class C = Some c \<and> super c = D)"
ZObis_raw_coind: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" assumes "theta \<subseteq> ZOretr theta" shows "theta \<subseteq> ZObis"
lhd_LCons: fixes x21 :: "'b"   and x22 :: "'b llist" shows "lhd (x21 ## x22) = x21"
divs_of_zero_UNIV: shows "divisors 0 = UNIV"
reduce_ivle_def: fixes Y :: "('c \<times> 'd) set"   and b :: "'d" shows "reduce_ivle Y b = scaleR2_rep Y \<bind> (\<lambda>((l, u), X). reduce_ivl X b \<bind> (\<lambda>R. scaleR2_rep R \<bind> (\<lambda>((l', u'), R). CHECK (\<lambda>_. ()) (0 < l' \<and> 0 < l \<and> 0 \<le> u \<and> l \<le> u \<and> l' \<le> u') \<bind> (\<lambda>_. scaleRe_ivl_spec (l' * l) (u' * u) R))))"
map_add_SomeD: fixes m :: "'b \<Rightarrow> 'a option"   and n :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and x :: "'a" assumes "(m ++ n) k = Some x" shows "n k = Some x \<or> n k = None \<and> m k = Some x"
formulas_var: fixes i :: "'a"   and S :: "'a set" assumes "i \<in> S" shows "var i \<in> formulas S"
f: shows "i < mult.row_length (mat_to_cols_list A) * mult.row_length (mat_to_cols_list B)"
altc_sample_option_def: fixes altc_sample :: "'c \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b"   and p :: "'c"   and f :: "'d \<Rightarrow> ('a, 'b) optionT" shows "altc_sample_option altc_sample p f = OptionT (altc_sample p (\<lambda>x. run_option (f x)))"
comp_null_2: fixes f :: "'a" shows "Left_a.comp f Left_a.null = Left_a.null"
inject: fixes x1 :: "'c"   and x2 :: "'d"   and y1 :: "'c"   and y2 :: "'d" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
interchange: fixes h :: "'a"   and g :: "'a"   and h' :: "'a"   and g' :: "'a" assumes "seq h g"   and "seq h' g'" shows "(h \<otimes>\<^sub>1 h') \<cdot> (g \<otimes>\<^sub>1 g') = h \<cdot> g \<otimes>\<^sub>1 h' \<cdot> g'"
xfer_reachable: fixes cs :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)" assumes "B.reachable cs" obtains as :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)" where "sim_rel1 as cs" and "A.reachable as"
while_type: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and e :: "'BExp"   and c :: "('Var, 'AExp, 'BExp) Stmt" assumes "\<Gamma> \<turnstile>\<^sub>b e \<in> Low "   and "\<turnstile> \<Gamma> {c} \<Gamma>" shows "\<turnstile> \<Gamma> {While e c} \<Gamma>"
conjugate_char_1_pp: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" shows "conjugate f g = (\<forall>x y. f (x \<sqinter> - g y) \<le> - - f x \<sqinter> - y \<and> g (y \<sqinter> - f x) \<le> - - g y \<sqinter> - x)"
DERIV_nonneg_imp_mono: fixes x :: "real"   and y :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real" assumes "\<And>t. t \<in> {x..y} \<Longrightarrow> (f has_real_derivative f' t) (at t)"   and "\<And>t. t \<in> {x..y} \<Longrightarrow> 0 \<le> f' t"   and "x \<le> y" shows "f x \<le> f y"
q0_def: shows "q0 = Cring_Poly.truncate R q"
lfp_exp_sound: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and R :: "'a \<Rightarrow> real" assumes "t R \<tturnstile> R"   and "sound R" shows "sound (lfp_exp t)"
tutm_uhalt: fixes tp :: "(action \<times> nat) list"   and l :: "nat"   and args :: "'a"   and m :: "nat"   and n :: "nat" assumes "tm_wf0 tp"   and "\<forall>stp. \<not> TSTD (steps0 (Suc 0, Bk \<up> l, <args>) tp stp)" shows "\<forall>stp. \<not> is_final (steps0 (Suc 0, Bk \<up> m, <[code tp, bl2wc (<args>)]> @ Bk \<up> n) t_utm stp)"
