hcju/isabelleps huggingface dataset:
subjects_reduction2: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and es :: "char list exp list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and es' :: "char list exp list"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and E :: "char list \<Rightarrow> ty option"   and Ts :: "ty list" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>es,(h, l)\<rangle> [\<rightarrow>] \<langle>es',(h', l')\<rangle>"   and "P,E \<turnstile> (h, l) \<surd>"   and "P,E,h \<turnstile> es [:] Ts" shows "\<exists>Ts'. P,E,h' \<turnstile> es' [:] Ts' \<and> P \<turnstile> Ts' [\<le>] Ts"
inter_is_a_intersec: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points" shows "is_a_intersec (Pascal_Property.inter (line A B) (line C D)) A B C D"
ty_dom_vrs: fixes G :: "binding list"   and x :: "vrs"   and Q :: "ty"   and D :: "binding list" shows "ty_dom (G @ [VarB x Q] @ D) = ty_dom (G @ D)"
less_5: shows "\<forall>j\<ge>i. x \<in> lnth Ys j"
Collect_S2P: fixes r :: "'c set" shows "Collect (S2P r) = r"
CFG_def: fixes \<alpha>e :: "'d \<Rightarrow> ('e \<times> 'f \<times> 'e) set"   and \<alpha>n :: "'d \<Rightarrow> 'e list"   and invar :: "'d \<Rightarrow> bool"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list"   and Entry :: "'d \<Rightarrow> 'e"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set"   and "uses" :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set" shows "CFG \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<equiv> graph_Entry \<alpha>e \<alpha>n invar inEdges' Entry \<and> CFG_axioms \<alpha>n invar defs uses"
rl: fixes q :: "nat" assumes "q < 2 ^ (LENGTH('a) - n)" shows "to_bl (2 ^ n * word_of_nat q) = drop n (to_bl (word_of_nat q)) @ replicate n False"
enat_ord_code_5: fixes n :: "nat" shows "(\<infinity> \<le> enat n) = False"
C_flds: shows "P \<turnstile> C has_fields FDTs"
assms: shows "a $ 0 \<noteq> (0::'a)"
eliminate_entries_i_def: fixes ops :: "'b arith_ops_record" shows "eliminate_entries_i ops \<equiv> eliminate_entries_gen_zero (arith_ops_record.minus ops) (arith_ops_record.times ops) (arith_ops_record.zero ops)"
drop_map_lift_map_eq: fixes i :: "nat"   and s :: "'c \<times> (nat \<Rightarrow> 'c) \<times> 'd" shows "drop_map i (lift_map i s) = s"
subset_Diff_insert: fixes A :: "'c set"   and B :: "'c set"   and x :: "'c"   and C :: "'c set" shows "(A \<subseteq> B - insert x C) = (A \<subseteq> B - C \<and> x \<notin> A)"
p_prod_iff: fixes a :: "'a"   and A :: "'a set"   and R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set" shows "((a, A) \<in> R \<parallel> S) = (\<exists>B C. A = B \<union> C \<and> (a, B) \<in> R \<and> (a, C) \<in> S)"
d0: shows "v1 = Matrix.mat 8 1 (\<lambda>(i, j). [[\<phi> $$ (0, 0), \<phi> $$ (1, 0), 0, 0, 0, 0, 0, 0]] ! j ! i)"
finite_greaterThanLessThan_int: fixes l :: "int"   and u :: "int" shows "finite {l<..<u}"
simps_50: fixes x7 :: "('a, 'b, 'c) expr"   and x4 :: "('a, 'b, 'c) expr" shows "Rfork x7 \<noteq> Ref x4"