orthogonal_scaleR: fixes c :: "real"   and x :: "'a" assumes "c \<noteq> 0" shows "orthogonal (c *\<^sub>R x) = orthogonal x"
len_hchop_left_left_border: fixes v :: "view"   and ts :: "traffic"   and c :: "cars"   and v1 :: "view"   and v2 :: "view" shows "len v ts c = ext v \<and> v=v1\<parallel>v2 \<longrightarrow> left (len v1 ts c) = left (ext v1)"
assms_1: shows "p = {l}receive(fmsg) . qq"
r_result_empty_stack_None: fixes i :: "nat"   and f :: "recf"   and x :: "nat"   and t :: "nat" assumes "i = encode f"   and "recfn (e_length x) f"   and "iterate t step ([(f, Partial_Recursive.list_decode x, [])], None) = ([], None)" shows "eval r_result [t, i, x] \<down>= 0"
not_splitted: shows "\<not> M.splitted (\<sigma> P)"
assms_7: shows "x1 \<notin> FvarsT t2"
A_def: shows "A = (\<Union>i\<in>I \<inter> {i. U i \<inter> {z<..T} \<noteq> {}}. U i \<inter> {z..T})"
mkProc_gState_progress_rel: fixes prog :: "program"   and g :: "'a gState_scheme"   and p :: "pState"   and name :: "String.literal"   and args :: "expr list"   and sidx :: "nat"   and pidN :: "nat" assumes "gState_inv prog g" shows "(g, fst (mkProc g p name args (processes prog !! sidx) pidN)) \<in> gState_progress_rel prog"
matched_Nil: fixes n :: "'node SDG_node" assumes "valid_SDG_node n" shows "matched n [] n"
push_precondI: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and u :: "nat"   and v :: "nat"   and l :: "nat \<Rightarrow> nat" assumes "(0::'capacity) < excess f u"   and "(u, v) \<in> cfE_of f"   and "l u = l v + 1" shows "push_precond f l (u, v)"
transitive_subset: fixes B :: "'a set" assumes "B \<subseteq> UNIV" shows "transitive B ((\<sqsubset>) \<restriction> A)"
extremum: fixes a :: "'g multiset" shows "{#} \<subseteq># a"
invar_NodeL: fixes l :: "('c \<times> nat) tree"   and x :: "'c"   and n :: "nat"   and r :: "('c \<times> nat) tree"   and l' :: "('c \<times> nat) tree" assumes "invar \<langle>l, (x, n), r\<rangle>"   and "invar l'"   and "lvl l' = lvl l" shows "invar \<langle>l', (x, n), r\<rangle>"
Expr2Data: fixes f :: "ZF"   and e :: "(ZF, ZF) Expression" shows "ZFData (Expr2ZF f E@ e) = Opair f (Expr2ZF e)"
process_set_def: shows "process_set \<equiv> {P. process_prop_1 P \<and> process_prop_2 P \<and> process_prop_3 P \<and> process_prop_4 P \<and> process_prop_5 P \<and> process_prop_6 P}"
p_def: fixes t :: "real" shows "other.p t = s\<^sub>o + v\<^sub>o * t + 1 / 2 * a\<^sub>o * t\<^sup>2"
eval_InsInitE: fixes G :: "prog"   and s :: "st"   and c :: "stmt"   and e :: "expr"   and v :: "val"   and s' :: "state" shows "G\<turnstile>Norm s \<midarrow>InsInitE c e-\<succ>v\<rightarrow> s' = False"
lsorted_coinduct': fixes X :: "'a llist \<Rightarrow> bool"   and xs :: "'a llist" assumes "X xs"   and "\<And>xs. \<lbrakk>X xs; \<not> lnull xs; \<not> lnull (ltl xs)\<rbrakk> \<Longrightarrow> lhd xs \<le> lhd (ltl xs) \<and> (X (ltl xs) \<or> lsorted (ltl xs))" shows "lsorted xs"
f'_cong: fixes Y :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "Y = Y"   and "\<And>x t. x \<in> Y \<Longrightarrow> f x = g x"   and "x \<in> Y" shows "(g has_derivative blinfun_apply (f' x)) (at x)"