problem_plan_bound_S_bound_2nd_step: fixes PROB :: "(('b, bool) fmap \<times> ('b, bool) fmap) set"   and vs :: "'b set"   and lss :: "('b, bool) fmap list"   and s :: "('b, bool) fmap"   and as :: "(('b, bool) fmap \<times> ('b, bool) fmap) list" assumes "finite PROB"   and "top_sorted_abs (\<lambda>x y. y \<in> state_successors (prob_proj PROB vs) x) lss"   and "set lss = valid_states (prob_proj PROB vs)"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "\<exists>as'. exec_plan s as' = exec_plan s as \<and> subseq as' as \<and> length as' \<le> S vs lss PROB (fmrestrict_set vs s)"
fmlookup_fmmap_keys: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and m :: "('b, 'c) fmap"   and x :: "'b" shows "fmlookup (fmmap_keys f m) x = map_option (f x) (fmlookup m x)"
iprev_nth_Suc_mono: fixes I :: "nat set"   and n :: "nat" shows "I \<leftarrow> Suc n \<le> I \<leftarrow> n"
E: shows "\<alpha> \<circ> r1 = \<alpha> \<circ> r2"
preserves_src: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "Span.src (SPN \<mu>) = SPN.map\<^sub>0 (src \<mu>)"
fls_shift_const_nth: fixes m :: "int"   and c :: "'a"   and n :: "int" shows "fls_nth (fls_shift m (fls_const c)) n = (if n = - m then c else (0::'a))"
eq: fixes z :: "complex" assumes "z \<in> S"   and "Im z \<le> 0"   and "0 \<le> Im z" shows "f z = cnj (f (cnj z))"
Col_def: 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" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Col Bet A B C \<equiv> Bet A B C \<or> Bet B C A \<or> Bet C A B"
indep_set_def: fixes A :: "'a set set"   and B :: "'a set set" shows "indep_set A B = indep_sets (case_bool A B) UNIV"
closed_in_cons: fixes S :: "ty"   and \<Gamma> :: "binding list"   and X :: "vrs"   and Q :: "ty"   and \<Delta> :: "binding list" assumes "S closed_in (\<Gamma> @ VarB X Q # \<Delta>)" shows "S closed_in (\<Gamma> @ \<Delta>)"
eval_monom_Cons: fixes c :: "real"   and e :: "real"   and es :: "real list"   and b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" shows "eval_monom (c, e # es) (b # basis) = (\<lambda>x. eval_monom (c, es) basis x * b x powr e)"
integrable_on_1_iff: fixes f :: "'a \<Rightarrow> (real, 1) vec"   and S :: "'a set" shows "(f integrable_on S) = ((\<lambda>x. f x $ 1) integrable_on S)"
crypts_hash: fixes KS :: "key list"   and X :: "msg"   and H :: "msg set" assumes "foldr Crypt KS (Hash X) \<in> crypts H" shows "foldr Crypt KS X \<in> crypts H"
abc_only_cba_2: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "betw a b c" shows "\<not> betw a c b"
induct: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and x1 :: "ty"   and x2 :: "val"   and x3 :: "val"   and Pa :: "ty \<Rightarrow> val \<Rightarrow> val \<Rightarrow> bool" assumes "P \<turnstile> x1 casts x2 to x3 "   and "\<And>T v. \<forall>C. T \<noteq> Class C \<Longrightarrow> Pa T v v"   and "\<And>C. Pa (Class C) Null Null"   and "\<And>Cs C Cs' Ds a. \<lbrakk>P \<turnstile> Path last Cs to C via Cs' ; Ds = Cs @\<^sub>p Cs'\<rbrakk> \<Longrightarrow> Pa (Class C) (Ref (a, Cs)) (Ref (a, Ds))" shows "Pa x1 x2 x3"
size_neq_size_imp_neq: fixes x :: "'d"   and y :: "'d" assumes "size x \<noteq> size y" shows "x \<noteq> y"