degree_map_poly_le: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b poly" shows "degree (map_poly f p) \<le> degree p"
fempty_fminus: fixes A :: "'b fset" shows "{||} |-| A = {||}"
certificate_valid_pubSK: fixes C :: "agent"   and SKi :: "nat"   and evs :: "event list" assumes "cert C SKi onlySig (priSK RCA) \<in> parts (knows Spy evs)"   and "evs \<in> set_cr" shows "SKi = pubSK C"
reflexiveE: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "reflexive A r" obtains "\<And>x. x \<in> A \<Longrightarrow> r x x"
psp_stable2: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set" assumes "F \<in> A \<longmapsto> A'"   and "F \<in> stable B" shows "F \<in> B \<inter> A \<longmapsto> B \<inter> A'"
ineq_model_singleI: fixes X :: "'a list"   and t :: "('b, 'a) Term.term"   and \<I> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and t' :: "('b, 'a) Term.term" assumes "\<forall>\<delta>. subst_domain \<delta> = set X \<and> ground (subst_range \<delta>) \<longrightarrow> t \<cdot> \<delta> \<cdot> \<I> \<noteq> t' \<cdot> \<delta> \<cdot> \<I>" shows "ineq_model \<I> X [(t, t')]"
Yr: fixes y :: "'a" assumes "y \<in> Y" shows "returns_to {x \<in> S. s x = 0} y"
map_removeAll_inj: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and xs :: "'b list" assumes "inj f" shows "map f (removeAll x xs) = removeAll (f x) (map f xs)"
eq: fixes x :: "'a" assumes "x \<in> set_pmf p" shows "pmf p x = pmf q (f x)"
are_open_subspaces: fixes i :: "real" assumes "i \<in> I" shows "is_open (C i)"
run_resource_def: fixes resource :: "('c, 'd) resource" shows "run_resource resource = (case resource of Resource x \<Rightarrow> x)"
Q_freelist_freeword': fixes xs :: "('w \<times> bool) list" assumes "proper_signed_list xs"   and "order.greater_eq S (fst ` set xs)"   and "Abs_freelistfst xs \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>" shows "Abs_freeword xs \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
set_cases: fixes e :: "'b"   and a :: "'b message_af" assumes "e \<in> set_message_af a" obtains "a = Msg e"
fdisjnt_subset_left: fixes N' :: "'c fset"   and N :: "'c fset"   and M :: "'c fset" assumes "N' |\<subseteq>| N"   and "fdisjnt N M" shows "fdisjnt N' M"
sglob_val_def: fixes s :: "s"   and i :: "nat"   and j :: "nat" shows "sglob_val s i j = g_val (sglob s i j)"
x: shows "(length (nat_to_bv (p * q)) - Suc 0) mod 8 \<noteq> 0"
lin_part_eq: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "f = lin_part f \<oplus>\<^bsub>P\<^esub> shift 2 f \<otimes>\<^bsub>P\<^esub> monom P \<one> 2"
starprime_def: shows "starprime = *s* {p. prime p}"
assms: shows "R\<^sup>+\<^sup>+ x y"
homI: fixes f :: "'a set_arrow"   and A :: "'a set"   and B :: "'a set" assumes "f \<in> Ar\<^bsub>BB\<^esub>"   and "Dom\<^bsub>BB\<^esub> f = A"   and "Cod\<^bsub>BB\<^esub> f = B" shows "f \<in> Hom\<^bsub>BB\<^esub> A B"
UN_constant: fixes c :: "'c set"   and A :: "'d set" shows "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
bigtheta_powr_def: fixes F :: "'a filter"   and A :: "('a \<Rightarrow> real) set"   and y :: "real" shows "bigtheta_powr F A y = (if y = 0 then {f. \<exists>g\<in>A. eventually_nonneg F g \<and> f \<in> \<Theta>[F](\<lambda>x. g x powr y)} else {f. \<exists>g\<in>A. eventually_nonneg F g \<and> (\<forall>x. norm (f x) = g x powr y)})"
S_def: shows "S = {Idl S' |S'. S' \<subseteq> I \<and> finite S'}"
\<Z>_srngD_4: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_srng \<alpha> \<SS>" shows "\<Z>_sgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vmult\<rparr>]\<^sub>\<circ>"
ord_resolve_sound: fixes CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and E :: "'a literal multiset"   and I :: "'a set"   and \<eta> :: "'s" assumes "ord_resolve CAs DA AAs As \<sigma> E"   and "\<And>\<sigma>. is_ground_subst \<sigma> \<Longrightarrow> I \<Turnstile>m (mset CAs + {#DA#}) \<cdot>cm \<sigma>"   and "is_ground_subst \<eta>" shows "I \<Turnstile> E \<cdot> \<eta>"
streams_iff_snth: fixes s :: "'c stream"   and X :: "'c set" shows "(s \<in> streams X) = (\<forall>n. s !! n \<in> X)"
nextl_Power_dfa: fixes qs :: "hf"   and u :: "'a list" assumes "qs \<in> dfa.states Power_dfa" shows "Power.nextl qs u = HF (\<Union>q\<in>hfset qs. nextl {q} u)"
SUP_least: fixes Inf :: "'h set \<Rightarrow> 'h"   and Sup :: "'h set \<Rightarrow> 'h"   and inf :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and sup :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and bot :: "'h"   and top :: "'h"   and A :: "'i set"   and f :: "'i \<Rightarrow> 'h"   and u :: "'h" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) u" shows "less_eq (Sup (f ` A)) u"
matrix_vector_mul_assoc: fixes A :: "(('e, 'g) vec, 'f) vec"   and B :: "(('e, 'h) vec, 'g) vec"   and x :: "('e, 'h) vec" shows "A *v (B *v x) = A ** B *v x"
ord_spmf_eqD_pmf_None: fixes x :: "'b spmf"   and y :: "'b spmf" assumes "ord_spmf (=) x y" shows "pmf y None \<le> pmf x None"
less_eq_node_ptr_def: fixes x :: "(_) node_ptr"   and y :: "(_) node_ptr" shows "x \<le> y \<equiv> case x of node_ptr.Ext i \<Rightarrow> case y of node_ptr.Ext j \<Rightarrow> i \<le> j"
column_transpose: fixes i :: "'c"   and A :: "(('a, 'b) vec, 'c) vec" shows "column i (Finite_Cartesian_Product.transpose A) = row i A"
compute_list_of_pdevs: fixes xs :: "(nat, 'd) slist" shows "list_of_pdevs (Pdevs xs) = list_of_slist xs"
two_composable_identity_arrows_of_spans_axioms: shows "two_composable_identity_arrows_of_spans (\<cdot>) prj0 prj1 (\<nu> \<star> \<pi>) \<rho>"
cases: fixes a :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and R' :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M :: "'b"   and N :: "'b" assumes "a \<in> ext R R' M N" obtains r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent" and seq :: "('a, 'b) sequent" where "a = extendRule seq__ r__" and "r__ \<in> R" and "case r__ of (x, xa) \<Rightarrow> (x, xa) \<in> Ax"   | r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent" and seq :: "('a, 'b) sequent" where "a = extendRule seqa__ ra__" and "ra__ \<in> R" and "case ra__ of (x, xa) \<Rightarrow> (x, xa) \<in> upRules"   | r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent" and seq :: "('a, 'b) sequent" where "a = extendConc seqb__ rb__" and "rb__ \<in> p_e R' M N" and "rb__ \<in> R"   | r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent" and seq :: "('a, 'b) sequent" where "a = extendRule seq r" and "r \<in> R" and "case r of (x, xa) \<Rightarrow> (x, xa) \<in> modRules2"