wf_osig_iff_exe_wf_osig: fixes oss :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" shows "wf_osig (translate_osig oss) = exe_wf_osig oss"
local_lipschitz_continuous_on: fixes T :: "'b set"   and X :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and t :: "'b" assumes "local_lipschitz T X f"   and "t \<in> T" shows "continuous_on X (f t)"
seql_def: fixes i :: "'e"   and P :: "'f \<times> 'g \<Rightarrow> bool" shows "seql i P \<equiv> \<lambda>(\<sigma>, p). P (\<sigma> i, p)"
iall_subset_imp_iall: fixes B :: "nat set"   and P :: "nat \<Rightarrow> bool"   and A :: "nat set" assumes "\<box> t B. P t"   and "A \<subseteq> B" shows "\<box> t A. P t"
degen_run_complete0: fixes r :: "nat \<Rightarrow> 'Q"   and m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b" assumes "num_acc = 0"   and "is_run r" shows "graph_defs.is_run (degeneralize_ext m) (\<lambda>i. (r i, 0))"
open_float_def: fixes S :: "float set" shows "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist (real_of_float y) (real_of_float x) < e \<longrightarrow> y \<in> S)"
funion_fempty_right: fixes A :: "'b fset" shows "A |\<union>| {||} = A"
equivP_pick_iff: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'c set"   and Y :: "'c set" assumes "equivP P \<phi>"   and "(P /// \<phi>) X"   and "(P /// \<phi>) Y" shows "\<phi> (pick X) (pick Y) = (X = Y)"
CFG_wf_def: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set" shows "CFG_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<equiv> CFG \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<and> CFG_wf_axioms \<alpha>n invar inEdges' Entry defs uses"
parallel_oracle_def: fixes left :: "'a \<Rightarrow> 'c \<Rightarrow> ('e \<times> 'a) spmf"   and right :: "'b \<Rightarrow> 'd \<Rightarrow> ('f \<times> 'b) spmf" shows "parallel_oracle left right = (\<lambda>(s1, s2). case_sum (\<lambda>a. map_spmf (map_prod Inl (\<lambda>s1'. (s1', s2))) (left s1 a)) (\<lambda>b. map_spmf (map_prod Inr (Pair s1)) (right s2 b)))"
simps_2: fixes l :: "('b \<times> bal) tree"   and a :: "'b"   and b :: "bal"   and r :: "('b \<times> bal) tree" shows "avl \<langle>l, (a, b), r\<rangle> = ((case b of Lh \<Rightarrow> height l = height r + 1 | Bal \<Rightarrow> height r = height l | Rh \<Rightarrow> height r = height l + 1) \<and> avl l \<and> avl r)"
span_subset_cspan: fixes A :: "'b set" shows "span A \<subseteq> cspan A"
\<beta>'''_def: shows "\<beta>''' = (\<lambda>\<alpha>. \<Sum>Y\<in>Roots_ms. \<beta>'' Y when \<Sum>\<^sub># Y = \<alpha>)"
pull_negpos_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "pull_negpos w1 w2" shows "kauff_mat w1 = kauff_mat w2"
intros: fixes M :: "('f \<times> 'l) set"   and N :: "('f \<times> 'l) set" assumes "M - N \<subseteq> Red_F_\<G> N" shows "M \<rhd>L N"