rel_distinct_26: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and y5 :: "'b val"   and x4 :: "'c val" shows "\<not> rel_val R (val.Left y5) (val.Right x4)"
mu_fusion_equal_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b \<Rightarrow> 'b"   and u :: "'b \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "class.order less_eq less"   and "ord.galois less_eq l u"   and "ord.isotone less_eq h"   and "order.has_least_prefixpoint less_eq g"   and "order.has_least_prefixpoint less_eq h" shows "less_eq (l (g (u (order.the_least_fixpoint less_eq h)))) (h (l (u (order.the_least_fixpoint less_eq h)))) \<and> l (g (order.the_least_prefixpoint less_eq g)) = h (l (order.the_least_prefixpoint less_eq g)) \<longrightarrow> order.the_least_prefixpoint less_eq h = l (order.the_least_prefixpoint less_eq g) \<and> order.the_least_fixpoint less_eq h = l (order.the_least_prefixpoint less_eq g)"
both_contains_arg1: fixes t :: "'a ttree"   and t' :: "'a ttree" shows "paths t \<subseteq> paths (t \<otimes>\<otimes> t')"
elims: fixes x :: "'c \<Rightarrow> 'b set"   and xa :: "'c option"   and y :: "'b set" assumes "\<gamma>_option x xa = y" obtains \<gamma> :: "'c \<Rightarrow> 'b set" where "x = \<gamma>'__" and "xa = None" and "y = {}"   | \<gamma> :: "'c \<Rightarrow> 'b set" and a :: "'c" where "x = \<gamma>" and "xa = Some a" and "y = \<gamma> a"
IH_1: fixes x21 :: "bexp"   and x22 :: "int"   and pc' :: "nat" assumes "pc < array_length ins"   and "array_get ins pc = TestI x21 x22"   and "bval x21 s"   and "pc + 1 < pc_bound ins"   and "pc' \<in> set (exec' ins s (pc + 1))" shows "pc' < pc_bound ins"
pseudominimal_inD1: fixes P :: "'a set"   and x :: "'a" assumes "domain.pseudominimal_in P x" shows "x \<in> P"
less_option_def: fixes x :: "'b option"   and y :: "'b option" shows "(x < y) = (case y of None \<Rightarrow> False | Some y \<Rightarrow> case x of None \<Rightarrow> True | Some x \<Rightarrow> x < y)"
path_connectedin_def: fixes X :: "'d topology"   and S :: "'d set" shows "path_connectedin X S \<equiv> S \<subseteq> topspace X \<and> path_connected_space (subtopology X S)"
stream_dec_def: fixes n :: "nat"   and FO :: "nat set"   and s :: "('a \<times> bool list) stream" shows "stream_dec n FO s = map (\<lambda>i. if i \<in> FO then Inl (the_elem (positions_in_row s i)) else Inr (positions_in_row s i)) [0..<n]"
g_def: shows "g = (\<lambda>p b. if b \<in> B then a_to_b (p (b_to_a b)) else b)"
Let_4: shows "\<turnstile> p' : T \<Rightarrow> \<Delta>"
dom_trm_def: fixes t :: "'a trm"   and C :: "'a literal set" shows "dom_trm t C = (\<exists>L u v p. L \<in> C \<and> decompose_literal L u v p \<and> (p = neg \<and> t = u \<or> (t, u) \<in> trm_ord))"
comp_ide_arr: fixes b :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "CC.ide b"   and "CC.seq b f" shows "CC.comp b f = f"
basic_gt_iff_gt: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" shows "(t >\<^sub>t s) = local.kbo_std.gt t s"
vlrestriction_on_vempty: fixes r :: "V" shows "r \<restriction>\<^sup>l\<^sub>\<circ> 0 = 0"
catalan_eq_I: fixes n :: "nat" shows "real (catalan n) = I n / (2 * pi)"
assms_2: shows "f \<in> stock_measure t \<rightarrow>\<^sub>M stock_measure t'"
Rep_Abs_mod: fixes m :: "int" shows "Rep (Abs (m mod n)) = m mod n"