induct: fixes P :: "v sem_env \<Rightarrow> 'b state \<Rightarrow> exp0 \<Rightarrow> bool"   and a0 :: "v sem_env"   and a1 :: "'b state"   and a2 :: "exp0" assumes "\<And>env s l. P env s (Lit l)"   and "\<And>env s e. P env s e \<Longrightarrow> P env s (Raise e)"   and "\<And>env s e pes. \<lbrakk>P env s e; \<And>x y x2 x1 x1a xa ya. \<lbrakk>(x, y) = Evaluate_Single.evaluate env s e; y = Rerr x2; x2 = Rraise x1; match_result env x x1 pes x1 = Rval x1a; (xa, ya) = x1a\<rbrakk> \<Longrightarrow> P (update_v (\<lambda>_. nsAppend (alist_to_ns ya) (sem_env.v env)) env) x xa\<rbrakk> \<Longrightarrow> P env s (Handle e pes)"   and "\<And>env s cn es. (\<And>e sa. \<lbrakk>do_con_check (c env) cn (length es); e \<in> set (rev es); clock sa \<le> clock s\<rbrakk> \<Longrightarrow> P env sa e) \<Longrightarrow> P env s (Con cn es)"   and "\<And>env s n. P env s (Var n)"   and "\<And>env s n e. P env s (Fun n e)"   and "\<And>env s op0 es. \<lbrakk>\<And>e sa. \<lbrakk>e \<in> set (rev es); clock sa \<le> clock s\<rbrakk> \<Longrightarrow> P env sa e; \<And>x y x1 x2 xa ya. \<lbrakk>(x, y) = Evaluate_Single.evaluate_list (Evaluate_Single.evaluate env) s (rev es); y = Rval x1; op0 = Opapp; do_opapp (rev x1) = Some x2; (xa, ya) = x2; clock x \<noteq> 0\<rbrakk> \<Longrightarrow> P xa (dec_clock x) ya\<rbrakk> \<Longrightarrow> P env s (App op0 es)"   and "\<And>env s op0 e1 e2. \<lbrakk>P env s e1; \<And>x y x1 x2 x1a. \<lbrakk>(x, y) = Evaluate_Single.evaluate env s e1; y = Rval x1; do_log op0 x1 e2 = Some x2; x2 = Exp x1a\<rbrakk> \<Longrightarrow> P env x x1a\<rbrakk> \<Longrightarrow> P env s (Log op0 e1 e2)"   and "\<And>env s e1 e2 e3. \<lbrakk>P env s e1; \<And>x y x1 x2. \<lbrakk>(x, y) = Evaluate_Single.evaluate env s e1; y = Rval x1; do_if x1 e2 e3 = Some x2\<rbrakk> \<Longrightarrow> P env x x2\<rbrakk> \<Longrightarrow> P env s (exp0.If e1 e2 e3)"   and "\<And>env s e pes. \<lbrakk>P env s e; \<And>x y x1 x1a xa ya. \<lbrakk>(x, y) = Evaluate_Single.evaluate env s e; y = Rval x1; match_result env x x1 pes Bindv = Rval x1a; (xa, ya) = x1a\<rbrakk> \<Longrightarrow> P (update_v (\<lambda>_. nsAppend (alist_to_ns ya) (sem_env.v env)) env) x xa\<rbrakk> \<Longrightarrow> P env s (Mat e pes)"   and "\<And>env s n e1 e2. \<lbrakk>P env s e1; \<And>x y x1. \<lbrakk>(x, y) = Evaluate_Single.evaluate env s e1; y = Rval x1\<rbrakk> \<Longrightarrow> P (update_v (\<lambda>_. nsOptBind n x1 (sem_env.v env)) env) x e2\<rbrakk> \<Longrightarrow> P env s (exp0.Let n e1 e2)"   and "\<And>env s funs e. (distinct (map (\<lambda>(x, y, z). x) funs) \<Longrightarrow> P (update_v (\<lambda>_. build_rec_env funs env (sem_env.v env)) env) s e) \<Longrightarrow> P env s (Letrec funs e)"   and "\<And>env s e t0. P env s e \<Longrightarrow> P env s (Tannot e t0)"   and "\<And>env s e l. P env s e \<Longrightarrow> P env s (Lannot e l)" shows "P a0 a1 a2"
op_inter_fst_coll_def: fixes XS :: "('c \<times> 'd) set"   and Y :: "'c set" shows "op_inter_fst_coll XS Y = sets_of_coll XS \<bind> (\<lambda>XS. FORWEAK XS (RETURN op_empty_coll) (\<lambda>X. RETURN (mk_coll (op_inter_fst X Y))) (\<lambda>X X'. RETURN (X' \<union> X)))"
actionPar2Dest: fixes \<alpha> :: "'d action"   and P :: "('d, 'e, 'f) psi"   and \<beta> :: "'d action"   and Q :: "('d, 'e, 'f) psi"   and R :: "('d, 'e, 'f) psi" assumes "\<alpha> \<prec> P = \<beta> \<prec> Q \<parallel> R"   and "bn \<alpha> \<sharp>* bn \<beta>" obtains T :: "('d, 'e, 'f) psi" and p :: "(name \<times> name) list" where "set p \<subseteq> set (bn \<alpha>) \<times> set (bn \<beta>)" and "P = (p \<bullet> Q) \<parallel> T" and "\<alpha> \<prec> T = \<beta> \<prec> R"
InitConfig_def: fixes c :: "('b, 't, 'loc) Combined.configuration" shows "InitConfig c = ((\<forall>p. init c p = False) \<and> cri.init_config (exchange_config c) \<and> (\<forall>p loc t. zcount (c_pts (prop_config c p) loc) t = zcount (c_glob (exchange_config c) p) (loc, t)) \<and> (\<forall>w. local.init_config (prop_config c w)))"
g2g: shows "g2 \<in> {{x} \<times> I |x. x \<in> bl}"
djf_Or: fixes vs :: "'b list"   and bs :: "'b list"   and f :: "'c \<Rightarrow> fm"   and p :: "'c"   and q :: "fm" shows "Ifm vs bs (djf f p q) = Ifm vs bs (Or (f p) q)"
WhileF: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and c :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>false,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>unit,s\<^sub>1\<rangle>"
comp_assoc: fixes h :: "'a"   and g :: "'a"   and f :: "'a" shows "(h \<cdot>\<^sub>A g) \<cdot>\<^sub>A f = h \<cdot>\<^sub>A g \<cdot>\<^sub>A f"
fold_A_endo_X: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z" shows "ChamberComplexEndomorphism X (ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C)"
strategyproofness: shows "(pmf (sds R36) d + pmf (sds R36) c < pmf (sds R39) d + pmf (sds R39) c \<or> pmf (sds R36) d + (pmf (sds R36) c + pmf (sds R36) a) < pmf (sds R39) d + (pmf (sds R39) c + pmf (sds R39) a)) \<or> pmf (sds R36) d + pmf (sds R36) c = pmf (sds R39) d + pmf (sds R39) c \<and> pmf (sds R36) d + (pmf (sds R36) c + pmf (sds R36) a) = pmf (sds R39) d + (pmf (sds R39) c + pmf (sds R39) a)"
in_latticeI: fixes f :: "'a vec"   and c :: "nat \<Rightarrow> int"   and fs :: "'a vec list" assumes "f = M.sumlist (map (\<lambda>i. of_int (c i) \<cdot>\<^sub>v fs ! i) [0..<length fs])" shows "f \<in> lattice_of fs"
tset_conv_tnth: fixes xs :: "('a, 'b) tllist" shows "tset xs = {tnth xs n |n. enat n < tlength xs}"
hyps_2: assumes "n'' \<in> sum_SDG_slice2 n" shows "n'' \<in> sum_SDG_slice2 nx"
build_first: fixes w :: "nat \<Rightarrow> 'a" shows "w 0 ## suffix (Suc 0) w = w"
simps_7: fixes Obj :: "'e set"   and Mor :: "'f set"   and Dom :: "'f \<Rightarrow> 'e"   and Cod :: "'f \<Rightarrow> 'e"   and Id :: "'e \<Rightarrow> 'f"   and Comp :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and more :: "'d" shows "Category.more \<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = more"
thetaParZOL2_ZObis: shows "thetaParZOL2 \<subseteq> ZObis"
powr_less_inverse_iff': fixes x :: "real"   and y :: "real"   and z :: "real" assumes "0 < x"   and "0 < y"   and "0 < z" shows "(z < x powr y) = (z powr inverse y < x)"
axioms_1: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and sample :: "'d pmf \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_state_prob return bind get put sample" shows "monad_state return bind get put"