hom_induced_reduced_hom: fixes p :: "int"   and X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "hom_induced p X {} Y {} f \<in> hom (reduced_homology_group p X) (reduced_homology_group p Y)"
force_enabled: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and \<omega> :: "'a stream" assumes "MC_syntax.enabled K x \<omega>" shows "MC_syntax.force_enabled K x \<omega> = \<omega>"
trace_eq_simcl_bind: fixes p :: "'a spmf"   and S :: "'b spmf \<Rightarrow> 'c spmf \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b spmf"   and g :: "'a \<Rightarrow> 'c spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> trace_eq_simcl S (f x) (g x)" shows "trace_eq_simcl S (p \<bind> f) (p \<bind> g)"
set_less_aux_insert_same: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "(x \<in> A) = (x \<in> B)" shows "ord.set_less_aux less_eq (insert x A) (insert x B) = ord.set_less_aux less_eq A B"
Aenc_Sign: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Aenc X Y \<noteq> Sign X' Y'"
ord_of_nat_succ: fixes a :: "nat" shows "ZFC_in_HOL.succ (a\<^sub>\<nat>) = Suc a\<^sub>\<nat>"
vproductD_3: fixes f :: "V"   and I :: "V"   and A :: "V \<Rightarrow> V" assumes "f \<in>\<^sub>\<circ> vproduct I A" shows "\<forall>i\<in>elts I. f\<lparr>i\<rparr> \<in>\<^sub>\<circ> A i"
face_of_Int_supporting_hyperplane_le_strong: fixes S :: "'b set"   and a :: "'b"   and b :: "real" assumes "convex (S \<inter> {x. a \<bullet> x = b})"   and "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b" shows "S \<inter> {x. a \<bullet> x = b} face_of S"
equal_real_def: fixes x :: "real"   and y :: "real" shows "equal_class.equal x y = (x - y = 0)"
exhaust: fixes y :: "('a, 'b) chain" obtains x1 :: "'a" where "y = \<lbrace> x1 \<rbrace>"   | x21 :: "'a" and x22 :: "'b" and x23 :: "('a, 'b) chain" where "y = \<lbrace> x21 \<rbrace> \<cdot> x22 \<cdot> x23"
has_integral_const_real: fixes c :: "'a"   and a :: "real"   and b :: "real" shows "((\<lambda>x. c) has_integral content {a..b} *\<^sub>R c) {a..b}"
simps_39: fixes x5 :: "msg"   and x4 :: "nat" shows "Hash x5 \<noteq> Key x4"
minim_in: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "B \<noteq> {}" shows "minim B \<in> B"
self_append_conv: fixes xs :: "'c list"   and ys :: "'c list" shows "(xs = xs @ ys) = (ys = [])"
ind: fixes as :: "nat"   and am :: "nat list" assumes "as < length ap"   and "abc_steps_l (0, lm) ap stp = (as, am)" shows "\<exists>stpaa\<ge>stp. crsp ly (as, am) (steps0 (Suc 0, l, r) tp stpaa) ires"
nadds_red_nonzero: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "\<And>q. q \<in> B \<Longrightarrow> \<not> lt q adds\<^sub>t lt p"   and "red B p r" shows "r \<noteq> 0"
upto_enum_triv: fixes x :: "'a" shows "[x .e. x] = [x]"
CACH_BodiesDerivable: fixes C :: "Class"   and m :: "Method"   and par :: "Var list"   and code :: "(int \<times> Instr) list"   and l :: "int"   and F :: "((Class \<times> Method) \<times> Var list \<times> Expr) list"   and \<Sigma> :: "((Class \<times> Method) \<times> nat) list" assumes "mbody_is C m (par, code, l)"   and "compileProg F"   and "TP_MST \<Sigma>"   and "TP \<Sigma> F" shows "\<exists>n. MST\<down>(C, m) = Some (mkSPEC (Cachera n) emp) \<and> deriv [] C m l (Cachera n)"