sep_dom2_aux2: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and y :: "((int \<times> int) set set, int) Combinators"   and z :: "((int \<times> int) set set, int) Combinators list"   and a :: "((int \<times> int) set set, int) Combinators" assumes "x \<in> dom (C (list2FWpolicy (separate (y # z)))) \<longrightarrow> x \<in> dom (C (list2FWpolicy (y # z)))"   and "x \<in> dom (C (list2FWpolicy (a # separate (y # z))))" shows "x \<in> dom (C (list2FWpolicy (a \<oplus> y # z)))"
wp_loop_sound: fixes P :: "'a \<Rightarrow> real"   and body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "sound P"   and "healthy (wp body)" shows "sound (wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) P)"
conjugation: fixes x :: "'b list"   and z :: "'b list"   and y :: "'b list" assumes "x \<cdot> z = z \<cdot> y"   and "x \<noteq> \<epsilon>" shows "\<exists>u v k. u \<cdot> v = x \<and> v \<cdot> u = y \<and> (u \<cdot> v) \<^sup>@ k \<cdot> u = z"
infsuff_nhd_base: fixes u :: "'a llist set"   and t :: "'a llist" assumes "u \<in> nhds t" obtains r :: "'a llist" where "r \<in> finpref A t" and "infsuff A r \<subseteq> u"
mod_pCons_eq: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" shows "pCons a p mod q = (if q = 0 then pCons a p else pCons a (p mod q) - smult (coeff (pCons a (p mod q)) (degree q) / lead_coeff q) q)"
Nom_6: fixes g :: "'a \<Rightarrow> 'c" assumes "M, g \<Turnstile>\<^sub>\<Theta> (p # ps, a) # branch" shows "False"
elims_1: fixes x :: "'d mat \<times> 'd mat \<times> 'd mat"   and xa :: "(('d, 'e) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec \<times> (('d, 'f) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec \<times> (('d, 'f) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and y :: "bool" assumes "HMA_M3 x xa = y" obtains P :: "'d mat" and A :: "'d mat" and Q :: "'d mat" and P' :: "(('d, 'e) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" and A' :: "(('d, 'f) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" and Q' :: "(('d, 'f) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" where "x = (P, A, Q)" and "xa = (P', A', Q')" and "y = (Mod_Type_Connect.HMA_M P P' \<and> Mod_Type_Connect.HMA_M A A' \<and> Mod_Type_Connect.HMA_M Q Q')"
elim: assumes "a divides b"   and "\<not> a \<sim> b" shows "P"
power_mult_distrib: fixes a :: "'a"   and b :: "'a"   and n :: "nat" shows "(a * b) ^ n = a ^ n * b ^ n"
fsmult_def: fixes smult :: "('b, 'c) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'b"   and v :: "'a" shows "aezfun_scalar_mult.fsmult smult a v \<equiv> smult (a \<delta>\<delta> (0::'c)) v"
assms_2: shows "\<bar>c\<bar> \<noteq> \<infinity>"
eq_fin_range: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list" assumes "traces ind"   and "(traces.eq_swap ind)\<^sup>*\<^sup>* w\<^sub>1 w\<^sub>2" shows "set w\<^sub>1 = set w\<^sub>2"
F1: shows "inverse \<bar>c\<bar> = \<bar>inverse c\<bar>"
run_alt_def_snth: fixes A :: "('c, 'd) nba"   and r :: "('c \<times> 'd) stream"   and p :: "'d" shows "run A r p = (\<forall>k. fst (r !! k) \<in> alphabet A \<and> snd (r !! k) \<in> transition A (fst (r !! k)) (target (stake k r) p))"
semilattice_order_axioms: shows "semilattice_order ord_term_lin.max (\<lambda>x y. y \<preceq>\<^sub>t x) (\<lambda>x y. y \<prec>\<^sub>t x)"
simps_15: fixes x21 :: "'b"   and x22 :: "'b list" shows "set (x21 # x22) = insert x21 (set x22)"
weight_random_oracle: fixes p :: "'a spmf"   and \<sigma> :: "'b \<Rightarrow> 'a option"   and x :: "'b" assumes "weight_spmf p = 1" shows "weight_spmf (random_function.random_oracle p \<sigma> x) = 1"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "inj_comm_ring_hom hom" shows "map_poly_inj_comm_ring_hom hom"
assms_3: shows "\<forall>\<^sub>F x in at_top. l x \<le> f x"
succ_chain_tableau_valuated: fixes l :: "('i, 'a) state list"   and i :: "nat"   and j :: "nat" assumes "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i \<le> j"   and "j < length l" shows "\<nabla> (l ! i) \<longrightarrow> \<nabla> (l ! j)"
red_in_Hom_2: fixes t :: "'c local.term" assumes "Ide t" shows "t\<^bold>\<down> \<in> VHom t \<^bold>\<lfloor>t\<^bold>\<rfloor>"
zero_float_def: shows "0 \<equiv> Abs_float (0, 0, 0)"
card_of_Plus_Times: fixes a1 :: "'a"   and a2 :: "'a"   and A :: "'a set"   and b1 :: "'b"   and b2 :: "'b"   and B :: "'b set" assumes "a1 \<noteq> a2 \<and> {a1, a2} \<subseteq> A"   and "b1 \<noteq> b2 \<and> {b1, b2} \<subseteq> B" shows "|A <+> B| \<le>o |A \<times> B|"
fp_cop_F_doctor_optimal_match: fixes ds :: "'b set" shows "TruePref_tax.doctor_optimal_match ds (TruePref_tax.cop ds)"
set_intros_2: fixes y :: "'x"   and x22 :: "'x list"   and x21 :: "'x" assumes "y \<in> set x22" shows "y \<in> set (x21 # x22)"
VDMAssign: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and x :: "Var"   and e :: "Expr" shows "G \<rhd> Assign x e : (\<lambda>s t. t = update s x (evalE e s))"
pow_inductl: fixes z :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set"   and i :: "nat" assumes "z \<union> x ; y \<subseteq> y" shows "rel_d.power x i ; z \<subseteq> y"
rel_simps_68: shows "(0::'a) < (1::'a)"
step: shows "\<Gamma>\<turnstile> (r, s) \<rightarrow> (r', s')"
UINF_image_eq: fixes f :: "'g \<Rightarrow> 'f"   and A :: "'g set"   and g :: "'f \<Rightarrow> ('d, 'e) uexpr" shows "(\<Sqinter> i \<in> f ` A \<bullet> g i) = (\<Sqinter> i \<in> A \<bullet> g (f i))"
ker_def: fixes F :: "('e, 'f) aGroup_scheme"   and G :: "('g, 'h) aGroup_scheme"   and f :: "'e \<Rightarrow> 'g" shows "ker\<^bsub>F,G\<^esub> f = {a \<in> carrier F. f a = \<zero>\<^bsub>G\<^esub>}"
sup_ge1: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.semilattice_sup sup less_eq less" shows "less_eq x (sup x y)"
inverse_arrows_hcomp: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and \<nu> :: "'b" assumes "horizontal_composition V H src trg"   and "category.iso V \<mu>"   and "category.iso V \<nu>"   and "src \<nu> = trg \<mu>" shows "category.inverse_arrows V (H \<nu> \<mu>) (H (category.inv V \<nu>) (category.inv V \<mu>))"
part: shows "UNIV = {0} \<union> {0<..<\<top>} \<union> {\<top>}"